Mathematical Thinking Quotes

The ‘Muse’ is not an artistic mystery, but a mathematical equation. The gift are those ideas you think of as you drift to sleep. The giver is that one you think of when you first awake.

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.

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

I am convinced that the act of thinking logically cannot possibly be natural to the human mind. If it were, then mathematics would be everybody's easiest course at school and our species would not have taken several millennia to figure out the scientific method.

The formulation of the problem is often more essential than its solution, which may be merely a matter of mathematical or experimental skill.

Whereas I think: I’m lying here in a haystack... The tiny space I occupy is so infinitesimal in comparison with the rest of space, which I don’t occupy and which has no relation to me. And the period of time in which I’m fated to live is so insignificant beside the eternity in which I haven’t existed and won’t exist... And yet in this atom, this mathematical point, blood is circulating, a brain is working, desiring something... What chaos! What a farce!

I think that modern physics has definitely decided in favor of Plato. In fact the smallest units of matter are not physical objects in the ordinary sense; they are forms, ideas which can be expressed unambiguously only in mathematical language.

Rohan, one of us is an unmarried man with superior mathematical abilities and no prospects for the evening. The other is a confirmed lecher in an amorous mood, with a willing and nubile young wife waiting at home. Who do you think should do the damned account books?" And, with a nonchalant wave, St. Vincent had left the office.

Knowing mathematics is like wearing a pair of X-ray specs that reveal hidden structures underneath the messy and chaotic surface of the world.

Improbable things happen a lot.

I think we need more math majors who don't become mathematicians. More math major doctors, more math major high school teachers, more math major CEOs, more math major senators. But we won't get there unless we dump the stereotype that math is only worthwhile for kid geniuses.

Dividing one number by another is mere computation ; knowing what to divide by what is mathematics.

if gambling is exciting, you’re doing it wrong.

I think that consciousness is the way information feels when being processed in certain complex ways.

Logic, n. The art of thinking and reasoning in strict accordance with the limitations and incapacities of the human misunderstanding. The basic of logic is the syllogism, consisting of a major and a minor premise and a conclusion - thus: Major Premise: Sixty men can do a piece of work sixty times as quickly as one man. Minor Premise: One man can dig a post-hole in sixty seconds; Therefore- Conclusion: Sixty men can dig a post-hole in one second. This may be called syllogism arithmetical, in which, by combining logic and mathematics, we obtain a double certainty and are twice blessed.

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

Genius is a thing that happens, not a kind of person.

I guess I think of lotteries as a tax on the mathematically challenged.

he presented me with a mathematical conundrum,” he said. “It’s a famous one, the P = NP problem. Basically, it asks whether it’s more difficult to think of the solution to a problem yourself or to ascertain if someone else’s answer to the same problem is correct.

Mathematics is the extension of common sense by other means.

Nonlinear thinking means which way you should go depends on where you already are.

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.

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.

It was all so very businesslike that one watched it fascinated. It was pork-making by machinery, pork-making by applied mathematics. And yet somehow the most matter-of-fact person could not help thinking of the hogs; they were so innocent, they came so very trustingly; and they were so very human in their protests - and so perfectly within their rights! They had done nothing to deserve it; and it was adding insult to injury, as the thing was done here, swinging them up in this cold-blooded, impersonal way, without pretence at apology, without the homage of a tear.

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.

Its a mathematical fact that two negatives make a positive so even under adverse circumstances think positively.

Scientists and inventors of the USA (especially in the so-called "blue state" that voted overwhelmingly against Trump) have to think long and hard whether they want to continue research that will help their government remain the world's superpower. All the scientists who worked in and for Germany in the 1930s lived to regret that they directly helped a sociopath like Hitler harm millions of people. Let us not repeat the same mistakes over and over again.

The only definition of rationality that I’ve found that is practically, empirically, and mathematically rigorous is the following: what is rational is that which allows for survival. Unlike modern theories by psychosophasters, it maps to the classical way of thinking. Anything that hinders one’s survival at an individual, collective, tribal, or general level is, to me, irrational.

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.

Working an integral or performing a linear regression is something a computer can do quite effectively. Understanding whether the result makes sense—or deciding whether the method is the right one to use in the first place—requires a guiding human hand. When we teach mathematics we are supposed to be explaining how to be that guide. A math course that fails to do so is essentially training the student to be a very slow, buggy version of Microsoft Excel.

It is an old maxim of mine that when you have excluded the impossible, whatever remains, however improbable, must be the truth, unless the truth is a hypothesis it didn’t occur to you to consider.

A math teacher’s least favorite thing to hear from a student is “I get the concept, but I couldn’t do the problems.” Though the student doesn’t know it, this is shorthand for “I don’t get the concept.

What I'm thinking is: here I am, lying under a haystack ... The tiny little place I occupy is so small in relation to the rest of space where I am not and where it's none of my business; and the amount of time which I'll succeed in living is so insignificant by comparison with the eternity where I haven't been and never will be ... And yet in this atom, in this mathematical point, the blood circulates, the brain works and even desires something as well .. What sheer ugliness! What sheer nonsense!

These are lives, Lada,” Radu said. “How can you speak of them like they are matters of simple mathematics, a problem to be solved?” “Because thinking like that is the only way to keep from losing our minds.” “What about our souls?” Mehmed whispered. “Souls and thrones are irreconcilable.

A man may possess a profound knowledge of history and mathematics; he may be an authority in psychology, biology, or astronomy; he may know all the discovered truths pertaining to geology and natural science; but if he has not with this knowledge that nobility of soul which prompts him to deal justly with his fellow men, to practice virtue and holiness in his personal life, he is not truly an educated man. Character is the aim of true education; and science, history, and literature are but means used to accomplish the desired end. Character is not the result of chance work but of continuous right thinking and right acting.

You know, people think mathematics is complicated. Mathematics is the simple bit. It's the stuff we can understand. It's cats that are complicated. I mean, what is it in those little molecules and stuff that make one cat behave differently than another, or that make a cat? And how do you define a cat? I have no idea.

In any case, do you really think kids even want something that is relevant to their daily lives? You think something practical like compound interest is going to get them excited? People enjoy fantasy, and that is just what mathematics can provide -- a relief from daily life, an anodyne to the practical workaday world.

A reasonable person believes, in short, that each of his beliefs is true and that some of them are false.

Life is a linear equation in which you can't cross multiply! If you think you can do it, you can do it. If you think you can't do it, you can't do it. It's a simple formula!

In the Bayesian framework, how much you believe something after you see the evidence depends not just on what the evidence shows, but on how much you believed it to begin with.

Physicists believe that the Gaussian law has been proved in mathematics while mathematicians think that it was experimentally established in physics.

Why don't we want our children to learn to do mathematics? Is it that we don't trust them, that we think it's too hard? We seem to feel that they are capable of making arguments and coming to their own conclusions about Napoleon. Why not about triangles?

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

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

When you reason correctly, as Silver does, you find that you always think you’re right, but you don’t think you’re always right.

...it's really more intelligent to be able to simplify things than to complicate them. Even if some people think it makes you look stupid.

Overstimulation has been the real drawback. I need to find ways to stop thinking about analysis of algorithms, in order to do various other things that human beings ought to do.

317 is a prime, not because we think so, or because our minds are shaped in one way rather than another, but because it is so, because mathematical reality is built that way.

Mathematicians can be persnickety about logical niceties. We're the kind of people who think it's funny, when asked, "Do you want soup or salad with that?" to reply, "Yes.

You ask me if an ordinary person—by studying hard—would get to be able to imagine these things like I imagine. Of course. I was an ordinary person who studied hard. There's no miracle people. It just happens they got interested in this thing, and they learned all this stuff. They're just people. There's no talent or special miracle ability to understand quantum mechanics or a miracle ability to imagine electromagnetic fields that comes without practice and reading and learning and study. So if you take an ordinary person who's willing to devote a great deal of time and study and work and thinking and mathematics, then he's become a scientist.

What drove me? I think most creative people want to express appreciation for being able to take advantage of the work that's been done by others before us. I didn't invent the language or mathematics I use. I make little of my own food, none of my own clothes. Everything I do depends on other members of our species and the shoulders that we stand on. And a lot of us want to contribute something back to our species and to add something to the flow. It's about trying to express something in the only way that most of us know how-because we can't write Bob Dylan songs or Tom Stoppard plays. We try to use the talents we do have to express our deep feelings, to show our appreciation of all the contributions that came before us, and to add something to that flow. That's what has driven me.

The aspirations of democracy are based on the notion of an informed citizenry, capable of making wise decisions. The choices we are asked to make become increasingly complex. They require the longer-term thinking and greater tolerance for ambiguity that science fosters. The new economy is predicated on a continuous pipeline of scientific and technological innovation. It can not exist without workers and consumers who are mathematically and scientifically literate.

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

One of the great joys of mathematics is the incontrovertible feeling that you've understood something the right way, all the way down to the bottom; it's a feeling I haven't experienced in any other sphere of mental life. And when you know how to do something the right way, it's hard-for some stubborn people, impossible-to make yourself explain it the wrong way.

We think of science as discovery, art as invention, but is there a “third world” of mathematics, which is somehow, mysteriously, both?

Safety warning: never divide by zero unless a licensed mathematician is present.

If you can't test it, it's not theorics -- it's metatheorics. A branch of philosophy. So, if you want to think of it this way, our test equipment is what defines the boundary separating theorics from philosophy.

A man craves ultimate truths. Every mortal mind, I think, is that way. But what is ultimate truth? It's the end of the road, where there is no more mystery, no more hope. And no more questions to ask, since all the answers have been given. But there is no such place. The Universe is a labyrinth made of labyrinths. Each leads to another. And wherever we cannot go ourselves, we reach with mathematics. Out of mathematics we build wagons to carry us into the nonhuman realms of the world.

A remarkably consistent finding, starting with elementary school students, is that males are better at math than females. While the difference is minor when it comes to considering average scores, there is a huge difference when it comes to math stars at the upper extreme of the distribution. For example, in 1983, for every girl scoring in the highest percentile in the math SAT, there were 11 boys. Why the difference? There have always been suggestions that testosterone is central. During development, testosterone fuels the growth of a brain region involved in mathematical thinking and giving adults testosterone enhances their math skills. Oh, okay, it's biological. But consider a paper published in science in 2008. The authors examined the relationship between math scores and sexual equality in 40 countries based on economic, educational and political indices of gender equality. The worst was Turkey, United States was middling, and naturally, the Scandinavians were tops. Low and behold, the more gender equal the country, the less of a discrepancy in math scores. By the time you get to the Scandinavian countries it's statistically insignificant. And by the time you examine the most gender equal country on earth at the time, Iceland, girls are better at math than boys. Footnote, note that the other reliable sex difference in cognition, namely better reading performance by girls than by boys doesn't disappear in more gender equal societies. It gets bigger. In other words, culture matters. We carry it with us wherever we go.

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.

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

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

People tend to think that mathematicians always work in sterile conditions, sitting around and staring at the screen of a computer, or at a ceiling, in a pristine office. But in fact, some of the best ideas come when you least expect them, possibly through annoying industrial noise.

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

Don’t you think he may be pursuing an ideal that is hidden in a cloud of unknowing—like an astronomer looking for a star that only a mathematical calculation tells him exists?

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

...whatever we do whatever way we move forward there will be damage carried from all previous rows and columns in this mathematical computation we make, all the multi-configured additions and subtractions in language America, in all bases, particularly I'm thinking about at the moment 1492, 1776, 1861, 1867, 1980, 2016, 2020.

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

Physics, mathematics, music, painting, my politics, my love for you, my work, the star-dust of my body, the spirit that impels it, clocks diurnal, time perpetual, the roll, rough, tender, swamping, liberating, breathing, moving, thinking nature, human nature and the cosmos are patterned together.

The afflicted are almost upon them. The air is a din of hypersonic bursts, snarls and empty shell casings. But still I hear him. As his people start to fall. As his pistol clicks empty. As he rises with only his knuckles left between him and the sheer brutality of mathematics. As the music swells above the carnage, still I hear him breathe the words. "Tell them I was thinking of them. At the end." They pile onto him. All snarls and teeth and fists. But as he falls, I am holding his hand. Easing him into his long good night. "I will tell them, David." The last words he will ever hear. 'I promise.

And I think that if God did exist, he had many children. I think Jesus proves this. Jesus must be the seventh son of God. A-sus, B-sus, C-sus, D-sus, E-sus, F-sus, G-sus. That's just logic. That's just mathematical. And T-sus would always be fucking about. And P-sus does deliveries. C-sus started the Roman Empire. Cae-sus. F-sus, City in Turkey. B-sus was covered in something. Some people applauding there; other people going, "What?" ... B-sus was covered in bees.

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

So, I convinced myself that, because the statistical likelihood of something happening in real life exactly the way I imagined it was so low, if I imagined the worst possible things in vivid detail, I could mathematically reduce the odds of them happening. I convinced myself that my brain had power over the probability projections of the universe. I’d lie awake at night thinking about all the worst stuff that could happen like it was my job, and I don’t know if I ever really broke the habit.

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

And why are you so firmly, so triumphantly, convinced that only the normal and the positive—in other words, only what is conducive to welfare—is for the advantage of man? Is not reason in error as regards advantage? Does not man, perhaps, love something besides well-being? Perhaps he is just as fond of suffering? Perhaps suffering is just as great a benefit to him as well-being? Man is sometimes extraordinarily, passionately, in love with suffering, and that is a fact. [. . .] And yet I think man will never renounce real suffering, that is, destruction and chaos. Why, suffering is the sole origin of consciousness. Though I did lay it down at the beginning that consciousness is the greatest misfortune for man, yet I know man prizes it and would not give it up for any satisfaction. Consciousness, for instance, is infinitely superior to twice two makes four. Once you have mathematical certainty there is nothing left to do or to understand. There will be nothing left but to bottle up your five senses and plunge into contemplation.

Maybe individual people seem irrational because they aren’t really individuals! Each one of us is a little nation-state, doing our best to settle disputes and broker compromises between the squabbling voices that drive us.

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

If you’ve ever run an experiment, you know scientific truth doesn’t pop out of the clouds blowing a flaming trumpet at you. Data is messy, and inference is hard.

Public opinion doesn’t exist. More precisely, it exists sometimes, concerning matters about which there’s a clear majority view.

Like works of literature, mathematical ideas help expand our circle of empathy, liberating us from the tyranny of a single, parochial point of view.

Analogy pervades all our thinking,

A good expository paper will benefit far more people than most research papers. A good text is worth a thousand of the usual trifles that appear in research journals.

Cauchy was not interested in the needs of engineers. Cauchy was interested in the truth.

Don't you think he may be pursuing an ideal that is hidden in a cloud of unknowing — like an astronomer looking for a star that only a mathematical calculation tells him exists?

I think there's something heavenly about numbers, anyway.

boys do not pursue mathematical activities at a higher rate than girls do because they are better at mathematics. They do so, at least partially, because they think they are better.

Before the work of Georg Cantor in the nineteenth century, the study of the infinite was as much theology as science; now, we understand Cantor’s theory of multiple infinities, each one infinitely larger than the last, well enough to teach it to first-year math majors. (To be fair, it does kind of blow their minds.)

Listen, Ducklescence, he whispered. You will never know what the world is made of. The only thing that’s certain is that it’s not made of the world. As you close upon some mathematical description of reality you cant help but lose what is being described. Every inquiry displaces what is addressed. A moment in time is a fact, not a possibility. The world will take your life. But above all and lastly the world does not know that you are here. You think that you understand this. But you dont. Not in your heart you dont. If you did you would be terrified. And you’re not. Not yet. And now, good night.

EB: Perhaps it's her way of rebelling. You know a thing or two about rebellion, I think. NW: Yes, but I did it the proper way. I drank and smoked and took lovers. Who rebels with mathematics?

People enjoy inventing slogans which violate basic arithmetic but which illustrate “deeper” truths, such as “1 and 1 make 1” (for lovers), or “1 plus 1 plus 1 equals 1” (the Trinity). You can easily pick holes in those slogans, showing why, for instance, using the plus-sign is inappropriate in both cases. But such cases proliferate. Two raindrops running down a window-pane merge; does one plus one make one? A cloud breaks up into two clouds -more evidence of the same? It is not at all easy to draw a sharp line between cases where what is happening could be called “addition”, and where some other word is wanted. If you think about the question, you will probably come up with some criterion involving separation of the objects in space, and making sure each one is clearly distinguishable from all the others. But then how could one count ideas? Or the number of gases comprising the atmosphere? Somewhere, if you try to look it up, you can probably fin a statement such as, “There are 17 languages in India, and 462 dialects.” There is something strange about the precise statements like that, when the concepts “language” and “dialect” are themselves fuzzy.

What is remarkable is that there are no traces of evolution from simple to sophisticated, and the same is true of mathematics, medicine, astronomy and architecture and of Egypt's amazingly rich and convoluted religio-mythological system (even the central content of such refined works as the Book of the Dead existed right at the start of the dynastic period). 7 The majority of Egyptologists will not consider the implications of Egypt's early sophistication. These implications are startling, according to a number of more daring thinkers. John Anthony West, an expert on the early dynastic period, asks: How does a complex civilization spring full-blown into being? Look at a 1905 automobile and compare it to a modern one. There is no mistaking the process of `development'. But in Egypt there are no parallels. Everything is right there at the start. The answer to the mystery is of course obvious but, because it is repellent to the prevailing cast of modern thinking, it is seldom considered. Egyptian civilization was not a `development', it was a legacy.

The notion of God, on the other hand, however inferior it may be in clearness to those mathematical notions so current in mechanical philosophy, has at least this practical superiority over them, that it guarantees an ideal order that shall be permanently preserved.

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

There is real danger that, by strengthening our abilities to analyze some questions mathematically, we acquire a general confidence in our beliefs, which extends unjustifiably to those things we’re still wrong about.

They were very upset when I said that the thing of greatest importance to mathematics in Europe was the discovery by Tartaglia that you can solve a cubic equation-which, altho it is very little used, must have been psychologically wonderful because it showed a modern man could do something no ancient Greek could do, and therefore helped in the renaissance which was the freeing of man from the intimidation of the ancients-what they are learning in school is to be intimidated into thinking they have fallen so far below their super ancestors.

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.

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

It's not wrong to say Hilbert was a genius. But it's more right to say that what Hilbert accomplished was genius. Genius is a thing that happens, not a kind of person.

You can explain it to a Congressman in six minutes and he can talk about it for six months.

Yes, the existence of human life is evidence for the existence of God; but it’s much better evidence that our world was programmed by people much smarter than us.

like literary fiction, mathematical imagination entertains pure possibilities.

They had chains which they fastened about the leg of the nearest hog, and the other end of the chain they hooked into one of the rings upon the wheel. So, as the wheel turned, a hog was suddenly jerked off his feet and borne aloft. At the same instant the ear was assailed by a most terrifying shriek; the visitors started in alarm, the women turned pale and shrank back. The shriek was followed by another, louder and yet more agonizing--for once started upon that journey, the hog never came back; at the top of the wheel he was shunted off upon a trolley and went sailing down the room. And meantime another was swung up, and then another, and another, until there was a double line of them, each dangling by a foot and kicking in frenzy--and squealing. The uproar was appalling, perilous to the ear-drums; one feared there was too much sound for the room to hold--that the walls must give way or the ceiling crack. There were high squeals and low squeals, grunts, and wails of agony; there would come a momentary lull, and then a fresh outburst, louder than ever, surging up to a deafening climax. It was too much for some of the visitors--the men would look at each other, laughing nervously, and the women would stand with hands clenched, and the blood rushing to their faces, and the tears starting in their eyes. Meantime, heedless of all these things, the men upon the floor were going about their work. Neither squeals of hogs nor tears of visitors made any difference to them; one by one they hooked up the hogs, and one by one with a swift stroke they slit their throats. There was a long line of hogs, with squeals and life-blood ebbing away together; until at last each started again, and vanished with a splash into a huge vat of boiling water. It was all so very businesslike that one watched it fascinated. It was pork-making by machinery, pork-making by applied mathematics. And yet somehow the most matter-of-fact person could not help thinking of the hogs; they were so innocent, they came so very trustingly; and they were so very human in their protests--and so perfectly within their rights! They had done nothing to deserve it; and it was adding insult to injury, as the thing was done here, swinging them up in this cold-blooded, impersonal way, without a pretence at apology, without the homage of a tear. Now and then a visitor wept, to be sure; but this slaughtering-machine ran on, visitors or no visitors. It was like some horrible crime committed in a dungeon, all unseen and unheeded, buried out of sight and of memory.

What would it be like, a world without snow? I cannot imagine such a place. It would be like a world devoid of numbers. Every snowflake, unique as every number, tells us something about complexity. Perhaps that is why we will never tire of its wonder.

The hoodlum-occultist is “sociopathic” enough to, see through the conventional charade, the social mythology of his species. “They’re all sheep,” he thinks. “Marks. Suckers. Waiting to be fleeced.” He has enough contact with some more-or-less genuine occult tradition to know a few of the gimmicks by which “social consciousness,” normally conditioned consciousness, can be suspended. He is thus able to utilize mental brutality in place of the simple physical brutality of the ordinary hooligan. He is quite powerless against those who realize that he is actually a stupid liar. He is stupid because spending your life terrorizing and exploiting your inferiors is a dumb and boring existence for anyone with more than five billion brain cells. Can you imagine Beethoven ignoring the heavenly choirs his right lobe could hear just to pound on the wall and annoy the neighbors? Gödel pushing aside his sublime mathematics to go out and cheat at cards? Van Gogh deserting his easel to scrawl nasty caricatures in the men’s toilet? Mental evil is always the stupidest evil because the mind itself is not a weapon but a potential paradise. Every kind of malice is a stupidity, but occult malice is stupidest of all. To the extent that the mindwarper is not 100 percent charlatan through-and-through (and most of them are), to the extent that he has picked up some real occult lore somewhere, his use of it for malicious purposes is like using Shakespeare’s sonnets for toilet tissue or picking up a Picasso miniature to drive nails. Everybody who has advanced beyond the barbarian stage of evolution can see how pre-human such acts are, except the person doing them. Genuine occult initiation confers “the philosopher’s stone,” “the gold of the wise” and “the elixir of life,” all of which are metaphors for the capacity to greet life with the bravery and love and gusto that it deserves. By throwing this away to indulge in spite, malice and the small pleasure of bullying the credulous, the mindwarper proves himself a fool and a dolt. And the psychic terrorist, besides being a jerk, is always a liar and a fraud. Healing is easier (and more fun) than cursing, to begin with, and cursing usually backfires or misfires. The mindwarper doesn’t want you to know that. He wants you to think he’s omnipotent.

Nobody ever looks in the mirror and says, “Let’s face it, I’m smarter than Gauss.” And yet, in the last hundred years, the joined effort of all these dummies-compared-to-Gauss has produced the greatest flowering of mathematical knowledge the world has ever seen.

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.

Math gives us a way of being unsure in a principled way: not just throwing up our hands and saying "huh", but rather making a firm assertion: "I'm not sure, this is why I'm not sure, and this is roughly how not sure I am." Or even more: "I'm unsure, and you should be too.

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.

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

There is no great religion without a great schism. All of them have it. And that's because you're dealing with something called faith. And faith is not something you can prove; faith is personal opinion. Uh, when you're dealing with something with certainty, like, y'know, science or logic, you don't have the--there's no wiggle room; that's why history is not filled with warring math cults, y'know, because you can settle the issue; you can prove something to be right or wrong, and that's the end of the argument: next case. Whereas, when you're dealing with faith, you can forever argue your point, or another point, because you're dealing with intangibles. Personally, I think, faith is what you ask of somebody when you don't have the goods to prove your point.

I was especially delighted with the mathematics, on account of the certitude and evidence of their reasonings; but I had not as yet a precise knowledge of their true use; and thinking that they but contributed to the advancement of the mechanical arts, I was astonished that foundations, so strong and solid, should have had no loftier superstructure reared on them.

Mathematics is one of the major modern mysteries. Perhaps it is the leading one, occupying a place in our society similar to the religious mysteries of another age. If we want to know something about what our age is all about, we should have some understanding of what mathematics is, and of how the mathematician operates and thinks.

When you think about an apple, you also think about the opposite of an apple. It’s a tool for understanding mathematics and things with two separate realities. But for creativity: That comes from a place of oneness. That’s not a duality consciousness.

She had had the idea that the mineral world was a world of perfect, inanimate forms, with an unchanging mathematical order of crystals and molecules beneath its sprouts and flows and branches. She had thought, when she started thinking, about her own transfiguration as something profoundly unnatural, a move from a world of warm change and decay to a world of cold permanence.But as she became mineral, and looked into the idea of minerals, she saw that there were reciprocities, both physical and figurative.

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

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

My lack of initiative was the root cause of all my troubles - of my inability to want something before having thought about it, of my inability to commit myself, of my inability to decide in the only way one can decide: by deciding, not by thinking. I'm like Buridan's donkey, dying at the mathematical midpoint between the water of emotion and the hay of action; if I didn't think, I might still die, but it wouldn't be from thirst or hunger.

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.

It's not like that, as we've seen. Mathematicians aren't crazy, and we aren't aliens, and we aren't mystics. What's true is that the sensation of mathematical understanding-of suddenly knowing what's going on, with total certainty, all the way to the bottom-is a special thing, attainable in few if any other places in life. You feel you've reached into the universe's guts and put your hand on the wire. It's hard to describe to people who haven't experienced it.

Knowing mathematics is like wearing a pair of X-ray specs that reveal hidden structures underneath the messy and chaotic surface of the world. Math is a science of not being wrong about things, its techniques and habits hammered out by centuries of hard work and argument. With the tools of mathematics in hand, you can understand the world in a deeper, sounder, and more meaningful way.

I think a strong claim can be made that the process of scientific discovery may be regarded as a form of art. This is best seen in the theoretical aspects of Physical Science. The mathematical theorist builds up on certain assumptions and according to well understood logical rules, step by step, a stately edifice, while his imaginative power brings out clearly the hidden relations between its parts. A well constructed theory is in some respects undoubtedly an artistic production. A fine example is the famous Kinetic Theory of Maxwell. ... The theory of relativity by Einstein, quite apart from any question of its validity, cannot but be regarded as a magnificent work of art.

Underlying our approach to this subject is our conviction that "computer science" is not a science and that its significance has little to do with computers. The computer revolution is a revolution in the way we think and in the way we express what we think. The essence of this change is the emergence of what might best be called procedural epistemology—the study of the structure of knowledge from an imperative point of view, as opposed to the more declarative point of view taken by classical mathematical subjects. Mathematics provides a framework for dealing precisely with notions of "what is". Computation provides a framework for dealing precisely with notions of "how to".

A man may possess a profound knowledge of history and mathematics; he may be an authority in psychology, biology, or astronomy; he may know all the discovered truths pertaining to geology and natural science; but if he has not with this knowledge that nobility of soul which prompts him to deal justly with his fellow men, to practice virtue and holiness in personal life, he is not a truly educated man. "Character is the aim of true education; and science, history, and literature are but means used to accomplish the desired end. Character is not the result of chance work but of continuous right thinking and right acting. "True education seeks, then, to make men and women not only good mathematicians, proficient linguists, profound scientists, or brilliant literary lights, but also honest men, combined with virtue, temperance, and brotherly love-men and women who prize truth, justice, wisdom, benevolence, and self-control as the choicest acquisitions of a successful life.

We teach our children the mathematics of certainty—geometry and trigonometry—but not the mathematics of uncertainty, statistical thinking. And we teach our children biology but not the psychology that shapes their fears and desires. Even experts, shockingly, are not trained how to communicate risks to the public in an understandable way. And there can be positive interest in scaring people: to get an article on the front page, to persuade people to relinquish civil rights, or to sell a product.

In mathematical modeling, as in all of science, we always have to make choices about what to stress and what to ignore. The art of abstraction lies in knowing what is essential and what is minutia, what is signal and what is noise, what is trend and what is wiggle. It's an art because such choices always involve an element of danger; they come close to wishful thinking and intellectual dishonesty.

Lagrange, in one of the later years of his life, imagined that he had overcome the difficulty (of the parallel axiom). He went so far as to write a paper, which he took with him to the Institute, and began to read it. But in the first paragraph something struck him that he had not observed: he muttered: 'Il faut que j'y songe encore', and put the paper in his pocket.' [I must think about it again].

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

It is a profoundly erroneous truism, repeated by all copy-books and by eminent people when they are making speeches, that we should cultivate the habit of thinking of what we are doing. The precise opposite is the case. Civilization advances by extending the number of important operations which we can perform without thinking about them. Operations of thought are like cavalry charges in a battle — they are strictly limited in number, they require fresh horses, and must only be made at decisive moments.

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

By the time the average person finishes college, he or she will have taken over 2,600 tests, quizzes, and exams. The right answer approach becomes deeply ingrained in our thinking. This may be fine for some mathematical problems where there is in fact only one right answer. The difficulty is that most of life isn’t this way. Life is ambiguous; there are many right answers- all depending on what you’re looking for. But if you think there is only one right answer, then you’ll stop looking as soon as you find one.

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.

Mutuality is accomplished by two whole persons; and if each partner truly intends to be but the fraction of a relationship (thinking my whole makes up half of us) he or she will soon discover that these halves do not fit perfectly together. The mathematics can work only if each subtracts something of himself or herself, shears it off, and lays it aside forever. There will come, then, a moment of shock when one spouse realizes, ‘you won’t want the whole of me? Not the whole of me, but only a part of me, makes up the whole of us?” P 45

There was only one hope left to me - Andre. But could he fill this emptiness within me? Had he loved me as I loved him? At the beginning I think he did; or rather the question never arose of either of us, for we were so happy together. But when his work no longer satisfied him, did he come to the conclusion that our love was not enough for him? Did it disappoint him? I think he looks upon me as a mathematical constant whose disappearance would take him very much aback without in any way altering his destiny, since the heart of the matter lies elsewhere.

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?

I do not think that G. H. Hardy was talking nonsense when he insisted that the mathematician was discovering rather than creating... The world for me is a necessary system, and in the degree to which the thinker can surrender his thought to that system and follow it, he is in a sense participating in that which is timeless or eternal.

For all the time schools devote to the teaching of mathematics, very little (if any) is spent trying to convey just what the subject is about. Instead, the focus is on learning and applying various procedures to solve math problems. That's a bit like explaining soccer by saying it is executing a series of maneuvers to get the ball into the goal. Both accurately describe various key features, but they miss the \what?" and the \why?" of the big picture.

I think I repeated the Heisenberg Uncertainty Principle in my head at least one thousand times: the mathematical product of the combined uncertainties of concurrent measurements of position and momentum in a specified direction could never be less than Planck’s constant, h, divided by 4π. This meant, rather encouragingly, that my uncertain position and zero momentum and the Beast Responsible for the Sound’s uncertain position and uncertain momentum had to sort of null each other out, leaving me with what is commonly known in the scientific world as “wide-ranging perplexity.

If you have a patient with a condition that's not understood why not ascribe it to a disorder that is also not understood? Autism occurs in males more than it does in females. So does higher order mathematical intuition. We think: What is this about? Dont know. What is at the heart of it? Dont know. All I can tell you is that I like numbers. I like their shapes and their colors and their smells and the way they taste. And I dont like to take people's word for things.

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

One thing the American defense establishment has traditionally understood very well is that countries don't win wars just by being braver than the other side, or freer, or slightly preferred by God. The winners are usually the guys who get 5% fewer of their planes shot down, or use 5% less fuel, or get 5% more nutrition into their infantry at 95% of the cost.

Do reflections also travel at the speed of light? What does your buddy Albert think? When the light hits the glass and starts back in the opposite direction doesnt it have to come to a full stop first? And so everything is supposed to hang on the speed of light but nobody wants to talk about the speed of dark. What’s in a shadow? Do they move along at the speed of the light that casts them? How deep do they get? How far down can you clamp your calipers? You scribbled somewhere in the margins that when you lose a dimension you’ve given up all claims to reality. Save for the mathematical. Is there a route here from the tangible to the numerical that hasnt been explored? I dont know. Me either. Photons are quantum particles. They’re not little tennisballs. Yeah, said the Kid. He dredged up his watch and checked the time. Maybe you’d better go eat. You need to keep your strength up if you aim to wrest the secrets of creation from the gods. They’re a testy lot by all accounts.

If you do happen to find yourself partially believing a crazy theory, don’t worry—probably the evidence you encounter will be inconsistent with it, driving down your degree of belief in the craziness until your beliefs come into line with everyone else’s. Unless, that is, the crazy theory is designed to survive this winnowing process. That’s how conspiracy theories work.

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

She may hear this story, someday, and wonder what it has to do with her. A girl hiding her scared in her prettiness, an unwanted baby, a dead mother. These are not her heartbreaks. Every heart is fractured differently and she knows the pattern of her cracks, she traces them like lines across her palm. She has a living mother and besides, she was always wanted. Prayed for, even. Now she's grown, or at least she thinks she is. But she hasn't yet learned from the mathematics of grief. The weight of what has been lost is always heavier than what remains. She's heard her granddaddy preach about the good shepherd who leaves the ninety-nine behind in search of the one lost sheep. But what about the flock he abandons? she wonders. Aren't they lost now too?

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

There was, I think, a feeling that the best science was that done in the simplest way. In experimental work, as in mathematics, there was 'style' and a result obtained with simple equipment was more elegant than one obtained with complicated apparatus, just as a mathematical proof derived neatly was better than one involving laborious calculations. Rutherford's first disintegration experiment, and Chadwick's discovery of the neutron had a 'style' that is different from that of experiments made with giant accelerators.

I don't know which is worse—to have a bad teacher or no teacher at all. In any case, I believe the teacher's work should be largely negative. He can't put the gift into you, but if he finds it there, he can try to keep it from going in an obviously wrong direction. We can learn how not to write, but this is a discipline that does not simply concern writing itself but concerns the whole intellectual life. A mind cleared of false emotion and false sentiment and egocentricity is going to have at least those roadblocks removed from its path. If you don't think cheaply, then there at least won't be the quality of cheapness in your writing, even though you may not be able to write well. The teacher can try to weed out what is positively bad, and this should be the aim of the whole college. Any discipline can help your writing: logic, mathematics, theology, and of course and particularly drawing. Anything that helps you to see, anything that makes you look. The writer should never be ashamed of staring. There is nothing that doesn't require his attention.

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

My other teachers did not seem to care about the challenge of being human and instead they taught us to think about mathematics and analyze different chemicals and as the months went by I felt further from myself. And the only thing that seemed to make sense was Ben Sweet and the way he talked to us and urged something in the deeps of us to come out—the way he looked, and listened, as if he had no other place on this Earth to be except with us, as if there were nothing more important in his life than what we had to say at just that moment in time.

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

No matter what I study, I can see patterns. I see the gestalt, the melody within the notes, in everything: mathematics and science, art and music, psychology and sociology. As I read the texts, I can think only that the authors are plodding along from one point to the next, groping for connections that they can’t see. They’re like a crowd of people unable to read music, peering at the score for a Bach sonata, trying to explain how one note leads to another. As glorious as these patterns are, they also whet my appetite for more. There are other patterns waiting to be discovered, gestalts of another scale entirely. With respect to those, I’m blind myself; all my sonatas are just isolated data points by comparison. I have no idea what form such gestalts might assume, but that’ll come in time. I want to find them, and comprehend them. I want this more than anything I’ve ever wanted before.

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

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

A distinguished writer [Siméon Denis Poisson] has thus stated the fundamental definitions of the science: 'The probability of an event is the reason we have to believe that it has taken place, or that it will take place.' 'The measure of the probability of an event is the ratio of the number of cases favourable to that event, to the total number of cases favourable or contrary, and all equally possible' (equally like to happen). From these definitions it follows that the word probability, in its mathematical acceptation, has reference to the state of our knowledge of the circumstances under which an event may happen or fail. With the degree of information which we possess concerning the circumstances of an event, the reason we have to think that it will occur, or, to use a single term, our expectation of it, will vary. Probability is expectation founded upon partial knowledge. A perfect acquaintance with all the circumstances affecting the occurrence of an event would change expectation into certainty, and leave neither room nor demand for a theory of probabilities.

I think; here I lie under a haystack.... The tiny space I occupy is so infinitely small in comparison with the rest of space, in which I am not, and which has nothing to do with me; and the period of time in which it is my lot to live is so petty beside the eternity in which I have not been, and shall not be.... And in this atom, this mathematical point, the blood is circulating, the brain is working and wanting something.... Isn't it loathsome? Isn't it petty?

We know next to nothing with any certainty about Pythagoras, except that he was not really called Pythagoras. The name by which he is known to us was probably a nickname bestowed by his followers. According to one source, it meant ‘He who spoke truth like an oracle’. Rather than entrust his mathematical and philosophical ideas to paper, Pythagoras is said to have expounded them before large crowds. The world’s most famous mathematician was also its first rhetorician.

Philosophy in its old form could exist only in the absence of engineering, but with engineering in existence and daily more active and far reaching, the old verbalistic philosophy and metaphysics have lost their reason to exist. They were no more able to understand the "production" of the universe and life than they are now able to understand or grapple with "production" as a means to provide a happier existence for humanity. They failed because their venerated method of "speculation" can not produce, and its place must be taken by mathematical thinking. Mathematical reasoning is displacing metaphysical reasoning. Engineering is driving verbalistic philosophy out of existence and humanity gains decidedly thereby. Only a few parasites and "speculators" will mourn the disappearance of their old companion "speculation." The world of producers -the predominating majority of human beings- will welcome a philosophy of ordered thought and production.

Ultimately Russell himself admitted that he made his greatest efforts in the field of traditional philosophy – in epistemology, the search for the ultimate grounds of our knowledge about the world. How can we be certain that what we claim to know is true? Where lies the certainty in our experience of the world? Can even the most precise knowledge – such as mathematics – be said to rest on any sure logical foundation? These were the questions that Russell sought to answer during the periods of his most profound philosophical thinking. They have remained the perennial questions of philosophy from Plato and Aristotle through Descartes, Hume, and Kant, to Russell and Wittgenstein.

But real-world questions aren't like word problems. A real-world problem is something like "Has the recession and its aftermath been especially bad for women in the workforce, and if so, to what extent is this the result of Obama administration policies?" Your calculator doesn't have a button for this. Because in order to give a sensible answer, you need to know more than just numbers. What shape do the job-loss curves for men and women have in a typical recession? Was this recession notably different in that respect? What kind of jobs are disproportionately held by women, and what decisions has Obama made that affect that sector of the economy? It's only after you've started to formulate these questions that you take out the calculator. But at that point the real mental work is already finished. Dividing one number by another is mere computation; figuring out what you should divide by what is mathematics.

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

Douglas R. Hofstadter, an American researcher, speculates that the human mind has consciousness because it has the capability of self-reference. Since we can think about ourselves and think about ourselves thinking about ourselves, etc., we are capable of feeling that we are an "I". Contrast that with what we have learned in this chapter. This chapter tries to show that the computer's ability to perform self-reference is the cause of its limitations. Can we say that self-reference in computers brings limitations while in humans it causes consciousness? Perhaps.

Aside from the encounter with the Sphinx, there is little in Oedipus to connect him to the common run of Greek heroic figures. He strikes us today as a modern tragic hero and political animal; it is hard to picture him shaking hands with Heracles or joining the crew of the Argo. many scholars and thinkers, most notably Friedrich Nietzsche in his book The Birth of Tragedy, have seen in Oedipus a character who works out on stage the tension in Athenians (and all of us) between the reasoning, mathematically literate citizen and the transgressive blood criminal; between the thinking and the instinctual being; between the superego and the id; between the Apollonian and the Dionysian impulses that contend within us. Oedipus is a detective who employs all the fields of enquiry of which the Athenians were so proud -- logic, numbers, rhetoric, order and discovery -- only to reveal a truth that is disordered, shameful, transgressive and bestial.

But it is evident that all analogies of this kind depend on principles of a more fundamental nature; and that, if we had a true mathematical classification of quantities, we should be able at once to detect the analogy between any system of quantities presented to us and other systems of quantities in known sciences, so that we should lose no time in availing ourselves of the mathematical labors of those who had already solved problems essentially the same. [...] At the same time, I think that the progress of science, both in the way of discovery, and in the way of diffusion, would be greatly aided if more attention were paid in a direct way to the classification of quantities. - Remarks on the mathematical classification of physical quantities Proceedings of the London Mathematical Society, 1871

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.

Things were changing; I was changing. All swelling limbs and sweating brain, suddenly I had more body than I knew what to do with. Arms and legs became the prey of low desktops and narrow corridors, were ambushed by sharp corners. Mr Baxter ignored my plight. Bodies were inimical to mathematics, or so we were led to believe. Bad hair, acrid breath, lumpy skin, all vanished for an hour every Tuesday and Thursday. Young minds in the buff soared into the sphere of pure reason. Pages turned to parallelograms; cities, circumferences; recipes, ratios. Shorn of our bearings, we groped our way around in this rarefied air.

There are two moments in the course of education where a lot of kids fall off the math train. The first comes in the elementary grades, when fractions are introduced. Until that moment, a number is a natural number, one of the figures 0, 1, 2, 3 . . . It is the answer to a question of the form “how many.”* To go from this notion, so primitive that many animals are said to understand it, to the radically broader idea that a number can mean “what portion of,” is a drastic philosophical shift. (“God made the natural numbers,” the nineteenth-century algebraist Leopold Kronecker famously said, “and all the rest is the work of man.”) The second dangerous twist in the track is algebra. Why is it so hard? Because, until algebra shows up, you’re doing numerical computations in a straightforwardly algorithmic way. You dump some numbers into the addition box, or the multiplication box, or even, in traditionally minded schools, the long-division box, you turn the crank, and you report what comes out the other side. Algebra is different. It’s computation backward. When you’re asked to solve

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

Like, okay. Everyone in history thought they were the ones who finally knew everything. In their naissance, right, they were positive they knew exactly how the universe worked. Til the next set of guys came along and proved they were missing like a hundred important things. and then that set of guys were sure they had it all down, til another set came along and showed them parts they were missing." He glances at Julia, checking if she's laughing at him, which she isn't, and if she's listening, which she is, completely. "So." he says, "it's pretty unlikely, mathematically, that we are living in the one single era that has everything figured out. Which means there's a decent possibility that the reason we can't explain how ghosts and stuff could exist is because we haven't figured it out yet, not because they don't. And it is pretty arrogant of us to think it definitely has to be the other way around.

The human mind is an incredible thing. It can conceive of the magnificence of the heavens and the intricacies of the basic components of matter. Yet for each mind to achieve its full potential, it needs a spark. The spark of enquiry and wonder. Often that spark comes from a teacher. Allow me to explain. I wasn’t the easiest person to teach, I was slow to learn to read and my handwriting was untidy. But when I was fourteen my teacher at my school in St Albans, Dikran Tahta, showed me how to harness my energy and encouraged me to think creatively about mathematics. He opened my eyes to maths as the blueprint of the universe itself. If you look behind every exceptional person there is an exceptional teacher. When each of us thinks about what we can do in life, chances are we can do it because of a teacher. [...] The basis for the future of education must lie in schools and inspiring teachers. But schools can only offer an elementary framework where sometimes rote-learning, equations and examinations can alienate children from science. Most people respond to a qualitative, rather than a quantitative, understanding, without the need for complicated equations. Popular science books and articles can also put across ideas about the way we live. However, only a small percentage of the population read even the most successful books. Science documentaries and films reach a mass audience, but it is only one-way communication.

Seldon, Hari—It is customary to think of Hari Seldon only in connection with psychohistory, to see him only as Mathematics and as social change personified. There is no doubt that he himself Encouraged this for at no time in his formal writings did he give any hint as to how he came to solve the various problems of Psychohistory. His leaps of thought might have all been plucked From Air, for all he tells us. Nor does he tell us of the blind alleys Into which he crept or the wrong turnings he may have made. …As for his private life, it is a blank. Concerning his parent and Siblings, We know a handful of factors, nor more. His only son, Raych Seldon, is known to have been adopted, but how that Came about is not known. Concerning his wife, we only Know that she existed. Clearly, Seldon wanted to a cipher Except where psychohistory was concerned. It is as though he felt-- Or wanted it to be felt—that he did not live, he merely psychohistorified.

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

Some could say it is the external world which has molded our thinking-that is, the operation of the human brain-into what is now called logic. Others-philosophers and scientists alike-say that our logical thought (thinking process?) is a creation of the internal workings of the mind as they developed through evolution "independently" of the action of the outside world. Obviously, mathematics is some of both. It seems to be a language both for the description of the external world, and possibly even more so for the analysis of ourselves. In its evolution from a more primitive nervous system, the brain, as an organ with ten or more billion neurons and many more connections between them must have changed and grown as a result of many accidents. The very existence of mathematics is due to the fact that there exist statements or theorems, which are very simple to state but whose proofs demand pages of explanations. Nobody knows why this should be so. The simplicity of many of these statements has both aesthetic value and philosophical interest.

Whether or not we subscribe to any particular religion or philosophy, it would be hard to deny that knowing our cosmic destiny must have some impact on how we think about our existence, or even how we live our lives. If we want to know whether what we do here ultimately matters, the first thing we ask is: how will it come out in the end? If we find the answer to that question, it leads immediately to the next: what does this mean for us now? Do we still have to take the trash out next Tuesday if the universe is going to die someday? I’ve done my own scouring of theological and philosophical texts, and while I learned many fascinating things from my studies, unfortunately the meaning of existence wasn’t one of them. I may just not have been cut out for it. The questions and answers that have always drawn me in most strongly are the ones that can be answered with scientific observation, mathematics, and physical evidence. As appealing as it sometimes seemed to have the whole story and meaning of life written down for me once and for all in a book, I knew I would only ever really be able to accept the kind of truth I could rederive mathematically.

Evolution endowed us with intuition only for those aspects of physics that had survival value for our distant ancestors, such as the parabolic orbits of flying rocks (explaining our penchant for baseball). A cavewoman thinking too hard about what matter is ultimately made of might fail to notice the tiger sneaking up behind and get cleaned right out of the gene pool. Darwin’s theory thus makes the testable prediction that whenever we use technology to glimpse reality beyond the human scale, our evolved intuition should break down. We’ve repeatedly tested this prediction, and the results overwhelmingly support Darwin. At high speeds, Einstein realized that time slows down, and curmudgeons on the Swedish Nobel committee found this so weird that they refused to give him the Nobel Prize for his relativity theory. At low temperatures, liquid helium can flow upward. At high temperatures, colliding particles change identity; to me, an electron colliding with a positron and turning into a Z-boson feels about as intuitive as two colliding cars turning into a cruise ship. On microscopic scales, particles schizophrenically appear in two places at once, leading to the quantum conundrums mentioned above. On astronomically large scales… weirdness strikes again: if you intuitively understand all aspects of black holes [then you] should immediately put down this book and publish your findings before someone scoops you on the Nobel Prize for quantum gravity… [also,] the leading theory for what happened [in the early universe] suggests that space isn’t merely really really big, but actually infinite, containing infinitely many exact copies of you, and even more near-copies living out every possible variant of your life in two different types of parallel universes.

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

Outsiders sometimes have an impression that mathematics consists of applying more and more powerful tools to dig deeper and deeper into the unknown, like tunnelers blasting through the rock with ever more powerful explosives. And that's one way to do it. But Grothendieck, who remade much of pure mathematics in his own image in the 1960's and 70's, had a different view: "The unknown thing to be known appeared to me as some stretch of earth or hard marl, resisting penetration...the sea advances insensibly in silence, nothing seems to happen, nothing moves, the water is so far off you hardly hear it...yet it finally surrounds the resistant substance." The unknown is a stone in the sea, which obstructs our progress. We can try to pack dynamite in the crevices of rock, detonate it, and repeat until the rock breaks apart, as Buffon did with his complicated computations in calculus. Or you can take a more contemplative approach, allowing your level of understanding gradually and gently to rise, until after a time what appeared as an obstacle is overtopped by the calm water, and is gone. Mathematics as currently practiced is a delicate interplay between monastic contemplation and blowing stuff up with dynamite.

Instead of proving all possible theorems in an axiomatic system (which Kurt Gödel showed is not possible), professional mathematicians continue to use a formal presentation of mathematics to specify and prove many theorems that are amenable to the formalist paradigm. This has generated a vast corpus of formal theory. Controversies continue unresolved. Some mathematicians continue to insist on giving explicit constructions of mathematical entities, and do not allow proof by contradiction. This is a valid approach in its own right with much to recommend it. In the end, however, the choice that is likely to lead to the greater conquests is the one that offers the greater power and at the moment, it is David Hilbert's formalism that continues to predominate, while steadily being expanded as mathematics expands." -David Tall (2013, p. 246) thinks though Formalism (mathematics) may have Lost the Battle it Still may Win the War.

"Ha! ha! ha! But after all, if you like, in reality there is not such thing as choice," you will interrupt with a laugh. "Science has even now succeeded in analysing man to such an extent that we know already that choice and what is called freedom of will are nothing other than--" Wait, gentlemen, I meant to begin with that myself. I admit I was even frightened. I was just going to shout that after all the devil only knows what choice depends on, and that perhaps that is a very good thing, but I remembered the teaching of science-- and pulled myself up. And here you have begun to speak. After all, really, well, if someday they truly discover a formula for all our desires and caprices-- that is, an explanations of what they depend upon, by what laws they arise, just how they develop, what they are aiming at in one case or another and so on, and so on, that is, a real mathematical formula-- then, after all, man would most likely at once stop to feel desire, indeed, he will be most certain to. For who would want to choose by rule? Besides, he will at once be transformed from a human being into an organ stop or something of the sort; for what is a man without desire, without free will and without choice, if not a stop in an organ? What do you think? Let us consider the probability-- can such a thing happen or not?

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

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

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

I always felt that someone, a long time ago, organized the affairs of the world into areas that made sense-catagories of stuff that is perfectible, things that fit neatly in perfect bundles. The world of business, for example, is this way-line items, spreadsheets, things that add up, that can be perfected. The legal system-not always perfect, but nonetheless a mind-numbing effort to actually write down all kinds of laws and instructions that cover all aspects of being human, a kind of umbrella code of conduct we should all follow. Perfection is crucial in building an aircraft, a bridge, or a high-speed train. The code and mathematics residing just below the surface of the Internet is also this way. Things are either perfectly right or they will not work. So much of the world we work and live in is based upon being correct, being perfect. But after this someone got through organizing everything just perfectly, he (or probably a she) was left with a bunch of stuff that didn't fit anywhere-things in a shoe box that had to go somewhere. So in desperation this person threw up her arms and said, 'OK! Fine. All the rest of this stuff that isn't perfectible, that doesn't seem to fit anywhere else, will just have to be piled into this last, rather large, tattered box that we can sort of push behind the couch. Maybe later we can come back and figure where it all is supposed to fit in. Let's label the box ART.' The problem was thankfully never fixed, and in time the box overflowed as more and more art piled up. I think the dilemma exists because art, among all the other tidy categories, most closely resembles what it is like to be human. To be alive. It is our nature to be imperfect. The have uncategorized feelings and emotions. To make or do things that don't sometimes necessarily make sense. Art is all just perfectly imperfect. Once the word ART enters the description of what you're up to , it is almost getting a hall pass from perfection. It thankfully releases us from any expectation of perfection. In relation to my own work not being perfect, I just always point to the tattered box behind the couch and mention the word ART, and people seem to understand and let you off the hook about being perfect a go back to their business.

In Tibet, we have a traditional image, the windhorse, which represents a balanced relationship between the wind and the mind. The horse represents wind and movement. On its saddle rides a precious jewel. That jewel is our mind. A jewel is a stone that is clear and reflects light. There is a solid, earthly element to it. You can pick it up in your hand, and at the same time you can see through it. These qualities represent the mind: it is both tangible and translucent. The mind is capable of the highest wisdom. It can experience love and compassion, as well as anger. It can understand history, philosophy, and mathematics—and also remember what’s on the grocery list. The mind is truly like a wish-fulfilling jewel. With an untrained mind, the thought process is said to be like a wild and blind horse: erratic and out of control. We experience the mind as moving all the time—suddenly darting off, thinking about one thing and another, being happy, being sad. If we haven’t trained our mind, the wild horse takes us wherever it wants to go. It’s not carrying a jewel on its back—it’s carrying an impaired rider. The horse itself is crazy, so it is quite a bizarre scene. By observing our own mind in meditation, we can see this dynamic at work. Especially in the beginning stages of meditation, we find it extremely challenging to control our mind. Even if we wish to control it, we have very little power to do so, like the infirm rider. We want to focus on the breathing, but the mind keeps darting off unexpectedly. That is the wild horse. The process of meditation is taming the horse so that it is in our control, while making the mind an expert rider.

As you know, there was a famous quarrel between Max Planck and Einstein, in which Einstein claimed that, on paper, the human mind was capable of inventing mathematical models of reality. In this he generalized his own experience because that is what he did. Einstein conceived his theories more or less completely on paper, and experimental developments in physics proved that his models explained phenomena very well. So Einstein says that the fact that a model constructed by the human mind in an introverted situation fits with outer facts is just a miracle and must be taken as such. Planck does not agree, but thinks that we conceive a model which we check by experiment, after which we revise our model, so that there is a kind of dialectic friction between experiment and model by which we slowly arrive at an explanatory fact compounded of the two. Plato-Aristotle in a new form! But both have forgotten something- the unconscious. We know something more than those two men, namely that when Einstein makes a new model of reality he is helped by his unconscious, without which he would not have arrived at his theories...But what role DOES the unconscious play?...either the unconscious knows about other realities, or what we call the unconscious is a part of the same thing as outer reality, for we do not know how the unconscious is linked with matter.

Economics is haunted by more fallacies than any other study known to man. This is no accident. The inherent difficulties of the subject would be great enough in any case, but they are multiplied a thousandfold by a factor that is insignificant in, say, physics, mathematics or medicine - the special pleading of selfish interests. While every group has certain economic interests identical with those of all groups, every group has also, as we shall see, interests antagonistic to those of all other groups. While certain public policies would in the long run benefit everybody, other policies would benefit one group only at the expense of all other groups. The group that would benefit by such policies, having such a direct interest in them, will argue for them plausibly and persistently. It will hire the best buyable minds to devote their whole time to presenting its case. And it will finally either convince the general public that its case is sound, or so befuddle it that clear thinking on the subject becomes next to impossible. In addition to these endless pleadings of self-interest, there is a second main factor that spawns new economic fallacies every day. This is the persistent tendency of man to see only the immediate effects of a given policy, or its effects only on a special group, and to neglect to inquire what the long-run effects of that policy will be not only on that special group but on all groups. It is the fallacy of overlooking secondary consequences.

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

Discreet as you are, Rohan, one can’t help but notice how ardently you are pursued. It seems you hold quite an appeal for the ladies of London. And from all appearances, you’ve taken full advantage of what’s been offered.” Cam stared at him without expression. “Pardon, but are you leading to an actual point, my lord?” Leaning back in his chair, St. Vincent made a temple of his elegant hands and regarded Cam steadily. “Since you’ve had no problem with lack of desire in the past, I can only assume that, as happens with other appetites, yours has been sated with an overabundance of sameness. A bit of novelty may be just the thing.” Considering the statement, which actually made sense, Cam wondered if the notorious former rake had ever been tempted to stray. Having known Evie since childhood, when she had come to visit her widowed father at the club from time to time, Cam felt as protective of her as if she’d been his younger sister. No one would have paired the gentle-natured Evie with such a libertine. And perhaps no one had been as surprised as St. Vincent himself to discover their marriage of convenience had turned into a passionate love match. “What of married life?” Cam asked softly. “Does it eventually become an overabundance of sameness?” St. Vincent’s expression changed, the light blue eyes warming at the thought of his wife. “It has become clear to me that with the right woman, one can never have enough. I would welcome an overabundance of such bliss—but I doubt such a thing is mortally possible.” Closing the account book with a decisive thud, he stood from the desk. “If you’ll excuse me, Rohan, I’ll bid you good night.” “What about finishing the accounting?” “I’ll leave the rest in your capable hands.” At Cam’s scowl, St. Vincent shrugged innocently. “Rohan, one of us is an unmarried man with superior mathematical abilities and no prospects for the evening. The other is a confirmed lecher in an amorous mood, with a willing and nubile young wife waiting at home. Who do you think should do the damned account books?” And, with a nonchalant wave, St. Vincent had left the office.

What is that other universe?" she said. "One of uncountable billions of parallel worlds. The witches have known about them for centuries, but the first theologians to prove their existence mathematically were excommunicated fifty or more years ago. However, it's true; there's no possible way of denying it. "But no one thought it would ever be possible to cross from one universe to another. That would violate fundamental laws, we thought. Well, we were wrong; we learned to see the world up there. If light can cross, so can we. And we had to learn to see it, Lyra, just as you learned to use the alethiometer. "Now that world, and every other universe, came about as a result of possibility. Take the example of tossing a coin: it can come down heads or tails, and we don't know before it lands which way it's going to fall. If it comes down heads, that means that the possibility of its coming down tails has collapsed. Until that moment the two possibilities were equal. "But on another world, it does come down tails. And when that happens, the two worlds split apart. I'm using the example of tossing a coin to make it clearer. In fact, these possibility collapses happen at the level of elementary particles, but they happen in just the same way: one moment several things are possible, the next moment only one happens, and the rest don't exist. Except that other worlds have sprung into being, on which they did happen. "And I'm going to that world beyond the Aurora," he said, "because I think that's where all the Dust in this universe comes from. You saw those slides I showed the Scholars in the retiring room. You saw Dust pouring into this world from the Aurora. You've seen that city yourself. If light can cross the barrier between the universes, if Dust can, if we can see that city, then we can build a bridge and cross. It needs a phenomenal burst of energy. But I can do it. Somewhere out there is the origin of all the Dust, all the death, the sin, the misery, the destructiveness in the world. Human beings can't see anything without wanting to destroy it, Lyra. That's original sin. And I'm going to destroy it. Death is going to die.

If you cannot drop a wrong problem, then the first time you meet one you will be stuck with it for the rest of your career. Einstein was tremendously creative in his early years, but once he began, in midlife, the search for a unified theory, he spent the rest of his life on it and had about nothing to show for all the effort. I have seen this many times while watching how science is done. It is most likely to happen to the very creative people; their previous successes convince them they can solve any problem, but there are other reasons besides overconfidence why, in many fields, sterility sets in with advancing age. Managing a creative career is not an easy task, or else it would often be done. In mathematics, theoretical physics, and astrophysics, age seems to be a handicap (all characterized by high, raw creativity), while in music composition, literature, and statesmanship, age and experience seem to be an asset. As valued by Bell Telephone Laboratories in the late 1970s, the first 15 years of my career included all they listed, and for my second 15 years they listed nothing I was very closely associated with! Yes, in my areas the really great things are generally done while the person is young, much as in athletics, and in old age you can turn to coaching (teaching), as I have done. Of course, I do not know your field of expertise to say what effect age will have, but I suspect really great things will be realized fairly young, though it may take years to get them into practice. My advice is if you want to do significant things, now is the time to start thinking (if you have not already done so) and not wait until it is the proper moment—which may never arrive!

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

Perhaps I don't know enough yet to find the right words for it, but I think I can describe it. It happened again just a moment ago. I don't know how to put it except by saying that I see things in two different ways-everything, ideas included. If I make an effort to find any difference in them, each of them is the same today as it was yesterday, but as soon as I shut my eyes they're suddenly transformed, in a different light. Perhaps I went wrong about the imaginary numbers. If I get to them by going straight along inside mathematics, so to speak, they seem quite natural. It's only if I look at them directly, in all their strangeness, that they seem impossible. But of course I may be all wrong about this, I know too little about it. But I wasn't wrong about Basini. I wasn't wrong when I couldn't turn my ear away from the faint trickling sound in the high wall or my eye from the silent, swirling dust going up in the beam of light from a lamp. No, I wasn't wrong when I talked about things having a second, secret life that nobody takes any notice of! I-I don't mean it literally-it's not that things are alive, it's not that Basini seemed to have two faces-it was more as if I had a sort of second sight and saw all this not with the eyes of reason. Just as I can feel an idea coming to life in my mind, in the same way I feel something alive in me when I look at things and stop thinking. There's something dark in me, deep under all my thoughts, something I can't measure out with thoughts, a sort of life that can't be expressed in words and which is my life, all the same. “That silent life oppressed me, harassed me. Something kept on making me stare at it. I was tormented by the fear that our whole life might be like that and that I was only finding it out here and there, in bits and pieces. . . . Oh, I was dreadfully afraid! I was out of my mind.. .” These words and these figures of speech, which were far beyond what was appropriate to Törless's age, flowed easily and naturally from his lips in this state of vast excitement he was in, in this moment of almost poetic inspiration. Then he lowered his voice and, as though moved by his own suffering, he added: “Now it's all over. I know now I was wrong after all. I'm not afraid of anything any more. I know that things are just things and will probably always be so. And I shall probably go on for ever seeing them sometimes this way and sometimes that, sometimes with the eyes of reason, and sometimes with those other eyes. . . . And I shan't ever try again to compare one with the other. .

prior probability that the sun will rise, since it’s prior to seeing any evidence. It’s not based on counting the number of times the sun has risen on this planet in the past, because you weren’t there to see it; rather, it reflects your a priori beliefs about what will happen, based on your general knowledge of the universe. But now the stars start to fade, so your confidence that the sun does rise on this planet goes up, based on your experience on Earth. Your confidence is now a posterior probability, since it’s after seeing some evidence. The sky begins to lighten, and the posterior probability takes another leap. Finally, a sliver of the sun’s bright disk appears above the horizon and perhaps catches “the Sultan’s turret in a noose of light,” as in the opening verse of the Rubaiyat. Unless you’re hallucinating, it is now certain that the sun will rise. The crucial question is exactly how the posterior probability should evolve as you see more evidence. The answer is Bayes’ theorem. We can think of it in terms of cause and effect. Sunrise causes the stars to fade and the sky to lighten, but the latter is stronger evidence of daybreak, since the stars could fade in the middle of the night due to, say, fog rolling in. So the probability of sunrise should increase more after seeing the sky lighten than after seeing the stars fade. In mathematical notation, we say that P(sunrise | lightening-sky), the conditional probability of sunrise given that the sky is lightening, is greater than P(sunrise | fading-stars), its conditional probability given that the stars are fading. According to Bayes’ theorem, the more likely the effect is given the cause, the more likely the cause is given the effect: if P(lightening-sky | sunrise) is higher than P(fading-stars | sunrise), perhaps because some planets are far enough from their sun that the stars still shine after sunrise, then P(sunrise | lightening sky) is also higher than P(sunrise | fading-stars).

A Puritan twist in our nature makes us think that anything good for us must be twice as good if it's hard to swallow. Learning Greek and Latin used to play the role of character builder, since they were considered to be as exhausting and unrewarding as digging a trench in the morning and filling it up in the afternoon. It was what made a man, or a woman -- or more likely a robot -- of you. Now math serves that purpose in many schools: your task is to try to follow rules that make sense, perhaps, to some higher beings; and in the end to accept your failure with humbled pride. As you limp off with your aching mind and bruised soul, you know that nothing in later life will ever be as difficult. What a perverse fate for one of our kind's greatest triumphs! Think how absurd it would be were music treated this way (for math and music are both excursions into sensuous structure): suffer through playing your scales, and when you're an adult you'll never have to listen to music again. And this is mathematics we're talking about, the language in which, Galileo said, the Book of the World is written. This is mathematics, which reaches down into our deepest intuitions and outward toward the nature of the universe -- mathematics, which explains the atoms as well as the stars in their courses, and lets us see into the ways that rivers and arteries branch. For mathematics itself is the study of connections: how things ideally must and, in fact, do sort together -- beyond, around, and within us. It doesn't just help us to balance our checkbooks; it leads us to see the balances hidden in the tumble of events, and the shapes of those quiet symmetries behind the random clatter of things. At the same time, we come to savor it, like music, wholly for itself. Applied or pure, mathematics gives whoever enjoys it a matchless self-confidence, along with a sense of partaking in truths that follow neither from persuasion nor faith but stand foursquare on their own. This is why it appeals to what we will come back to again and again: our **architectural instinct** -- as deep in us as any of our urges.

It was all so very businesslike that one watched it fascinated. It was porkmaking by machinery, porkmaking by applied mathematics. And yet somehow the most matter-of-fact person could not help thinking of the hogs; they were so innocent, they came so very trustingly; and they were so very human in their protests—and so perfectly within their rights! They had done nothing to deserve it; and it was adding insult to injury, as the thing was done here, swinging them up in this cold-blooded, impersonal way, without a pretense of apology, without the homage of a tear. Now and then a visitor wept, to be sure; but this slaughtering machine ran on, visitors or no visitors. It was like some horrible crime committed in a dungeon, all unseen and unheeded, buried out of sight and of memory. One could not stand and watch very long without becoming philosophical, without beginning to deal in symbols and similes, and to hear the hog squeal of the universe. Was it permitted to believe that there was nowhere upon the earth, or above the earth, a heaven for hogs, where they were requited for all this suffering? Each one of these hogs was a separate creature. Some were white hogs, some were black; some were brown, some were spotted; some were old, some young; some were long and lean, some were monstrous. And each of them had an individuality of his own, a will of his own, a hope and a heart’s desire; each was full of self-confidence, of self-importance, and a sense of dignity. And trusting and strong in faith he had gone about his business, the while a black shadow hung over him and a horrid Fate waited in his pathway. Now suddenly it had swooped upon him, and had seized him by the leg. Relentless, remorseless, it was; all his protests, his screams, were nothing to it—it did its cruel will with him, as if his wishes, his feelings, had simply no existence at all; it cut his throat and watched him gasp out his life. And now was one to believe that there was nowhere a god of hogs, to whom this hog personality was precious, to whom these hog squeals and agonies had a meaning? Who would take this hog into his arms and comfort him, reward him for his work well done, and show him the meaning of his sacrifice?

We must consider also whether soul is divisible or is without parts, and whether it is everywhere homogeneous or not; and if not homogeneous, whether its various forms are different specifically or generically; up to the present time those who have discussed and investigated soul seem to have confined themselves to the human soul. We must be careful not to ignore the question whether soul can be defined in a single account, as is the case with animal, or whether we must not give a separate account of each sort of it, as we do for horse, dog, man, god (in the latter case the universal, animal—and so too every other common predicate—is either nothing or posterior). Further, if what exists is not a plurality of souls, but a plurality of parts of one soul, which ought we to investigate first, the whole soul or its parts? It is also a difficult problem to decide which of these parts are in nature distinct from one another. Again, which ought we to investigate first, these parts or their functions, mind or thinking, the faculty or the act of sensation, and so on? If the investigation of the functions precedes that of the parts, the further question suggests itself: ought we not before either to consider the correlative objects, e.g. of sense or thought? It seems not only useful for the discovery of the causes of the incidental proprieties of substances to be acquainted with the essential nature of those substances (as in mathematics it is useful for the understanding of the property of the equality of the interior angles of a triangle to two right angles to know the essential nature of the straight and the curved or of the line and (the plane) but also conversely, for the knowledge of the essential nature of a substance is largely promoted by an acquaintance with its properties: for, when we are able to give an account conformable to experience of all or most of the properties of a substance, we shall be in the most favourable position to say something worth saying about the essential nature of that subject: in all demonstration a definition of the essence is required as a starting point, so that definitions which do not enable us to discover the incidental properties, or which fail to facilitate even a conjecture about them, must obviously, one and all, be dialectical and futile.

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

Computational models of the mind would make sense if what a computer actually does could be characterized as an elementary version of what the mind does, or at least as something remotely like thinking. In fact, though, there is not even a useful analogy to be drawn here. A computer does not even really compute. We compute, using it as a tool. We can set a program in motion to calculate the square root of pi, but the stream of digits that will appear on the screen will have mathematical content only because of our intentions, and because we—not the computer—are running algorithms. The computer, in itself, as an object or a series of physical events, does not contain or produce any symbols at all; its operations are not determined by any semantic content but only by binary sequences that mean nothing in themselves. The visible figures that appear on the computer’s screen are only the electronic traces of sets of binary correlates, and they serve as symbols only when we represent them as such, and assign them intelligible significances. The computer could just as well be programmed so that it would respond to the request for the square root of pi with the result “Rupert Bear”; nor would it be wrong to do so, because an ensemble of merely material components and purely physical events can be neither wrong nor right about anything—in fact, it cannot be about anything at all. Software no more “thinks” than a minute hand knows the time or the printed word “pelican” knows what a pelican is. We might just as well liken the mind to an abacus, a typewriter, or a library. No computer has ever used language, or responded to a question, or assigned a meaning to anything. No computer has ever so much as added two numbers together, let alone entertained a thought, and none ever will. The only intelligence or consciousness or even illusion of consciousness in the whole computational process is situated, quite incommutably, in us; everything seemingly analogous to our minds in our machines is reducible, when analyzed correctly, only back to our own minds once again, and we end where we began, immersed in the same mystery as ever. We believe otherwise only when, like Narcissus bent above the waters, we look down at our creations and, captivated by what we see reflected in them, imagine that another gaze has met our own.

In every area of thought we must rely ultimately on our judgments, tested by reflection, subject to correction by the counterarguments of others, modified by the imagination and by comparison with alternatives. Antirealism is always a conjectural possibility: the question can always be posed, whether there is anything more to truth in a certain domain than our tendency to reach certain conclusions in this way, perhaps in convergence with others. Sometimes, as with grammar or etiquette, the answer is no. For that reason the intuitive conviction that a particular domain, like the physical world, or mathematics, or morality, or aesthetics, is one in which our judgments are attempts to respond to a kind of truth that is independent of them may be impossible to establish decisively. Yet it may be very robust all the same, and not unjustified. To be sure, there are competing subjectivist explanations of the appearance of mind-independence in the truth of moral and other value judgments. One of the things a sophisticated subjectivism allows us to say when we judge that infanticide is wrong is that it would be wrong even if none of us thought so, even though that second judgment too is still ultimately grounded in our responses. However, I find those quasi-realist, expressivist accounts of the ground of objectivity in moral judgments no more plausible than the subjectivist account of simpler value judgments. These epicycles are of the same kind as the original proposal: they deny that value judgments can be true in their own right, and this does not accord with what I believe to be the best overall understanding of our thought about value. There is no crucial experiment that will establish or refute realism about value. One ground for rejecting it, the type used by Hume, is simply question-begging: if it is supposed that objective moral truths can exist only if they are like other kinds of facts--physical, psychological, or logical--then it is clear that there aren't any. But the failure of this argument doesn't prove that there are objective moral truths. Positive support for realism can come only from the fruitfulness of evaluative and moral thought in producing results, including corrections of beliefs formerly widely held and the development of new and improved methods and arguments over time. The realist interpretation of what we are doing in thinking about these things can carry conviction only if it is a better account than the subjectivist or social-constructivist alternatives, and that is always going to be a comparative question and a matter of judgment, as it is about any other domain, whether it be mathematics or science or history or aesthetics.

Chapter 13 - 1 Sometimes a kind of glory lights up the mind of a man. It happens to nearly everyone. You can feel it growing or preparing like a fuse burning toward dynamite. It is a feeling in the stomach, a delight of the nerves, of the forearms. The skin tastes the air, and every deep-drawn breath is sweet. Its beginning has the pleasure of a great stretching yawn; it flashes in the brain and the whole world glows outside your eyes. A man may have lived all of his life in the gray, and the land and trees of him dark and somber. The events, even the important ones, may have trooped by faceless and pale. And then—the glory—so that a cricket song sweetens his ears, the smell of the earth rises chanting to his nose, and dappling light under a tree blesses his eyes. Then a man pours outward, a torrent of him, and yet he is not diminished. And I guess a man’s importance in the world can be measured by the quality and number of his glories. It is a lonely thing but it relates us to the world. It is the mother of all creativeness, and it sets each man separate from all other men. I don’t know how it will be in the years to come. There are monstrous changes taking place in the world, forces shaping a future whose face we do not know. Some of these forces seem evil to us, perhaps not in themselves but because their tendency is to eliminate other things we hold good. It is true that two men can lift a bigger stone than one man. A group can build automobiles quicker and better than one man, and bread from a huge factory is cheaper and more uniform. When our food and clothing and housing all are born in the complication of mass production, mass method is bound to get into our thinking and to eliminate all other thinking. In our time mass or collective production has entered our economics, our politics, and even our religion, so that some nations have substituted the idea collective for the idea God. This in my time is the danger. There is great tension in the world, tension toward a breaking point, and men are unhappy and confused. At such a time it seems natural and good to me to ask myself these questions. What do I believe in? What must I fight for and what must I fight against? Our species is the only creative species, and it has only one creative instrument, the individual mind and spirit of a man. Nothing was ever created by two men. There are no good collaborations, whether in music, in art, in poetry, in mathematics, in philosophy. Once the miracle of creation has taken place, the group can build and extend it, but the group never invents anything. The preciousness lies in the lonely mind of a man. And now the forces marshaled around the concept of the group have declared a war of extermination on that preciousness, the mind of man. By disparagement, by starvation, by repressions, forced direction, and the stunning hammerblows of conditioning, the free, roving mind is being pursued, roped, blunted, drugged. It is a sad suicidal course our species seems to have taken. And this I believe: that the free, exploring mind of the individual human is the most valuable thing in the world. And this I would fight for: the freedom of the mind to take any direction it wishes, undirected. And this I must fight against: any idea, religion, or government which limits or destroys the individual. This is what I am and what I am about. I can understand why a system built on a pattern must try to destroy the free mind, for this is one thing which can by inspection destroy such a system. Surely I can understand this, and I hate it and I will fight against it to preserve the one thing that separates us from the uncreative beasts. If the glory can be killed, we are lost.