Mathematics Day Quotes

Once upon a time, there was a girl who talked to the moon. And she was mysterious and she was perfect, in that way that girls who talk to moons are. In the house next door, there lived a boy. And the boy watched the girl grow more and more perfect, more and more beautiful with each passing year. He watched her watch the moon. And he began to wonder if the moon would help him unravel the mystery of the beautiful girl. So the boy looked into the sky. But he couldn't concentrate on the moon. He was too distracted by the stars. And it didn't matter how many songs or poems had already been written about them, because whenever he thought about the girl, the stars shone brighter. As if she were the one keeping them illuminated. One day, the boy had to move away. He couldn't bring the girl with him, so he brought the stars. When he'd look out his window at night, he would start with one. One star. And the boy would make a wish on it, and the wish would be her name. At the sound of her name, a second star would appear. And then he'd wish her name again, and the stars would double into four. And four became eight, and eight became sixteen, and so on, in the greatest mathematical equation the universe had ever seen. And by the time an hour had passed, the sky would be filled with so many stars that it would wake the neighbors. People wondered who'd turned on the floodlights. The boy did. By thinking about the girl.

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.

Time grabs you by the scruff of your neck and drags you forward. You get over it, of course. Everyone was right about that. One mathematically insignificant day, you stop hoping for happiness and become actually happy.

[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.

Life has puffed and blown itself into a summer day, and clouds and spring billow over the heavens as if calendars were a listing of mathematical errors.

Naturally, we are inclined to be so mathematical and calculating that we look upon uncertainty as a bad thing...Certainty is the mark of the common-sense life. To be certain of God means that we are uncertain in all our ways, we do not know what a day may bring forth. This is generally said with a sigh of sadness; it should rather be an expression of breathless expectation.

The only reason I don’t know more about love is because there just isn’t more to know. In fact, I’ve reduced love to a mathematical formula: Hdgk(X)=H2k(X,Q)∩Hk,k(X). Actually, that’s not right. That’s the statement piece of the Hodge conjecture, but I’m sure you already knew that.


I am never forget the day I first meet the great Lobachevsky. In one word he told me secret of success in mathematics: Plagiarize! Plagiarize! Let no one else's work evade your eyes! Remember why the good Lord made your eyes! So don't shade your eyes, But plagiarize, plagiarize, plagiarize - Only be sure always to call it please 'research'." [Lobachevsky]

In truth, we will only perform any action a certain number of times, and to know that can never be helpful. There is, in my opinion, no use in demanding to know the number, in demanding to know upon waking the number of boxes to be ticked off every single day. After all, why would it help to be shown the mathematics of things, when instead we could simply imagine that whatever time we have is limitless.

How happy is the lot of the mathematician! He is judged solely by his peers, and the standard is so high that no colleague or rival can ever win a reputation he does not deserve. No cashier writes a letter to the press complaining about the incomprehensibility of Modern Mathematics and comparing it unfavorably with the good old days when mathematicians were content to paper irregularly shaped rooms and fill bathtubs without closing the waste pipe.

What I love about cooking is that after a hard day, there is something comforting about the fact that if you melt butter and add flour and then hot stock, it will get thick! It’s a sure thing! It’s sure thing in a world where nothing is sure; it has a mathematical certainty in a world where those of us who long for some kind of certainty are forced to settle for crossword puzzles.

Each pleasure we feel is a pleasure less; each day a stroke on a calendar. What we will not accept is that the joy in the day and the passing of the day are inseparable. What makes our existence worthwhile is precisely that its worth and its while - its quality and duration - are as impossible to unravel as time and space in mathematics of relativity.

Give a man a fish, he’ll eat for a day. Teach a man to fish, he’ll eat for a lifetime.”? It’s the same for mathematics education for twenty-first century life.

The calculative exactness of practical life which the money economy has brought about corresponds to the ideal of natural science: to transform the world by mathematical formulas. Only money economy has filled the days of so many people with weighing, calculating, with numerical determinations, with a reduction of qualitative values to quantitative ones.

Miri took genuine comfort in studying Mathematics that day. She could sort numbers into two simple ideas: true and not true. Unlike numbers, words were rarely just one thing. They moved and changed, camouflaging and leaping out unexpectedly. Words were slippery and alive; words wrestled out of her grip and became something new. Words were dangerous.

You know,” he said, “this design begins to appeal to me after all. Sea slugs aren’t the least bit arousing, but logarithms . . . I’ve always thought that word sounded splendidly naughty.” He let it roll off his tongue with ribald inflection. “Logarithm.” He gave an exaggerated shiver. “Ooh. Yes and thank you and may I have some more.” “Lots of mathematical terms sound that way. I think it’s because they were all coined by men. ‘Hypotenuse’ is downright lewd.” “ ‘Quadrilateral’ brings rather carnal images to mind.” She was silent for a long time. Then one of her dark eyebrows arched. “Not so many as ‘rhombus.’ ” Good Lord. That word was wicked. Her pronunciation of it did rather wicked things to him. He had to admire the way she didn’t shrink from a challenge, but came back with a new and surprising retort. One day, she’d make some fortunate man a very creative lover.

There is no branch of mathematics, however abstract, which may not some day be applied to phenomena of the real world.

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)

She would not have cared to confess how infinitely she preferred the exactitude, the star-like impersonality, of figures to the confusion, agitation, and vagueness of the finest prose.

The Greeks were the first mathematicians who are still ‘real’ to us to-day. Oriental mathematics may be an interesting curiosity, but Greek mathematics is the real thing. The Greeks first spoke a language which modern mathematicians can understand: as Littlewood said to me once, they are not clever schoolboys or ‘scholarship candidates’, but ‘Fellows of another college’. So Greek mathematics is ‘permanent’, more permanent even than Greek literature. Archimedes will be remembered when Aeschylus is forgotten, because languages die and mathematical ideas do not. ‘Immortality’ may be a silly word, but probably a mathematician has the best chance of whatever it may mean.

And so to read is, in truth, to be in the constant act of creation. The old lady on the bus with her Orwell, the businessman on the Tube with Patricia Cornwell, the teenager roaring through Capote -- they are not engaged in idle pleasure. Their heads are on fire. Their hearts are flooding. With a book, you are the landscape, the sets, the snow, the hero, the kiss -- you are the mathematical calculation that plots the trajectory of the blazing, crashing zeppelin. You -- pale, punchable reader -- are terraforming whole worlds in your head, which will remain with you until the day you die. These books are as much a part of you as your guts and your bone. And when your guts fail and your bones break, Narnia, or Jamaica Inn, or Gormenghast will still be there; as pin-sharp and bright as the day you first imagined them -- hiding under the bedclothes, sitting on the bus. Exhausted, on a rainy day, weeping over the death of someone you never met, and who was nothing more than words until you transfused them with your time, and your love, and the imagination you constantly dismiss as "just being a bit of a bookworm.

In the past, Cameron, worrying about Ian, would go make sure that he wasn’t sitting alone in a huddle, or staring for hours at a Ming bowl, or pouring over some endless mathematical exercise. These days, Cameron knew that Ian used the excuse of not liking crowds to spend more time alone with his wife – in bed.

At night I would return home, set out a lamp before me, and devote myself to reading and writing. Whenever sleep overcame me or I became conscious of weakening, I would turn aside to drink a cup of wine, so that my strength would return to me. Then I would return to reading. And whenever sleep seized me I would see those very problems in my dream; and many questions became clear to me in my sleep. I continued in this until all of the sciences were deeply rooted within me and I understood them as is humanly possible. Everything which I knew at the time is just as I know it now; I have not added anything to it to this day. Thus I mastered the logical, natural, and mathematical sciences, and I had now reached the science.

They held the funeral on the second day, with the town coming to look at Miss Emily beneath a mass of bought flowers with the crayon face of her father musing profoundly above the bier and the ladies sibilant and macabre; and the very old men - some in their brushed Confederate uniforms - on the porch and the lawn, talking of Miss Emily as if she had been a contemporary of theirs, believing that they had danced with her and courted her perhaps, confusing time with its mathematical progression, as the old do, to whom all the past is not a diminishing road but, instead, a huge meadow which no winter ever quite touches, divided from them now by the narrow bottle-neck of the most recent decade of years.

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.

Whenever I meet in Laplace with the words 'Thus it plainly appears', I am sure that hours and perhaps days, of hard study will alone enable me to discover how it plainly appears.

God loves an enquiring mind, a fact that has been a great encouragement to me in my study of mathematics and the history and philosophy of science.

Astrology is the study of man’s response to planetary stimuli. The stars have no conscious benevolence or animosity; they merely send forth positive and negative radiations. Of themselves, these do not help or harm humanity, but offer a lawful channel for the outward operation of cause-effect equilibriums which each man has set into motion in the past. “A child is born on that day and at that hour when the celestial rays are in mathematical harmony with his individual karma. His horoscope is a challenging portrait, revealing his unalterable past and its probable future results. But the natal chart can be rightly interpreted only by men of intuitive wisdom: these are few.

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.

There was a fine mathematics for how to mete out sedation. The goal for most days was to get to a point where I could drift off easily, and come to without being startled. My thoughts were banal. My pulse was casual. Only the coffee made my heart work a bit harder. Caffeine was my exercise.

The tantalizing and compelling pursuit of mathematical problems offers mental absorption, peace of mind amid endless challenges, repose in activity, battle without conflict, “refuge from the goading urgency of contingent happenings," and the sort of beauty changeless mountains present to sense tried by the present-day kaleidoscope of events.

Seven is my favorite number," he said. "Why?" He nuzzled gently at her stomach. "There are seven colors in a rainbow, seven days of the week, and..." His voice lowered seductively, "...seven is the lowest natural number that can't be represented as the sum of the squares of three integers." "Mathematics," she exclaimed, laughing breathlessly. "How stirring.

Jesus probably studied this same information, in his youth. The apostle Paul probably studied this same information. How can I make such a bold assertion? Because, without this knowledge, much of the New Testament would make no sense. Many of the idioms used in the New Testament are the result of lessons learned from this ancient Hebrew education system. Unfortunately, what was common in their day, has become forgotten in ours. For a Hebrew, math doesn’t get in the way. It blazes the way. Other languages are disconnected from this mathematical relationship . . . and it shows.

...if there really is some day discovered a formula for all our desires and caprices - that is, an explanation of what they depend upon, by what laws they arise, how they develop, what they are aiming at in one case and in another and so on, that is a real mathematical formula - then, most likely, man will at once cease to feel desire, indeed, he will be certain to. For who would want to choose by rule? Besides, he will at once be transformed from a human being into an organ-stop or something of that sort; for what is a man without desires, without freewill and without choice, if not a stop in an organ?

Something else gets under your skin, keeps you working days and nights at the sacrifice of your sleeping and eating and attention to your family and friends, something beyond the love of puzzle solving. And that other force is the anticipation of understanding something about the world that no one has ever understood before you. Einstein wrote that when he first realized that gravity was equivalent to acceleration -- an idea that would underlie his new theory of gravity -- it was the "happiest thought of my life." On projects of far smaller weight, I have experienced that pleasure of discovering something new. It is an exquisite sensation, a feeling of power, a rush of the blood, a sense of living forever. To be the first vessel to hold this new thing. All of the scientists I've known have at least one more quality in common: they do what they do because they love it, and because they cannot imagine doing anything else. In a sense, this is the real reason a scientist does science. Because the scientist must. Such a compulsion is both blessing and burden. A blessing because the creative life, in any endeavor, is a gift filled with beauty and not given to everyone, a burden because the call is unrelenting and can drown out the rest of life. This mixed blessing and burden must be why the astrophysicist Chandrasekhar continued working until his mid-80's, why a visitor to Einstein's apartment in Bern found the young physicist rocking his infant with one hand while doing mathematical calculations with the other. This mixed blessing and burden must have been the "sweet hell" that Walt Whitman referred to when he realized at a young age that he was destined to be a poet. "Never more," he wrote, "shall I escape.

. . . 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.

For example, knowing that it takes only about eleven and a half days for a million seconds to tick away, whereas almost thirty-two years are required for a billion seconds to pass, gives one a better grasp of the relative magnitudes of these two common numbers.

You can FEEL the wave of emotion online when something is about to go viral, good or bad. A scientist I met once mathematically compared internet behavior to swarm behavior seen in starlings or locusts.

[It] wants you to believe there are foreseeable trends and forces. When in fact it's all random phenomena. You apply mathematics and other disciplines, yes. But in the end you're dealing with a system that's out of control. Hysteria at high speeds, day to day, minute to minute. "People in free societies don't have to fear the pathology of the state. We create our own frenzy, our own mass convulsions, driven by thinking machines that we have no final authority over. The frenzy is barely noticeable most of the time. It's simply how we live.

Our generation is witness to a development of physical knowledge such as has not been seen since the days of Kepler, Galileo and Newton, and mathematics has scarcely ever experienced such a stormy epoch. Mathematical thought removes the spirit from its worldly haunts to solitude and renounces the unveiling of the secrets of Nature. But as recompense, mathematics is less bound to the course of worldly events than physics.

Meanwhile, someplace in the world, somebody is making love and another a poem. Elsewhere in the universe, a star manyfold the mass of our third-rate sun is living out its final moments in a wild spin before collapsing into a black hole, its exhale bending spacetime itself into a well of nothingness that can swallow every atom that ever touched us and every datum we ever produced, every poem and statue and symphony we’ve ever known—an entropic spectacle insentient to questions of blame and mercy, devoid of why. “In four billion years, our own star will follow its fate, collapsing into a white dwarf. We exist only by chance, after all. The Voyager will still be sailing into the interstellar shorelessness on the wings of the “heavenly breezes” Kepler had once imagined, carrying Beethoven on a golden disc crafted by a symphonic civilization that long ago made love and war and mathematics on a distant blue dot. But until that day comes, nothing once created ever fully leaves us. Seeds are planted and come abloom generations, centuries, civilizations later, migrating across coteries and countries and continents. Meanwhile, people live and people die—in peace as war rages on, in poverty and disrepute as latent fame awaits, with much that never meets its more, in shipwrecked love. I will die. You will die. The atoms that huddled for a cosmic blink around the shadow of a self will return to the seas that made us. What will survive of us are shoreless seeds and stardust.

Plenty of mathematicians, Hardy knew, could follow a step-by-step discursus unflaggingly—yet counted for nothing beside Ramanujan. Years later, he would contrive an informal scale of natural mathematical ability on which he assigned himself a 25 and Littlewood a 30. To David Hilbert, the most eminent mathematician of the day, he assigned an 80. To Ramanujan he gave 100.

But that would mean it was originally a sideways number eight. That makes no sense at all. Unless..." She paused as understanding dawned. "You think it was the symbol for infinity?" "Yes, but not the usual one. A special variant. Do you see how one line doesn't fully connect in the middle? That's Euler's infinity symbol. Absolutus infinitus." "How is it different from the usual one?" "Back in the eighteenth century, there were certain mathematical calculations no one could perform because they involved series of infinite numbers. The problem with infinity, of course, is that you can't come up with a final answer when the numbers keep increasing forever. But a mathematician named Leonhard Euler found a way to treat infinity as if it were a finite number- and that allowed him to do things in mathematical analysis that had never been done before." Tom inclined his head toward the date stone. "My guess is whoever chiseled that symbol was a mathematician or scientist." "If it were my date stone," Cassandra said dryly, "I'd prefer the entwined hearts. At least I would understand what it means." "No, this is much better than hearts," Tom exclaimed, his expression more earnest than any she'd seen from him before. "Linking their names with Euler's infinity symbol means..." He paused, considering how best to explain it. "The two of them formed a complete unit... a togetherness... that contained infinity. Their marriage had a beginning and end, but every day of it was filled with forever. It's a beautiful concept." He paused before adding awkwardly, "Mathematically speaking.

At Ge 1:1 God used a matrix of sevens: (1) Seven words. (2) 28 letters (28 ÷ 4 = 7). (3) First three words contain 14 letters (14 ÷ 2 = 7). (4) Last four words contain 14 letters (14 ÷ 2 = 7). (5) Fourth and fifth words have seven letters. (6) Sixth and seventh words have seven letters. (7) Key words (God, heaven, earth) contain 14 letters (14 ÷ 2 = 7). (8) Remaining words contain 14 letters (14 ÷ 2 = 7). (9) Numeric value of first, middle and last letters equal, 133 (133 ÷ 19 = 7). (10) Numeric value of the first and last letters of all seven words equal 1,393 (1,393 ÷ 199 = 7). (11) The book of Genesis has 78,064 letters (78,064 ÷ 11,152 = 7). So, what is the big deal about seven? Jesus is our Shiva (7), our Shabbat (7th day). (Lu 6:5) You couldn’t see this messianic reference, however, unless you are reading in Hebrew. This book is the beginning of an amazing pilgrimage.

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

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.

He'd met other prodigies in mathematical competitions. In fact he'd been thoroughly trounced by competitors who probably spent literally all day practising maths problems and who'd never read a science-fiction book and who would burn out completely before puberty and never amount to anything in their future lives because they'd just practised known techniques instead of learning to think creatively. (Harry was something of a sore loser.)

Is it possible that the Pentateuch could not have been written by uninspired men? that the assistance of God was necessary to produce these books? Is it possible that Galilei ascertained the mechanical principles of 'Virtual Velocity,' the laws of falling bodies and of all motion; that Copernicus ascertained the true position of the earth and accounted for all celestial phenomena; that Kepler discovered his three laws—discoveries of such importance that the 8th of May, 1618, may be called the birth-day of modern science; that Newton gave to the world the Method of Fluxions, the Theory of Universal Gravitation, and the Decomposition of Light; that Euclid, Cavalieri, Descartes, and Leibniz, almost completed the science of mathematics; that all the discoveries in optics, hydrostatics, pneumatics and chemistry, the experiments, discoveries, and inventions of Galvani, Volta, Franklin and Morse, of Trevithick, Watt and Fulton and of all the pioneers of progress—that all this was accomplished by uninspired men, while the writer of the Pentateuch was directed and inspired by an infinite God? Is it possible that the codes of China, India, Egypt, Greece and Rome were made by man, and that the laws recorded in the Pentateuch were alone given by God? Is it possible that Æschylus and Shakespeare, Burns, and Beranger, Goethe and Schiller, and all the poets of the world, and all their wondrous tragedies and songs are but the work of men, while no intelligence except the infinite God could be the author of the Pentateuch? Is it possible that of all the books that crowd the libraries of the world, the books of science, fiction, history and song, that all save only one, have been produced by man? Is it possible that of all these, the bible only is the work of God?

I then began to study arithmetical questions without any great apparent result, and without suspecting that they could have the least connexion with my previous researches. Disgusted at my want of success, I went away to spend a few days at the seaside, and thought of entirely different things. One day, as I was walking on the cliff, the idea came to me, again with the same characteristics of conciseness, suddenness, and immediate certainty, that arithmetical transformations of indefinite ternary quadratic forms are identical with those of non-Euclidian geometry.

It is impossible to derive happiness from the company of those whom we deprive of happiness. To be happy in old age it is necessary that we accustom ourselves to objects than can accompany the mind all the way through life, and that we take the rest as good in their day. The mere man of pleasure is miserable in old age; and the mere drudge in business is but little better: whereas, natural philosophy, mathematical and mechanical science, are a continual source of tranquil pleasure, and in spite of the gloomy study of the true theology; it teaches man to know and to admire the Creator, for the principles of science are in the creation, and are unchangeable, and of divine origin.

Dictionaries, manuals, grammars, study guides and topic notes, classical authors and the entire book trade in de Viris, Quintus-Curtius, Sallust, and Livy peacefully crumbled to dust on the shelves of the old Hachette publishing house; but introductions to mathematics, textbooks on civil engineering, mechanics, physics, chemistry, astronomy, courses in commerce, finance, industrial arts- whatever concerned the market tendencies of the day - sold by the millions of copies.

Are there Laws of Humanics as there are Laws of Robotics? How many Laws of Humanics might there be and how can they be expressed mathematically? I don’t know. “Perhaps, though, there may come a day when someone will work out the Laws of Humanics and then be able to predict the broad strokes of the future, and know what might be in store for humanity, instead of merely guessing as I do, and know what to do to make things better, instead of merely speculating. I dream sometimes of founding a mathematical science which I think of as ‘psychohistory,’ but I know I can’t and I fear no one ever will.

At the end of the day, bitcoin is programmable money. When you have programmable money, the possibilities are truly endless. We can take many of the basic concepts of the current system that depend on legal contracts, and we can convert these into algorithmic contracts, into mathematical transactions that can be enforced on the bitcoin network. As I’ve said, there is no third party, there is no counterparty. If I choose to send value from one part of the network to another, it is peer-to-peer with no one in between. If I invent a new form of money, I can deploy it to the entire world and invite others to come and join me. Bitcoin is not just money for the internet. Yes, it’s perfect money for the internet. It’s instant, it’s safe, it’s free. Yes, it is money for the internet, but it’s so much more. Bitcoin is the internet of money. Currency is only the first application. If you grasp that, you can look beyond the price, you can look beyond the volatility, you can look beyond the fad. At its core, bitcoin is a revolutionary technology that will change the world forever. Join

I had been so terrorized by scientific statistics (if ten million people each leave over three grains of rice from their lunch, how many sacks of rice are wasted in one day; if ten million people each economize one paper handkerchief a day, how much pulp will be saved?) that whenever I left over a single grain of rice, whenever I blew my nose, I imagined that I was wasting mountains of rice, tons of paper, and I fell prey to a mood dark as if I had committed some terrible crime. But these were the lies of science, the lies of statistics and mathematics: you can't collect three grains of rice from everybody.

Mathematically, maybe,” I said. “But trust isn’t one of those things that lends itself well to math.” “Sure it does,” Bob said. “You trust somebody, they betray you, you get a negative value. You never trust, they can never disappoint you, you break even.

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.

Philosophers and psychiatrists should explain why it is that we mathematicians are in the habit of systematically erasing our footsteps. Scientists have always looked askance at this strange habit of mathematicians, which has changed little from Pythagoras to our day.

Nothing is more fecund, all the mathematicians know it, than those obscure analogies, the blurred reflections from one theory to another ... nothing gives more pleasure to the researcher. One day the illusion drifts away, the premonition changes to a certitude: the twin theories reveal their common source before disappearing; as the Gita teaches it, knowledge and indifference are reached at the same time. The metaphysics has become mathematics, ready to form the subject matter of a treatise, the cold beauty of which cannot move us anymore.

Her husband once said that he believed some sort of mathematical equation could be applied to life - since the longer you lived, the greater its seeming velocity. She always attributed this to familiarity. If you kept the same habits - and if you lived in the same place, worked in the same place - then you no longer spent a lot of time noticing. Noticing things - and trying to make sense of them - is what makes time remarkable. Otherwise, life blurs by, as it does now, so that she has difficulty keeping track of time at all, one day evaporating into the next.

I stretched my arms towards the sky like blades of tall grass The sun beat between my shoulders like carnival drums I sat still in hopes that it would help my wings grow So then I could really be fly And then she arrived Like day break inside a railway tunnel Like the new moon, like a diamond in the mines Like high noon to a drunkard, sudden She made my heart beat in a now-now time signature Her skinny canvas for ultraviolet brushstrokes She was the sun's painting She was a deep cognac color Her eyes sparkled like lights along the new city She lips pursed as if her breath was too sweet And full for her mouth to hold I said, "You are the beautiful, the stress of mathematics." I said, "For you, I would peel open the clouds like new fruit And give you lightning and thunder as a dowry I would make the sky shed all of it's stars, light and rain And I would clasp the constellations across your waist And I would make the heavens your cape And they would be pleased to cover you They would be pleased to cover you May I please, cover you, please

In the beginning of the year 1665 I found the Method of approximating series & the Rule for reducing any dignity of any Binomial into such a series. The same year in May I found the method of Tangents of Gregory & Slusius, & in November had the direct method of fluxions & the next year in January had the Theory of Colours & in May following I had entrance into ye inverse method of fluxions. And the same year I began to think of gravity extending to ye orb of the Moon & (having found out how to estimate the force with wch [a] globe revolving within a sphere presses the surface of the sphere) from Kepler's rule of the periodic times of the Planets being in sesquialterate proportion of their distances from the center of their Orbs, I deduced that the forces wch keep the Planets in their Orbs must [be] reciprocally as the squares of their distances from the centers about wch they revolve: & thereby compared the force requisite to keep the Moon in her Orb with the force of gravity at the surface of the earth, & found them answer pretty nearly. All this was in the two plague years of 1665-1666. For in those days I was in the prime of my age for invention & minded Mathematicks & Philosophy more then than at any time since.

You think you have it all figured out—you’ve timed your commute, you’ve fit in your weekend run or you haven’t, you’ve got life down to a science, a mathematical equation of time, interest, and energy. But one day something stands up to you, surprises you in a place where you’ve determined never to be surprised. And that’s when you run. You move fast the wrong way through traffic. You think it’s working. But something deep inside, driving the rhythm of your steps, tells you that it isn’t. So you try again. You search for that tiny space hidden in you, untouched by everything that you’ve experienced or survived.

It puzzled him that she did not mourn all the things she could have been. Was it a quality inherent in women, or did they just learn to shield their personal regrets, to suspend their lives, subsume themselves in child care? She browsed online forums about tutoring and music and schools, and she told him what she had discovered as though she truly felt the rest of the world should be as interested as she was in how music improved the mathematics skills of nine-year-olds. Or she would spend hours on the phone talking to her friends, about which violin teacher was good and which tutorial was a waste of money. One day, after

And so to read is, in truth, to be in the constant act of creation. The old lady on the bus with her Orwell, the businessman on the Tube with Patricia Cornwell, the teenager roaring through Capote -- they are not engaged in idle pleasure. Their heads are on fire. Their hearts are flooding. With a book, you are the landscape, the sets, the snow, the hero, the kiss -- you are the mathematical calculation the plots the trajectory of the blazing, crashing zeppelin. You -- pale, punchable reader -- are terraforming whole worlds in your head, which will remain with you until the day you die. These books are as much a part of you as your guts and your bone. And when your guts fail and your bones break, Narnia, or Jamaica Inn, or Gormenghast will still be there; as pin-sharp and bright as the day you first imagined them -- hiding under the bedclothes, sitting on the bus. Exhausted, on a rainy day, weeping over the death of someone you never met, and who was nothing more than words until you transformed them with your time, and your love, and the imagination you constantly dismiss as "just being a bit of a bookworm.

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.

This is a common theme in human progress. We make things beyond what we understand, and we always have done. Steam engines worked before we had a theory of thermodynamics; vaccines were developed before we knew how the immune system works; aircraft continue to fly to this day, despite the many gaps in our understanding of aerodynamics. When theory lags behind application, there will always be mathematical surprises lying in wait. The important thing is that we learn from these inevitable mistakes and don’t repeat them.

Now we can see what makes mathematics unique. Only in mathematics is there no significant correction-only extension. Once the Greeks had developed the deductive method, they were correct in what they did, correct for all time. Euclid was incomplete and his work has been extended enormously, but it has not had to be corrected. His theorems are, every one of them, valid to this day. Ptolemy may have developed an erroneous picture of the planetary system, but the system of trigonometry he worked out to help him with his calculations remains correct forever. Each great mathematician adds to what came previously, but nothing needs to be uprooted. Consequently, when we read a book like A History of Mathematics, we get the picture of a mounting structure, ever taller and broader and more beautiful and magnificent and with a foundation, moreover, that is as untainted and as functional now as it was when Thales worked out the first geometrical theorems nearly 26 centuries ago. Nothing pertaining to humanity becomes us so well as mathematics. There, and only there, do we touch the human mind at its peak.

Always preoccupied with his profound researches, the great Newton showed in the ordinary-affairs of life an absence of mind which has become proverbial. It is related that one day, wishing to find the number of seconds necessary for the boiling of an egg, he perceived, after waiting a minute, that he held the egg in his hand, and had placed his seconds watch (an instrument of great value on account of its mathematical precision) to boil! This absence of mind reminds one of the mathematician Ampere, who one day, as he was going to his course of lectures, noticed a little pebble on the road; he picked it up, and examined with admiration the mottled veins. All at once the lecture which he ought to be attending to returned to his mind; he drew out his watch; perceiving that the hour approached, he hastily doubled his pace, carefully placed the pebble in his pocket, and threw his watch over the parapet of the Pont des Arts.

The manner of Demoivre's death has a certain interest for psychologists. Shortly before it, he declared that it was necessary for him to sleep some ten minutes or a quarter of an hour longer each day than the preceding one: the day after he had thus reached a total of something over twenty-three hours he slept up to the limit of twenty-four hours, and then died in his sleep.

What did we know? This was early days. We had no idea what was out there. How dangerous it might be. It was just a school maths problem. They never asked that in the exams, did they? Like, “If John walks at three miles an hour from London to Brighton, and he's attacked by rabid grown-ups four times, and they bite his right leg off, how long will it take him to bleed to death?

One day Shizuo Kakutani was teaching a class at Yale. He wrote down a lemma on the blackboard and announced that the proof was obvious. One student timidly raised his hand and said that it wasn't obvious to him. Could Kakutani explain? After several moments' thought, Kakutani realized that he could not himself prove the lemma. He apologized, and said that he would report back at their next class meeting. After class, Kakutani, went straight to his office. He labored for quite a time and found that he could not prove the pesky lemma. He skipped lunch and went to the library to track down the lemma. After much work, he finally found the original paper. The lemma was stated clearly and succinctly. For the proof, the author had written, 'Exercise for the reader.

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.

Magic is like a lot of other disciplines that people have recently begun developing, in historic terms. Working with magic is a way of understanding the universe and how it functions. You can approach it from a lot of different angles, applying a lot of different theories and mental models to it. You can get to the same place through a lot of different lines of theory and reasoning, kind of like really advanced mathematics. There's no truly right or wrong way to get there, either--there are just different ways, some more or less useful than others for a given application. And new vistas of thought, theory, and application open up on a pretty regular basis, as the Art develops and expands through the participation of multiple brilliant minds. But that said, once you have a good grounding in it,you get a pretty solid idea of what's possible and what isn't. No matter how much circumlocution you do with your formulae, two plus two doesn't equal five. (Except maybe very, very rarely, sometimes, in extremely specific and highly unlikely circumstances.)

Mathematics enjoys the greatest reputation as a diversion from sexuality. This had been the very advice to which Jean-Jacques Rousseau was obliged to listen from a lady who was dissatisfied with him: 'Lascia le donne e studia la matematica!' So too our fugitive threw himself with special eagerness into the mathematics and geometry which he was taught at school, till suddenly one day his powers of comprehension were paralysed in the face of some apparently innocent problems. It was possible to establish two of these problems; 'Two bodies come together, one with a speed of ... etc' and 'On a cylinder, the diameter of whose surface is m, describe a cone ... etc' Other people would certainly not have regarded these as very striking allusions to sexual events; but he felt that he had been betrayed by mathematics as well, and took flight from it too.

And are we not guilty of offensive disparagement in calling chess a game? Is it not also a science and an art, hovering between those categories as Muhammad’s coffin hovered between heaven and earth, a unique link between pairs of opposites: ancient yet eternally new; mechanical in structure, yet made effective only by the imagination; limited to a geometrically fixed space, yet with unlimited combinations; constantly developing, yet sterile; thought that leads nowhere; mathematics calculating nothing; art without works of art; architecture without substance – but nonetheless shown to be more durable in its entity and existence than all books and works of art; the only game that belongs to all nations and all eras, although no one knows what god brought it down to earth to vanquish boredom, sharpen the senses and stretch the mind. Where does it begin and where does it end? Every child can learn its basic rules, every bungler can try his luck at it, yet within that immutable little square it is able to bring forth a particular species of masters who cannot be compared to anyone else, people with a gift solely designed for chess, geniuses in their specific field who unite vision, patience and technique in just the same proportions as do mathematicians, poets, musicians, but in different stratifications and combinations. In the old days of the enthusiasm for physiognomy, a physician like Gall might perhaps have dissected a chess champion’s brain to find out whether some particular twist or turn in the grey matter, a kind of chess muscle or chess bump, is more developed in such chess geniuses than in the skulls of other mortals. And how intrigued such a physiognomist would have been by the case of Czentovic, where that specific genius appeared in a setting of absolute intellectual lethargy, like a single vein of gold in a hundredweight of dull stone. In principle, I had always realized that such a unique, brilliant game must create its own matadors, but how difficult and indeed impossible it is to imagine the life of an intellectually active human being whose world is reduced entirely to the narrow one-way traffic between black and white, who seeks the triumphs of his life in the mere movement to and fro, forward and back of thirty-two chessmen, someone to whom a new opening, moving knight rather than pawn, is a great deed, and his little corner of immortality is tucked away in a book about chess – a human being, an intellectual human being who constantly bends the entire force of his mind on the ridiculous task of forcing a wooden king into the corner of a wooden board, and does it without going mad!

You'll stay," he said firmly. "But-" He crossed his arms. "Do I look like a man in the mood to be argued with?" She stared at him mutinously. "If you run," he warned, "I will catch you." Sophie eyed the distance between them, then tried to judge the distance back to My Cottage.If he stopped to pull on his clothing she might have a chance of escaping, but if he didn't... "Sophie," he said, "I can practically see the steam coming out of your ears. Stop taxing your brain with useless mathematical computations and do as I asked." One of her feet twitched. Whether it was itching to run home or merely turn around, she'd never know. "Now," he ordered. With a loud sigh and grumble, Sophie crossed her arms and turned around to stare at a knothole in the tree trunk in front of her as if her very life depended on it The inferal man wasn't being particularly quiet as he went about his business, and she couldn't seem to keep herself from listening to and trying to identify every sound that rustled and splashed behind her.Now he was emerging from the water, now he was reaching for his breeches, now he was... It was no use.She had a dreadfully wicked imagination, and there was no getting around it. He should have just let her return to the house. Instead she was forced to wait, utterly mortified, while he dressed. Her skin felt like it was on fire, and she was certain her cheeks must be eight different shades of red. A gentleman would have let her weasle out of her embarrassment and hole up in her room back at the house for at least three days in hopes that he'd just forget about the entire affair.

You don't understand that one can be an atheist, one can not know whether God exists or why, and at the same time know that man does not live in nature but in history, and that in present-day understanding it was founded by Christ, that its foundation is the Gospel. And what is history? It is the setting in motion of centuries of work at the gradual unriddling of death and its eventual overcoming. Hence the discovery of mathematical infinity and electromagnetic waves, hence the writing of symphonies. It is impossible to move on in that direction without a certain uplift. These discoveries call for spiritual equipment. The grounds for it are contained in the Gospel. They are these. First, love of one's neighbor, that highest form of living energy, overflowing man's heart and demanding to be let out and spent, and then the main component parts of modern man, without which he is unthinkable-- namely, the idea of the free person and the idea of life as sacrifice.

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. While every group has certain economic interests identical with those of all groups, every group has also, as we shall see, interests antagonistic to those of all other groups. While certain public policies would in the long run benefit everybody, other policies would benefit one group only at the expense of all other groups. The group that would benefit by such policies, having such a direct interest in them, will argue for them plausibly and persistently. It will hire the best buyable minds to devote their whole time to presenting its case. And it will finally either convince the general public that its case is sound, or so befuddle it that clear thinking on the subject becomes next to impossible. In addition to these endless pleadings of self-interest, there is a second main factor that spawns new economic fallacies every day. This is the persistent tendency of man to see only the immediate effects of a given policy, or its effects only on a special group, and to neglect to inquire what the long-run effects of that policy will be not only on that special group but on all groups. It is the fallacy of overlooking secondary consequences.

Not the Happiness but the Consequence of Happiness He wakes up in the silence of the winter woods, the silence of birds not singing, knowing he will not hear his voice all day. He remembers what the brown owl sounded like while he was sleeping. The man wakes in the frigid morning thinking about women. Not with desire so much as with a sense of what is not. The January silence is the sound of his feet in the snow, a squirrel scolding, or the scraping calls of a single blue jay. Something of him dances there, apart and gravely mute. Many days in the woods he wonders what it is that he has for so long hunted down. We go hand in hand, he thinks, into the dark pleasure, but we are rewarded alone, just as we are married into aloneness. He walks the paths doing the strange mathematics of the brain, multiplying the spirit. He thinks of caressing her feet as she kept dying. For the last four hours, watching her gradually stop as the hospital slept. Remembers the stunning coldness of her head when he kissed her just after. There is light or more light, darkness and less darkness. It is, he decides, a quality without definition. How strange to discover that one lives with the heart as one lives with a wife. Even after many years, nobody knows what she is like. The heart has a life of its own. It gets free of us, escapes, is ambitiously unfaithful. Dies out unaccountably after eight years, blooms unnecessarily and too late. Like the arbitrary silence in the white woods, leaving tracks in the snow he cannot recognize.

Generalizations in biology are almost invariably of a probabilistic nature. As one wit has formulated it, there is only one universal law in biology: 'All biological laws have exceptions.' This probabilistic conceptualization contrasts strikingly with the view during the early period of the scientific revolution that causation in nature is regulated by laws that can be stated in mathematical terms. Actually, this idea occurred apparently first to Pythagoras. It has remained a dominant idea, particularly in the physical sciences, up to the present day. Again and again it was made the basis of some comprehensive philosophy, but taking very different forms in the hands of various authors. With Plato it gave rise to essentialism, with Galileo to a mechanistic world picture, and with Descartes to the deductive method. All three philosophies had a fundamental impact on biology.

Simplify? Let me try. In school days, we are taught that if there are four animals in a room and you add two more, the total will be six. That is logic. But behind this logic, there are underlying assumptions. Now, if somebody tells you, there are four rats in the room and if you add two more cats in the room, how many animals in total exist in the room now? The answer will depend upon assumption. If you just use your mathematical brain, you will say six animals. If you use your human brain, you will say two animals. Why two animals? Because the two cats will eat the four rats in no time.

Day and far into the opalescent Embelyon night [Turjan of Miir] worked under Pandelume's unseen tutelage. He learned the secret of renewed youth, many spells of the ancients, and a strange abstract lore that Pandelume termed 'Mathematics.' "Within this instrument," said Pandelume, "resides the Universe. Passive in itself and not of sorcery, it elucidates every problem, each phase of existence, all the secrets of time and space. Your spells and runes are built upon its power and codified according to a great underlying mosaic of magic. The design of this mosaic we cannot surmise; our knowledge is didactic, empirical, arbitrary. Phandaal glimpsed the pattern and so was able to formulate many of the spells which bear his name. I have endeavored through the ages to break the clouded glass, but so far my research has failed. He who discovers the pattern will know all of sorcery and be a man powerful beyond comprehension.

I’ll be honest with you. The variables that construct my existence are confusing. Like handwritten math equations jammed together on a sloppy page of homework. They don’t make any sense. One math problem leads to another, and then another and so it goes. One day you realize that your life is one whole page of problems and nothing ever gets solved. One ongoing equation with no equal sign at the end. But it occurred to me, beneath the canopy of a starlight heaven, that I’d been looking at my life all wrong. It wasn’t a math equation. Things weren’t supposed to add up. There was no solution. In fact, there was no problem. Life’s variables and numbers and pages of chicken scratch weren’t mathematical marks. They were art. A drawing. An abstract painting. It was meant to be beautiful, not sensical. And embedded within the mess of it all were miracles. Small ones. I’d never paid attention to them because I was too busy, but it didn’t make them less real.

You know why I’m raising you kids to be Cubs fans?” Buddy shakes his head. “Any mook can be a fan of a winning team,” Dad says. “It takes character to root for the doomed. You show up, you watch your boys take their swings, and you watch ’em go down in flames—every damn day. You think Jack Brickhouse is an optimist? No-siree. He may sound happy, but he’s dying inside. There’s no seat in Wrigley Field for a God damn Pollyanna. You root-root-root for the home team, and they lose anyway. It teaches you how the world works, kid. Sure, start every spring with your hopes and dreams, but in the universe in which we live, you will be mathematically eliminated by Labor Day. Count on it.

What I am talking about may be difficult for you to understand, but it is really quite important. You see, technicians are not creators; and there are more and more technicians in the world, people who know what to do and how to do it, but who are not creators. In America there are calculating machines capable of solving in a few minutes mathematical problems which would take a man, working ten hours every day, a hundred years to solve. These extraordinary machines are being developed. But machines can never be creators—and human beings are becoming more and more like machines. Even when they rebel, their rebellion is within the limits of the machine and is therefore no rebellion at all.

There was a great king in ancient India who was once asked four questions, of which one was: "What is the most wonderful thing in the world?" "Hope," was the answer. This is the most wonderful thing. Day and nights we see people dying around us, and yet we think we shall not die; we never think that we shall die, or that we shall suffer. Each man thinks that success will be his, hoping against hope, against all odds, against all mathematical reasoning. Nobody is ever really happy here. If a man be wealthy and have plenty to eat, his digestion is: out of order, and he cannot eat. If a man's digestion be good, and he have the digestive power of a cormorant, he has nothing to put into his mouth. If he be rich, he has no children. If he be hungry and poor, he has a whole regiment of children, and does not know what to do with them. Why is it so? Because happiness and misery are the obverse and reverse of the same coin; he who takes happiness, must take misery also. We all have this foolish idea that we can have happiness without misery, and it has taken such possession of us that we have no control over the senses.

It as mathematical, marriage, not, as one might expect, additional; it was exponential. This one man, nervous in a suite a size too small for his long, lean self, this woman, in a green lace dress cut to the upper thigh, with a white rose behind her ear. Christ, so young. The woman before them was a unitarian minister, and on her buzzed scalp, the grey hairs shone in a swab of sun through the lace in the window. Outside, Poughkeepsie was waking. Behind them, a man in a custodian's uniform cried softly beside a man in pajamas with a Dachshund, their witnesses, a shine in everyone's eye. One could taste the love on the air, or maybe that was sex, or maybe that was all the same then. 'I do,' she said. 'I do,' he said. They did. They would. Our children will be so fucking beautiful, he thought, looking at her. Home, she thought, looking at him. 'You may kiss,' said the officiant. They did, would. Now they thanked everyone and laughed, and papers were signed and congratulations offered, and all stood for a moment, unwilling to leave this gentile living room where there was such softness. The newlyweds thanked everyone again, shyly, and went out the door into the cool morning. They laughed, rosy. In they'd come integers, out they came, squared. Her life, in the window, the parakeet, scrap of blue midday in the London dusk, ages away from what had been most deeply lived. Day on a rocky beach, creatures in the tide pool. All those ordinary afternoons, listening to footsteps in the beams of the house, and knowing the feeling behind them. Because it was so true, more than the highlights and the bright events, it was in the daily where she'd found life. The hundreds of time she'd dug in her garden, each time the satisfying chew of spade through soil, so often that this action, the pressure and release and rich dirt smell delineated the warmth she'd felt in the cherry orchard. Or this, each day they woke in the same place, her husband waking her with a cup of coffee, the cream still swirling into the black. Almost unremarked upon this kindness, he would kiss her on the crown of her head before leaving, and she'd feel something in her rising in her body to meet him. These silent intimacies made their marriage, not the ceremonies or parties or opening nights or occasions, or spectacular fucks. Anyway, that part was finished. A pity...

Countries measured their success by the size of their territory, the increase in their population and the growth of their GDP – not by the happiness of their citizens. Industrialised nations such as Germany, France and Japan established gigantic systems of education, health and welfare, yet these systems were aimed to strengthen the nation rather than ensure individual well-being. Schools were founded to produce skilful and obedient citizens who would serve the nation loyally. At eighteen, youths needed to be not only patriotic but also literate, so that they could read the brigadier’s order of the day and draw up tomorrow’s battle plans. They had to know mathematics in order to calculate the shell’s trajectory or crack the enemy’s secret code. They needed a reasonable command of electrics, mechanics and medicine in order to operate wireless sets, drive tanks and take care of wounded comrades. When they left the army they were expected to serve the nation as clerks, teachers and engineers, building a modern economy and paying lots of taxes. The same went for the health system. At the end of the nineteenth century countries such as France, Germany and Japan began providing free health care for the masses. They financed vaccinations for infants, balanced diets for children and physical education for teenagers. They drained festering swamps, exterminated mosquitoes and built centralised sewage systems. The aim wasn’t to make people happy, but to make the nation stronger. The country needed sturdy soldiers and workers, healthy women who would give birth to more soldiers and workers, and bureaucrats who came to the office punctually at 8 a.m. instead of lying sick at home. Even the welfare system was originally planned in the interest of the nation rather than of needy individuals. When Otto von Bismarck pioneered state pensions and social security in late nineteenth-century Germany, his chief aim was to ensure the loyalty of the citizens rather than to increase their well-being. You fought for your country when you were eighteen, and paid your taxes when you were forty, because you counted on the state to take care of you when you were seventy.30 In 1776 the Founding Fathers of the United States established the right to the pursuit of happiness as one of three unalienable human rights, alongside the right to life and the right to liberty. It’s important to note, however, that the American Declaration of Independence guaranteed the right to the pursuit of happiness, not the right to happiness itself. Crucially, Thomas Jefferson did not make the state responsible for its citizens’ happiness. Rather, he sought only to limit the power of the state.

Many other physicists get a little blasé about the vastness of the cosmos and forces too powerful to comprehend. You can reduce it all to mathematics, tweak some equations, and get on with your day. But the shock and vertigo of the recognition of the fragility of everything, and my own powerlessness in it, has left its mark on me. There’s something about taking the opportunity to wade into that cosmic perspective that is both terrifying and hopeful, like holding a newborn infant and feeling the delicate balance of the tenuousness of life and the potential for not-yet-imagined greatness. It is said that astronauts returning from space carry with them a changed perspective on the world, the “overview effect,” in which, having seen the Earth from above, they can fully perceive how fragile our little oasis is and how unified we ought to be as a species, as perhaps the only thinking beings in the cosmos.

At first all was well. In fact, all was terrific. The Takers were pedaling away and the wings of their craft were flapping beautifully. They felt wonderful, exhilarated. They were experiencing the freedom of the air: freedom from restraints that bind and limit the rest of the biological community. And with that freedom came marvels—all the things you mentioned the other day: urbanization, technology, literacy, mathematics, science. “Their flight could never end, it could only go on becoming more and more exciting. They couldn’t know, couldn’t even have guessed that, like our hapless airman, they were in the air but not in flight. They were in free fall, because their craft was simply not in compliance with the law that makes flight possible. But their disillusionment is far away in the future, and so they’re pedaling away and having a wonderful time. Like our airman, they see strange sights in the course of their fall. They see the remains of craft very like their own—not destroyed, merely abandoned—by the Maya, by the Hohokam, by the Anasazi, by the peoples of the Hopewell cult, to mention only a few of those found here in the New World. ‘Why,’ they wonder, ‘are these craft on the ground instead of in the air? Why would any people prefer to be earthbound when they could have the freedom of the air, as we do?’ It’s beyond comprehension, an unfathomable mystery.

No salvation comes from exhumed gods; we must penetrate deeper into substance. If I take a fossil, say, a trilobite, in my hand (marvelously preserved specimens are found in the quarries at the foot of the Casbah), I am transfixed by the impact of mathematical harmony. Purpose and beauty, as fresh as on the first day, are still seamlessly united in a medal engraved by a master's hand. The bios must have discovered the secret of tripartition in this primordial crab. Tripartition then frequently recurs, even without any natural kinship; figures, in transversal symmetry, dwell in the triptych. How many millions of years ago might this creature have animated an ocean that no longer exists? I hold its impression, a seal of imperishable beauty, in my hand. Some day, this seal, too, will decay or else burn out in cosmic conflagrations of the future. The matrix that formed it remains concealed in and operative from the law, untouched by death or fire.

...the following parable may be useful. Long ago, when shepherds wanted to see if two herds of sheep were isomorphic, they would look for an explicit isomorphism. In other words, they would line up both herds and try to match each sheep in one herd with a sheep in the other. But one day, along came a shepherd who invented decategorification. She realized one could take each herd and ‘count’ it, setting up an isomorphism between it and some set of ‘numbers’, which were nonsense words like ‘one, two, three, . . . ’ specially designed for this purpose. By comparing the resulting numbers, she could show that two herds were isomorphic without explicitly establishing an isomorphism! In short, by decategorifying the category of finite sets, the set of natural numbers was invented. According to this parable, decategorification started out as a stroke of mathematical genius. Only later did it become a matter of dumb habit, which we are now struggling to overcome by means of categorification.

Listen, Google,’ I will say, ‘both John and Paul are courting me. I like both of them, but in a different way, and it’s so hard to make up my mind. Given everything you know, what do you advise me to do?’ And Google will answer: ‘Well, I know you from the day you were born. I have read all your emails, recorded all your phone calls, and know your favourite films, your DNA and the entire history of your heart. I have exact data about each date you went on, and if you want, I can show you second-by-second graphs of your heart rate, blood pressure and sugar levels whenever you went on a date with John or Paul. If necessary, I can even provide you with accurate mathematical ranking of every sexual encounter you had with either of them. And naturally enough, I know them as well as I know you. Based on all this information, on my superb algorithms, and on decades’ worth of statistics about millions of relationships – I advise you to go with John, with an 87 per cent probability of being more satisfied with him in the long run. Indeed, I know you so well that I also know you don’t like this answer. Paul is much more handsome than John, and because you give external appearances too much weight, you secretly wanted me to say “Paul”. Looks matter, of course; but not as much as you think. Your biochemical algorithms – which evolved tens of thousands of years ago in the African savannah – give looks a weight of 35 per cent in their overall rating of potential mates. My algorithms – which are based on the most up-to-date studies and statistics – say that looks have only a 14 per cent impact on the long-term success of romantic relationships. So, even though I took Paul’s looks into account, I still tell you that you would be better off with John.

Simply put, within AS, there is a wide range of function. In truth, many AS people will never receive a diagnosis. They will continue to live with other labels or no label at all. At their best, they will be the eccentrics who wow us with their unusual habits and stream-of-consciousness creativity, the inventors who give us wonderfully unique gadgets that whiz and whirl and make our life surprisingly more manageable, the geniuses who discover new mathematical equations, the great musicians and writers and artists who enliven our lives. At their most neutral, they will be the loners who never now quite how to greet us, the aloof who aren't sure they want to greet us, the collectors who know everyone at the flea market by name and date of birth, the non-conformists who cover their cars in bumper stickers, a few of the professors everyone has in college. At their most noticeable, they will be the lost souls who invade our personal space, the regulars at every diner who carry on complete conversations with the group ten tables away, the people who sound suspiciously like robots, the characters who insist they wear the same socks and eat the same breakfast day in and day out, the people who never quite find their way but never quite lose it either.

There are some mysteries in this world," Yukawa said suddenly, "that cannot be unraveled with modern science. However, as science develops, we will one day be able to understand them. The question is, is there a limit to what science can know? If so, what creates that limit?" Kyohei looked at Yukawa. He couldn't figure out why the professor was telling him this, except he had a feeling it was very important. Yukawa pointed a finger at Kyohei's forehead. "People do." he said. "People's brains, to be more precise. For example, in mathematics, when somebody discovers a new theorem, they may have other mathematicians verify it to see if it's correct. The problem is, the theorems getting discovered are becoming more and more complex. That limits the number of mathematicians who can properly verify them. What happens when someone comes up with a theorem so hard to understand that there isn't anyone else who can understand it? In order for that theorem to be accepted as fact, they have to wait until another genius comes along. That's the limit the human brain imposes on the progress of scientific knowledge. You understand?" Kyohei nodded, still having no idea where he was going with this. "Every problem has a solution," Yukawa said, staring straight at Kyohei through his glasses. "But there's no guarantee that the solution will be found immediately. The same holds true in our lives. We encounter several problems to which the solutions are not immediately apparent in life. There is value to be had in worrying about those problems when you get to them. But never feel rushed. Often, in order to find the answer, you need time to grow first. That's why we apply ourselves, and learn as we go." Kyohei chewed on that for a moment, then his mouth opened a little and he looked up with sudden understanding. "You have questions now, I know, and until you find your answers, I'll be working on those questions too, and worrying with you. So don't forget, you're never alone.

The mathematician is only too willing to admit that he is dealing exclusively with acts of the mind. To be sure, he is aware that the ingenious artifices which form his stock in trade had their genesis in the sense impressions which he identifies with crude reality, and he is not surprised to find that at times these artifices fit quite neatly the reality in which they were born. But this neatness the mathematician refuses to recognize as a criterion of his achievement: the value of the beings which spring from his creative imagination shall not be measured by the scope of their application to physical reality. No! Mathematical achievement shall be measured by standards which are peculiar to mathematics. These standards are independent of the crude reality of our senses. They are: freedom from logical contradictions, the generality of the laws governing the created form, the kinship which exists between this new form and those that have preceded it. The mathematician may be compared to a designer of garments, who is utterly oblivious of the creatures whom his garments may fit. To be sure, his art originated in the necessity for clothing such creatures, but this was long ago; to this day a shape will occasionally appear which will fit into the garment as if the garment had been made for it. Then there is no end of surprise and of delight!

Most people-all, in fact, who regard the whole heaven as finite-say it lies at the centre. But the Italian philosophers known as Pythagoreans take the contrary view. At the centre, they say, is fire, and the earth is one of the stars, creating night and day by its circular motion about the centre. They further construct another earth in opposition to ours to which they give the name counterearth. In all this they are not seeking for theories and causes to account for observed facts, but rather forcing their observations and trying to accommodate them to certain theories and opinions of their own. But there are many others who would agree that it is wrong to give the earth the central position, looking for confirmation rather to theory than to the facts of observation. Their view is that the most precious place befits the most precious thing: but fire, they say, is more precious than earth, and the limit than the intermediate, and the circumference and the centre are limits. Reasoning on this basis they take the view that it is not earth that lies at the centre of the sphere, but rather fire. The Pythagoreans have a further reason. They hold that the most important part of the world, which is the centre, should be most strictly guarded, and name it, or rather the fire which occupies that place, the 'Guardhouse of Zeus', as if the word 'centre' were quite unequivocal, and the centre of the mathematical figure were always the same with that of the thing or the natural centre. But it is better to conceive of the case of the whole heaven as analogous to that of animals, in which the centre of the animal and that of the body are different. For this reason they have no need to be so disturbed about the world, or to call in a guard for its centre: rather let them look for the centre in the other sense and tell us what it is like and where nature has set it. That centre will be something primary and precious; but to the mere position we should give the last place rather than the first. For the middle is what is defined, and what defines it is the limit, and that which contains or limits is more precious than that which is limited, see ing that the latter is the matter and the former the essence of the system. (2-13-1) There are similar disputes about the shape of the earth. Some think it is spherical, others that it is flat and drum-shaped. For evidence they bring the fact that, as the sun rises and sets, the part concealed by the earth shows a straight and not a curved edge, whereas if the earth were spherical the line of section would have to be circular. In this they leave out of account the great distance of the sun from the earth and the great size of the circumference, which, seen from a distance on these apparently small circles appears straight. Such an appearance ought not to make them doubt the circular shape of the earth. But they have another argument. They say that because it is at rest, the earth must necessarily have this shape. For there are many different ways in which the movement or rest of the earth has been conceived. (2-13-3)

The European and the African have an entirely different concept of time. In the European worldview, time exists outside man, exists objectively, and has measurable and linear characteristics. According to Newton, time is absolute: “Absolute, true, mathematical time of itself and from its own nature, it flows equitably and without relation to anything external.” The European feels himself to be time’s slave, dependent on it, subject to it. To exist and function, he must observe its ironclad, inviolate laws, its inflexible principles and rules. He must heed deadlines, dates, days, and hours. He moves within the rigors of time and cannot exist outside them. They impose upon him their requirements and quotas. An unresolvable conflict exists between man and time, one that always ends with man’s defeat—time annihilates him. Africans apprehend time differently. For them, it is a much looser concept, more open, elastic, subjective. It is man who influences time, its shape, course, and rhythm (man acting, of course, with the consent of gods and ancestors ). Time is even something that man can create outright, for time is made manifest through events, and whether an event takes place or not depends, after all, on man alone. If two armies do not engage in a battle, then that battle will not occur (in other words, time will not have revealed its presence, will not have come into being). Time appears as a result of our actions, and vanishes when we neglect or ignore it. It is something that springs to life under our influence, but falls into a state of hibernation, even nonexistence, if we do not direct our energy toward it. It is a subservient, passive essence, and, most importantly, one dependent on man.

I saw a guy the other day at a wedding, and I told him my theory on why we’ve seen this explosion in comedies in the past fifteen years. Number one, America is tacking hard to the right. That sort of extremism always kind of kicks up the need to create comedy. But the second thing is Avid. What’s Avid? It’s a digital movie-editing program that directors use, and it’s incredibly helpful. I think Avid is hugely responsible for this boom in comedy. In the past, one would have to shoot the film and edit it, which was a big deal. Now, filmmakers can record the laughs from a test audience at a screening, and we can then cut to the rhythm of those laughs, the rhythm of the audience. We synchronize the laughs with the film. We can really get our timing down to a hundredth of a second. You can decide where you want your story to kick in, where you want a little bit of mood, where you want a hard laugh line. All of this can really be calibrated to these test screenings that we do. It doesn’t mean that it becomes mathematical. It still ultimately means that you have to make creative choices, but you can just really get a lot out of it. Sort of like surgery with a laser compared with a regular scalpel. We’re able to download a movie onto the computer and literally do all our edits in minutes. The precision is incredible. You play back the audio of the test screening and get everything timed just right. Like, “This laugh is losing this next line; let’s split the difference here.” You’re able to achieve this rolling energy. You can try experimental edits, and do multiple test screenings, and it’s all because you can move so fast with this program. Comedy is the one genre that I think has just really benefited from this more than any other.

Separated from everyone, in the fifteenth dungeon, was a small man with fiery brown eyes and wet towels wrapped around his head. For several days his legs had been black, and his gums were bleeding. Fifty-nine years old and exhausted beyond measure, he paced silently up and down, always the same five steps, back and forth. One, two, three, four, five, and turn . . . an interminable shuffle between the wall and door of his cell. He had no work, no books, nothing to write on. And so he walked. One, two, three, four, five, and turn . . . His dungeon was next door to La Fortaleza, the governor’s mansion in Old San Juan, less than two hundred feet away. The governor had been his friend and had even voted for him for the Puerto Rican legislature in 1932. This didn’t help much now. The governor had ordered his arrest. One, two, three, four, five, and turn . . . Life had turned him into a pendulum; it had all been mathematically worked out. This shuttle back and forth in his cell comprised his entire universe. He had no other choice. His transformation into a living corpse suited his captors perfectly. One, two, three, four, five, and turn . . . Fourteen hours of walking: to master this art of endless movement, he’d learned to keep his head down, hands behind his back, stepping neither too fast nor too slow, every stride the same length. He’d also learned to chew tobacco and smear the nicotined saliva on his face and neck to keep the mosquitoes away. One, two, three, four, five, and turn . . . The heat was so stifling, he needed to take off his clothes, but he couldn’t. He wrapped even more towels around his head and looked up as the guard’s shadow hit the wall. He felt like an animal in a pit, watched by the hunter who had just ensnared him. One, two, three, four, five, and turn . . . Far away, he could hear the ocean breaking on the rocks of San Juan’s harbor and the screams of demented inmates as they cried and howled in the quarantine gallery. A tropical rain splashed the iron roof nearly every day. The dungeons dripped with a stifling humidity that saturated everything, and mosquitoes invaded during every rainfall. Green mold crept along the cracks of his cell, and scarab beetles marched single file, along the mold lines, and into his bathroom bucket. The murderer started screaming. The lunatic in dungeon seven had flung his own feces over the ceiling rail. It landed in dungeon five and frightened the Puerto Rico Upland gecko. The murderer, of course, was threatening to kill the lunatic. One, two, three, four, five, and turn . . . The man started walking again. It was his only world. The grass had grown thick over the grave of his youth. He was no longer a human being, no longer a man. Prison had entered him, and he had become the prison. He fought this feeling every day. One, two, three, four, five, and turn . . . He was a lawyer, journalist, chemical engineer, and president of the Nationalist Party. He was the first Puerto Rican to graduate from Harvard College and Harvard Law School and spoke six languages. He had served as a first lieutenant in World War I and led a company of two hundred men. He had served as president of the Cosmopolitan Club at Harvard and helped Éamon de Valera draft the constitution of the Free State of Ireland.5 One, two, three, four, five, and turn . . . He would spend twenty-five years in prison—many of them in this dungeon, in the belly of La Princesa. He walked back and forth for decades, with wet towels wrapped around his head. The guards all laughed, declared him insane, and called him El Rey de las Toallas. The King of the Towels. His name was Pedro Albizu Campos.