Mathematics Sayings Quotes

I'm sorry to say that the subject I most disliked was mathematics. I have thought about it. I think the reason was that mathematics leaves no room for argument. If you made a mistake, that was all there was to it.

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

You have a roommate." "Yeah." He sounds confused. "The, um, picture on your door surprised me." "NO. No. I prefer my women with...fewer carnivorous beasts and less weaponry." He pauses and smiles. "Naked is okay. What she needs are a golden retriever and a telescope. Maybe then it would do it for me." I laugh. "A squirrel and a laboratory beaker?" "A bunny rabbit and a flip chart," I say. "Only if the flip chart has mathematical equations on it." I fake swoon onto his bed. "Too much, too much!

Soon after I began working for the Professor, I realized that he talked about numbers whenever he was unsure of what to say or do. Numbers were also his way of reaching out to the world. They were safe, a source of comfort.

whatever you are feeling right now, there is a mathematical certainty that someone is feeling that exact thing. This is not to say you aren’t special. This is to say thank god you aren’t special.

The law of gravity and gravity itself did not exist before Isaac Newton." ...and what that means is that that law of gravity exists nowhere except in people's heads! It 's a ghost!" Mind has no matter or energy but they can't escape its predominance over everything they do. Logic exists in the mind. numbers exist only in the mind. I don't get upset when scientists say that ghosts exist in the mind. it's that only that gets me. science is only in your mind too, it's just that that doesn't make it bad. or ghosts either." Laws of nature are human inventions, like ghosts. Law of logic, of mathematics are also human inventions, like ghosts." ...we see what we see because these ghosts show it to us, ghosts of Moses and Christ and the Buddha, and Plato, and Descartes, and Rousseau and Jefferson and Lincoln, on and on and on. Isaac Newton is a very good ghost. One of the best. Your common sense is nothing more than the voices of thousands and thousands of these ghosts from the past.

[The Old Astronomer to His Pupil] Reach me down my Tycho Brahe, I would know him when we meet, When I share my later science, sitting humbly at his feet; He may know the law of all things, yet be ignorant of how We are working to completion, working on from then to now. Pray remember that I leave you all my theory complete, Lacking only certain data for your adding, as is meet, And remember men will scorn it, 'tis original and true, And the obloquy of newness may fall bitterly on you. But, my pupil, as my pupil you have learned the worth of scorn, You have laughed with me at pity, we have joyed to be forlorn, What for us are all distractions of men's fellowship and smiles; What for us the Goddess Pleasure with her meretricious smiles. You may tell that German College that their honor comes too late, But they must not waste repentance on the grizzly savant's fate. Though my soul may set in darkness, it will rise in perfect light; I have loved the stars too fondly to be fearful of the night. What, my boy, you are not weeping? You should save your eyes for sight; You will need them, mine observer, yet for many another night. I leave none but you, my pupil, unto whom my plans are known. You 'have none but me,' you murmur, and I 'leave you quite alone'? Well then, kiss me, -- since my mother left her blessing on my brow, There has been a something wanting in my nature until now; I can dimly comprehend it, -- that I might have been more kind, Might have cherished you more wisely, as the one I leave behind. I 'have never failed in kindness'? No, we lived too high for strife,-- Calmest coldness was the error which has crept into our life; But your spirit is untainted, I can dedicate you still To the service of our science: you will further it? you will! There are certain calculations I should like to make with you, To be sure that your deductions will be logical and true; And remember, 'Patience, Patience,' is the watchword of a sage, Not to-day nor yet to-morrow can complete a perfect age. I have sown, like Tycho Brahe, that a greater man may reap; But if none should do my reaping, 'twill disturb me in my sleep So be careful and be faithful, though, like me, you leave no name; See, my boy, that nothing turn you to the mere pursuit of fame. I must say Good-bye, my pupil, for I cannot longer speak; Draw the curtain back for Venus, ere my vision grows too weak: It is strange the pearly planet should look red as fiery Mars,-- God will mercifully guide me on my way amongst the stars.

The first rule of style is to have something to say. The second rule of style is to control yourself when, by chance, you have two things to say; say first one, then the other, not both at the same time.

To enjoy life, we might have to stop thinking about what we will never be able to read and watch and say and do, and start to think of how to enjoy the world within our boundaries. To live on a human scale. To focus on the few things we can do, rather than the millions of things we can't. To not crave parallel lives. To find a smaller mathematics. To be a proud and singular one. An indivisible prime.

I would say, if you like, that the party is like an out-moded mathematics...that is to say, the mathematics of Euclid. We need to invent a non-Euclidian mathematics with respect to political discipline.

. . . the world in which we live has an increasing number of feedback loops, causing events to be the cause of more events (say, people buy a book because other people bought it), thus generating snowballs and arbitrary and unpredictable planet-wide winner-take-all effects.

Those who assert that the mathematical sciences say nothing of the beautiful or the good are in error. For these sciences say and prove a great deal about them; if they do not expressly mention them, but prove attributes which are their results or definitions, it is not true that they tell us nothing about them. The chief forms of beauty are order and symmetry and definiteness, which the mathematical sciences demonstrate in a special degree.

I would say that the five most important skills are of course, reading, writing, arithmetic, and then as you’re adding in, persuasion, which is talking. And then finally, I would add computer programming just because it’s an applied form of arithmetic that just gets you so much leverage for free in any domain that you operate in. If you’re good with computers, if you’re good at basic mathematics, if you’re good at writing, if you’re good at speaking, and if you like reading, you’re set for life.

While the individual man is an insoluble puzzle, in the aggregate he becomes a mathematical certainty. You can, for example, never foretell what any one man will do, but you can say with precision what an average number will be up to. Individuals vary, but the percentages remain constant

We love men because they can never fake orgasms, even if they wanted to. Because they write poems, songs, and books in our honor. Because they never understand us, but they never give up. Because they can see beauty in women when women have long ceased to see any beauty in themselves. Because they come from little boys. Because they can churn out long, intricate, Machiavellian, or incredibly complex mathematics and physics equations, but they can be comparably clueless when it comes to women. Because they are incredible lovers and never rest until we’re happy. Because they elevate sports to religion. Because they’re never afraid of the dark. Because they don’t care how they look or if they age. Because they persevere in making and repairing things beyond their abilities, with the naïve self-assurance of the teenage boy who knew everything. Because they never wear or dream of wearing high heels. Because they’re always ready for sex. Because they’re like pomegranates: lots of inedible parts, but the juicy seeds are incredibly tasty and succulent and usually exceed your expectations. Because they’re afraid to go bald. Because you always know what they think and they always mean what they say. Because they love machines, tools, and implements with the same ferocity women love jewelry. Because they go to great lengths to hide, unsuccessfully, that they are frail and human. Because they either speak too much or not at all to that end. Because they always finish the food on their plate. Because they are brave in front of insects and mice. Because a well-spoken four-year old girl can reduce them to silence, and a beautiful 25-year old can reduce them to slobbering idiots. Because they want to be either omnivorous or ascetic, warriors or lovers, artists or generals, but nothing in-between. Because for them there’s no such thing as too much adrenaline. Because when all is said and done, they can’t live without us, no matter how hard they try. Because they’re truly as simple as they claim to be. Because they love extremes and when they go to extremes, we’re there to catch them. Because they are tender they when they cry, and how seldom they do it. Because what they lack in talk, they tend to make up for in action. Because they make excellent companions when driving through rough neighborhoods or walking past dark alleys. Because they really love their moms, and they remind us of our dads. Because they never care what their horoscope, their mother-in-law, nor the neighbors say. Because they don’t lie about their age, their weight, or their clothing size. Because they have an uncanny ability to look deeply into our eyes and connect with our heart, even when we don’t want them to. Because when we say “I love you” they ask for an explanation.

And that is not all: even if man really were nothing but a piano-key, even if this were proved to him by natural science and mathematics, even then he would not become reasonable, but would purposely do something perverse out of simple ingratitude, simply to gain his point. And if he does not find means he will contrive destruction and chaos, will contrive sufferings of all sorts, only to gain his point! He will launch a curse upon the world, and as only man can curse (it is his privilege, the primary distinction between him and other animals), may be by his curse alone he will attain his object--that is, convince himself that he is a man and not a piano-key! If you say that all this, too, can be calculated and tabulated--chaos and darkness and curses, so that the mere possibility of calculating it all beforehand would stop it all, and reason would reassert itself, then man would purposely go mad in order to be rid of reason and gain his point!

Academics are only useful when they try to be useless (say, as in mathematics and philosophy) and dangerous when they try to be useful.

You call a star a star, and say it is just a ball of matter moving on a mathematical course. But that is merely how you see it. By so naming things and describing them you are only inventing your own terms about them. And just as speech is invention about objects and ideas, so myth is invention about truth.

A mathematician says that an electromagnetic wave travels from Andromeda to your eye and that it also extends from Andromeda to your eye.

There is another physical law that teases me, too: the Doppler Effect. The sound of anything coming at you- a train, say, or the future- has a higher pitch than the sound of the same thing going away. If you have perfect pitch and a head for mathematics you can compute the speed of the object by the interval between its arriving and departing sounds. I have neither perfect pitch nor a head for mathematics, and anyway who wants to compute the speed of history? Like all falling bodies, it constantly accelerates. But I would like to hear your life as you heard it, coming at you, instead of hearing it as I do, a somber sound of expectations reduced, desires blunted, hopes deferred or abandoned, chances lost, defeats accepted, griefs borne.

Economics is haunted by more fallacies than any other study known to man. This is no accident. The inherent difficulties of the subject would be great enough in any case, but they are multiplied a thousandfold by a factor that is insignificant in, say, physics, mathematics or medicine - the special pleading of selfish interests.

The Couple Overfloweth We sometimes go on as though people can’t express themselves. In fact they’re always expressing themselves. The sorriest couples are those where the woman can’t be preoccupied or tired without the man saying “What’s wrong? Say something…,” or the man, without the woman saying … and so on. Radio and television have spread this spirit everywhere, and we’re riddled with pointless talk, insane quantities of words and images. Stupidity’s never blind or mute. So it’s not a problem of getting people to express themselves but of providing little gaps of solitude and silence in which they might eventually find something to say. Repressive forces don’t stop people expressing themselves but rather force them to express themselves; What a relief to have nothing to say, the right to say nothing, because only then is there a chance of framing the rare, and ever rarer, thing that might be worth saying. What we’re plagued by these days isn’t any blocking of communication, but pointless statements. But what we call the meaning of a statement is its point. That’s the only definition of meaning, and it comes to the same thing as a statement’s novelty. You can listen to people for hours, but what’s the point? . . . That’s why arguments are such a strain, why there’s never any point arguing. You can’t just tell someone what they’re saying is pointless. So you tell them it’s wrong. But what someone says is never wrong, the problem isn’t that some things are wrong, but that they’re stupid or irrelevant. That they’ve already been said a thousand times. The notions of relevance, necessity, the point of something, are a thousand times more significant than the notion of truth. Not as substitutes for truth, but as the measure of the truth of what I’m saying. It’s the same in mathematics: Poincaré used to say that many mathematical theories are completely irrelevant, pointless; He didn’t say they were wrong – that wouldn’t have been so bad. (Negotiations)

In my own professional work I have touched on a variety of different fields. I’ve done work in mathematical linguistics, for example, without any professional credentials in mathematics; in this subject I am completely self-taught, and not very well taught. But I’ve often been invited by universities to speak on mathematical linguistics at mathematics seminars and colloquia. No one has ever asked me whether I have the appropriate credentials to speak on these subjects; the mathematicians couldn’t care less. What they want to know is what I have to say. No one has ever objected to my right to speak, asking whether I have a doctor’s degree in mathematics, or whether I have taken advanced courses in the subject. That would never have entered their minds. They want to know whether I am right or wrong, whether the subject is interesting or not, whether better approaches are possible… the discussion dealt with the subject, not with my right to discuss it. But on the other hand, in discussion or debate concerning social issues or American foreign policy…. The issue is constantly raised, often with considerable venom. I’ve repeatedly been challenged on grounds of credentials, or asked, what special training do I have that entitles you to speak on these matters. The assumption is that people like me, who are outsiders from a professional viewpoint, are not entitled to speak on such things. Compare mathematics and the political sciences… it’s quite striking. In mathematics, in physics, people are concerned with what you say, not with your certification. But in order to speak about social reality, you must have the proper credentials, particularly if you depart from the accepted framework of thinking. Generally speaking, it seems fair to say that the richer the intellectual substance of a field, the less there is a concern for credentials, and the greater is the concern for content.

Cindy, have you heard of the second law of thermodynamics?” “Yes. Something about heat energy can never be created or destroyed?” “That’s the first law of thermodynamics. The second one is this…all organized systems tend to slide slowly into chaos and disorder. Energy tends to run down. The universe itself heads inevitably towards darkness and stasis. Our own star system eventually will die, the sun will become a red giant, and the earth will be swallowed by the red giant.” “Cheery thought.” “But mathematics has altered this concept; rather one particular mathematician. His name was Ilya Prigogine, a Belgian mathematician.” “Who and what does that have to do with your being a PI and a great psychologist?” “Are you being sarcastic? Of course you are. Anyway, what I was trying to say was that Prigogine used the analogy of a walled city and open city. The walled city is isolated from its surroundings and will run down, decay, and die. The open city will have an exchange of materials and energy with its surroundings and will become larger and more complex; capable of dissipating energy even as it grows. So my point is, this analogy very much pertains to a certain female. The walled person versus the open person. The walled person will eventually decline, fade, and decay.

Pure mathematics consists entirely of assertions to the effect that, if such and such a proposition is true of anything, then such and such another proposition is true of that thing. It is essential not to discuss whether the first proposition is really true, and not to mention what the anything is, of which it is supposed to be true. [...] Thus mathematics may be defined as the subject in which we never know what we are talking about, nor whether what we are saying is true. People who have been puzzled by the beginnings of mathematics will, I hope, find comfort in this definition, and will probably agree that it is accurate.

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

economics was largely a form of intellectual prostitution where you got rewarded for saying what the powers that be wanted to hear.

He called the scientific subjects ‘low cunning’, and would sniff and say, ‘This room smells of mathematics! Go out and fetch a disinfectant spray!

Our craving for generality has [as one] source … our preoccupation with the method of science. I mean the method of reducing the explanation of natural phenomena to the smallest possible number of primitive natural laws; and, in mathematics, of unifying the treatment of different topics by using a generalization. Philosophers constantly see the method of science before their eyes, and are irresistibly tempted to ask and answer in the way science does. This tendency is the real source of metaphysics, and leads the philosopher into complete darkness. I want to say here that it can never be our job to reduce anything to anything, or to explain anything. Philosophy really is “purely descriptive.

Be honest: did you actually read [the above geometric proof]? Of course not. Who would want to? The effect of such a production being made over something so simple is to make people doubt their own intuition. Calling into question the obvious by insisting that it be 'rigorously proved' ... is to say to a student 'Your feelings and ideas are suspect. You need to think and speak our way.

I have come to believe that the merits of a death custom are not based on mathematics (e.g., 36.7 percent a "barbarous act"), but on emotions, a belief in the unique nobility of one's own culture. That is to say, we consider death rituals savage only when they don't match our own.

Music and Dancing, not only give great pleasure but have the honour of depending on Mathematics, for they consist in number and in measure.....Therefore, whatever the old doctors may say, to employ oneself at all this is to be a Philosopher and a Mathematician.

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.

Mr Baley", said Quemot, "you can't treat human emotions as though they were built about a positronic brain". "I'm not saying you can. Robotics is a deductive science and sociology an inductive one. But mathematics can be made to apply in either case.

ECONOMICS IS HAUNTED by more fallacies than any other study known to man. This is no accident. The inherent difficulties of the subject would be great enough in any case, but they are multiplied a thousandfold by a factor that is insignificant in, say, physics, mathematics or medicine—the special pleading of selfish interests.

One of the most painful parts of teaching mathematics is seeing students damaged by the cult of the genius. The genius cult tells students it’s not worth doing mathematics unless you’re the best at mathematics, because those special few are the only ones whose contributions matter. We don’t treat any other subject that way! I’ve never heard a student say, “I like Hamlet, but I don’t really belong in AP English—that kid who sits in the front row knows all the plays, and he started reading Shakespeare when he was nine!” Athletes don’t quit their sport just because one of their teammates outshines them. And yet I see promising young mathematicians quit every year, even though they love mathematics, because someone in their range of vision was “ahead” of them.

You never say what I wish you’d say, and you frequently say nothing at all when it’s clear you should say something, so it’s not entirely fantastical that you’d say a certain thing when you mean something else entirely.” He opened his mouth, shut it, and considered the ground briefly before responding. “I remember studying Fleet Admiral Starcrest’s Mathematical Probabilities Applied to Military Strategies as a young boy and finding that less confusing than what you just said.” Now it was her turn for a stunned pause before answering. “Sicarius?” She laid a tentative hand on his shoulder. “Was that a joke?” “A statement of fact.

In mathematics, in physics, people are concerned with what you say, not with your certification. But in order to speak about social reality, you must have the proper credentials, particularly if you depart from the accepted framework of thinking. Generally speaking, it seems fair to say that the richer the intellectual substance of a field, the less there is a concern for credentials, and the greater is concern for content.

Within mathematics, assertions must always be proven mathematically and expressed in a valid and scientifically correct formula. The mathematician must be able to stand on a podium and say the words 'This is so because …

People who don't like math always accuse mathematicians of trying to make math complicated. (...) But anyone who does love math knows it's really the opposite: math rewards simplicity, and mathematicians value it above all else. So it's no surprise that Walter's favourite axiom was also the most simple in the realm of mathematics: the axiom of the empty set. The axiom of the empty set is the axiom of zero. it states that there must be a concept of nothingness, that there must be the concept of zero: zero value, zero items. Math assumes there's a concept of nothingness, but is it proven? No. But it must exist. And if we're being philosophical—which we today are—we can say that life itself is the axiom of the empty set. It begins in zero and ends in zero. We know that both states exist, but we will not be conscious of either experience: they are states that are necessary parts of life, even as they cannot be experienced as life. We assume the concept of nothingness, but we cannot prove it. But it must exist. So I prefer to think that Walter has not died but has instead proven for himself the axiom of the empty set, that he has proven the concept of zero. I know nothing else would have made him happier. An elegant mind wants elegant endings, and Walter had the most elegant mind. So I wish him goodbye; I wish him the answer to the axiom he so loved.

People assume you have to have some special talent to do mathematics,” Sylvia has said. “They think you’re either born with it, or you’re not. But Rhonda and I keep saying, ‘You actually develop the ability to do mathematics. Don’t give up!

You call a tree a tree, he said, and you think nothing more of the word. But it was not a 'tree' until someone gave it that name. You call a star a star, and say it is just a ball of matter moving on a mathematical course. But that is merely how you see it. By so naming things and describing them you are only inventing your own terms about them. And just as speech is invention about objects and ideas, so myth is invention about truth. We have come from God (continued Tolkien), and inevitably the myths woven by us, though they contain error, will also reflect a splintered fragment of the true light, the eternal truth that is with God. Indeed only by myth-making, only by becoming a 'sub-creator' and inventing stories, can Man aspire to the state of perfection that he knew before the Fall. Out myths may be misguided, but they steer however shakily towards the true harbor, while materialistic 'progress' leads only to a yawning abyss and the Iron Crown of evil.

It is generally recognized that women are better than men at languages, personal relations and multitasking, but less good at map-reading and spatial awareness. It is therefore not unreasonable to suppose that women might be less good at mathematics and physics. It is not politically correct to say such things....But it cannot be denied that there are differences between men and women. Of course, these are differences between the averages only. There are wide variations about the mean.

I noticed that the [drawing] teacher didn't tell people much... Instead, he tried to inspire us to experiment with new approaches. I thought of how we teach physics: We have so many techniques - so many mathematical methods - that we never stop telling the students how to do things. On the other hand, the drawing teacher is afraid to tell you anything. If your lines are very heavy, the teacher can't say, "Your lines are too heavy." because *some* artist has figured out a way of making great pictures using heavy lines. The teacher doesn't want to push you in some particular direction. So the drawing teacher has this problem of communicating how to draw by osmosis and not by instruction, while the physics teacher has the problem of always teaching techniques, rather than the spirit, of how to go about solving physical problems.

A famous Japanese Zen master, Hakuun Yasutani Roshi, said that unless you can explain Zen in words that a fisherman will comprehend, you don’t know what you’re talking about. Some fifty years ago a UCLA professor told me the same thing about applied mathematics. We like to hide from the truth behind foreign-sounding words or mathematical lingo. There’s a saying: The truth is always encountered but rarely perceived. If we don’t perceive it, we can’t help ourselves and we can’t much help anyone else.

When we say that the ancestors of the Blacks, who today live mainly in Black Africa, were the first to invent mathematics, astronomy, the calendar, sciences in general, arts, religion, agriculture, social organization, medicine, writing, technique, architecture; that they were the first to erect buildings out of 6 million tons of stone (the Great Pyramid) as architects and engineers—not simply as unskilled laborers; that they built the immense temple of Karnak, that forest of columns with its famed hypostyle hall large enough to hold Notre-Dame and its towers; that they sculpted the first colossal statues (Colossi of Memnon, etc.)—when we say all that we are merely expressing the plain unvarnished truth that no one today can refute by arguments worthy of the name.

Going back to the basis, the phrase ‘Fight Like A Girl’, and we’ve all heard that growing up. And by that they mean that you’re some kind of weakling and have no skills as a male. It’s said to little boys when they can’t fight yet, and it ridicules us. By the time we were born, the most of us hear things which program you to accept and know that you are less than your male counter part. It comes apparent in the way you’re paid for your job, it comes apparent when yóu are not allowed to go outside after a certain hour because you stand a good chance of getting raped while no one says that to your boyfriend. While women, anywhere, live in some kind of fear, there is no equality and that is mathematically impossible. We cannot see that change or solved in our lifetimes, but we have to do everything that we can. We should remind ourselves that we are fifty-one percent. Everyone should know that fighting like a girl is a positive thing and that there is not inherently anything wrong with us by the fact that we are born like ladies. That is a beautiful thing that we should never be put down because of. Being compared to a woman should only make a man feel stronger. It should be a compliment. In this world we’re creating it actually is. I remember this one guy who came to our show in Texas or something and he had painted his shirt “real men fight like a girl”, and I cried, because he was going away in the army next day. He bought my book because he wanted something he could read over there. I just hoped that this men, fully straight and fully male can maintain and retain all of those things that make him understand us, and what makes him so beautiful. A lot of military training is step one: you take all those guys and put them in front of bunch of hardcore videogames where you kill a bunch of people and become desensitised. But that is NOT power! I will not do that. I will not become less of a human being and I refuse to give up my femininity because that’s bullshit. I’m not going to have to shave my head and become all buff and all that to be able to say “now I’m powerful” because that’s bullshit. All of this, all of us, we are power. You don’t have to change anything to be strong.

My sub doesn't pay for me,” he says, pulling me to my feet. “That just doesn't happen.” “But we ordered so much,” I say helplessly. “It made you happy,” he says simply. “Now I get to play with you. And that makes me happy.” “I don't think it's that simple an equation.” “Maybe not,” he concedes. “But then, if if sex were the same thing as math, a lot more people would be lining up to take calculus.

Saint Bartleby's School for Young Gentlemen Annual Report Student: Artemis Fowl II Year: First Fees: Paid Tutor: Dr Po Language Arts As far as I can tell, Artemis has made absolutely no progress since the beginning of the year. This is because his abilities are beyond the scope of my experience. He memorizes and understands Shakespeare after a single reading. He finds mistakes in every exercise I administer, and has taken to chuckling gently when I attempt to explain some of the more complex texts. Next year I intend to grant his request and give him a library pass during my class. Mathematics Artemis is an infuriating boy. One day he answers all my questions correctly, and the next every answer is wrong. He calls this an example of the chaos theory, and says that he is only trying to prepare me for the real world. He says the notion of infinity is ridiculous. Frankly, I am not trained to deal with a boy like Artemis. Most of my pupils have trouble counting without the aid of their fingers. I am sorry to say, there is nothing I can teach Artemis about mathematics, but someone should teach him some manners. Social Studies Artemis distrusts all history texts, because he says history was written by the victors. He prefers living history, where survivors of certain events can actually be interviewed. Obviously this makes studying the Middle Ages somewhat difficult. Artemis has asked for permission to build a time machine next year during double periods so that the entire class may view Medieval Ireland for ourselves. I have granted his wish and would not be at all surprised if he succeeded in his goal. Science Artemis does not see himself as a student, rather as a foil for the theories of science. He insists that the periodic table is a few elements short and that the theory of relativity is all very well on paper but would not hold up in the real world, because space will disintegrate before lime. I made the mistake of arguing once, and young Artemis reduced me to near tears in seconds. Artemis has asked for permission to conduct failure analysis tests on the school next term. I must grant his request, as I fear there is nothing he can learn from me. Social & Personal Development Artemis is quite perceptive and extremely intellectual. He can answer the questions on any psychological profile perfectly, but this is only because he knows the perfect answer. I fear that Artemis feels that the other boys are too childish. He refuses to socialize, preferring to work on his various projects during free periods. The more he works alone, the more isolated he becomes, and if he does not change his habits soon, he may isolate himself completely from anyone wishing to be his friend, and, ultimately, his family. Must try harder.

These rules, the sign language and grammar of the Game, constitute a kind of highly developed secret language drawing upon several sciences and arts, but especially mathematics and music (and/or musicology), and capable of expressing and establishing interrelationships between the content and conclusions of nearly all scholarly disciplines. The Glass Bead Game is thus a mode of playing with the total contents and values of our culture; it plays with them as, say, in the great age of the arts a painter might have played with the colours on his palette.

I'm angry because you think everything happened by chance but there are billions of people on this planet and I found you so if you're saying I could just as well have found someone else then I can't bear your bloody mathematics!" Her fists had been clenched. He stood there looking at her for several minutes. Then he said that he loved her. It was the first time. They never stopped arguing and they never slept apart; he spent an entire working life calculating probabilities and she was the most improbable person he ever met. She turned him upside-down.

No human investigation can be called real knowledge if it does not pass through mathematical demonstrations; and if you say that the kinds of knowledge that begin and end in the mind have any value as truth, this cannot be conceded, but rather must be denied for many reasons, and first of all because in such mental discussions there is no experimentation, without which nothing provides certainty of itself.

Compare mathematics and the political sciences—it’s quite striking. In mathematics, in physics, people are concerned with what you say, not with your certification. But in order to speak about social reality, you must have the proper credentials, particularly if you depart from the accepted framework of thinking. Generally speaking, it seems fair to say that the richer the intellectual substance of a field, the less there is a concern for credentials, and the greater is the concern for content.

I don't mind nothing happening in a book, but nothing happening in a phony way--characters saying things people never say, doing jobs that don't fit, the whole works--is simply asking too much of a reader. Something happening in a phony way must beat nothing happening in a phony way every time, right? I mean, you could prove that, mathematically, in an equation, and you can't often apply science to literature.

If a mathematician wishes to disparage the work of one of his colleagues, say, A, the most effective method he finds for doing this is to ask where the results can be applied. The hard pressed man, with his back against the wall, finally unearths the researches of another mathematician B as the locus of the application of his own results. If next B is plagued with a similar question, he will refer to another mathematician C. After a few steps of this kind we find ourselves referred back to the researches of A, and in this way the chain closes.

The philosophers make still another objection: "What you gain in rigour," they say, "you lose in objectivity. You can rise toward your logical ideal only by cutting the bonds which attach you to reality. Your science is infallible, but it can only remain so by imprisoning itself in an ivory tower and renouncing all relation with the external world. From this seclusion it must go out when it would attempt the slightest application.

Is everyone with one face called a Milo?" "Oh no," Milo replied; "some are called Henry or George or Robert or John or lots of other things." "How terribly confusing," he cried. "Everything here is called exactly what it is. The triangles are called triangles, the circles are called circles, and even the same numbers have the same name. Why, can you imagine what would happen if we named all the twos Henry or George or Robert or John or lots of other things? You'd have to say Robert plus John equals four, and if the four's name were Albert, things would be hopeless." "I never thought of it that way," Milo admitted. "Then I suggest you begin at once," admonished the Dodecahedron from his admonishing face, "for here in Digitopolis everything is quite precise.

. . . we come astonishingly close to the mystical beliefs of Pythagoras and his followers who attempted to submit all of life to the sovereignty of numbers. Many of our psychologists, sociologists, economists and other latter-day cabalists will have numbers to tell them the truth or they will have nothing. . . . We must remember that Galileo merely said that the language of nature is written in mathematics. He did not say that everything is. And even the truth about nature need not be expressed in mathematics. For most of human history, the language of nature has been the language of myth and ritual. These forms, one might add, had the virtues of leaving nature unthreatened and of encouraging the belief that human beings are part of it. It hardly befits a people who stand ready to blow up the planet to praise themselves too vigorously for having found the true way to talk about nature.

You may have to go with English,” Bailey says. “Einstein said, So far as the theories of mathematics are about reality, they are not certain; so far as they are certain, they are not about reality.” Bailey tilts her head. “Still waiting on the English there, Professor,” she says. “It basically means, we don’t know shit about anything,” he says.

I started studying law, but this I could stand just for one semester. I couldn't stand more. Then I studied languages and literature for two years. After two years I passed an examination with the result I have a teaching certificate for Latin and Hungarian for the lower classes of the gymnasium, for kids from 10 to 14. I never made use of this teaching certificate. And then I came to philosophy, physics, and mathematics. In fact, I came to mathematics indirectly. I was really more interested in physics and philosophy and thought about those. It is a little shortened but not quite wrong to say: I thought I am not good enough for physics and I am too good for philosophy. Mathematics is in between.

But then, the sky! Blue, untainted by a single cloud (the Ancientes had such barbarous tastes given that their poets could have been inspired by such stupid, sloppy, silly-lingering clumps of vapour). I love - and i'm certain that i'm not mistaken if i say we love - skies like this, sterile and flawless! On days like these, the whole world is blown from the same shatterproof, everlasting glass as the glass of the Green Wall and of all our structures. On days like these, you can see to the very blue depths of things, to their unknown surfaces, those marvelous expressions of mathematical equality - which exist in even the most usual and everyday objects.

Don't be scared by the word authority. Believing things on authority only means believing them because you've been told them by someone you think trustworthy. Ninety-nine per cent of the things you believe are believed on authority. I believe there is such a place as New York. I haven't seen it myself. I couldn't prove by abstract reasoning that there must be such a place. I believe it because reliable people have told me so. The ordinary man believes in the Solar System, atoms, evolution, and the circulation of the blood on authority -because the scientists say so. Every historical statement in the world is believed on authority. None of us has seen the Norman Conquest or the defeat of the Armada. None of us could prove them by pure logic as you prove a thing in mathematics. We believe them simply because people who did see them have left writings that tell us about them: in fact, on authority. A man who jibbed at authority in other things as some people do in religion would have to be content to know nothing all his life.

If Henry Adams, whom you knew slightly, could make a theory of history by applying the second law of thermodynamics to human affairs, I ought to be entitled to base one on the angle of repose, and may yet. There is another physical law that teases me, too: the Doppler Effect. The sound of anything coming at you -- a train, say, or the future -- has a higher pitch than the sound of the same thing going away. If you have perfect pitch and a head for mathematics you can compute the speed of the object by the interval between its arriving and departing sounds. I have neither perfect pitch nor a head for mathematics, and anyway who wants to compute the speed of history? Like all falling bodies, it constantly accelerates. But I would like to hear your life as you heard it, coming at you, instead of hearing it as I do, a sober sound of expectations reduced, desires blunted, hopes deferred or abandoned, chances lost, defeats accepted, griefs borne. I don't find your life uninteresting, as Rodman does. I would like to hear it as it sounded while it was passing. Having no future of my own, why shouldn't I look forward to yours.

I cannot understand the principle at all,' said Stephen. 'I should very much like to show it to Captain Aubrey, who is so very well versed in the mathematics and dynamics of sailing. Landlord, pray ask him whether he is willing to part with the instrument.' Not on your fucking life,' said the Aboriginal, snatching the boomerang and clasping it to his bosom. He says he does not choose to dispose of it, your honour,' said the landlord. 'But never fret. I have a dozen behind the bar that I sell to ingenious travelers for half a guinea. Choose any one that takes your fancy, sit, and Bennelong will throw it to prove it comes back, a true homing pigeon, as we say. Won't you?' This much louder, in the black man's ear. Won't I what?' Throw it for the gentleman.' Give um dram.' Sir, he says he will be happy to throw it for you; and hopes you will encourage him with a tot of rum. (pp. 353-354)

He did not believe in luck at all, good or bad. Gamblers believed in luck, and he was not a gambler. Never had been, never would be. John Henry Holliday believed in mathematics, in statistics, in the computation of odds. Fifty-two cards in a deck. Make it easy. Say it's fifty. Any card has a 2 percent chance of being dealt from a full deck. Keep track of what's out. Adjust the probabilities as the hand progresses.

When I described Madame de T's night, I recalled the well-known equation from one of the first chapters of the textbook of existential mathematics: the degree of speed is directly proportional to the intensity of forgetting. From that equation we can deduce various corrollaries, for instance this one: our period is given over to the demon of speed, and that is the reason it so easily forgets its own self. Now I would reverse that statement and say: our period is obsessed by the desire to forget, and it is to fulfill that desire that it gives over to the demon of speed; it picks up the pace to show us that it no longer wishes to be remembered; that it is tired of itself; sick of itself; that it wants to blow out the tiny trembling flame of memory.

For the next half hour it continued. Dr. Joseph would call on someone who looked half bright, then he would call on someone whom he felt was just the opposite. In the upper grades—fourth, fifth, and sixth—he asked grammatical, mathematical, and geographical questions. And besides looking at hands, now he began inspecting teeth. Open wide, say “Ahhh”—and he would have the poor children spreading out their lips as far as they could while he peered into their mouths. At the university I had read about slave masters who had done the same when buying new slaves, and I had read of cattlemen doing it when purchasing horses and cattle. At least Dr. Joseph had graduated to the level where he let the children spread out their own lips, rather than using some kind of crude metal instrument. I appreciated his humanitarianism.

What is the proper justification of a mathematician’s life? My answers will be, for the most part, such as are expected from a mathematician: I think that it is worthwhile, that there is ample justification. But I should say at once that my defense of mathematics will be a defense of myself, and that my apology is bound to be to some extent egotistical. I should not think it worth while to apologize for my subject if I regarded myself as one of its failures. Some egotism of this sort is inevitable, and I do not feel that it really needs justification. Good work is no done by "humble" men. It is one of the first duties of a professor, for example, in any subject, to exaggerate a little both the importance of his subject and his own importance in it. A man who is always asking "Is what I do worth while?" and "Am I the right person to do it?" will always be ineffective himself and a discouragement to others. He must shut his eyes a little and think a little more of his subject and himself than they deserve. This is not too difficult: it is harder not to make his subject and himself ridiculous by shutting his eyes too tightly.

Congratulations,” he says. Uh? “On your Ph.D.” “W-what?” “A noteworthy accomplishment,” he continues, serious, calm, “given that less than twenty-four hours ago you weren’t even working on one.” I exhale deeply. “Listen, it’s not what you—” “Will you be leaving your post at the library, or are you planning on a dual career? I’d be worried for your schedule, but I hear that theoretical physics often consists of staring into the void and jotting down the occasional mathematical symbol—

Doing a geographic” is a term alcoholics often use for acting on the impulse to start over by moving to a new town, or state, instead of making any internal changes. It’s the anywhere-but-here part of the disease that says, “Remove yourself from this, go someplace new, and everything will be better.” Two years into our Florida stint, my mother pulled a geographic as radical as the move from Rochester. The new plan was to head for California. She enrolled in the mathematics graduate program at the University of California’s shiny new campus in San Diego, and as soon as our elementary school let out for the summer, she put us into a new Buick station wagon – a gift from her parents – and drove us across the country. You’d think we’d have protested at yet another move. After all, having been duped before, we were in no position to believe that the next move would be any different. But I have no memory of being unhappy about the news. Because that’s what often happens when an alcoholic parent is doing a geographic. She pulls you in and, before you know it, you, too, believe in the promise of the new place.

Because, you see, everything you know about the way this universe works is correct—except for the little problem that this isn’t the only universe we have to worry about. Information can leak between one universe and another. And in a vanishingly small number of the other universes there are things that listen, and talk back—see Al-Hazred, Nietzsche, Lovecraft, Poe, et cetera. The many-angled ones, as they say, live at the bottom of the Mandelbrot set, except when a suitable incantation in the platonic realm of mathematics—computerised or otherwise—draws them forth. (And you thought running that fractal screen-saver was good for your computer?)

Russell is reputed at a dinner party once to have said, ‘Oh, it is useless talking about inconsistent things, from an inconsistent proposition you can prove anything you like.’ Well, it is very easy to show this by mathematical means. But, as usual, Russell was much cleverer than this. Somebody at the dinner table said, 'Oh, come on!’ He said, 'Well, name an inconsistent proposition,’ and the man said, 'Well, what shall we say, 2 = 1.’ 'All right,’ said Russell, 'what do you want me to prove?’ The man said, 'I want you to prove that you’re the pope.’ 'Why,’ said Russell, 'the pope and I are two, but two equals one, therefore the pope and I are one.

It seems to be one of the fundamental features of nature that fundamental physical laws are described in terms of a mathematical theory of great beauty and power, needing quite a high standard of mathematics for one to understand it. You may wonder: Why is nature constructed along these lines? One can only answer that our present knowledge seems to show that nature is so constructed. We simply have to accept it. One could perhaps describe the situation by saying that God is a mathematician of a very high order, and He used very advanced mathematics in constructing the universe. Our feeble attempts at mathematics enable us to understand a bit of the universe, and as we proceed to develop higher and higher mathematics we can hope to understand the universe better.

Another explanation for the failure of logic and observation alone to advance medicine is that unlike, say, physics, which uses a form of logic - mathematics - as its natural language, biology does not lend itself to logic. Leo Szilard, a prominent physicist, made this point when he complained that after switching from physics to biology he never had a peaceful bath again. As a physicist he would soak in the warmth of a bathtub and contemplate a problem, turn it in his mind, reason his way through it. But once he became a biologist, he constantly had to climb out of the bathtub to look up a fact.

Another time somebody gave a talk about poetry. He talked about the structure of the poem and the emotions that come with it; he divided everything up into certain kinds of classes. In the discussion that came afterwards, he said, “Isn’t that the same as in mathematics, Dr. Eisenhart?” Dr. Eisenhart was the dean of the graduate school and a great professor of mathematics. He was also very clever. He said, “I’d like to know what Dick Feynman thinks about it in reference to theoretical physics.” He was always putting me on in this kind of situation. I got up and said, “Yes, it’s very closely related. In theoretical physics, the analog of the word is the mathematical formula, the analog of the structure of the poem is the interrelationship of the theoretical bling-bling with the so-andso”–and I went through the whole thing, making a perfect analogy. The speaker’s eyes were _beaming_ with happiness. Then I said, “It seems to me that no matter _what_ you say about poetry, I could find a way of making up an analog with _any_ subject, just as I did for theoretical physics. I don’t consider such analogs meaningful.

I don’t know,” he says. “I guess so. In space, everything has its relative position. Space is an entity, right, but also limitless. It’s less dense the farther out you go, but you can always keep going. There’s no definitive border between the start and the end. We’ll never fully understand or know it. We can’t.” “You don’t think?” “Dark matter makes up the majority of all matter, and it’s still a mystery.” “Dark matter?” “It’s invisible. It’s all the extra mass we can’t see that makes the formation of galaxies and the rotational velocities of stars around galaxies mathematically possible.” “I’m glad we don’t know everything.

Moreover: then, you say, science itself will teach man (though this is really a luxury in my opinion) that in fact he has neither will nor caprice, and never did have any, and that he himself is nothing but a sort of piano key or a sprig in an organ;14 and that, furthermore, there also exist in the world the laws of nature; so that whatever he does is done not at all according to his own wanting, but of itself, according to the laws of nature. Consequently, these laws of nature need only be discovered, and then man will no longer be answerable for his actions, and his life will become extremely easy. Needless to say, all human actions will then be calculated according to these laws, mathematically, like a table of logarithms, up to 108,000, and entered into a calendar; or, better still, some well-meaning publications will appear, like the present-day encyclopedic dictionaries, in which everything will be so precisely calculated and designated that there will no longer be any actions or adventures in the world.

And that is not all: even if man really were nothing but a piano-key, even if this were proved to him by natural science and mathematics, even then he would not become reasonable, but would purposely do something perverse out of simple ingratitude, simply to gain his point. And if he does not find means he will contrive destruction and chaos, will contrive sufferings of all sorts, only to gain his point! He will launch a curse upon the world, and as only man can curse (it is his privilege, the primary distinction between him and other animals), may be by his curse alone he will attain his object--that is, convince himself that he is a man and not a piano-key! If you say that all this, too, can be calculated and tabulated--chaos and darkness and curses, so that the mere possibility of calculating it all beforehand would stop it all, and reason would reassert itself, then man would purposely go mad in order to be rid of reason and gain his point! I believe in it, I answer for it, for the whole work of man really seems to consist in nothing but proving to himself every minute that he is a man and not a piano-key! It may be at the cost of his skin, it may be by cannibalism! And this being so, can one help being tempted to rejoice that it has not yet come off, and that desire still depends on something we don't know?

But man is a frivolous and incongruous creature, and perhaps, like a chess player, loves the process of the game, not the end of it. And who knows (there is no saying with certainty), perhaps the only goal on earth to which mankind is striving lies in this incessant process of attaining, in other words, in life itself, and not in the thing to be attained, which must always be expressed as a formula, as positive as twice two makes four, and such positiveness is not life, gentlemen, but is the beginning of death. Anyway, man has always been afraid of this mathematical certainty, and I am afraid of it now. Granted that man does nothing but seek that mathematical certainty, he traverses oceans, sacrifices his life in the quest, but to succeed, really to find it, dreads, I assure you. He feels that when he has found it there will be nothing for him to look for. When workmen have finished their work they do at least receive their pay, they go to the tavern, then they are taken to the police-station–and there is occupation for a week. But where can man go? Anyway, one can observe a certain awkwardness about him when he has attained such objects. He loves the process of attaining, but does not quite like to have attained, and that, of course, is very absurd. In fact, man is a comical creature; there seems to be a kind of jest in it all. But yet mathematical certainty is after all, something insufferable. Twice two makes four seems to me simply a piece of insolence. Twice two makes four is a pert coxcomb who stands with arms akimbo barring your path and spitting. I admit that twice two makes four is an excellent thing, but if we are to give everything its due, twice two makes five is sometimes a very charming thing too.

If you say that everything—chaos, darkness, anathema— can be reduced to mathematical formulae—then man will go insane on purpose to have no judgement, and to behave as he likes. I believe this because it appears that man’s whole business is to prove that he is a man and not a cog-wheel... And perhaps, who knows, the striving of man on earth may consist in this uninterrupted striving for something ahead, that is, in life itself rather than some real end which obviously must be a static formula of the same kind as two and two make four—I am sure that man will never renounce the genuine suffering that comes of ruin and chaos. Why, suffering is the one and only source of knowledge

The only gain of civilisation for mankind is the greater capacity for variety of sensations--and absolutely nothing more. And through the development of this many-sidedness man may come to finding enjoyment in bloodshed. In fact, this has already happened to him. Have you noticed that it is the most civilised gentlemen who have been the subtlest slaughterers, to whom the Attilas and Stenka Razins could not hold a candle, and if they are not so conspicuous as the Attilas and Stenka Razins it is simply because they are so often met with, are so ordinary and have become so familiar to us. In any case civilisation has made mankind if not more bloodthirsty, at least more vilely, more loathsomely bloodthirsty. In old days he saw justice in bloodshed and with his conscience at peace exterminated those he thought proper. Now we do think bloodshed abominable and yet we engage in this abomination, and with more energy than ever. Which is worse? Decide that for yourselves. They say that Cleopatra (excuse an instance from Roman history) was fond of sticking gold pins into her slave-girls' breasts and derived gratification from their screams and writhings. You will say that that was in the comparatively barbarous times; that these are barbarous times too, because also, comparatively speaking, pins are stuck in even now; that though man has now learned to see more clearly than in barbarous ages, he is still far from having learnt to act as reason and science would dictate. But yet you are fully convinced that he will be sure to learn when he gets rid of certain old bad habits, and when common sense and science have completely re-educated human nature and turned it in a normal direction. You are confident that then man will cease from INTENTIONAL error and will, so to say, be compelled not to want to set his will against his normal interests. That is not all; then, you say, science itself will teach man (though to my mind it's a superfluous luxury) that he never has really had any caprice or will of his own, and that he himself is something of the nature of a piano-key or the stop of an organ, and that there are, besides, things called the laws of nature; so that everything he does is not done by his willing it, but is done of itself, by the laws of nature. Consequently we have only to discover these laws of nature, and man will no longer have to answer for his actions and life will become exceedingly easy for him. All human actions will then, of course, be tabulated according to these laws, mathematically, like tables of logarithms up to 108,000, and entered in an index; or, better still, there would be published certain edifying works of the nature of encyclopaedic lexicons, in which everything will be so clearly calculated and explained that there will be no more incidents or adventures in the world.

Now I ask you: what can be expected of man since he is a being endowed with strange qualities? Shower upon him every earthly blessing, drown him in a sea of happiness, so that nothing but bubbles of bliss can be seen on the surface; give him economic prosperity, such that he should have nothing else to do but sleep, eat cakes and busy himself with the continuation of his species, and even then out of sheer ingratitude, sheer spite, man would play you some nasty trick. He would even risk his cakes and would deliberately desire the most fatal rubbish, the most uneconomical absurdity, simply to introduce into all this positive good sense his fatal fantastic element. It is just his fantastic dreams, his vulgar folly that he will desire to retain, simply in order to prove to himself—as though that were so necessary—that men still are men and not the keys of a piano, which the laws of nature threaten to control so completely that soon one will be able to desire nothing but by the calendar. And that is not all: even if man really were nothing but a piano-key, even if this were proved to him by natural science and mathematics, even then he would not become reasonable, but would purposely do something perverse out of simple ingratitude, simply to gain his point. And if he does not find means he will contrive destruction and chaos, will contrive sufferings of all sorts, only to gain his point! He will launch a curse upon the world, and as only man can curse (it is his privilege, the primary distinction between him and other animals), may be by his curse alone he will attain his object—that is, convince himself that he is a man and not a piano-key! If you say that all this, too, can be calculated and tabulated—chaos and darkness and curses, so that the mere possibility of calculating it all beforehand would stop it all, and reason would reassert itself, then man would purposely go mad in order to be rid of reason and gain his point! I believe in it, I answer for it, for the whole work of man really seems to consist in nothing but proving to himself every minute that he is a man and not a piano-key! It may be at the cost of his skin, it may be by cannibalism! And this being so, can one help being tempted to rejoice that it has not yet come off, and that desire still depends on something we don’t know?

In other words, our conscious representations are sometimes ordered (or arranged in a pattern) before they have become conscious to us. The 18th-century German mathematician Karl Friedrich Gauss gives an example of an experience of such an unconscious order of ideas: He says that he found a certain rule in the theory of numbers "not by painstaking research, but by the Grace of God, so to speak. The riddle solved itself as lightning strikes, and I myself could not tell or show the connection between what I knew before, what I last used to experiment with, and what produced the final success." The French scientist Henri Poincare is even more explicit about this phenomenon; he describes how during a sleepless night he actually watched his mathematical representations colliding in him until some of them "found a more stable connection. One feels as if one could watch one's own unconscious at work, the unconscious activity partially becoming manifest to consciousness without losing its own character. At such moments one has an intuition of the difference between the mechanisms of the two egos.

From the moment I bought my ticket, I had a premonition I wasn’t returning to New York anytime soon. You Know, this happens a lot to Russians. The Soviet Union is gone, and the borders are as free and passable as they’ve ever been. And yet, when a Russian moves between the two universes, this feeling of finality persists, the logical impossibility of a place like Russia existing alongside the civilized world, of Ann Arbor, Michigan, sharing the same atmosphere with, say, Vladivostok. It was like those mathematical concepts I could never understand in high school: if, then. If Russia exists, then the West is a mirage; conversely, if Russia does not exist, then and only then is the West real and tangible. No wonder young people talk about “going beyond the cordon” when they talk of emigrating, as if Russia were ringed by a vast cordon sanitaire. Either you stay in the leper colony or you get out into the wider world and maybe try to spread your disease to others.

Questioner: I am full of hate. Will you please teach me how to love? KRISHNAMURTI: No one can teach you how to love. If people could be taught how to love, the world problem would be very simple, would it not? If we could learn how to love from a book as we learn mathematics, this would be a marvellous world; there would be no hate, no exploitation, no wars, no division of rich and poor, and we would all be really friendly with each other. But love is not so easily come by. It is easy to hate, and hate brings people together after a fashion; it creates all kinds of fantasies, it brings about various types of cooperation as in war. But love is much more difficult. You cannot learn how to love, but what you can do is to observe hate and put it gently aside. Don’t battle against hate, don’t say how terrible it is to hate people, but see hate for what it is and let it drop away; brush it aside, it is not important. What is important is not to let hate take root in your mind. Do you understand? Your mind is like rich soil, and if given sufficient time any problem that comes along takes root like a weed, and then you have the trouble of pulling it out; but if you do not give the problem sufficient time to take root, then it has no place to grow and it will wither away. If you encourage hate, give it time to take root, to grow, to mature, it becomes an enormous problem. But if each time hate arises you let it go by, then you will find that your mind becomes very sensitive without being sentimental; therefore it will know love. The mind can pursue sensations, desires, but it cannot pursue love. Love must come to the mind. And, when once love is there, it has no division as sensuous and divine: it is love. That is the extraordinary thing about love: it is the only quality that brings a total comprehension of the whole of existence.

This book is an essay in what is derogatorily called "literary economics," as opposed to mathematical economics, econometrics, or (embracing them both) the "new economic history." A man does what he can, and in the more elegant - one is tempted to say "fancier" - techniques I am, as one who received his formation in the 1930s, untutored. A colleague has offered to provide a mathematical model to decorate the work. It might be useful to some readers, but not to me. Catastrophe mathematics, dealing with such events as falling off a height, is a new branch of the discipline, I am told, which has yet to demonstrate its rigor or usefulness. I had better wait. Econometricians among my friends tell me that rare events such as panics cannot be dealt with by the normal techniques of regression, but have to be introduced exogenously as "dummy variables." The real choice open to me was whether to follow relatively simple statistical procedures, with an abundance of charts and tables, or not. In the event, I decided against it. For those who yearn for numbers, standard series on bank reserves, foreign trade, commodity prices, money supply, security prices, rate of interest, and the like are fairly readily available in the historical statistics.

Says the Cardinal: "Freethought leads to Atheism, to the destruction of social and civil order, and to the overthrow of government." I accept the gentleman's statement; I credit him with much intellectual acumen for perceiving that which many freethinkers have failed to perceive: accepting it, I shall do my best to prove it, and then endeavor to show that this very iconoclastic principle is the salvation of the economic slave and the destruction of the economic tyrant. ... Hence the freethinker who recognizes the science of astronomy, the science of mathematics, and the equally positive and exact science of justice, is logically forced to the denial of supreme authority. For no human being who observes and reflects can admit a supreme tyrant and preserve his self-respect. No human mind can accept the dogma of divine despotism and the doctrine of eternal justice at the same time; they contradict each other, and it takes two brains to hold them. The cardinal is right: freethought does logically lead to atheism, if by atheism he means the denial of supreme authority.

Sometimes it's hard because I don't like to hurt people's feelings. So there have been times when a friend will get a haircut and I will see it and my initial reaction is "Oh my God, you look like a streetwalker who got caught in a wind tunnel." But I obviously can't say that because that would be an insult to streetwalkers. So I have to say, "I love it! It looks great!" But when I say it my voice goes up about three octaves. "It looks greee-aaattt!" So I'm certain they know I'm lying. How come when we lie our voices go up so many octaves? It's a dead giveaway. It happens when we dole out compliments we don't mean and it happens when we say things like "You didn't have to get my anything!" or "What do you mean you weren't invited to my party? You're always invited!" Everyone knows what those mean. "You definitely had to get me something" and "You haven't been invited back to the house since the urn incident of '04." And it's a mathematical fact: the higher the octave, the bigger the lie. "I didn't even hear my phone ring!" is usually like a four on the scale. "You think I'm sleeping with someone else?" is off the charts. I can tell when people are lying to me when they start their sentence with "I have to be honest with you." They may as well say, "Listen, I'm about to lie straight to your face." Why do people need to clarify when they're being honest? Does that mean everything else they've ever said has been a lie? Yesterday they said they liked my sweater but they didn't say they were being honest. Does that mean they hated it? It's so strange to me. It almost feels like they're giving me the option to not hear the truth. As if when they say, "I have to be honest with you," I might say, "No, no. Please. Only lies right now.

Q. Would you repeat, Dr. Seldon, your thoughts concerning the future of Trantor? A. I have said, and I say again, that Trantor will lie in ruins within the next three centuries. Q. You do not consider your statement a disloyal one? A. No, sir. Scientific truth is beyond loyalty and disloyalty." Q. You are sure that your statement represents scientific truth? A. I am. Q. On what basis? A. On the basis of the mathematics of psychohistory. Q. Can you prove that this mathematics is valid? A. Only to another mathematician. Q. ( with a smile) Your claim then is that your truth is of so esoteric a nature that it is beyond the understanding of a plain man. It seems to me that truth should be clearer than that, less mysterious, more open to the mind. A. It presents no difficulties to some minds. The physics of energy transfer, which we know as thermodynamics, has been clear and true through all the history of man since the mythical ages, yet there may be people present who would find it impossible to design a power engine. People of high intelligence, too. I doubt if the learned Commissioners— At this point, one of the Commissioners leaned toward the Advocate. His words were not heard but the hissing of the voice carried a certain asperity. The Advocate flushed and interrupted Seldon. Q. We are not here to listen to speeches, Dr. Seldon. Let us assume that you have made your point. Let me suggest to you that your predictions of disaster might be intended to destroy public confidence in the Imperial Government for purposes of your own! A. That is not so. Q. Let me suggest that you intend to claim that a period of time preceding the so-called ruin of Trantor will be filled with unrest of various types. A. That is correct. Q. And that by the mere prediction thereof, you hope to bring it about, and to have then an army of a hundred thousand available. A. In the first place, that is not so. And if it were, investigation will show you that barely ten thousand are men of military age, and none of these has training in arms. Q. Are you acting as an agent for another? A. I am not in the pay of any man, Mr. Advocate. Q. You are entirely disinterested? You are serving science? A. I am.

Of course, I’ve only brought up two examples. Other universal laws of physics have been used as weapons as well, though we don’t know all of them. It’s very possible that every law of physics has been weaponized. It’s possible that in some parts of the universe, even … Forget it, I don’t even believe that.” “What were you going to say?” “The foundation of mathematics.” Cheng Xin tried to imagine it, but it was simply impossible. “That’s … madness.” Then she asked, “Will the universe turn into a war ruin? Or, maybe it’s more accurate to ask: Will the laws of physics turn into war ruins?” “Maybe they already are.… The physicists and cosmologists of the new world are focused on trying to recover the original appearance of the universe before the wars more than ten billion years ago. They’ve already constructed a fairly clear theoretical model describing the pre-war universe. That was a really lovely time, when the universe itself was a Garden of Eden. Of course, the beauty could only be described mathematically. We can’t picture it: Our brains don’t have enough dimensions.” Cheng Xin thought back to the conversation with the Ring again. Did you build this four-dimensional fragment? You told me that you came from the sea. Did you build the sea? “You are saying that the universe of the Edenic Age was four-dimensional, and that the speed of light was much higher?” “No, not at all. The universe of the Edenic Age was ten-dimensional. The speed of light back then wasn’t only much higher—rather, it was close to infinity. Light back then was capable of action at a distance, and could go from one end of the cosmos to the other within a Planck time.… If you had been to four-dimensional space, you would have some vague hint of how beautiful that ten-dimensional Garden must have been.” “You’re saying—” “I’m not saying anything.” Yifan seemed to have awakened from a dream. “We’ve only seen small hints; everything else is just guessing. You should treat it as a guess, just a dark myth we’ve made up.” But Cheng Xin continued to follow the course of the discussion taken so far. “—that during the wars after the Edenic Age, one dimension after another was imprisoned from the macroscopic into the microscopic, and the speed of light was reduced again and again.…” “As I said, I’m not saying anything, just guessing.” Yifan’s voice grew softer. “But no one knows if the truth is even darker than our guesses.… We are certain of only one thing: The universe is dying.” The

The artistically inclined delight in the Game because it provides opportunities for improvisation and fantasy. The strict scholars and scientists despise it – and so do some musicians also – because, they say, it lacks that degree of strictness which their specialties can achieve. Well and good, you will encounter these antinomies, and in time you will discover that they are subjective, not objective – that, for example, a fancy-free artist avoids pure mathematics or logic not because he understands them and could say something about them if he wished, but because he instinctively inclines toward other things. Such instinctive and violent inclinations and disinclinations are signs by which you can recognize the pettier souls. In great souls and superior minds, these passions are not found. Each of us is merely one human being, merely an experiment, a way station. But each of us should be on the way toward perfection, should be striving to reach the center, not the periphery. Remember this: one can be a strict logician or grammarian, and at the same time full of imagination and music. One can be a musician or Glass Bead Game player and at the same time wholly devoted to rule and order. The kind of person we want to develop, the kind of person we aim to become, would at any time be able to exchange his discipline or art for any other. He would infuse the Glass Bead Game with crystalline logic, and grammar with creative imagination. That is how we ought to be. We should be so constituted that we can at any time be placed in a different position without offering resistance or losing our heads.

In adolescence, I hated life and was continually on the verge of suicide, from which, however, I was restrained by the desire to know more mathematics. Now, on the contrary, I enjoy life; I might almost say that with every year that passes I enjoy it more…very largely it is due to a diminishing preoccupation with myself. Like others who had a Puritan education, I had the habit of meditating on my sins, follies, and shortcomings. I seemed to myself - no doubt justly - a miserable specimen. Gradually I learned to be indifferent to myself and my deficiencies; I came to center my attention increasingly upon external objects: the state of the world, various branches of knowledge, individuals for whom I felt affection…And every external interest inspires some activity which, so long as the interest remains alive, is a complete preventive of ennui. Interest in oneself, on the contrary, leads to no activity of a progressive kind. It may lead to the keeping of a diary, to getting psychoanalyzed, or perhaps to becoming a monk. But the monk will not be happy until the routine of the monastery has made him forget his own soul. The happiness which he attributes to religion he could have obtained from becoming a crossing-sweeper, provided he were compelled to remain one. External discipline is the only road to happiness for those unfortunates whose self-absorption is too profound to be cured in any other way.

In learning any subject of a technical nature where mathematics plays a role, one is confronted with the task of understanding and storing away in the memory a huge body of facts and ideas, held together by certain relationships which can be “proved” or “shown” to exist between them. It is easy to confuse the proof itself with the relationship which it establishes. Clearly, the important thing to learn and to remember is the relationship, not the proof. In any particular circumstance we can either say “it can be shown that” such and such is true, or we can show it. In almost all cases, the particular proof that is used is concocted, first of all, in such form that it can be written quickly and easily on the chalkboard or on paper, and so that it will be as smooth-looking as possible. Consequently, the proof may look deceptively simple, when in fact, the author might have worked for hours trying different ways of calculating the same thing until he has found the neatest way, so as to be able to show that it can be shown in the shortest amount of time! The thing to be remembered, when seeing a proof, is not the proof itself, but rather that it can be shown that such and such is true. Of course, if the proof involves some mathematical procedures or “tricks” that one has not seen before, attention should be given not to the trick exactly, but to the mathematical idea involved.

Do you know what the foundation of mathematics is?" I ask. "The foundation of mathematics is numbers. If anyone asked me what makes me truly happy, I would say: numbers. Snow and ice and numbers. And do you know why?" He splits the claws with a nutcracker and pulls out the meat with curved tweezers. "Because the number system is like human life. First you have the natural numbers. The ones that are whole and positive. The numbers of a small child. But human consciousness expands. The child discovers a sense of longing, and do you know what the mathematical expression is for longing?" He adds cream and several drops of orange juice to the soup. "The negative numbers. The formalization of the feeling that you are missing something. And human consciousness expands and grows even more, and the child discovers the in between spaces. Between stones, between pieces of moss on the stones, between people. And between numbers. And do you know what that leads to? It leads to fractions. Whole numbers plus fractions produce rational numbers. And human consciousness doesn't stop there. It wants to go beyond reason. It adds an operation as absurd as the extraction of roots. And produces irrational numbers." He warms French bread in the oven and fills the pepper mill. "It's a form of madness.' Because the irrational numbers are infinite. They can't be written down. They force human consciousness out beyond the limits. And by adding irrational numbers to rational numbers, you get real numbers.

We know, however, that the mind is capable of understanding these matters in all their complexity and in all their simplicity. A ball flying through the air is responding to the force and direction with which it was thrown, the action of gravity, the friction of the air which it must expend its energy on overcoming, the turbulence of the air around its surface, and the rate and direction of the ball's spin. And yet, someone who might have difficulty consciously trying to work out what 3 x 4 x 5 comes to would have no trouble in doing differential calculus and a whole host of related calculations so astoundingly fast that they can actually catch a flying ball. People who call this "instinct" are merely giving the phenomenon a name, not explaining anything. I think that the closest that human beings come to expressing our understanding of these natural complexities is in music. It is the most abstract of the arts - it has no meaning or purpose other than to be itself. Every single aspect of a piece of music can be represented by numbers. From the organization of movements in a whole symphony, down through the patterns of pitch and rhythm that make up the melodies and harmonies, the dynamics that shape the performance, all the way down to the timbres of the notes themselves, their harmonics, the way they change over time, in short, all the elements of a noise that distinguish between the sound of one person piping on a piccolo and another one thumping a drum - all of these things can be expressed by patterns and hierarchies of numbers. And in my experience the more internal relationships there are between the patterns of numbers at different levels of the hierarchy, however complex and subtle those relationships may be, the more satisfying and, well, whole, the music will seem to be. In fact the more subtle and complex those relationships, and the further they are beyond the grasp of the conscious mind, the more the instinctive part of your mind - by which I mean that part of your mind that can do differential calculus so astoundingly fast that it will put your hand in the right place to catch a flying ball- the more that part of your brain revels in it. Music of any complexity (and even "Three Blind Mice" is complex in its way by the time someone has actually performed it on an instrument with its own individual timbre and articulation) passes beyond your conscious mind into the arms of your own private mathematical genius who dwells in your unconscious responding to all the inner complexities and relationships and proportions that we think we know nothing about. Some people object to such a view of music, saying that if you reduce music to mathematics, where does the emotion come into it? I would say that it's never been out of it.

I had better say something here about this question of age, since it is particularly important for mathematicians. No mathematician should ever allow himself to forget that mathematics, more than any other art or science, is a young man's game. To take a simple illustration at a comparatively humble level, the average age of election to the Royal Society is lowest in mathematics. We can naturally find much more striking illustrations. We may consider, for example, the career of a man who was certainly one of the world's three greatest mathematicians. Newton gave up mathematics at fifty, and had lost his enthusiasm long before; he had recognized no doubt by the time he was forty that his greatest creative days were over. His greatest idea of all, fluxions and the law of gravitation, came to him about 1666 , when he was twentyfour—'in those days I was in the prime of my age for invention, and minded mathematics and philosophy more than at any time since'. He made big discoveries until he was nearly forty (the 'elliptic orbit' at thirty-seven), but after that he did little but polish and perfect. Galois died at twenty-one, Abel at twenty-seven, Ramanujan at thirty-three, Riemann at forty. There have been men who have done great work a good deal later; Gauss's great memoir on differential geometry was published when he was fifty (though he had had the fundamental ideas ten years before). I do not know an instance of a major mathematical advance initiated by a man past fifty. If a man of mature age loses interest in and abandons mathematics, the loss is not likely to be very serious either for mathematics or for himself.

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.

In South Texas I saw three interesting things. The first was a tiny girl, maybe ten years old, driving in a 1965 Cadillac. She wasn't going very fast, because I passed her, but still she was cruising right along, with her head tilted back and her mouth open and her little hands gripping the wheel. Then I saw an old man walking up the median strip pulling a wooden cross behind him. It was mounted on something like a golf cart with two spoked wheels. I slowed down to read the hand-lettered sign on his chest. JACKSONVILLE FLA OR BUST I had never been to Jacksonville but I knew it was the home of the Gator Bowl and I had heard it was a boom town, taking in an entire county or some such thing. It seemed an odd destination for a religious pilgrim. Penance maybe for some terrible sin, or some bargain he had worked out with God, or maybe just a crazed hiker. I waved and called out to him, wishing him luck, but he was intent on his marching and had no time for idle greetings. His step was brisk and I was convinced he wouldn't bust. The third interesting thing was a convoy of stake-bed trucks all piled high with loose watermelons and cantaloupes. I was amazed. I couldn't believe that the bottom ones weren't crushed under all that weight, exploding and spraying hazardous melon juice onto the highway. One of nature's tricks with curved surfaces. Topology! I had never made it that far in mathematics and engineering studies, and I knew now that I never would, just as I knew that I would never be a navy pilot or a Treasury agent. I made a B in Statics but I was failing in Dynamics when I withdrew from the field. The course I liked best was one called Strength of Materials. Everybody else hated it because of all the tables we had to memorize but I loved it, the sheared beam. I had once tried to explain to Dupree how things fell apart from being pulled and compressed and twisted and bent and sheared but he wouldn't listen. Whenever that kind of thing came up, he would always say - boast, the way those people do - that he had no head for figures and couldn't do things with his hands, slyly suggesting the presence of finer qualities.

If we analyse the classes of life, we readily find that there are three cardinal classes which are radically distinct in function. A short analysis will disclose to us that, though minerals have various activities, they are not "living." The plants have a very definite and well known function-the transformation of solar energy into organic chemical energy. They are a class of life which appropriates one kind of energy, converts it into another kind and stores it up; in that sense they are a kind of storage battery for the solar energy; and so I define THE PLANTS AS THE CHEMISTRY-BINDING class of life. The animals use the highly dynamic products of the chemistry-binding class-the plants-as food, and those products-the results of plant-transformation-undergo in animals a further transformation into yet higher forms; and the animals are correspondingly a more dynamic class of life; their energy is kinetic; they have a remarkable freedom and power which the plants do not possess-I mean the freedom and faculty to move about in space; and so I define ANIMALS AS THE SPACE-BINDING CLASS OF LIFE. And now what shall we say of human beings? What is to be our definition of Man? Like the animals, human beings do indeed possess the space-binding capacity but, over and above that, human beings possess a most remarkable capacity which is entirely peculiar to them-I mean the capacity to summarise, digest and appropriate the labors and experiences of the past; I mean the capacity to use the fruits of past labors and experiences as intellectual or spiritual capital for developments in the present; I mean the capacity to employ as instruments of increasing power the accumulated achievements of the all-precious lives of the past generations spent in trial and error, trial and success; I mean the capacity of human beings to conduct their lives in the ever increasing light of inherited wisdom; I mean the capacity in virtue of which man is at once the heritor of the by-gone ages and the trustee of posterity. And because humanity is just this magnificent natural agency by which the past lives in the present and the present for the future, I define HUMANITY, in the universal tongue of mathematics and mechanics, to be the TIME-BINDING CLASS OF LIFE.

The relationship between the Sophotechs and the men as depicted in that tale made no sense. How could they be hostile to each other?” Diomedes said, “Aren’t men right to fear machines which can perform all tasks men can do, artistic, intellectual, technical, a thousand or a million times better than they can do? Men become redundant.” Phaethon shook his head, a look of distant distaste on his features, as if he were once again confronted with a falsehood that would not die no matter how often it was denounced. In a voice of painstaking patience, he said: “Efficiency does not harm the inefficient. Quite the opposite. That is simply not the way it works. Take me, for example. Look around: I employed partials to do the thought-box junction spotting when I built this ship. My employees were not as skilled as I was in junction spotting. It took them three hours to do the robopsychology checks and hierarchy links I could have done in one hour. But they were in no danger of competition from me. My time is too valuable. In that same hour it would have taken me to spot their thought-box junction, I can earn far more than their three-hour wages by writing supervision architecture thought flows. And it’s the same with me and the Sophotechs. “Any midlevel Sophotech could have written in one second the architecture it takes me, even with my implants, an hour to compose. But if, in that same one second of time, that Sophotech can produce something more valuable—exploring the depth of abstract mathematics, or inventing a new scientific miracle, anything at all (provided that it will earn more in that second than I earn in an hour)—then the competition is not making me redundant. The Sophotech still needs me and receives the benefit of my labor. Since I am going to get the benefit of every new invention and new miracle put out on the market, I want to free up as many of those seconds of Sophotech time as my humble labor can do. “And I get the lion’s share of the benefit from the swap. I only save him a second of time; he creates wonder upon wonder for me. No matter what my fear of or distaste for Sophotechs, the forces in the marketplace, our need for each other, draw us together. “So you see why I say that not a thing the Silent One said about Sophotechs made sense. I do not understand how they could have afforded to hate each other. Machines don’t make us redundant; they increase our efficiency in every way. And the bids of workers eager to compete for Sophotech time creates a market for merely human work, which it would not be efficient for Sophotechs to underbid.