Mathematical Beauty Quotes

Mathematics expresses values that reflect the cosmos, including orderliness, balance, harmony, logic, and abstract beauty.

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.

The axiom of equality states that x always equals x: it assumes that if you have a conceptual thing named x, that it must always be equivalent to itself, that it has a uniqueness about it, that it is in possession of something so irreducible that we must assume it is absolutely, unchangeably equivalent to itself for all time, that its very elementalness can never be altered. But it is impossible to prove. Always, absolutes, nevers: these are the words, as much as numbers, that make up the world of mathematics. Not everyone liked the axiom of equality––Dr. Li had once called it coy and twee, a fan dance of an axiom––but he had always appreciated how elusive it was, how the beauty of the equation itself would always be frustrated by the attempts to prove it. It was the kind of axiom that could drive you mad, that could consume you, that could easily become an entire life. But now he knows for certain how true the axiom is, because he himself––his very life––has proven it. The person I was will always be the person I am, he realizes. The context may have changed: he may be in this apartment, and he may have a job that he enjoys and that pays him well, and he may have parents and friends he loves. He may be respected; in court, he may even be feared. But fundamentally, he is the same person, a person who inspires disgust, a person meant to be hated.

This most beautiful system of the sun, planets and comets, could only proceed from the counsel and dominion of an intelligent and powerful Being... This Being governs all things, not as the soul of the world, but as Lord over all; and on account of his dominion he is wont, to be called Lord God παντοκρατωρ or Universal Ruler.

Mathematics, rightly viewed, possesses not only truth, but supreme beauty—a beauty cold and austere, like that of sculpture, without appeal to any part of our weaker nature, without the gorgeous trappings of painting or music, yet sublimely pure, and capable of a stern perfection such as only the greatest art can show.

Maybe that is what beauty was, for humans. Accidents, imperfections, placed inside a pretty pattern. Asymmetry. The defiance of mathematics.

[When asked why are numbers beautiful?] It’s like asking why is Ludwig van Beethoven’s Ninth Symphony beautiful. If you don't see why, someone can't tell you. I know numbers are beautiful. If they aren't beautiful, nothing is.

Mathematics reveals its secrets only to those who approach it with pure love, for its own beauty.

Mathematics rightly viewed possesses not only truth but supreme beauty.

The harmony of the world is made manifest in Form and Number, and the heart and soul and all the poetry of Natural Philosophy are embodied in the concept of mathematical beauty.

[Grief] is everything. It is the fabric of selfhood, and beautifully chaotic. It shares mathematical characteristics with many natural forms.

It's possible to understand the world from studying a leaf. You can comprehend the laws of aerodynamics, mathematics, poetry and biology through the complex beauty of such a perfect structure. It's also possible to travel the whole globe and learn nothing.

Doing mathematics should always mean finding patterns and crafting beautiful and meaningful explanations.

Beauty is startling. She wears a gold shawl in the summer and sells seven kinds of honey at the flea market. She is young and old at once, my daughter and my grandmother. In school she excelled in mathematics and poetry. . . Beauty will dance with anyone who is brave enough to ask her.

I wanted very much to learn to draw, for a reason that I kept to myself: I wanted to convey an emotion I have about the beauty of the world. It's difficult to describe because it's an emotion. It's analogous to the feeling one has in religion that has to do with a god that controls everything in the whole universe: there's a generality aspect that you feel when you think about how things that appear so different and behave so differently are all run "behind the scenes" by the same organization, the same physical laws. It's an appreciation of the mathematical beauty of nature, of how she works inside; a realization that the phenomena we see result from the complexity of the inner workings between atoms; a feeling of how dramatic and wonderful it is. It's a feeling of awe — of scientific awe — which I felt could be communicated through a drawing to someone who had also had this emotion. It could remind him, for a moment, of this feeling about the glories of the universe.

[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.

My beautiful proof lies all in ruins.

Séverin crossed his arms. “Zofia, tell him he’s pretty.” Zofia didn’t look up from her tea. “I am personally undecided, but if we’re assessing based on objectivity, then according to the principles of the golden ratio, also known as phi, which is approximately 1.618, your facial beauty is mathematically pleasing.

It is impossible to explain honestly the beauties of the laws of nature in a way that people can feel, without their having some deep understanding of mathematics. I am sorry, but this seems to be the case.

If one is working from the point of view of getting beauty into one's equation, ... one is on a sure line of progress.

To those who do not know mathematics it is difficult to get across a real feeling as to the beauty, the deepest beauty, of nature ... If you want to learn about nature, to appreciate nature, it is necessary to understand the language that she speaks in.

The mathematical sciences particularly exhibit order symmetry and limitations; and these are the greatest forms of the beautiful.

Beauty in mathematics is seeing the truth without effort.

Mathematics is the music of reason. To do mathematics is to engage in an act of discovery and conjecture, intuition and inspiration; to be in a state of confusion—not because it makes no sense to you, but because you gave it sense and you still don't understand what your creation is up to; to have a break-through idea; to be frustrated as an artist; to be awed and overwhelmed by an almost painful beauty; to be alive, damn it.

Mathematics has beauty and romance. It's not a boring place to be, the mathematical world. It's an extraordinary place; it's worth spending time there.

The mathematician’s patterns, like the painter’s or the poet’s must be beautiful; the ideas like the colours or the words, must fit together in a harmonious way. Beauty is the first test: there is no permanent place in the world for ugly mathematics.

No doubt there are some who, when confronted with a line of mathematical symbols, however simply presented, can only see the face of a stern parent or teacher who tried to force into them a non-comprehending parrot-like apparent competence--a duty and a duty alone--and no hint of magic or beauty of the subject might be allowed to come through.

The mathematician does not study pure mathematics because it is useful; he studies it because he delights in it and he delights in it because it is beautiful.

Let's face it, the universe is messy. It is nonlinear, turbulent, and chaotic. It is dynamic. It spends its time in transient behavior on its way to somewhere else, not in mathematically neat equilibria. It self-organizes and evolves. It creates diversity, not uniformity. That's what makes the world interesting, that's what makes it beautiful, and that's what makes it work.

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

Last time I asked: 'What does mathematics mean to you?', and some people answered: "The manipulation of numbers, the manipulation of structures.' And if I had asked what music means to you, would you have answered: 'The manipulation of notes?

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

If nature leads us to mathematical forms of great simplicity and beauty—by forms, I am referring to coherent systems of hypotheses, axioms, etc.—to forms that no one has previously encountered, we cannot help thinking that they are “true,” that they reveal a genuine feature of nature…. You must have felt this too: the almost frightening simplicity and wholeness of the relationships which nature suddenly spreads out before us and for which none of us was in the least prepared.

Mathematics and poetry are the two ways to drink the beauty of truth.

This most beautiful system [The Universe] could only proceed from the dominion of an intelligent and powerful Being.

For the most part, physicists and mathematicians have settled on a fine division of labour in which the former complain about the finickiness of the latter, and the latter complain about the sloppiness of the former.

The greatest mathematics has the simplicity and inevitableness of supreme poetry and music, standing on the borderland of all that is wonderful in Science, and all that is beautiful in Art.

There are other reasons we use math in physics. Besides keeping us honest, math is also the most economical and unambiguous terminology that we know of. Language is malleable; it depends on context and interpretation. But math doesn’t care about culture or history. If a thousand people read a book, they read a thousand different books. But if a thousand people read an equation, they read the same equation.

Mathematics, rightly viewed, possesses not only truth, but supreme beauty— a beauty cold and austere, like that of sculpture, without appeal to any part of our weaker nature, without the gorgeous trappings of painting or music, yet sublimely pure, and capable of a stern perfection such as only the greatest art can show. The true spirit of delight, the exaltation, the sense of being more than Man, which is the touchstone of the highest excellence, is to be found in mathematics as surely as poetry.

A businessman is someone who buys at ten and is happy to get out at twelve. The other kind of man buys at ten, sees it rise to eighteen and does nothing. He is waiting for it to get to twenty. The beauty of numbers. When it drops to ten again he waits for it to get back to eighteen. When it drops to two he waits for it to get back to ten. Well, it gets back there. But he has wasted a quarter of his life. And all he's got out of his money is a little mathematical excitement.

Tom felt his darkness. His father was beautiful and clever, his mother was short and mathematically sure. Each of his brothers and sisters had looks or gifts or fortune. Tom loved all of them passionately, but he felt heavy and earth-bound. He climbed ecstatic mountains and floundered in the rocky darkness between the peaks. He had spurts of bravery but they were bracketed in battens of cowardice.

It might seem that the empirical philosopher is the slave of his material, but that the pure mathematician, like the musician, is a free creator of his world of ordered beauty.

We especially need imagination in science. It is not all logic, nor all mathematics, but is somewhat beauty and poetry.

Mathematical solutions are selected by the subliminal self on the basis of “mathematical beauty,” of the harmony of numbers and forms, of geometric elegance.

A physical law must possess mathematical beauty

It is convenient for the old men to blame Eve. To insist we are damned because a country girl talked to the snake one afternoon long ago. Children must starve in Somalia for that, and old women be abandoned in our greatest cities. It’s why we will finally be thrown into the lakes of molten lead. Because she was confused by happiness that first time anyone said she was beautiful. Nevertheless, she must be the issue, so people won’t notice that rocks and galaxies, mathematics and rust are also created in His image.

As he soars, he thinks, suddenly, of Dr. Kashen. Or not of Dr. Kashen, necessarily, but the question he had asked him when he was applying to be his advisee: What's your favorite axiom? (The nerd pickup line, CM had once called it.) "The axiom of equality," he'd said, and Kashen had nodded, approvingly. "That's a good one," he'd said. The axiom of equality states that x always equals x: it assumes that if you have a conceptual thing named x, that it must always be equivalent to itself, that it has a uniqueness about it, that it is in possession of something so irreducible that we must assume it is absolutely, unchangeably equivalent to itself for all time, that its very elementalness can never be altered. But it is impossible to prove. Always, absolutes, nevers: these are the words, as much as numbers, that make up the world of mathematics. Not everyone liked the axiom of equality––Dr. Li had once called it coy and twee, a fan dance of an axiom––but he had always appreciated how elusive it was, how the beauty of the equation itself would always be frustrated by the attempts to prove it. I was the kind of axiom that could drive you mad, that could consume you, that could easily become an entire life. But now he knows for certain how true the axiom is, because he himself––his very life––has proven it. The person I was will always be the person I am, he realizes. The context may have changed: he may be in this apartment, and he may have a job that he enjoys and that pays him well, and he may have parents and friends he loves. He may be respected; in court, he may even be feared. But fundamentally, he is the same person, a person who inspires disgust, a person meant to be hated. And in that microsecond that he finds himself suspended in the air, between ecstasy of being aloft and the anticipation of his landing, which he knows will be terrible, he knows that x will always equal x, no matter what he does, or how many years he moves away from the monastery, from Brother Luke, no matter how much he earns or how hard he tries to forget. It is the last thing he thinks as his shoulder cracks down upon the concrete, and the world, for an instant, jerks blessedly away from beneath him: x = x, he thinks. x = x, x = x.

This most beautiful system of the sun, planets, and comets, could only proceed from the counsel and dominion of an intelligent and powerful Being. And if the fixed stars are the centers of other like systems, these, being formed by the like wise counsel, must be all subject to the dominion of One.

In fact, most of these comrades who I gauged to be more brilliant than I have gone on to become distinguished mathematicians. Still, from the perspective of thirty or thirty-five years, I can state that their imprint upon the mathematics of our time has not been very profound. They've all done things, often beautiful things, in a context that was already set out before them, which they had no inclination to disturb. Without being aware of it, they've remained prisoners of those invisible and despotic circles which delimit the universe of a certain milieu in a given era. To have broken these bounds they would have had to rediscover in themselves that capability which was their birthright, as it was mine: the capacity to be alone.

[Mathematics] is security. Certainty. Truth. Beauty. Insight. Structure. Architecture. I see mathematics, the part of human knowledge that I call mathematics, as one thing—one great, glorious thing. Whether it is differential topology, or functional analysis, or homological algebra, it is all one thing. ... They are intimately interconnected, they are all facets of the same thing. That interconnection, that architecture, is secure truth and is beauty. That's what mathematics is to me.

All nature has come to expect from God a sense of orderliness. Whatever God does carries with it His fingerprint. And in the world around us His fingerprint of orderliness is evident to anybody who is honest with the facts. If you look at nature, you will discover a mathematical exactness. Without this precision, the entire world would be in utter confusion. One plus one always equals two no matter what part of the universe you happen to be in. And the laws of nature operate in beautiful harmony, a harmony that is ordered by God Himself.

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

The mathematical order is beautiful precisely because it has no effect on the real world. Life isn't going to be easier, not is anyone going to make a fortune, just because they know something about prime numbers.

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.

It struck me at some point that the whole basis of education was memory. A list of names, a column of numbers, a mathematical formula, a beautiful poem—to learn it you had to upload it to the part of the brain that stored stuff, but that was the same part of my brain I was resisting. My memory had been spotty since Mummy disappeared, by design, and I didn’t want to fix it, because memory equaled grief. Not remembering was balm.

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.

Perfection. That's what it's about. It's those moments. When you can feel the perfection of creation. The beauty the physics you know the wonder of mathematics. The elations of action and reaction and that is the kind of perfection that I want to be connected to.

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

We see that music, like the world, is formed from unchanging mathematical principles deployed in time, creating complexity, variety and beauty.

Like a Shakespearean sonnet that captures the very essence of love, or a painting that brings out the beauty of the human form that is far more than just skin deep, Euler’s equation reaches down into the very depths of existence.

The detective story, as created by Poe, is something as specialised and as intellectual as a chess problem, whereas the best English detective fiction has relied less on the beauty of the mathematical problem and much more on the intangible human element. [...] In The Moonstone the mystery is finally solved, not altogether by human ingenuity, but largely by accident. Since Collins, the best heroes of English detective fiction have been, like Sergeant Cuff, fallible.

I knew that the languages which one learns there are necessary to understand the works of the ancients; and that the delicacy of fiction enlivens the mind; that famous deeds of history ennoble it and, if read with understanding, aid in maturing one's judgment; that the reading of all the great books is like conversing with the best people of earlier times; it is even studied conversation in which the authors show us only the best of their thoughts; that eloquence has incomparable powers and beauties; that poetry has enchanting delicacy and sweetness; that mathematics has very subtle processes which can serve as much to satisfy the inquiring mind as to aid all the arts and diminish man's labor; that treatises on morals contain very useful teachings and exhortations to virtue; that theology teaches us how to go to heaven; that philosophy teaches us to talk with appearance of truth about things, and to make ourselves admired by the less learned; that law, medicine, and the other sciences bring honors and wealth to those who pursue them; and finally, that it is desirable to have examined all of them, even to the most superstitious and false in order to recognize their real worth and avoid being deceived thereby

She was his comet, his shooting star. She sparkled like the heavens, and when she smiled it felt like mathematical equations sliding into place. The world in balance, each side properly weighted. She was everything that was beauty, and everything that was brilliant, and he was – He was not well.

If there is anything like a unifying aesthetic principle in mathematics, it is this: simple is beautiful. Mathematicians enjoy thinking about the simplest possible things, and the simplest possible things are imaginary.

How can I number the worlds to which the eye gives me entry? - the world of light, of colour, of shape, of shadow: of mathematical precision in the snowflake, the ice formation, the quartz crystal, the patterns of stamen and petal: of rhythm in the fluid curve and plunging line of the mountain faces. Why some blocks of stone, hacked into violent and tortured shapes, should so profoundly tranquillise the mind I do not know. Perhaps the eye imposes its own rhythm on what is only a confusion: one has to look creatively to see this mass of rock as more than jag and pinnacle - as beauty. Else why did men for so many centuries think mountains repulsive? A certain kind of consciousness interacts with the mountain-forms to create this sense of beauty. Yet the forms must be there for the eye to see. And forms of a certain distinction: mere dollops won't do it. It is, as with all creation, matter impregnated with mind: but the resultant issue is a living spirit, a glow in the consciousness, that perishes when the glow is dead. It is something snatched from non-being, that shadow which creeps in on us continuously and can be held off by continuous creative act. So, simply to look on anything, such as a mountain, with the love that penetrates to its essence, is to widen the domain of being in the vastness of non-being. Man has no other reason for his existence.

The association of multiplication with vector rotation was one of the geometric interpretation's most important elements because it decisively connected the imaginaries with rotary motion. As we'll see, that was a big deal.

That's the big difference between [the BookWorld] and [the RealWorld]," said Plum. "When things happen after a randomly pointless event, all that follows is simply unintended consequences, not a coherent narrative thrust that propels the story forward." I rolled the idea of unintended consequences around in my head. "Nope, I said finally, "you've got me on that one." "It confuses me, too," admitted Plum, "but that's the RealWorld for you. A brutal and beautiful place, run for the most part on passion, fads, incentives, and mathematics. A lot of mathematics.

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

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.

Nature seems to take advantage of the simple mathematical representations of the symmetry laws. When one pauses to consider the elegance and the beautiful perfection of the mathematical reasoning involved and contrast it with the complex and far-reaching physical consequences, a deep sense of respect for the power of the symmetry laws never fails to develop.

Réduites à des théories générales, les mathématiques seraient une belle forme sans contenu. Reduced to general theories, mathematics would be a beautiful form without content.

It is everything. It is the fabric of selfhood, and beautifully chaotic. It shares mathematical characteristics with many natural forms.

Beauty is the first test: there is no permanent place in the world for ugly mathematics.

Music can have wonderful, formal, quasi-mathematical perfection, and it can have heartbreaking tenderness, poignancy, and beauty. But it does not have to have any 'meaning' whatever. One may recall music, give it the life of imagination simply because one likes it - this is reason enough. Or perhaps there may be no reason at all, as Rodolfo Llinas points out.

Ada’s ability to appreciate the beauty of mathematics is a gift that eludes many people, including some who think of themselves as intellectual. She realized that math was a lovely language, one that describes the harmonies of the universe and can be poetic at times.

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.

... those who seek the lost Lord will find traces of His being and beauty in all that men have made, from music and poetry and sculpture to the gingerbread men in the pâtisseries, from the final calculation of the pure mathematician to the first delighted chalk drawing of a small child.

Quantum mechanics can be appreciated, to some degree, on a purely qualitative level. But mathematics is what brings its beauty into sharp focus. We have tried to make this amazing body of work fully accessible to mathematically literate nonphysicists.

When will you ask for your post back?” he whispered in her ear. “I miss the smell of industrial-strength solvents.” She laughed softly. “Soon. And when will you have papers read at the mathematical society again? I rather like having my husband called a genius for reasons that are not clear to me.” My husband. The words rolled off her tongue, easy and beautiful. He kissed her fervently. “Soon. My brilliance quite overflowed on the way home. I have four notebooks to show for it.” “Good. We don’t want people to think I love you for your looks alone.” “In that case we should also put you in some rather revealing gowns once in a while, so that people don’t think I married you for your accomplishments alone.

Thus far our meditation on quantum reality has revealed that the world of everyday matter, when properly understood, embodies concepts of extraordinary beauty. Indeed, ordinary matter is built up from atoms that are, in a rich and precise sense, tiny musical instruments. In their interplay with light, they realize a mathematical Music of the Spheres that surpasses the visions of Pythagoras, Plato, and Kepler. In molecules and ordered materials, those atomic instruments play together as harmonious ensembles and synchronized orchestras.

If a thousand people read a book, they read a thousand different books. But if a thousand people read an equation, they read the same equation.

I only know that when I study mathematics, I transport myself to another world, a world of exquisite beauty and truth. And in that world I am the person I like to be.

The best mathematics is serious as well as beautiful—‘important’ if you like, but the word is very ambiguous, and ‘serious’ expresses what I mean much better

It is more important to have beauty in one’s equation than to have them fit experiment

We should not be afraid of mathematics. If you already understand nature, you understand mathematics.

One of the beautiful things about mathematical physics is that equations contain stories.

To emphasize only the beautiful seems to me to be like a mathematical system that only concerns itself with positive numbers.

It is a funny paradox of design: utility breeds beauty. There is elegance in efficiency, a visual pleasure in things that just barely work.

What makes a mathematician is not technical skill or encyclopedic knowledge but insatiable curiosity and a desire for simple beauty.

The chief forms of beauty are order and symmetry and definiteness, which the mathematical sciences demonstrate in a special degree.

I know numbers are beautiful. If they aren't beautiful, nothing is.

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

Why do I know I exist if I also know I will not? Why was I given access to logical space and the mathematical structure of the world? Just to lose them when my body is destroyed? Why do I wake up in the night with the thought that I will die, why do I sit up, drenched in sweat, and scream and slap myself and try to suppress the thought that I will disappear for all eternity, that I will never be again, to the end of time? Why will the world end with me? We age: we stand quietly in line with those condemned to death. We are executed one after the other in a sinister extermination camp. We are first stripped of our beauty, youth, and hope. We are next wrapped in the penitential robe of illness, weariness, and decay. Our grandparents die, our parents are executed in front of us, and suddenly time gets short, you suddenly see your reflection in the axeblade. And only then do you realize you are living in a slaughterhouse, that generations are butchered and swallowed by the earth, that billions are pushed down the throat of hell, that no one, absolutely no one escapes. That not one person that you see coming out of the factory gates in a Mélies film is still alive. That absolutely everyone in an eighty-year-old sepia photograph is dead. That we all come into this world from a frightening abyss without our memories, that we suffer unimaginably on a speck of dust, and that we then perish, all in a nanosecond, as though we had never lived, as though we had never been.

Golden Ratio is a powerful mathematical constant woven into the very fabric of biology. It is the unique visual tension between comforting symmetry and compelling asymmetry, and its thoughtful application can bring beauty and harmony and intrigue to all manner of designed things.

Beauty magnetizes curiosity and wonder, beckoning us to discover—in the literal sense, to uncover and unconceal—what lies beneath the surface of the human label. What we recognize as beauty may be a language for encoding truth, a memetic mechanism for transmitting it, as native to the universe as mathematics—the one perceived by the optical eye, the other by the mind's eye.

There is a certain tyranny about perfection, a certain exhaustion about it even, something that denies the viewer a role in its creation and that asserts itself with all the dogmatism of an unambiguous statement. True beauty cannot be measured because it is fluctuating, it has only a few angles from which it may be seen, and then not in all lights and at all times. It flirts dangerously with ugliness, it takes risks with itself, it does not side comfortably with mathematical rules of proportion, it draws its appeal from precisely those areas that will also lend themselves to ugliness. Nothing can be beautiful that does not take a calculated risk with ugliness.

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

I could quote enough Nietzsche to bore someone into a coma, solve mathematical problems so beautiful they'd make Pythagoras cry; I could talk so much bullshit the listener didn't know if I was coming or going, but I had gotten to know Rickie well enough to realize that I had no idea how a woman's mind worked.

There is no such thing as beauty, especially in the human face… what we call the physiognomy. It’s all a mathematical and imagined alignment of features. Like, if the nose doesn’t stick out too much, the sides are in fashion, if the earlobes aren’t too large, if the hair is long… It’s kind of a mirage of generalization. People think of certain faces as beautiful, but, truly, in the final measure, they are not. It’s a mathematical equation of zero. “True beauty” comes, of course, of character. Not through how the eyebrows are shaped. So many women that I’m told are beautiful… hell, it’s like looking into a soup bowl.

Look you wanna know the truth? I don’t really care about the stats or the cup or the trophy or anything like that. In fact even the games aren’t that important to me. What matters to me is the perfect throw, making the perfect catch, the perfect step and block. Perfection. That's what it's about. It's those moments. When you can feel the perfection of creation. The beauty the physics you know the wonder of mathematics. The elations of action and reaction and that is the kind of perfection that I want to be connected to.

Steenrod was a careful, methodical man who chose his suits and sports coats according to a mathematical formula and had a mania for thinking up highly logical, if impractical, solutions to social problems like crime.

The Night Dances       A smile fell in the grass. Irretrievable!   And how will your night dances Lose themselves. In mathematics?   Such pure leaps and spirals—— Surely they travel   The world forever, I shall not entirely Sit emptied of beauties, the gift   Of your small breath, the drenched grass Smell of your sleeps, lilies, lilies.   Their flesh bears no relation. Cold folds of ego, the calla,   And the tiger, embellishing itself——

The description of this proportion as Golden or Divine is fitting perhaps because it is seen by many to open the door to a deeper understanding of beauty and spirituality in life. That’s an incredible role for one number to play, but then again this one number has played an incredible role in human history and the universe at large.

Quantum mechanics. What a repository, a dump, of human aspiration it was, the borderland where mathematical rigor defeated common sense, and reason and fantasy irrationally merged. Here the mystically inclined could find whatever they required and claim science as their proof. And for these ingenious men in their spare time, what ghostly and beautiful music it must be--spectral asymmetry, resonances, entanglement, quantum harmonic oscillators--beguiling ancient airs, the harmony of the spheres that might transmute a lead wall into gold and bring into being the engine that ran on virtually nothing, on virtual particles, that emitted no harm and would power the human enterprise as well as save it. Beard was stirred by the yearnings of these lonely men. And why should he think they were lonely? It was not, or not only, condescension that made him think them so. They did not know enough, but they knew too much to have anyone to talk to. What mate waiting down the pub or in the British Legion, what hard-pressed wife with job and kids and housework, was going to follow them down these warped funnels in the space-time continuum, into the wormhole, the shortcut to a single, final answer to the global problem of energy?

Neither parents nor schools are very effective at teaching the young to find pleasure in the right things. Adults, themselves often deluded by infatuation with fatuous models, conspire in the deception. They make serious tasks seem dull and hard, and frivolous ones exciting and easy. Schools generally fial to teach how exciting, how mesmerizingly beautiful science or mathematics can be; they teach the routine of literature or history rather than the adventure.

His fighting technique was a study in trigonometry, a beautiful composition of trajectories and rebounded forces. He won consistently because he had perfect control of distance and torque. He had the mathematics of fighting down to a science.

The modern science of nature is significant for many other reasons, beyond the obvious setting of conservation priorities and actions. Foremost in my mind being the fact that it is beautiful. Its wondrous mathematical synchronicities, the specifics of its chemical analyses, the complexity of its physics are beyond both the practical and intuitive knowledge of most lay naturalists (or mystics), no matter how seasoned. When mingled with the wildness of the natural world and the creativity of the human mind, good science reveals its center, its story, its deeper teaching. The science has its own poetic force.

Nature already uses the language of mathematics, so why not work with the environment instead of against it. We need to start mimicking the mathematical logic that occurs in the landscape, identify existing systems, and out of those concepts create new ones.

The mathematician’s patterns, like the painter’s or the poet’s, must be beautiful; the ideas, like the colours or the words, must fit together in a harmonious way. Beauty is the first test: there is no permanent place in the world for ugly mathematics. G.H. Hardy

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.

mathematician’s patterns, like the painter’s or the poet’s, must be beautiful; the ideas, like the colors or the words, must fit together in a harmonious way. Beauty is the first test; there is no permanent place in the world for ugly mathematics.” And that beauty is symmetry.

Geometry exist everywhere.It is necessary, however, to have eyes to see it, intelligence to understand it , and spirit to wonder at it.The wild Bedouin sees geometric forms but doesn't understand them ; the Sunni understands them but does not admire them; the artist, finally, perceives the perfection of figures, understands beauty, and admires order and harmony.God was the Great Geometer.He geometrized heaven and earth.

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 physics, theories are made of math. We don’t use math because we want to scare away those not familiar with differential geometry and graded Lie algebras; we use it because we are fools. Math keeps us honest—it prevents us from lying to ourselves and to each other. You can be wrong with math, but you can’t lie.

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.

Were he now still among the living, Dr. Incandenza would now describe tennis in the paradoxical terms of what’s now called ‘Extra-Linear Dynamics.’ And Schtitt, whose knowledge of formal math is probably about equivalent to that of a Taiwanese kindergartner, nevertheless seemed to know what Hopman and van der Meer and Bollettieri seemed not to know: that locating beauty and art and magic and improvement and keys to excellence and victory in the prolix flux of match play is not a fractal matter of reducing chaos to a pattern. Seemed intuitively to sense that it was a matter not of reduction at all, but — perversely — of expansion, the aleatory flutter of uncontrolled, metastatic growth — each well-shot ball admitting of n possible responses, n² responses to those responses, and on into what Incandenza would articulate to anyone who shared both his backgrounds as a Cantorian continuum of infinities of possible move and response, Cantorian and beautiful because infoliating, contained, this diagnate infinity of infinities of choice and execution, mathematically uncontrolled but humanly contained, bounded by the talent and imagination of self and opponent, bent in on itself by the containing boundaries of skill and imagination that brought one player finally down, that kept both from winning, that made it, finally, a game, these boundaries of self.

The profound ability to use aural and written language has enabled our species to collectively explore the concepts of science and mathematics, to capture the beauty of intricate thought, experience and philosophy, and indeed to venture beyond our tiny planet with the desire to expand our understanding of the very nature of existence itself.

I must study politics and war," wrote John Adams, "that my sons may have the liberty to study mathematics, and philosophy, geography, natural history, and naval architecture, navigation, commerce, and agriculture, in order to give their children the right to study painting, poetry, music, architecture, statuary, tapestry, and porcelain." Adams saw clearly that politics is the indispensable foundation for things elegant and beautiful. First and above all else, you must secure life, liberty and the right to pursue your own happiness.

This world is of a single piece; yet, we invent nets to trap it for our inspection. Then we mistake our nets for the reality of the piece. In these nets we catch the fishes of the intellect but the sea of wholeness forever eludes our grasp. So, we forget our original intent and then mistake the nets for the sea. Three of these nets we have named Nature, Mathematics, and Art. We conclude they are different because we call them by different names. Thus, they are apt to remain forever separated with nothing bonding them together. It is not the nets that are at fault but rather our misunderstanding of their function as nets. They do catch the fishes but never the sea, and it is the sea that we ultimately desire.

Other mathematical geniuses, Einstein and Bertrand Russell among them, recount similarly revelatory experiences in early adolescence.

Carrero’s First Law of Landscape Mathematics Mathematics is the language of nature. If you understand nature, you understand mathematics.

Mathematics is the most beautiful and most powerful creation of the human spirit.

The mathematical sciences particularly exhibit order, symmetry, and limitation; and these are the greatest forms of the beautiful.

Bertrand Russell once said, “Mathematics, rightly viewed, possesses not only truth, but supreme beauty. . . .

several studies have since shown that basic military training during peacetime can precipitate schizophrenia in men with a hitherto unsuspected vulnerability to the illness.15

When we see new ways of displaying data, we should be careful not to confuse presentation with substance

As for everything else, so for a mathematical theory: beauty can be perceived but not explained.

Sense of beauty, perception, and the mathematical universe are all part of the same texture.

Tom felt his darkness. His father was beautiful and clever, his mother was short and mathematically sure. Each of his brothers and sisters had looks or gifts or fortune. Tom loved all of them passionately, but he felt heavy and earth-bound. He climbed ecstatic mountains and floundered in the rocky darkness between the peaks. He had spurts of bravery but they were bracketed in battens of cowardice. Samuel said that Tom was quavering over greatness, trying to decide whether he could take the cold responsibility. Samuel knew his son’s quality and felt the potential of violence, and it frightened him, for Samuel had no violence—even when he hit Adam Trask with his fist he had no violence. And the books that came into the house, some of them secretly—well, Samuel rode lightly on top of a book and he balanced happily among ideas the way a man rides white rapids in a canoe. But Tom got into a book, crawled and groveled between the covers, tunneled like a mole among the thoughts, and came up with the book all over his face and hands. John Steinbeck. East of Eden (Kindle Locations 4766-4770). Viking.

The Golden Proportion, sometimes called the Divine Proportion, has come down to us from the beginning of creation. The harmony of this ancient proportion, built into the very structure of creation, can be unlocked with the 'key' ... 528, opening to us its marvelous beauty. Plato called it the most binding of all mathematical relations, and the key to the physics of the cosmos.

As always when he worked with this much concentration he began to feel a sense of introverting pressure. There was no way out once he was in, no genuine rest, no one to talk to who was capable of understanding the complexity (simplicity) of the problem or the approaches to a tentative solution. There came a time in every prolonged effort when he had a moment of near panic, or "terror in a lonely place," the original semantic content of the word. The lonely place was his own mind. As a mathematician he was free from subjection to reality, free to impose his ideas and designs on his own test environment. The only valid standard for his work, its critical point (zero or infinity), was the beauty it possessed, the deft strength of his mathematical reasoning. THe work's ultimate value was simply what it revealed about the nature of his intellect. What was at stake, in effect, was his own principle of intelligence or individual consciousness; his identity, in short. This was the infalling trap, the source of art's private involvement with obsession and despair, neither more nor less than the artist's self-containment, a mental state that led to storms of overwork and extended stretches of depression, that brought on indifference to life and at times the need to regurgitate it, to seek the level of expelled matter. Of course, the sense at the end of a serious effort, if the end is reached successfully, is one of lyrical exhilaration. There is air to breathe and a place to stand. The work gradually reveals its attachment to the charged particles of other minds, men now historical, the rediscovered dead; to the main structure of mathematical thought; perhaps even to reality itself, the so-called sum of things. It is possible to stand in time's pinewood dust and admire one's own veronicas and pavanes.

Mathematics, which doesn’t include the tawdry efforts of statistics or probability, pure mathematics, the product of the human mind turning to face itself, turning into itself, and finding in the realm of necessary consequences, where no contingent fact is to be seen or heard or smelled or tasted or touched – it discloses a beauty that exhausts human comprehension and a certainty the senses can never touch.

He was beautiful when he sat alone, he was like me, he had wide lapels, he was holding the mug in the hardest possible way so that his fingers were all twisted but still long and beautiful, he didn’t like to sit alone all the time, but this time, I swear, he didn’t care on way or the other. I’ll tell you why I like to sit alone, because I’m a sadist, that’s why we like to sit alone, because we’re the sadists who like to sit alone. He sat alone because he was beautifully dressed for the occasion and because he was not a civilian. We are the sadists you don’t have to worry about, you think, and we have no opinion on the matter of whether you have to worry about us, and we don’t even like to think about the matter because it baffles us. Maybe he doesn’t mean a thing to me any more but I think he was like me. You didn’t expect to fall in love, I said to myself and at the same time I answered gently, Do you think so? I heard you humming beautifully, your hum said that I can’t ignore you, that I’d finally come around for a number of delicious reasons that only you knew about, and here I am, Miss Blood. And you won’t come back, you won’t come back to where you left me, and that’s why you keep my number, so you don’t dial it by mistake when you’re fooling with the dial not even dialing numbers. You begin to bore us with your pain and we have decided to change your pain. You said you were happiest when you danced, you said you were happiest when you danced with me, now which do you mean? And so we changed his pain, we threw the idea of a body at him and we told him a joke, and then he thought a great deal about laughing and about the code. And he thought that she thought that he thought that she thought the worst thing a woman could do was to take a man away from his work because that made her what, ugly or beautiful? And now you’ve entered the mathematical section of your soul which you claimed you never had. I suppose that this, plus the broken heart, makes you believe that now you have a perfect right to go out and tame the sadists. He had the last line of each verse of the song but he didn’t have any of the other lines, the last line was always the same, Don’t call yourself a secret unless you mean to keep it. He thought he knew, or he actually did know too much about singing to be a singer; and if there is actually such a condition, is anybody in it, and are sadists born there? It is not a question mark, it is not an exclamation point, it is a full stop by the man who wrote Parasites of Heaven. Even if we stated our case very clearly and all those who held as we do came to our side, all of them, we would still be very few.

If you were given to thinking of numbers as having human-like qualities, you might picture e^i*pi as a guru into transcendental meditation who'd achieved infinite enlightenment. But there's a problem with that-Euler's formula shows that e^i*pi can never free itslef from worldly concerns. Recall, e^i*pi is really -1 in disguise, and -1 is just a mathism for owing a dollar to your friend, Steve. One hand clapping.

Finding a taxi, she felt like a child pressing her nose to the window of a candy store as she watched the changing vista pass by while the twilight descended and the capital became bathed in a translucent misty lavender glow. Entering the city from that airport was truly unique. Charles de Gaulle, built nineteen miles north of the bustling metropolis, ensured that the final point of destination was veiled from the eyes of the traveller as they descended. No doubt, the officials scrupulously planned the airport’s location to prevent the incessant air traffic and roaring engines from visibly or audibly polluting the ambience of their beloved capital, and apparently, they succeeded. If one flew over during the summer months, the visitor would be visibly presented with beautifully managed quilt-like fields of alternating gold and green appearing as though they were tilled and clipped with the mathematical precision of a slide rule. The countryside was dotted with quaint villages and towns that were obviously under meticulous planning control. When the aircraft began to descend, this prevailing sense of exactitude and order made the visitor long for an aerial view of the capital city and its famous wonders, hoping they could see as many landmarks as they could before they touched ground, as was the usual case with other major international airports, but from this point of entry, one was denied a glimpse of the city below. Green fields, villages, more fields, the ground grew closer and closer, a runway appeared, a slight bump or two was felt as the craft landed, and they were surrounded by the steel and glass buildings of the airport. Slightly disappointed with this mysterious game of hide-and-seek, the voyager must continue on and collect their baggage, consoled by the reflection that they will see the metropolis as they make their way into town. For those travelling by road, the concrete motorway with its blue road signs, the underpasses and the typical traffic-logged hubbub of industrial areas were the first landmarks to greet the eye, without a doubt, it was a disheartening first impression. Then, the real introduction began. Quietly, and almost imperceptibly, the modern confusion of steel and asphalt was effaced little by little as the exquisite timelessness of Parisian heritage architecture was gradually unveiled. Popping up like mushrooms were cream sandstone edifices filigreed with curled, swirling carvings, gently sloping mansard roofs, elegant ironwork lanterns and wood doors that charmed the eye, until finally, the traveller was completely submerged in the glory of the Second Empire ala Baron Haussmann’s master plan of city design, the iconic grand mansions, tree-lined boulevards and avenues, the quaint gardens, the majestic churches with their towers and spires, the shops and cafés with their colourful awnings, all crowded and nestled together like jewels encrusted on a gold setting.

The fundamental theorem of calculus is one of the simplest and most beautiful results in mathematics. It asserts a deep connection between integrals and derivatives. What it says is that if F(T) = ∫ f(t) d t, then

In my entire scientific life, extending over forty-five years, the most shattering experience has been the realization that an exact solution of Einstein's equations of general relativity, discovered by the New Zealand mathematician Roy Kerr, provides the absolute exact representation of untold numbers of massive black holes that populate the universe. This "shuddering before the beautiful," this incredible fact that a discovery motivated by a search after the beautiful in mathematics should find its exact replica in Nature, persuades me to say that beauty is that to which the human mind responds at its deepest and most profound level.

Because we are forced to learn about numbers and operations with numbers while we are still too young to appreciate them—a preparation for life which hardly excites our interest in the future—we grow up believing that numbers are drab and uninteresting. But the number system warrants attention not only as the basis of mathematics, but because it contains weighty and beautiful ideas which lend themselves to powerful applications.

Physics is the ultimate intellectual adventure, the quest to understand the deepest mysteries of our Universe. Physics doesn’t take something fascinating and make it boring. Rather, it helps us see more clearly, adding to the beauty and wonder of the world around us. When I bike to work in the fall, I see beauty in the trees tinged with red, orange and gold. But seeing these trees through the lens of physics reveals even more beauty.

The axiom of equality states that x always equals x: it assumes that if you have a conceptual thing named x, that it must always be equivalent to itself, that it has a uniqueness about it, that it is in possession of something so irreducible that we must assume it is absolutely, unchangeably equivalent to itself for all time, that its very elementalness can never be altered. But it is impossible to prove. Always, absolutes, nevers: these are words, as much as numbers, that make up the world of mathematics. Not everyone liked the axiom of equality - Dr. Li had once called it coy and twee, a fan dance of an axiom - but he had always appreciated how elusive it was, how the beauty of the equation itself would always be frustrated by the attempts to prove it. It was the kind of axiom that could drive you mad, that could consume you, that could easily become an entire life.

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

When Euler discovered that i^i is real, he exclaimed in a letter to a friend that this "seems extraordinary to me"-part of his genius, as well as of his charm, was an inexhaustible capacity to be surprised and delighted by his discoveries.

I wanted very much to learn to draw, for a reason that I kept to myself: I wanted to convey an emotion I have about the beauty of the world. It’s difficult to describe because it’s an emotion. It’s analogous to the feeling one has in religion that has to do with a god that controls everything in the whole universe: there’s a generality aspect that you feel when you think about how things that appear so different and behave so differently are all run “behind the scenes” by the same organization, the same physical laws. It’s an appreciation of the mathematical beauty of nature, of how she works inside; a realization that the phenomena we see result from the complexity of the inner workings between atoms; a feeling of how dramatic and wonderful it is. It’s a feeling of awe—of scientific awe—which I felt could be communicated through a drawing to someone who had also had this emotion. It could remind him, for a moment, of this feeling about the glories of the universe.

Nash’s genius was of that mysterious variety more often associated with music and art than with the oldest of all sciences. It wasn’t merely that his mind worked faster, that his memory was more retentive, or that his power of concentration was greater. The flashes of intuition were non-rational. Like other great mathematical intuitionists — Georg Friedrich Bernhard Riemann, Jules Henri Poincaré, Srinivasa Ramanujan — Nash saw the vision first, constructing the laborious proofs long afterward.

I grew up in Brooklyn, New York.. a city neighborhood that included houses, lampposts, walls, and bushes. But with an early bedtime in the winter, I could look out my window and see the stars, and the stars were not like anything else in my neighborhood. [At age 5] I didn’t know what they were. [At age 9] my mother … said to me, “You have a library card now, and you know how to read. Take the streetcar to the library and get a book on stars.” … I stepped up to the big librarian and asked for a book on stars. … I sat down and found out the answer, which was something really stunning. I found out that the stars are glowing balls of gas. I also found out that the Sun is a star but really close and that the stars are all suns except really far away I didn’t know any physics or mathematics at that time, but I could imagine how far you’d have to move the Sun away from us till it was only as bright as a star. It was in that library, reading that book, that the scale of the universe opened up to me. There was something beautiful about it. At that young age, I already knew that I’d be very happy if I could devote my life to finding out more about the stars and the planets that go around them. And it’s been my great good fortune to do just that.

Mathematics, rightly viewed, possesses not only truth, but supreme beauty, a beauty cold and austere, like that of sculpture, without appeal to any part of our weaker nature, without the gorgeous trappings of painting or music, yet sublimely pure, and capable of a stern perfection such as only the greatest art can show. The true spirit of delight, the exaltation, the sense of being more than Man, which is the touchstone of the highest excellence, is to be found in mathematics as surely as in poetry.

The shadows on the flat screen of a shadow play are projections from more complex objects. Our three-dimensional hands can cast a variety of two-dimensional shadows to delight the audience. In the same way there are fabulous beasts that swim in the seas of mathematics. Multidimensional behemoths of incredible beauty that even the best of minds struggle to glimpse. The equations we battle with, the proofs that we use to nibble at the edges of such wonders: these are the shadows cast by those we hunt.

The imaginaries finally lost their air of impossibility when nineteenth-century mathematicians realized that they're actually perfectly ordinary, law-abiding numerical beings-it's just that they hail from a different dimension. We'll go there later on.

In my opinion, the black hole is incomparably the most exciting and the most important consequence of general relativity. Black holes are the places in the universe where general relativity is decisive. But Einstein never acknowledged his brainchild. Einstein was not merely skeptical, he was actively hostile to the idea of black holes. He thought that the black hole solution was a blemish to be removed from his theory by a better mathematical formulation, not a consequence to be tested by observation. He never expressed the slightest enthusiasm for black holes, either as a concept or as a physical possibility. Oddly enough, Oppenheimer too in later life was uninterested in black holes, although in retrospect we can say that they were his most important contribution to science. The older Einstein and the older Oppenheimer were blind to the mathematical beauty of black holes, and indifferent to the question whether black boles actually exist. How did this blindness and this indifference come about?

Cusps can be found at many places on the body. Am obvious example is a cleavage, when viewed obliquely. It is a little-known fact that there is a two-thirds power law at the focus of many a historical period drama, center stage in the heroine's decolletage.

One of them, Emmanuel Giroux, is a true giant of mathematics and currently heads a laboratory of sixty people at the École normale supérieure in Lyon. Blind since the age of eleven, he is most well-known for his beautiful proof of an important theorem of contact geometry.

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.

In my opinion, defining intelligence is much like defining beauty, and I don’t mean that it’s in the eye of the beholder. To illustrate, let’s say that you are the only beholder, and your word is final. Would you be able to choose the 1000 most beautiful women in the country? And if that sounds impossible, consider this: Say you’re now looking at your picks. Could you compare them to each other and say which one is more beautiful? For example, who is more beautiful— Katie Holmes or Angelina Jolie? How about Angelina Jolie or Catherine Zeta-Jones? I think intelligence is like this. So many factors are involved that attempts to measure it are useless. Not that IQ tests are useless. Far from it. Good tests work: They measure a variety of mental abilities, and the best tests do it well. But they don’t measure intelligence itself.

we must not forget that the restful experience of enjoyable beauty is not limited to the contemplation of sensible objects. We can experience it as well in the contemplation of purely intelligible objects—the contemplation of truths we understand. “Mathematics,” wrote Bertrand Russell, “rightly viewed, possesses not only truth, but supreme beauty—a beauty cold and austere … without appeal to any part of our weaker nature, without the gorgeous trappings of painting or music …” Or, as the poet Edna St. Vincent Millay wrote in the opening line of her sonnet on Euclid, “Euclid alone has looked on beauty bare.

pretty sure it’s a mathematic symbol.” He removes a beautiful fountain pen from his pocket, turns over his coaster, and writes. He holds up the coaster. Ethan looks at it and says, “The Kronecker delta symbol?” Leo points at him, delighted. “Exactly! I guess you studied it as part of computer science, right?

Philosophy, a love of wisdom, is both a desire for a good and an appreciation of the admirable. The good is an object of desire and love, the admirable is an object of contemplation. If we focus too exclusively on what is useful or even on what is good, we lose the capacity for admiration: “We become blind to the beauty that completes the good.” The admirable manifests itself in all the works of intelligence: in the elegance of well-formed mathematical systems, in deeply moving political speeches, in a life well lived, and in a well-ordered city. What is admirable in all of these things is the way they have to be. Their forms express this necessity, not in the sense of something relentless and overpowering, but in the sense of a fullness that displays their perfection. Philosophy is to remind us of the necessity in things: not just the necessities to which we have to resign ourselves, but those we can find splendid.

If we increase r [in a logistic map] even more, we will eventually force the system into a period-8 limit cycle, then a period-16 cycle, and so on. The amount that we have to increase r to get another period doubling gets smaller and smaller for each new bifurcation. This cascade of period doublings is reminiscent of the race between Achilles and the tortoise, in that an infinite number of bifurcations (or time steps in the race) can be confined to a local region of finite size. At a very special critical value, the dynamical system will fall into what is essentially an infinite-period limit cycle. This is chaos.

We human beings are moved by music as no other animal is. Stranger still, it moves us rational animals apart from whether we can play it, read it, or even much understand it. Music reaches the passions without passing through the mind. Although some music calls forth enormous, in truth, life-long diligence from those who play it, those who have devoted no study whatever to listening to it are moved by it. As a consequence music is unique among human pursuits in being able to overcome the vast gulf between rare virtue and common aptitude. It is the most mathematical of the fine arts. It is science and fun together.19

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.

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

David watched Sol finish the can of beans. Sol always ate fast. He overate. Since he was eight. Sol leaned back in the wooden chair against the wall, under the window. Baby brother. Sol could have been anything he wanted to be. Nothing mathematical, nothing quantitative, of course, but anything else. A beautiful boy, a wonderful brother, they got along well. Then when Sol turned eight years old, age of a new beginning, their mother obsessed over him, ignoring David. Obsessed over Sol and his underwear. Over and over, a regeneration, a newness. Changed his clothes constantly, had him on her lap every minute possible. She put him in bed and tucked the covers in every night. She refused to let their father do it, so he always took care of David. But, of course, David needed no help and Sol always needed Ruth to take care of him. Sol was still being breast fed even when he finally went to kindergarten. Then Ruth slowly increased regular, solid food. But before bed, Sol had a nightcap, mother’s milk. Their special time. Their unique closeness took a turn from breastfeeding to something else. By the time Sol was in third grade, he was one of the fattest kids in class. Then the brothers became a real team. Now here he was. David and Sol still together, on a mission given by the Creator. It was perfect.

The mathematical order is beautiful precisely because it has no effect on the real world. Life isn't going to be easier, nor is anyone going to make a fortune, just because they know something about prime numbers. Of course, lots of mathematical discoveries have practical applications, no matter how esoteric they may seem. Research on ellipses made it possible to determine the orbits of the planets, and Einstein used non-Euclidean geometry to describe the form of the universe. Even prime numbers were used during the war to create codes—to cite a regrettable example. But those things aren't the goal of mathematics. The only goal is to discover the truth.

Equality (outside mathematics) is a purely social conception. It applies to man as a political and economic animal. It has no place in the world of the mind. Beauty is not democratic; she reveals herself more to the few than to the many, more to the persistent and disciplined seekers than to the careless. Virtue is not democratic; she is achieved by those who pursue her more hotly than most men. Truth is not democratic; she demands special talents and special industry in those to whom she gives her favours. Political democracy is doomed if it tries to extend its demand for equality into these higher spheres. Ethical, intellectual, or aesthetic democracy is death.

Euler's general equation stands out because it forged a fundamental link between different areas of math, and because of its versatility in applied mathematics. After Euler's time it came to be regarded as a cornerstone in "complex analysis," a fertile branch of mathematics concerned with functions whose variables stand for complex numbers.

If you needed to see someone’s true colors before you could love them, you could never love anyone. For love to exist, people must be able to love lies and masks, veneers and artifice. How many people are loved for who they are? Most people are loved for who they are not! People hide their true colors. They are afraid to show them. They don’t find their true colors beautiful. They find them ugly. They are afraid of them. If your true colors are the colors of the rainbow, what are your false colors? They are the colors of the anti-rainbow. People are broken. Who will fix them? Where is the Cosmic Repair Shop? Don’t worry, mathematics is designed to cure every single problem you have.

Quantum mechanics. What a repository, a dump, of human aspiration it was, the borderland where mathematical rigor defeated common sense, and reason and fantasy irrationally merged. Here the mystically inclined could find whatever they required and claim science as their proof. And for these ingenious men in their spare time, what ghostly and beautiful music it must be—spectral asymmetry, resonances, entanglement, quantum harmonic oscillators—beguiling ancient airs, the harmony of the spheres that might transmute a lead wall into gold and bring into being the engine that ran on virtually nothing, on virtual particles, that emitted no harm and would power the human enterprise as well as save it.

The genius of Laplace was a perfect sledge hammer in bursting purely mathematical obstacles; but, like that useful instrument, it gave neither finish nor beauty to the results. In truth, in truism if the reader please, Laplace was neither Lagrange nor Euler, as every student is made to feel. The second is power and symmetry, the third power and simplicity; the first is power without either symmetry or simplicity. But, nevertheless, Laplace never attempted investigation of a subject without leaving upon it the marks of difficulties conquered: sometimes clumsily, sometimes indirectly, always without minuteness of design or arrangement of detail; but still, his end is obtained and the difficulty is conquered.

Multiplying an infinitesimal times an infinitely large number yields a finite number. There's no analogue to this rule in regular arithmetic. However, it accords with the intuitive idea that when the infinitely large is pitted against the infinitely small, the two basically cancel each other out in a titanic clash, and after the dust clears a finite number remains.

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.

Zen master is not trying to give you ideas about life; he is trying to give you life itself, to make you realize life in and around you, to make you live it instead of being a mere spectator, a mere pedant absorbed in the dry bones of something which the life has long deserted. A symphony is not explained by a mathematical analysis of its notes; the mystery of a woman’s beauty is not revealed by a postmortem dissection; and no one ever understood the wonder of a bird on the wing by stuffing it and putting it in a glass case. To understand these things, you must live and move with them as they are alive. The same is true of the universe: no amount of intellectual analysis will explain it, for philosophy and science can only reveal its mechanism, never its meaning or, as the Chinese say, its Tao. “What is the Tao?” A Zen master answers, “Usual life is the very Tao.” “How does one bring oneself into accord with it?” “If you try to accord with it, you will get away from it.” For to imagine that there is a “you” separate from life which somehow has to accord with life is to fall straight into the trap.

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...

The mathematical order is beautiful precisely because it has no effect on the real world. Life isn’t going to be easier, nor is anyone going to make a fortune, just because they know something about prime numbers. Of course, lots of mathematical discoveries have practical applications, no matter how esoteric they may seem. …. But those things aren’t the goal of mathematics. The only goal is to discover the truth.

Scientific thinking explores and redraws the world, gradually offering us better and better images of it, teaching us to think in ever more effective ways. Science is a continual exploration of ways of thinking. Its strength is its visionary capacity to demolish preconceived ideas, to reveal new regions of reality, and to construct new and more effective images of the world. This adventure rests upon the entirety of past knowledge, but at its heart is change. The world is boundless and iridescent; we want to go and see it. We are immersed in its mystery and in its beauty, and over the horizon there is unexplored territory. The incompleteness and the uncertainty of our knowledge, our precariousness, suspended over the abyss of the immensity of what we don't know, does not render life meaningless: it makes it interesting and precious.

We talk about the fact that the universe is designed in this way and you can find all of these patterns; do you think that that's a coincidence? One of the things I love to point out is we call the universe the cosmos which means ordered and structured and designed, as opposed to chaos, and the reason why we can find these mathematical principles is because there was a Designer. We didn't just spring into being. It has immense beauty.

Möbius Strips. Year of the Whopper. Latrodectus Mactans Productions. 'Hugh G. Section,' Pam Heath, 'Bunny Day,' 'Taffy Appel'; 35 mm.; 109 minutes black and white; sound. Pornography-parody, possible parodic homage to Fosse's All That Jazz, in which a theoretical physicist ('Reaction'), who can only achieve mathematical insight during coitus, conceives of Death as a lethally beautiful woman (Heath). INTERLACE TALENT FEATURE CARTRIDGE #357-65-32 (Y.W.)

Nash’s lifelong quest for meaning, control, and recognition in the context of a continuing struggle, not just in society, but in the warring impulses of his paradoxical self, was now reduced to a caricature. Just as the overconcreteness of a dream is related to the intangible themes of waking life, Nash’s search for a piece of paper, a carte d’identité, mirrored his former pursuit of mathematical insights. Yet the gulf between the two recognizably related Nashes was as great as that between Kafka, the controlling creative genius, struggling between the demands of his self-chosen vocation and ordinary life, and K, a caricature of Kafka, the helpless seeker of a piece of paper that will validate his existence, rights, and duties. Delusion is not just fantasy but compulsion. Survival, both of the self and the world, appears to be at stake. Where once he had ordered his thoughts and modulated them, he was now subject to their peremptory and insistent commands.

locating beauty and art and magic and improvement and keys to excellence and victory in the prolix flux of match play is not a fractal matter of reducing chaos to pattern. Seemed intuitively to sense that it was a matter not of reduction at all, but—perversely—of expansion, the aleatory flutter of uncontrolled, metastatic growth—each well-shot ball admitting of n possible responses, n2 possible responses to those responses, and on into what Incandenza would articulate to anyone who shared both his backgrounds as a Cantorian 35 continuum of infinities of possible move and response, Cantorian and beautiful because infoliating, contained, this diagnate infinity of infinities of choice and execution, mathematically uncontrolled but humanly contained, bounded by the talent and imagination of self and opponent, bent in on itself by the containing boundaries of skill and imagination that brought one player finally down, that kept both from winning, that made it, finally, a game, these boundaries of self. ‘You

But just what are imaginary numbers, you may now be asking yourself, and what on earth could it mean to raise e to an imaginary-number power? This chapter concerns mathematicians' long struggle to answer the first of these two questions. Later we'll take up the second one , which inspired Euler to devise the most radical expansion of the concept of exponents in math history. At this point, suffice it to say that affixing an imaginary exponent to a number has a dramatic effect on it-something lime what happens to a frog when it's tapped by a standard-issue magic wand.

Euclid alone has looked on Beauty bare. Let all who prate of Beauty hold their peace, And lay them prone upon the earth and cease To ponder on themselves, the while they stare At nothing, intricately drawn nowhere In shapes of shifting lineage; let geese Gabble and hiss, but heroes seek release From dusty bondage into luminous air. O blinding hour, O holy, terrible day, When first the shaft into his vision shone Of light anatomized! Euclid alone Has looked on Beauty bare. Fortunate they Who, though once only and then but far away, Have heard her massive sandal set on stone

In my view, Euler's tranquil temperament, fairness, and generosity were integral to his greatness as a mathematician and scientist- he was never inclined to waste time and energy engaging in petty one-upmanship (like his mentor, Johann Bernoulli, who was known for getting into the eighteenth-century version of flame wars with his older brother, mathematician Jakob Bernoulli, and even with his own son, Daniel, over technical disputes), brooding about challenges to his authority (like Newton), or refusing to publish important findings because of the fear that they might be disputed (like Gauss).

In fact, the last great, and often dark, wizard was Isaac Newton. Newton’s work had beauty, simplicity, and elegance. He is widely thought to have made the greatest work of science ever created. Newton was the 17th century British natural philosopher who first uncovered the laws of physics that govern the cosmos. He made up new branches of mathematics, conquered the composition of light, and divined the laws of gravity and motion, which hold sway across the entire universe. Newton ushered in an age, the Newtonian Age, based on the notion that all things in the cosmos were open to rational understanding.

Every culture has its own creation myth, its own cosmology. And in some respects every cosmology is true, even if I might flatter myself in assuming mine is somehow truer because it is scientific. But it seems to me that no culture, including scientific culture, has cornered the market on definitive answers when it comes to the ultimate questions. Science may couch its models in the language of mathematics and observational astronomy, while other cultures use poetry and sacrificial propitiations to defend theirs. But in the end, no one knows, at least not yet. The current flux in the state of scientific cosmology attests to this, as we watch physicists and astronomers argue over string theory and multiverses and the cosmic inflation hypothesis. Many of the postulates of modern cosmology lie beyond, or at least at the outer fringes, of what can be verified through observation. As a result, aesthetics—as reflected by the “elegance” of the mathematical models—has become as important as observation in assessing the validity of a cosmological theory. There is the assumption, sometimes explicit and sometimes not, that the universe is rationally constructed, that it has an inherent quality of beauty, and that any mathematical model that does not exemplify an underlying, unifying simplicity is to be considered dubious if not invalid on such criteria alone. This is really nothing more than an article of faith; and it is one of the few instances where science is faith-based, at least in its insistence that the universe can be understood, that it “makes sense.” It is not entirely a faith-based position, in that we can invoke the history of science to support the proposition that, so far, science has been able to make sense, in a limited way, of much of what it has scrutinized. (The psychedelic experience may prove to be an exception.)

First, let me frame what I'm calling beautiful. It's not simply the equation's neat little string of symbols. Rather, it's the entire nimbus of meaning surrounding the formula, including its funneling of many concepts into a statement of stunning brevity, its arresting combination of apparent simplicity and hidden complexity, the way its derivation bridges disparate topics in mathematics, and the fact that it's rich with implications, some of which weren't apparent until many years after it was proved to be true. I think most mathematicians would agree that the equation's beauty concerns something like this nimbus.

(William) Hamilton recast the central ideas (of the evolutionary theory of aging) in mathematical form. Though this work tells us a good deal about why human lives take the course they do, Hamilton was a biologist whose great love was insects and their relatives, especially insects which make both our lives and an octopus’s life seem rather humdrum. Hamilton found mites in which the females hang suspended in the air with their swollen bodies packed with newly hatched young, and the males in the brood search out and copulate with their sisters there inside the mother. He found tiny beetles in which the males produce “and manhandle sperm cells longer than their whole bodies. Hamilton died in 2000, after catching malaria on a trip to Africa to investigate the origins of HIV. About a decade before his death, he wrote about how he would like his own burial to go. He wanted his body carried to the forests of Brazil and laid out to be eaten from the inside by an enormous winged Coprophanaeus beetle using his body to nurture its young, who would emerge from him and fly off. 'No worm for me nor sordid fly, I will buzz in the dusk like a huge bumble bee. I will be many, buzz even as a swarm of motorbikes, be borne, body by flying body out into the Brazilian wilderness beneath the stars, lofted under those beautiful and un-fused elytra [wing covers] which we will all hold over our “backs. So finally I too will shine like a violet ground beetle under a stone.

It was this situation that led mathematician Chris Hauert and his colleagues to consider another possibility in an important evolutionary model published in Science in 2002. In Axelrod's study and in most previous theoretical models, individuals were forced to interact with each other. But what if they could choose not to interact? Rather than attempting to cooperate and risking being taken advantage of, a person could fend for herself. In other words, she could sever her connections to others in the network. Hauert called the people who adopt this strategy "loners." Using some beautiful mathematics, Hauert and his colleagues showed that in a world full of loners it is easy for cooperation to evolve because there are no people to take advantage of the cooperators that appear. The loners fend for themselves, and the cooperators form networks with other cooperators. Soon, the cooperators take over the population because they always do better together than the loners. But once the world is full of cooperators, it is very easy for free riders to evolve and enjoy the fruits of cooperation without contributing (like parasites). As the free riders become the dominant type in the population, there is no one left for them to take advantage of; then, the loners once again take over -- because they want nothing to do, as it were, with those bastards. In short, cooperating can emerge because we can do more together than we can apart. But because of the free-rider problem, cooperation is not guaranteed to succeed.

Chinese philosopher Tsang Lap-Chuen is a leading modern exponent of the idea that the sublime involves this kind of experience. In The Sublime: Groundwork towards a Theory, published in 1998, he wrote that the sublime "evokes our awareness of our being on the threshold from the human to that which transcends the human; which borders on the possible and the impossible; the knowable and the unknowable; the meaningful and the fortuitous; the finite and the infinite." In his view, there is no single essential common property possessed by sublime works or sublime natural objects, nor is there a single emotional state evoked by all of them. But he argues that there's a common thread in experiences of the sublime, which is that they take us "to the limit of some human possibility.

I’ve been sitting here and thinking about God. I don’t think I believe in God any more. It is not only me, I think of all the millions who must have lived like this in the war. The Anne Franks. And back through history. What I feel I know now is that God doesn’t intervene. He lets us suffer. If you pray for liberty then you may get relief just because you pray, or because things happen anyhow which bring you liberty. But God can’t hear. There’s nothing human like hearing or seeing or pitying or helping about him. I mean perhaps God has created the world and the fundamental laws of matter and evolution. But he can’t care about the individuals. He’s planned it so some individuals are happy, some sad, some lucky, some not. Who is sad, who is not, he doesn’t know, and he doesn’t care. So he doesn’t exist, really. These last few days I’ve felt Godless. I’ve felt cleaner, less muddled, less blind. I still believe in a God. But he’s so remote, so cold, so mathematical. I see that we have to live as if there is no God. Prayer and worship and singing hymns—all silly and useless. I’m trying to explain why I’m breaking with my principles (about never committing violence). It is still my principle, but I see you have to break principles sometimes to survive. It’s no good trusting vaguely in your luck, in Providence or God’s being kind to you. You have to act and fight for yourself. The sky is absolutely empty. Beautifully pure and empty. As if the architects and builders would live in all the houses they built! Or could live in them all. It’s obvious, it stares you in the face. There must be a God and he can’t know anything about us.

Some people object to such a view of music, saying that if you reduce music to mathematics, where does the emotion come into it? I would say that it’s never been out of it. The things by which our emotions can be moved—the shape of a flower or a Grecian urn, the way a baby grows, the way the wind brushes across your face, the way clouds move, their shapes, the way light dances on the water, or daffodils flutter in the breeze, the way in which the person you love moves their head, the way their hair follows that movement, the curve described by the dying fall of the last chord of a piece of music—all these things can be described by the complex flow of numbers. That’s not a reduction of it, that’s the beauty of it. Ask Newton. Ask Einstein. Ask the poet (Keats) who said that what the imagination seizes as beauty must be truth.

The predominant thoughts and feelings of a pregnant woman are lodged in some of the major chakras of the unborn baby. They will therefore affect the character of the unborn baby. To produce better babies, it is very important for a pregnant woman to see and hear things that are beautiful, inspiring, and strong. The feelings and thoughts should be harmonious and progressive or positive. Anger, pessimism, hopelessness, injurious words, negative feelings and thoughts should be avoided. It is advisable for a pregnant mother to read books that are inspirational like the biographies of great yogis or great people, books on spiritual teachings, mathematics, sciences, business and languages. All of these will have beneficial effects on the unborn baby and will tend to make the baby not only spiritual, but also sharp-minded and practical.

It is the best of times in physics. Physicists are on the verge of obtaining the long-sought theory of everything. In a few elegant equations, perhaps concise enough to be emblazoned on a T-shirt, this theory will reveal how the universe began and how it will end. The key insight is that the smallest constituents of the world are not particles, as had been supposed since ancient times, but “strings”—tiny strands of energy. By vibrating in different ways, these strings produce the essential phenomena of nature, the way violin strings produce musical notes. String theory isn’t just powerful; it’s also mathematically beautiful. All that remains to be done is to write down the actual equations. This is taking a little longer than expected. But, with almost the entire theoretical-physics community working on the problem—presided over by a sage in Princeton, New Jersey—the millennia-old dream of a final theory is sure to be realized before long. It is the worst of times in physics. For more than a generation, physicists have been chasing a will-o’-the-wisp called string theory. The beginning of this chase marked the end of what had been three-quarters of a century of progress. Dozens of string-theory conferences have been held, hundreds of new Ph.D.’s have been minted, and thousands of papers have been written. Yet, for all this activity, not a single new testable prediction has been made; not a single theoretical puzzle has been solved. In fact, there is no theory so far—just a set of hunches and calculations suggesting that a theory might exist. And, even if it does, this theory will come in such a bewildering number of versions that it will be of no practical use: a theory of nothing. Yet the physics establishment promotes string theory with irrational fervor, ruthlessly weeding dissenting physicists from the profession. Meanwhile, physics is stuck in a paradigm doomed to barrenness.

Supersymmetry was (and is) a beautiful mathematical idea. The problem with applying supersymmetry is that it is too good for this world. We simply do not find particles of the sort it predicts. We do not, for example, see particles with the same charge and mass as electrons, but a different amount of spin. However, symmetry principles that might help to unify fundamental physics are hard to come by, so theoretical physicists do not give up on them easily. Based on previous experience with other forms of symmetry, we have developed a fallback strategy, called spontaneous symmetry breaking. In this approach, we postulate that the fundamental equations of physics have the symmetry, but the stable solutions of these equations do not. The classic example of this phenomenon occurs in an ordinary magnet. In the basic equations that describe the physics of a lump of iron, any direction is equivalent to any other, but the lump becomes a magnet with some definite north-seeking pole.

Now be honest-wouldn't you have expected e^i*pi to be (a) gibberish aling the lines of "elephant inkpie," or, if it were mathematically meaningful, to be (b) an infinitely complicated irrational number? Indeed, e^i*pi is a transcendental number raised to an imaginary transcendental power. And if (b) were the case, surely e^i*pi would not compute no matter how much computer power were available to try to pin down its value. As you know, neither (a) nor (b) is true, because e^i*pi = -1. (I suspect the fact that both (a) and (b) are provably false is the reason that Benjamin Peirce, the nineteenth-century mathematician, found Euler's formula (or a closely rekated formula) "absolutely paradoxical.") In other words, when the three enigmatic numbers are combined in this form, e^i*pi , they react together to carve out a wormhole that spirals through the infinite depths of number space to emerge smack dab in the heartland of integers. It's as if greenish-pink androids rocketing toward Alpha Centauri in 2370 had hit a space time anomaly and suddenly found themselves sitting in a burger joint in Topeka, Kansas, in 1956. Elvis, of course , was playing on the jukebox.

I taught mathematics here at the university for many years. But before that time, I used to create beautiful music with my beloved wife, who passed away. Her name was Sarah.” He paused for a moment to gather himself after uttering her name, but then continued. “I didn’t play for many years after she died, but one thing that loss has taught me is that keeping a gift locked away only brings harm. It is essential to express ourselves. Walls and doors keep out not just the bad but also the good. It is our job to own the keys of our freedom, and to be able to open those doors. I spent many years behind a closed door without knowing where to find the key to my life, but thanks to my music, I am now finding my way out. To quote an extraordinary but lesser-known philosopher, my late wife Sarah Held, ‘What is life without the beauty of art or music or poetry to help us interpret it, encourage us to know how to feel, how to love and how to live?’ In music I see the darkness, the light, the messiness, the beauty, and the complexities of life that simply can’t be summed up like an equation. Music, for me, helps bring down the walls, to open those locked doors. And I hope it has been that way for you too, tonight.

And Schtitt, whose knowledge of formal math is probably about equivalent to that of a Taiwanese kindergartner, nevertheless seemed to know what Hopman and van der Meer and Bollettieri seemed not to know: that locating beauty and art and magic and improvement and keys to excellence and victory in the prolix flux of match play is not a fractal matter of reducing chaos to pattern. Seemed intuitively to sense that it was a matter not of reduction at all, but — perversely — of expansion, the aleatory flutter of uncontrolled, metastatic growth — each well-shot ball admitting of n possible responses, 2n possible responses to those responses, and on into what Incandenza would articulate to anyone who shared both his backgrounds as a Cantorian 35 continuum of infinities of possible move and response, Cantorian and beautiful because infoliating, contained, this diagnate infinity of infinities of choice and execution, mathematically uncontrolled but humanly contained, bounded by the talent and imagination of self and opponent, bent in on itself by the containing boundaries of skill and imagination that brought one player finally down, that kept both from winning, that made it, finally, a game, these boundaries of self.

Finally, some people tell me that they avoid science fiction because it’s depressing. This is quite understandable if they happened to hit a streak of post-holocaust cautionary tales or a bunch of trendies trying to outwhine each other, or overdosed on sleaze-metal-punk-virtual-noir Capitalist Realism. But the accusation often, I think, reflects some timidity or gloom in the reader’s own mind: a distrust of change, a distrust of the imagination. A lot of people really do get scared and depressed if they have to think about anything they’re not perfectly familiar with; they’re afraid of losing control. If it isn’t about things they know all about already they won’t read it, if it’s a different color they hate it, if it isn’t McDonald’s they won’t eat at it. They don’t want to know that the world existed before they were, is bigger than they are, and will go on without them. They do not like history. They do not like science fiction. May they eat at McDonald’s and be happy in Heaven." Pro: "But what I like in and about science fiction includes these particular virtues: vitality, largeness, and exactness of imagination; playfulness, variety, and strength of metaphor; freedom from conventional literary expectations and mannerism; moral seriousness; wit; pizzazz; and beauty. Let me ride a moment on that last word. The beauty of a story may be intellectual, like the beauty of a mathematical proof or a crystalline structure; it may be aesthetic, the beauty of a well-made work; it may be human, emotional, moral; it is likely to be all three. Yet science fiction critics and reviewers still often treat the story as if it were a mere exposition of ideas, as if the intellectual “message” were all. This reductionism does a serious disservice to the sophisticated and powerful techniques and experiments of much contemporary science fiction. The writers are using language as postmodernists; the critics are decades behind, not even discussing the language, deaf to the implications of sounds, rhythms, recurrences, patterns—as if text were a mere vehicle for ideas, a kind of gelatin coating for the medicine. This is naive. And it totally misses what I love best in the best science fiction, its beauty." "I am certainly not going to talk about the beauty of my own stories. How about if I leave that to the critics and reviewers, and I talk about the ideas? Not the messages, though. There are no messages in these stories. They are not fortune cookies. They are stories.

Let me return from history and draw my conclusion. What all this means to us at the present time is this: Our system has already passed its flowering. Some time ago it reached that summit of blessedness which the mysterious game of world history sometimes allows to things beautiful and desirable in themselves. We are on the downward slope. Our course may possible stretch out for a very long time, but in any case nothing finer, ore beautiful, and more desirable than what we have already had can henceforth be expected. The road leads downhill. Historically we are, I believe, ripe for dismantling. And there is no doubt that such will be our fate, not today or tomorrow, but the day after tomorrow. I do not draw this conclusion from any excessively moralistic estimate of our accomplishments and our abilities: I draw it far more from the movements which I see already on the way in the outside world. Critical times are approaching; the omens can be sensed everywhere; the world is once again about to shift its center of gravity. Displacements of power are in the offing. They will not take place without war and violence. From the Far East comes a threat not only to peace, but to life and liberty. Even if our country remains politically neutral, even if our whole nation unanimously abides by tradition (which is not the case) and attempts to remain faithful to Castalian ideals, that will be in vain. Some of our representatives in Parliament are already saying that Castalia is a rather expensive luxury for our country. The country may very soon be forced into a serious rearmament - armaments for defensive purposes only, of course - and great economies will be necessary. In spite of the government's benevolent disposition towards us, much of the economizing will strike us directly. We are proud that our Order and the cultural continuity it provides have cost the country as little as they have. In comparison with other ages, especially the early period of the Feuilletonistic Age with its lavishly endowed universities, its innumerable consultants and opulent institutes, this toll is really not large. It is infinitesimal compared with the sums consumed for war and armaments during the Century of Wars. But before too long this kind of armament may once again be the supreme necessity; the generals will again dominate Parliament; and if the people are confronted with the choice of sacrificing Castalia or exposing themselves to the danger of war and destruction, we know how they will choose. Undoubtedly a bellicose ideology will burgeon. The rash of propaganda will affect youth in particular. Then scholars and scholarship, Latin and mathematics, education and culture, will be considered worth their salt only to the extent that they can serve the ends of war.

That same day we drove to Seville to celebrate. I asked someone for the name of the smartest hotel in Seville. Alfonso XIII, came the reply. It is where the King of Spain always stays. We found the hotel and wandered in. It was amazing. Shara was a little embarrassed as I was dressed in shorts and an old holey jersey, but I sought out a friendly-looking receptionist and told her our story. “Could you help us out? I have hardly any money.” She looked us up and down, paused--then smiled. “Just don’t tell my manager,” she whispered. So we stayed in a $1,000-a-night room for $100 and celebrated--like the King of Spain. The next morning we went on a hunt for a ring. I asked the concierge in my best university Spanish where I would find a good (aka well-priced) jeweler. He looked a little surprised. I tried speaking slower. Eventually I realized that I had actually been asking him where I might find a good mustache shop. I apologized that my Spanish was a little rusty. Shara rolled her eyes again, smiling. When we eventually found a small local jeweler, I had to do some nifty subcounter mathematics, swiftly converting Spanish pesetas into British pounds, to work out whether or not I could afford each ring Shara tried on. We eventually settled on one that was simple, beautiful--and affordable. Just. Love doesn’t require expensive jewelry. And Shara has always been able to make the simple look exquisite. Luckily.

The theory of relativity is a beautiful example of the basic character of the modern development of theory. That is to say, the hypotheses from which one starts become ever more abstract and more remote from experience. But in return one comes closer to the preeminent goal of science, that of encompassing a maximum of empirical contents through logical deduction with a minimum of hypotheses or axioms. The intellectual path from the axioms to the empirical contents or to the testable consequences becomes, thereby, ever longer and more subtle. The theoretician is forced, ever more, to allow himself to be directed by purely mathematical, formal points of view in the search for theories, because the physical experience of the experimenter is not capable of leading us up to the regions of the highest abstraction. Tentative deduction takes the place of the predominantly inductive methods appropriate to the youthful state of science. Such a theoretical structure must be quite thoroughly elaborated in order for it to lead to consequences that can be compared with experience. It is certainly the case that here, as well, the empirical fact is the all-powerful judge. But its judgment can be handed down only on the basis of great and difficult intellectual effort that first bridges the wide space between the axioms and the testable consequences. The theorist must accomplish this Herculean task with the clear understanding that this effort may only be destined to prepare the way for a death sentence for his theory. One should not reproach the theorist who undertakes such a task by calling him a fantast; instead, one must allow him his fantasizing, since for him there is no other way to his goal whatsoever. Indeed, it is no planless fantasizing, but rather a search for the logically simplest possibilities and their consequences.

When I first started coming to the seminar, Gelfand had a young physicist, Vladimir Kazakov, present a series of talks about his work on so-called matrix models. Kazakov used methods of quantum physics in a novel way to obtain deep mathematical results that mathematicians could not obtain by more conventional methods. Gelfand had always been interested in quantum physics, and this topic had traditionally played a big role at his seminar. He was particularly impressed with Kazakov’s work and was actively promoting it among mathematicians. Like many of his foresights, this proved to be golden: a few years later this work became famous and fashionable, and it led to many important advances in both physics and math. In his lectures at the seminar, Kazakov was making an admirable effort to explain his ideas to mathematicians. Gelfand was more deferential to him than usual, allowing him to speak without interruptions longer than other speakers. While these lectures were going on, a new paper arrived, by John Harer and Don Zagier, in which they gave a beautiful solution to a very difficult combinatorial problem.6 Zagier has a reputation for solving seemingly intractable problems; he is also very quick. The word was that the solution of this problem took him six months, and he was very proud of that. At the next seminar, as Kazakov was continuing his presentation, Gelfand asked him to solve the Harer–Zagier problem using his work on the matrix models. Gelfand had sensed that Kazakov’s methods could be useful for solving this kind of problem, and he was right. Kazakov was unaware of the Harer–Zagier paper, and this was the first time he heard this question. Standing at the blackboard, he thought about it for a couple of minutes and immediately wrote down the Lagrangian of a quantum field theory that would lead to the answer using his methods. Everyone in the audience was stunned.

[T]o look back on our life and also to discover something that can no longer be made good: the squandering of our youth when our educators failed to employ those eager, hot and thirsty years to lead us towards knowledge of things but used them for a so-called 'classical education'! The squandering of our youth when we had a meagre knowledge of the Greeks and Romans and their languages drummed into us in a way as clumsy as it was painful and one contrary to the supreme principle of all education, that one should offer food only to him who hungers for it ! When we had mathematics and physics forced upon us instead of our being led into despair at our ignorance and having our little daily life, our activities, and all that went on at home, in the work-place, in the sky, in the countryside from morn to night, reduced to thousands of problems, to annoying, mortifying, irritating problems so as to show us that we needed a knowledge of mathematics and mechanics, and then to teach us our first delight in science through showing us the absolute consistency of this knowledge! If only we had been taught to revere these sciences, if only our souls had even once been made to tremble at the way in which the great men of the past had struggled and been defeated and had struggled anew, at the martyrdom which constitutes the history of rigorous science! What we felt instead was the breath of a certain disdain for the actual sciences in favour of history, of 'formal education' and of 'the classics'! And we let ourselves be deceived so easily! Formal education! Could we not have pointed to the finest teachers at our grammar schools, laughed at them and asked: 'are they the products of formal education? And if not, how can they teach it?' And the classics! Did we learn anything of that which these same ancients taught their young people? Did we learn to speak or write as they did? Did we practise unceasingly the fencing-art of conversation, dialectics? Did we learn to move as beautifully and proudly as they did, to wrestle, to throw, to box as they did? Did we learn anything of the asceticism practised by all Greek philosophers? Were we trained in a single one of the antique virtues and in the manner in which the ancients practised it? Was all reflection on morality not utterly lacking in our education not to speak of the only possible critique of morality, a brave and rigorous attempt to live in this or that morality? Was there ever aroused in us any feeling that the ancients regarded more highly than the moderns? Were we ever shown the divisions of the day and of life, and goals beyond life, in the spirit of antiquity? Did we learn even the ancient languages in the way we learn those of living nations namely, so as to speak them with ease and fluency? Not one real piece of ability, of new capacity, out of years of effort! Only a knowledge of what men were once capable of knowing!