Mathematics Importance Quotes

You’re probably better at math than I am, because pretty much everyone’s better at math than I am, but it’s okay, I’m fine with it. See, I excel at other, more important things—guitar, sex, and consistently disappointing my dad, to name a few.

Yes, we have to divide up our time like that, between our politics and our equations. But to me our equations are far more important, for politics are only a matter of present concern. A mathematical equation stands forever.

The important thing to remember about mathematics is not to be frightened

One must divide one's time between politics and equations. But our equations are much more important to me, because politics is for the present, while our equations are for eternity.

The Hitchhiker's Guide to the Galaxy's definition of "Universe": The Universe is a very big thing that contains a great number of planets and a great number of beings. It is Everything. What we live in. All around us. The lot. Not nothing. It is quite difficult to actually define what the Universe means, but fortunately the Guide doesn't worry about that and just gives us some useful information to live in it. Area: The area of the Universe is infinite. Imports: None. This is a by product of infinity; it is impossible to import things into something that has infinite volume because by definition there is no outside to import things from. Exports: None, for similar reasons as imports. Population: None. Although you might see people from time to time, they are most likely products of your imagination. Simple mathematics tells us that the population of the Universe must be zero. Why? Well given that the volume of the universe is infinite there must be an infinite number of worlds. But not all of them are populated; therefore only a finite number are. Any finite number divided by infinity is zero, therefore the average population of the Universe is zero, and so the total population must be zero. Art: None. Because the function of art is to hold a mirror up to nature there can be no art because the Universe is infinite which means there simply isn't a mirror big enough. Sex: None. Although in fact there is quite a lot, given the zero population of the Universe there can in fact be no beings to have sex, and therefore no sex happens in the Universe.

What is important is to deeply understand things and their relations to each other. This is where intelligence lies. The fact of being quick or slow isn’t really relevant.

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

Nothing is more important than to see the sources of invention which are, in my opinion, more interesting than the inventions themselves.

And, most important of all," added the Mathemagician, "here is your own magic staff. Use it well and there is nothing it cannot do for you." He placed in Milo's breast pocket a small gleaming pencil which, except for the size, was much like his own.

No one shall expel us from the paradise which Cantor has created for us. {Expressing the importance of Georg Cantor's set theory in the development of mathematics.}

Mathematical problems, or puzzles, are important to real mathematics (like solving real-life problems), just as fables, stories and anecdotes are important to the young in understanding real life

The key point to keep in mind, however, is that symmetry is one of the most important tools in deciphering nature's design.

(1) Use mathematics as shorthand language, rather than as an engine of inquiry. (2) Keep to them till you have done. (3) Translate into English. (4) Then illustrate by examples that are important in real life (5) Burn the mathematics. (6) If you can’t succeed in 4, burn 3. This I do often.

Questions that pertain to the foundations of mathematics, although treated by many in recent times, still lack a satisfactory solution. Ambiguity of language is philosophy's main source of problems. That is why it is of the utmost importance to examine attentively the very words we use.

It is India that gave us the ingenious method of expressing all numbers by means of ten symbols, each symbol receiving a value of position as well as an absolute value; a profound and important idea which appears so simple to us now that we ignore its true merit. But its very simplicity and the great ease which it has lent to computations put our arithmetic in the first rank of useful inventions; and we shall appreciate the grandeur of the achievement the more when we remember that it escaped the genius of Archimedes and Apollonius, two of the greatest men produced by antiquity.

Math is hard work and it occupies your mind—and it doesn’t hurt to learn all you can of it, no matter what rank you are; everything of any importance is founded on mathematics.

Memory cannot be understood, either, without a mathematical approach. The fundamental given is the ratio between the amount of time in the lived life and the amount of time from that life that is stored in memory. No one has ever tried to calculate this ratio, and in fact there exists no technique for doing so; yet without much risk of error I could assume that the memory retains no more than a millionth, a hundred-millionth, in short an utterly infinitesimal bit of the lived life. That fact too is part of the essence of man. If someone could retain in his memory everything he had experienced, if he could at any time call up any fragment of his past, he would be nothing like human beings: neither his loves nor his friendships nor his angers nor his capacity to forgive or avenge would resemble ours. We will never cease our critique of those persons who distort the past, rewrite it, falsify it, who exaggerate the importance of one event and fail to mention some other; such a critique is proper (it cannot fail to be), but it doesn't count for much unless a more basic critique precedes it: a critique of human memory as such. For after all, what can memory actually do, the poor thing? It is only capable of retaining a paltry little scrap of the past, and no one knows why just this scrap and not some other one, since in each of us the choice occurs mysteriously, outside our will or our interests. We won't understand a thing about human life if we persist in avoiding the most obvious fact: that a reality no longer is what it was when it was; it cannot be reconstructed.

The possible, as it was presented in her Health textbook (a mathematical progression of dating, "career," marriage, and motherhood), did not interest Harriet. Of all the heroes on her list, the greatest of them all was Sherlock Holmes, and he wasn’t even a real person. Then there was Harry Houdini. He was the master of the impossible; more importantly, for Harriet, he was a master of escape. No prison in the world could hold him: he escaped from straitjackets, from locked trunks dropped in fast rivers and from coffins buried six feet underground. And how had he done it? He wasn’t afraid. Saint Joan had galloped out with the angels on her side but Houdini had mastered fear on his own. No divine aid for him; he’d taught himself the hard way how to beat back panic, the horror of suffocation and drowning and dark. Handcuffed in a locked trunk in the bottom of a river, he squandered not a heartbeat on being afraid, never buckled to the terror of the chains and the dark and the icy water; if he became lightheaded, for even a moment, if he fumbled at the breathless labor before him– somersaulting along a river-bed, head over heels– he would never come up from the water alive. A training program. This was Houdini’s secret.

The importance of C.F. Gauss for the development of modern physical theory and especially for the mathematical fundament of the theory of relativity is overwhelming indeed; also his achievement of the system of absolute measurement in the field of electromagnetism. In my opinion it is impossible to achieve a coherent objective picture of the world on the basis of concepts which are taken more or less from inner psychological experience.

Certainly one of the most important things I learned is that numbers can be deceiving. There is a logic to mathematics, but there is also the underlying human element that must be considered. Numbers can't lie, but the people who create those numbers can and do. As so many people have learned, forgetting to include human nature in an equation can be devastating.

Just as it is in golf, procedure practice is important in math. But when it comprises the entire math training strategy, it’s a problem. “Students do not view mathematics as a system,” Richland and her colleagues wrote. They view it as just a set of procedures.

Deep Blue didn't win by being smarter than a human; it won by being millions of times faster than a human. Deep Blue had no intuition. An expert human player looks at a board position and immediately sees what areas of play are most likely to be fruitful or dangerous, whereas a computer has no innate sense of what is important and must explore many more options. Deep Blue also had no sense of the history of the game, and didn't know anything about its opponent. It played chess yet didn't understand chess, in the same way a calculator performs arithmetic bud doesn't understand mathematics.

Our schools will not improve if we continue to focus only on reading and mathematics while ignoring the other studies that are essential elements of a good education. Schools that expect nothing more of their students than mastery of basic skills will not produce graduates who are ready for college or the modern workplace. *** Our schools will not improve if we value only what tests measure. The tests we have now provide useful information about students' progress in reading and mathematics, but they cannot measure what matters most in education....What is tested may ultimately be less important that what is untested... *** Our schools will not improve if we continue to close neighborhood schools in the name of reform. Neighborhood schools are often the anchors of their communities, a steady presence that helps to cement the bond of community among neighbors. *** Our schools cannot improve if charter schools siphon away the most motivated students and their families in the poorest communities from the regular public schools. *** Our schools will not improve if we continue to drive away experienced principals and replace them with neophytes who have taken a leadership training course but have little or no experience as teachers. *** Our schools cannot be improved if we ignore the disadvantages associated with poverty that affect children's ability to learn. Children who have grown up in poverty need extra resources, including preschool and medical care.

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.

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

The geometer offers to the physicist a whole set of maps from which to choose. One map, perhaps, will fit the facts better than others, and then the geometry which provides that particular map will be the geometry most important for applied mathematics.

there is a physical problem that is common to many fields, that is very old, and that has not been solved. It is not the problem of finding new fundamental particles, but something left over from a long time ago—over a hundred years. Nobody in physics has really been able to analyze it mathematically satisfactorily in spite of its importance to the sister sciences. It is the analysis of circulating or turbulent fluids.

The ingenious method of expressing every possible number using a set of ten symbols (each symbol having a place value and an absolute value) emerged in India. The idea seems so simple nowadays that its significance and profound importance is no longer appreciated ... The importance of this invention is more readily appreciated when one considers that it was beyod the two greatest men of antiquity, Archimedes and Apollonius.

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

line of reasoning by which the detective solves the mystery is more important than the identity of the murderer.

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

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

...successful research doesn't depend on mathematical skill, or even the deep understanding of theory. It depends to a large degree on choosing an important problem and finding a way to solve it, even if imperfectly at first. Very often ambition and entrepreneurial drive, in combination, beat brilliance.

Just as it's important to take the changing value of a dollar into account when comparing spending over time, it's important to take doctors' changing diagnoses into account when looking at disease trends

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.

It must be this overarching commitment to what is really an abstraction, to one's children right or wrong, that can be even more fierce than the commitment to them as explicit, difficult people, and that can consequently keep you devoted to them when as individuals they disappoint. On my part it was this broad covenant with children-in-theory that I may have failed to make and to which I was unable to resort when Kevin finally tested my maternal ties to a perfect mathematical limit on Thursday. I didn't vote for parties, but for candidates. My opinions were as ecumenical as my larder, then still chock full of salsa verde from Mexico City, anchovies from Barcelona, lime leaves from Bangkok. I had no problem with abortion but abhorred capital punishment, which I suppose meant that I embraced the sanctity of life only in grown-ups. My environmental habits were capricious; I'd place a brick in our toilet tank, but after submitting to dozens of spit-in-the-air showers with derisory European water pressure, I would bask under a deluge of scalding water for half an hour. My closet wafter with Indian saris, Ghanaian wraparounds, and Vietnamese au dais. My vocabulary was peppered with imports -- gemutlich, scusa, hugge, mzungu. I so mixed and matched the planet that you sometimes worried I had no commitments to anything or anywhere, though you were wrong; my commitments were simply far-flung and obscenely specific. By the same token, I could not love a child; I would have to love this one. I was connected to the world by a multitude of threads, you by a few sturdy guide ropes. It was the same with patriotism: You loved the idea of the United States so much more powerfully than the country itself, and it was thanks to your embrace of the American aspiration that you could overlook the fact that your fellow Yankee parents were lining up overnight outside FAO Schwartz with thermoses of chowder to buy a limited release of Nintendo. In the particular dwells the tawdry. In the conceptual dwells the grand, the transcendent, the everlasting. Earthly countries and single malignant little boys can go to hell; the idea of countries and the idea of sons triumph for eternity. Although neither of us ever went to church, I came to conclude that you were a naturally religious person.

Primary causes are unknown to us; but are subject to simple and constant laws, which may be discovered by observation, the study of them being the object of natural philosophy. Heat, like gravity, penetrates every substance of the universe, its rays occupy all parts of space. The object of our work is to set forth the mathematical laws which this element obeys. The theory of heat will hereafter form one of the most important branches of general physics.

THE IMPORTANCE OF CHUNKING “Mathematics is amazingly compressible: you may struggle a long time, step by step, to work through the same process or idea from several approaches. But once you really understand it and have the mental perspective to see it as a whole, there is often a tremendous mental compression. You can file it away, recall it quickly and completely when you need it, and use it as just one step in some other mental process. The insight that goes with this compression is one of the real joys of mathematics.”26 —William Thurston, winner of the Fields Medal, the top award in mathematics

Calculus succeeds by breaking complicated problems down into simpler parts. That strategy, of course, is not unique to calculus. All good problem-solvers know that hard problems become easier when they’re split into chunks. The truly radical and distinctive move of calculus is that it takes this divide-and-conquer strategy to its utmost extreme — all the way out to infinity.

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.

This skipping is another important point. It should be done whenever a proof seems too hard or whenever a theorem or a whole paragraph does not appeal to the reader. In most cases he will be able to go on and later he may return to the parts which he skipped.

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

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

Where There’s Pattern, There’s Reason The key thought in the preceding few lines is the article of faith that this pattern cannot merely be a coincidence. A mathematician who finds a pattern of this sort will instinctively ask, “Why? What is the reason behind this order?” Not only will all mathematicians wonder what the reason is, but even more importantly, they will all implicitly believe that whether or not anyone ever finds the reason, there must be a reason for it. Nothing happens “by accident” in the world of mathematics.

Mathematical truth is not determined arbitrarily by the rules of some 'man-made' formal system, but has an absolute nature, and lies beyond any such system of specifiable rules. Support for the Platonic viewpoint ...was an important part of Godel's initial motivations.

Well, suppose we use our brains. We see things solid. Solidities are important to us in nature. In solidities, there are measures that greatly affect us. There are rhythms in the ins and outs of form. Music, the forest and to many the most impressive of arts deals in measures which seem to go in every direction. They combine, they move together, they deflect and they oppose. Music is a structure of highly mathematical measures. According to the selection and relative value of these measures the music is great or small in its effect on us.

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.

I have stressed this distinction because it is an important one. It defines the fundamental difference between probability and statistics: the former concerns predictions based on fixed probabilities; the latter concerns the inference of those probabilities based on observed data.

Dear friends & fellow characters, you all know the importance we attach to the power of collective prayer in this our desperate struggle for survival. Some of us have more existence than others, at various times according to fashion. But even this is becoming extremely shadowy & precarious, for we are not read, & when read , we are read badly, we are not lived as we used to be, we are not identified with & fantasized, we are rapidly forgotten. Those of us who have the good fortune to be read by teachers, scholars, & students are not read as we used to be read, but analyzed as schemata, structures, functions within structures, logical & mathematical formulae, aporia, psychic movements, social significances & so forth.

And so, since Archimedes led more than anyone else to the formation of the calculus and since he was the pioneer of the application of mathematics to the physical world, it turns out that Western science is but a series of footnotes to Archimedes. Thus, it turns out that Archimedes is the most important scientist who ever lived.

You’re probably better at math than I am, because pretty much everyone’s better at math than I am, but it’s okay, I’m fine with it. See, I excel at other, more important things—guitar, sex, and consistently disappointing my dad, to name a few. By the way, it's apparently true that you'll never use it in the real world. Math, I mean.

Altruism is not an improbable achievement against the individualizing forces of natural selection; rather, it is an integral part of the social lives of all beings that live with others interdependently—up to a (mathematical) point. Everyone helps and gets helped, up to a point, because everyone is important to someone in some way, up to a point.

understood the infamous spiritual terror which this movement exerts, particularly on the bourgeoisie, which is neither morally nor mentally equal to such attacks; at a given sign it unleashes a veritable barrage of lies and slanders against whatever adversary seems most dangerous, until the nerves of the attacked persons break down… This is a tactic based on precise calculation of all human weaknesses, and its result will lead to success with almost mathematical certainty… I achieved an equal understanding of the importance of physical terror toward the individual and the masses… For while in the ranks of their supporters the victory achieved seems a triumph of the justice of their own cause, the defeated adversary in most cases despairs of the success of any further resistance.49 No more precise analysis of Nazi tactics, as Hitler was eventually to develop them, was ever written.

First comes intuition. Rigor comes later.

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.

Well, of course one must have concentration. Courage. Self-control. That goes without saying. But more important than these, one must have... I don't know how to say it. One must be both a mathematician and a poet. As though poetry were a science; or mathematics an art. One must have an affection for proportion to play Go at all well.Ah... what Go is to philosophers and warriors, chess is to accountants and merchants.

Looking down from the heavens, she saw how small, and yet how important each human life is. Drops in the bucket of eternity. She saw her minute place in the organic machine of the Cosmos, witnessed the give and take and the slow, steady swinging of life's pendulum. The world relies on order, pattern, and repetition. The earth spins and swings around the sun with rational, mathematical predictability. But she also saw the chaotic nature of things. No matter what, you can never know with certainty what will happen. Lightening can strike, the ground can open up and swallow you, and the very air you breathe can tear your life away.

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 aborted research project wasn’t important in and of itself. What mattered was the instruction that Ye Wenjie had given him, so that’s where Luo Ji’s mind was stuck. Over and over again he recalled her words: Suppose a vast number of civilizations are distributed throughout the universe, on the order of the number of detectable stars. Lots and lots of them. The mathematical structure of cosmic sociology is far clearer than that of human sociology. The factors of chaos and randomness in the complex makeups of every civilized society in the universe get filtered out by the immense distance, so those civilizations can act as reference points that are relatively easy to manipulate mathematically. First: Survival is the primary need of civilization. Second: Civilization continuously grows and expands, but the total matter in the universe remains constant. One more thing: To derive a basic picture of cosmic sociology from these two axioms, you need two other important concepts: chains of suspicion and the technological explosion. I’m afraid there won’t be that opportunity.… Well, you might as well just forget I said anything. Either way, I’ve fulfilled my duty. He

The more vast the amount of time we've left behind us, the more irresistible is the voice calling us to return to it. This pronouncement seems to state the obvious and yet it is false. Men grow old, the end grows near, each moment becomes more and more valuable and there is no time to waste on recollection. It's important to understand the mathematical paradox in nostalgia, that it is most powerful in early youth , when the volume of life that has passed is quite small.

The transmission of SARS, Dwyer said, seems to depend much on super spreaders—and their behavior, not to mention the behavior of people around them, can be various. The mathematical ecologist’s term for variousness of behavior is “heterogeneity,” and Dwyer’s models have shown that heterogeneity of behavior, even among forest insects, let alone among humans, can be very important in damping the spread of infectious disease. “If you hold mean transmission rate constant,” he told me, “just adding heterogeneity by itself will tend to reduce the overall infection rate.” That sounds dry. What it means is that individual effort, individual discernment, individual choice can have huge effects in averting the catastrophes that might otherwise sweep through a herd. An individual gypsy moth may inherit a slightly superior ability to avoid smears of NPV as it grazes on a leaf. An individual human may choose not to drink the palm sap, not to eat the chimpanzee, not to pen the pig beneath mango trees, not to clear the horse’s windpipe with his bare hand, not to have unprotected sex with the prostitute, not to share the needle in a shooting gallery, not to cough without covering her mouth, not to board a plane while feeling ill, or not to coop his chickens along with his ducks. “Any tiny little thing that people do,” Dwyer said, if it makes them different from one another, from the idealized standard of herd behavior, “is going to reduce infection rates.

Mesoamerica would deserve its place in the human pantheon if its inhabitants had only created maize, in terms of harvest weight the world’s most important crop. But the inhabitants of Mexico and northern Central America also developed tomatoes, now basic to Italian cuisine; peppers, essential to Thai and Indian food; all the world’s squashes (except for a few domesticated in the United States); and many of the beans on dinner plates around the world. One writer has estimated that Indians developed three-fifths of the crops now in cultivation, most of them in Mesoamerica. Having secured their food supply, Mesoamerican societies turned to intellectual pursuits. In a millennium or less, a comparatively short time, they invented their own writing, astronomy, and mathematics, including the zero.

The thesis of the book,” he writes in response, “when correctly interpreted, is essentially trivial. . . . To ‘prove’ such a mathematical result by a costly and prolonged numerical study of many kinds of business profit and expense ratios is analogous to proving the multiplication table by arranging elephants in rows and columns, and then doing the same for numerous other kinds of animals. The performance, though perhaps entertaining, and having a certain pedagogical value, is not an important contribution either to zoölogy or mathematics.

One is the notion that knowledge is worth acquiring, all knowledge, and that a solid grounding in mathematics provides one with the essential language of many of the most important forms of knowledge. The third theme is that, while it is desirable to live peaceably, there are things worth fighting for and values worth dying for—and that it is far better for a man to die than to live under circumstances that call for such sacrifice. The fourth theme is that individual human freedoms are of basic value, without which mankind is less than human.63

An even more important philosophical contact was with the Austrian philosopher Ludwig Wittgenstein, who began as my pupil and ended as my supplanter at both Oxford and Cambridge. He had intended to become an engineer and had gone to Manchester for that purpose. The training for an engineer required mathematics, and he was thus led to interest in the foundations of mathematics. He inquired at Manchester whether there was such a subject and whether anybody worked at it. They told him about me, and so he came to Cambridge. He was queer, and his notions seemed to me odd, so that for a whole term I could not make up my mind whether he was a man of genius or merely an eccentric. At the end of his first term at Cambridge he came to me and said: “Will you please tell me whether I am a complete idiot or not?” I replied, “My dear fellow, I don’t know. Why are you asking me?” He said, “Because, if I am a complete idiot, I shall become an aeronaut; but, if not, I shall become a philosopher.” I told him to write me something during the vacation on some philosophical subject and I would then tell him whether he was complete idiot or not. At the beginning of the following term he brought me the fulfillment of this suggestion. After reading only one sentence, I said to him: “No, you must not become an aeronaut.” And he didn’t. The collected papers of Bertrand Russell: Last Philosophical Testament

But what is the use of the humanities as such? Admittedly they are not practical, and admittedly they concern themselves with the past. Why, it may be asked, should we engage in impractical investigations, and why should we be interested in the past? The answer to the first question is: because we are interested in reality. Both the humanities and the natural sciences, as well as mathematics and philosophy, have the impractical outlook of what the ancients called vita contemplativa as opposed to vita activa. But is the contemplative life less real or, to be more precise, is its contribution to what we call reality less important, than that of the active life? The man who takes a paper dollar in exchange for twenty-five apples commits an act of faith, and subjects himself to a theoretical doctrine, as did the mediaeval man who paid for indulgence. The man who is run over by an automobile is run over by mathematics, physics and chemistry. For he who leads the contemplative life cannot help influencing the active, just as he cannot prevent the active life from influencing his thought. Philosophical and psychological theories, historical doctrines and all sorts of speculations and discoveries, have changed, and keep changing, the lives of countless millions. Even he who merely transmits knowledge or learning participates, in his modest way, in the process of shaping reality - of which fact the enemies of humanism are perhaps more keenly aware than its friends. It is impossible to conceive of our world in terms of action alone. Only in God is there a "Coincidence of Act and Thought" as the scholastics put it. Our reality can only be understood as an interpenetration of these two.

A horoscope is a specific map, or picture, of the heavens that is cast for the date, time, and location of your birth. The positions of the sun, moon, and planets, as well as the sign that hovers at the horizon, are all placed around the wheel of the zodiac to reveal the intricate mathematical relationships that portray your personal blueprint and potential for development. This map can reveal your physical, mental, emotional, and spiritual gifts and challenges, and you are always free to grow and change, according to your own volition. Also noteworthy are the nodal points, or the locations where the path of Earth and the path of the moon intersect, forming what is known as the “head and tail of the sky dragon,” or the north and south nodes. The location of the celestial dragon in a chart is of utmost importance, for it indicates the direction in which you are moving to achieve the fulfillment of your personal destiny, as well as the place in the past that you are emerging from. Once you are born into physical reality, you unfold your life within an imprint of cosmic energy that embodies a plan of intent and purpose, a plan designed and approved by you. Throughout

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

Godel showed how a statement about any mathematical formal system (such as the assertion that Principia Mathematica is contradiction-free) can be translated into a mathematical statement inside number theory (the study of whole numbers). In other words, any metamathematical statement can be imported into mathematics, and in its new guise the statement simply asserts (as do all statements of number theory) that certain whole numbers have certain properties or relationships to each other. But on another level, it also has a vastly different meaning that, on its surface, seems as far removed from a statement of number theory as would be a sentence in a Dostoevsky novel.

a result, the most efficient way for evolutionary forces to spread beneficial mutations has often been to invent mutations anew rather than to import them from other populations.44 The limited migration rates between some regions of Africa over the last few thousand years has resulted in what Ralph and Coop have described as a “tessellated” pattern of population structure in Africa. Tessellation is a mathematical term for a landscape of tiles—regions of genetic homogeneity demarcated by sharp boundaries—that is expected to form when the process of homogenization due to gene exchanges among neighbors competes with the process of generating new advantageous variations in each region.

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

Ohm found that the results could be summed up in such a simple law that he who runs may read it, and a schoolboy now can predict what a Faraday then could only guess at roughly. By Ohm's discovery a large part of the domain of electricity became annexed by Coulomb's discovery of the law of inverse squares, and completely annexed by Green's investigations. Poisson attacked the difficult problem of induced magnetisation, and his results, though differently expressed, are still the theory, as a most important first approximation. Ampere brought a multitude of phenomena into theory by his investigations of the mechanical forces between conductors supporting currents and magnets. Then there were the remarkable researches of Faraday, the prince of experimentalists, on electrostatics and electrodynamics and the induction of currents. These were rather long in being brought from the crude experimental state to a compact system, expressing the real essence. Unfortunately, in my opinion, Faraday was not a mathematician. It can scarcely be doubted that had he been one, he would have anticipated much later work. He would, for instance, knowing Ampere's theory, by his own results have readily been led to Neumann's theory, and the connected work of Helmholtz and Thomson. But it is perhaps too much to expect a man to be both the prince of experimentalists and a competent mathematician.

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

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

Suppose a vast number of civilizations are distributed throughout the universe, on the order of the number of detectable stars. Lots and lots of them. The mathematical structure of cosmic sociology is far clearer than that of human sociology. The factors of chaos and randomness in the complex makeups of every civilized society in the universe get filtered out by the immense distance, so those civilizations can act as reference points that are relatively easy to manipulate mathematically. First: Survival is the primary need of civilization. Second: Civilization continuously grows and expands, but the total matter in the universe remains constant. One more thing: To derive a basic picture of cosmic sociology from these two axioms, you need two other important concepts: chains of suspicion and the technological explosion.

I want economists to quit concerning themselves with allocation problems, per se, with the problem, as it has been traditionally defined. The vocabulary of science is important here, and as T. D. Weldon once suggested, the very word "problem" in and of itself implies the presence of "solution." Once the format has been established in allocation terms, some solution is more or less automatically suggested. Our whole study becomes one of applied maximization of a relatively simple computational sort. Once the ends to be maximized are provided by the social welfare function, everything becomes computational, as my colleague, Rutledge Vining, has properly noted. If there is really nothing more to economics than this, we had as well turn it all over to the applied mathematicians. This does, in fact, seem to be the direction in which we are moving, professionally, and developments of note, or notoriety, during the past two decades consist largely in improvements in what are essentially computing techniques, in the mathematics of social engineering. What I am saying is that we should keep these contributions in perspective; I am urging that they be recognized for what they are, contributions to applied mathematics, to managerial science if you will, but not to our chosen subject field which we, for better or for worse, call "economics.

Some books are to be tasted, others to be swallowed, and some few to be chewed and digested; that is, some books are to be read only in parts; others to be read, but not curiously; and some few to be read wholly, and with diligence and attention. Some books also may be read by deputy, and extracts made of them by others; but that would be only in the less important arguments, and the meaner sort of books, else distilled books are like common distilled waters, flashy things. Reading maketh a full man; conference a ready man; and writing an exact man. And therefore, if a man write little, he had need have a great memory; if he confer little, he had need have a present wit: and if he read little, he had need have much cunning, to seem to know, that he doth not. Histories make men wise; poets witty; the mathematics subtile; natural philosophy deep; moral grave; logic and rhetoric able to contend. ---- Alcuni libri devono essere gustati, altri masticati e digeriti, vale a dire che alcuni libri vanno letti solo in parte, altri senza curiosità, e altri per intero, con diligenza ed attenzione. Alcuni libri possono essere letti da altri e se ne possono fare degli estratti, ma ciò riguarderebbe solo argomenti di scarsa importanza o di libri secondari perché altrimenti i libri sintetizzati sono come l’acqua distillata, evanescente. La lettura completa la formazione di un uomo; il parlare lo fa abile, e la scrittura lo trasforma in un uomo preciso. E, pertanto, se un uomo scrive poco, deve avere una grande memoria, se parla poco ha bisogno di uno spirito arguto; se legge poco deve avere bisogno di molta astuzia in modo da far sembrare di sapere quello che non sa. Le storie fanno gli uomini saggi; i poeti arguti; la matematica sottile; la filosofia naturale profondi; la logica e la retorica abili nella discussione.

What about the inside of the earth? A great deal is known about the speed of earthquake waves through the earth and the density of distribution of the earth. However, physicists have been unable to get a good theory as to how dense a substance should be at the pressures that would be expected at the center of the earth. In other words, we cannot figure out the properties of matter very well in these circumstances. We do much less well with the earth than we do with the conditions of matter in the stars. The mathematics involved seems a little too difficult, so far, but perhaps it will not be too long before someone realizes that it is an important problem, and really work it out. The other aspect, of course, is that even if we did know the density, we cannot figure out the circulating currents. Nor can we really work out the properties of rocks at high pressure. We cannot tell how fast the rocks should "give"; that must all be worked out by experiment.

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

Now and then, teaching may approach poetry, and now and then it may approach profanity. May I tell you a little story about the great Einstein? I listened once to Einstein as he talked to a group of physicists in a party. "Why have all the electrons the same charge?" said he. "Well, why are all the little balls in the goat dung of the same size?" Why did Einstein say such things? Just to make some snobs to raise their eyebrows? He was not disinclined to do so, I think. Yet, probably, it went deeper. I do not think that the overheard remark of Einstein was quite casual. At any rate, I learnt something from it: Abstractions are important; use all means to make them more tangible. Nothing is too good or too bad, too poetical or too trivial to clarify your abstractions. As Montaigne put it: The truth is such a great thing that we should not disdain any means that could lead to it. Therefore, if the spirit moves you to be a little poetical, or a little profane, in your class, do not have the wrong kind of inhibition." - George Polya's Mathematical Discovery, Volume 11, pp 102, 1962.

The propositions that accompany most of the chapters . . . are not as snappy as I would prefer—but there’s a reason for their caution and caveats. On certain important points, the clamor of genuine scientific dispute has abated and we don’t have to argue about them anymore. But to meet that claim requires me to state the propositions precisely. I am prepared to defend all of them as “things we don’t have to argue about anymore”—but exactly as I worded them, not as others may paraphrase them. Here they are: 1. Sex differences in personality are consistent worldwide and tend to widen in more gender-egalitarian cultures. 2. On average, females worldwide have advantages in verbal ability and social cognition while males have advantages in visuospatial abilities and the extremes of mathematical ability. 3. On average, women worldwide are more attracted to vocations centered on people and men to vocations centered on things. 4. Many sex differences in the brain are coordinate with sex differences in personality, abilities, and social behavior. 5. Human populations are genetically distinctive in ways that correspond to self-identified race and ethnicity. 6. Evolutionary selection pressure since humans left Africa has been extensive and mostly local. 7. Continental population differences in variants associated with personality, abilities, and social behavior are common. 8. The shared environment usually plays a minor role in explaining personality, abilities, and social behavior. 9. Class structure is importantly based on differences in abilities that have a substantial genetic component. 10. Outside interventions are inherently constrained in the effects they can have on personality, abilities, and social behavior.

The revolution caused by the sharing of experience and the spread of knowledge had begun. The Chinese, a thousand years ago, gave it further impetus by devising mechanical means of reproducing such marks in great numbers. In Europe, Johann Gutenberg independently, though much later, developed the technique of printing from movable type. Today, our libraries, the descendants of those mud tablets, can be regarded as immense communal brains, memorising far more than any one human brain could hold. More than that, they can be seen as extra-corporeal DNA, adjuncts to our genetic inheritance as important and influential in determining the way we behave as the chromosomes in our tissues are in determining the physical shape of our bodies. It was this accumulated wisdom that eventually enabled us to devise ways of escaping the dictates of the environment. Our knowledge of agricultural techniques and mechanical devices, of medicine and engineering, of mathematics and space travel, all depend on stored experience. Cut off from our libraries and all they represent and marooned on a desert island, any one of us would be quickly reduced to the life of a hunter-gatherer.

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

The Sexual plight of these children [those adolescents experimenting sexually] is officially not mentioned. The revolutionary attack on hypocrisy by Ibsen, Freud, Ellis, Dreiser, did not succeed this far. Is it an eccentric opinion that an important part of the kids' restiveness in school from the onset of puberty has to do with puberty? The teachers talk about it among themselves, all right. (In his school, Bertrand Russell thought it was better if they had sex, so they could give their undivided attention to mathematics, which was the main thing.) But since the objective factor does not exist in our schools, the school itself begins to be irrelevant. The question here is not whether sexuality should be discouraged or encouraged. That is an important issue, but far more important is that it is hard to grow up when existing facts are treated as though they do not exist. For then there is no dialogue, it is impossible to be taken seriously, to be understood, to make a bridge between oneself and society. In American society we have perfected a remarkable form of censorship: to allow every one his political right to say what he believes, but to swamp his little boat with literally thousands of millions of newspapers, mass-circulation magazines, best-selling books, broadcasts, and public pronouncements that disregard what he says and give the official way of looking at things.

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?

When he applied this approach to a gas of quantum particles, Einstein discovered an amazing property: unlike a gas of classical particles, which will remain a gas unless the particles attract one another, a gas of quantum particles can condense into some kind of liquid even without a force of attraction between them. This phenomenon, now called Bose-Einstein condensation,* was a brilliant and important discovery in quantum mechanics, and Einstein deserves most of the credit for it. Bose had not quite realized that the statistical mathematics he used represented a fundamentally new approach. As with the case of Planck’s constant, Einstein recognized the physical reality, and the significance, of a contrivance that someone else had devised.

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

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)

It is a curious paradox that several of the greatest and most creative spirits in science, after achieving important discoveries by following their unfettered imaginations, were in their later years obsessed with reductionist philosophy and as a result became sterile. Hilbert was a prime example of this paradox. Einstein was another. Like Hilbert, Einstein did his great work up to the age of forty without any reductionist bias. His crowning achievement, the general relativistic theory of gravitation, grew out of a deep physical understanding of natural processes. Only at the very end of his ten-year struggle to understand gravitation did he reduce the outcome of his understanding to a finite set of field equations. But like Hilbert, as he grew older he concentrated his attention more and more on the formal properties of his equations, and he lost interest in the wider universe of ideas out of which the equations arose. His last twenty years were spent in a fruitless search for a set of equations that would unify the whole of physics, without paying attention to the rapidly proliferating experimental discoveries that any unified theory would finally have to explain. I do not need to say more about this tragic and well-known story of Einstein's lonely attempt to reduce physics to a finite set of marks on paper. His attempt failed as dismally as Hilbert's attempt to do the same thing with mathematics. I shall instead discuss another aspect of Einstein's later life, an aspect that has received less attention than his quest for the unified field equations: his extraordinary hostility to the idea of black holes.

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

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

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

What’s more, AI researchers have begun to realize that emotions may be a key to consciousness. Neuroscientists like Dr. Antonio Damasio have found that when the link between the prefrontal lobe (which governs rational thought) and the emotional centers (e.g., the limbic system) is damaged, patients cannot make value judgments. They are paralyzed when making the simplest of decisions (what things to buy, when to set an appointment, which color pen to use) because everything has the same value to them. Hence, emotions are not a luxury; they are absolutely essential, and without them a robot will have difficulty determining what is important and what is not. So emotions, instead of being peripheral to the progress of artificial intelligence, are now assuming central importance. If a robot encounters a raging fire, it might rescue the computer files first, not the people, since its programming might say that valuable documents cannot be replaced but workers always can be. It is crucial that robots be programmed to distinguish between what is important and what is not, and emotions are shortcuts the brain uses to rapidly determine this. Robots would thus have to be programmed to have a value system—that human life is more important than material objects, that children should be rescued first in an emergency, that objects with a higher price are more valuable than objects with a lower price, etc. Since robots do not come equipped with values, a huge list of value judgments must be uploaded into them. The problem with emotions, however, is that they are sometimes irrational, while robots are mathematically precise. So silicon consciousness may differ from human consciousness in key ways. For example, humans have little control over emotions, since they happen so rapidly and because they originate in the limbic system, not the prefrontal cortex of the brain. Furthermore, our emotions are often biased.

Music is an art form whose medium is sound and silence. Its common elements are pitch (which governs melody and harmony), rhythm (and its associated concepts tempo, meter, and articulation), dynamics, and the sonic qualities of timbre and texture. The word derives from Greek μουσική (mousike; "art of the Muses"). The creation, performance, significance, and even the definition of music vary according to culture and social context. Music ranges from strictly organized compositions (and their recreation in performance), through improvisational music to aleatoric forms. Music can be divided into genres and subgenres, although the dividing lines and relationships between music genres are often subtle, sometimes open to personal interpretation, and occasionally controversial. Within the arts, music may be classified as a performing art, a fine art, and auditory art. It may also be divided among art music and folk music. There is also a strong connection between music and mathematics. Music may be played and heard live, may be part of a dramatic work or film, or may be recorded. To many people in many cultures, music is an important part of their way of life. Ancient Greek and Indian philosophers defined music as tones ordered horizontally as melodies and vertically as harmonies. Common sayings such as "the harmony of the spheres" and "it is music to my ears" point to the notion that music is often ordered and pleasant to listen to. However, 20th-century composer John Cage thought that any sound can be music, saying, for example, "There is no noise, only sound. Musicologist Jean-Jacques Nattiez summarizes the relativist, post-modern viewpoint: "The border between music and noise is always culturally defined—which implies that, even within a single society, this border does not always pass through the same place; in short, there is rarely a consensus ... By all accounts there is no single and intercultural universal concept defining what music might be.

C. P. Snow was right about the need to respect both of “the two cultures,” science and the humanities. But even more important today is understanding how they intersect. Those who helped lead the technology revolution were people in the tradition of Ada, who could combine science and the humanities. From her father came a poetic streak and from her mother a mathematical one, and it instilled in her a love for what she called “poetical science.” Her father defended the Luddites who smashed mechanical looms, but Ada loved how punch cards instructed those looms to weave beautiful patterns, and she envisioned how this wondrous combination of art and technology could be manifest in computers. (...) This innovation will come from people who are able to link beauty to engineering, humanity to technology, and poetry to processors. In other words, it will come from the spiritual heirs of Ada Lovelace, creators who can flourish where the arts intersect with the sciences and who have a rebellious sense of wonder that opens them to the beauty of both.

According to chaos theory, although it is impossible to predict the individual behavior of each element in a complex dynamic system (for instance, the individual neurons or neuronal groups in the primary visual cortex), patterns can be discerned at a higher level by using mathematical models and computer analyses. There are “universal behaviors” which represent the ways such dynamic, nonlinear systems self-organize. These tend to take the form of complex reiterative patterns in space and time—indeed the very sorts of networks, whorls, spirals, and webs that one sees in the geometrical hallucinations of migraine. Such chaotic, self-organizing behaviors have now been recognized in a vast range of natural systems, from the eccentric motions of Pluto to the striking patterns that appear in the course of certain chemical reactions to the multiplication of slime molds or the vagaries of weather. With this, a hitherto insignificant or unregarded phenomenon like the geometrical patterns of migraine aura suddenly assumes a new importance. It shows us, in the form of a hallucinatory display, not only an elemental activity of the cerebral cortex but an entire self-organizing system, a universal behavior, at work.*3

Even male children of affluent white families think that history as taught in high school is “too neat and rosy.” 6 African American, Native American, and Latino students view history with a special dislike. They also learn history especially poorly. Students of color do only slightly worse than white students in mathematics. If you’ll pardon my grammar, nonwhite students do more worse in English and most worse in history.7 Something intriguing is going on here: surely history is not more difficult for minorities than trigonometry or Faulkner. Students don’t even know they are alienated, only that they “don’t like social studies” or “aren’t any good at history.” In college, most students of color give history departments a wide berth. Many history teachers perceive the low morale in their classrooms. If they have a lot of time, light domestic responsibilities, sufficient resources, and a flexible principal, some teachers respond by abandoning the overstuffed textbooks and reinventing their American history courses. All too many teachers grow disheartened and settle for less. At least dimly aware that their students are not requiting their own love of history, these teachers withdraw some of their energy from their courses. Gradually they end up going through the motions, staying ahead of their students in the textbooks, covering only material that will appear on the next test. College teachers in most disciplines are happy when their students have had significant exposure to the subject before college. Not teachers in history. History professors in college routinely put down high school history courses. A colleague of mine calls his survey of American history “Iconoclasm I and II,” because he sees his job as disabusing his charges of what they learned in high school to make room for more accurate information. In no other field does this happen. Mathematics professors, for instance, know that non-Euclidean geometry is rarely taught in high school, but they don’t assume that Euclidean geometry was mistaught. Professors of English literature don’t presume that Romeo and Juliet was misunderstood in high school. Indeed, history is the only field in which the more courses students take, the stupider they become. Perhaps I do not need to convince you that American history is important. More than any other topic, it is about us. Whether one deems our present society wondrous or awful or both, history reveals how we arrived at this point. Understanding our past is central to our ability to understand ourselves and the world around us. We need to know our history, and according to sociologist C. Wright Mills, we know we do.8

In the absence of expert [senior military] advice, we have seen each successive administration fail in the business of strategy - yielding a United States twice as rich as the Soviet Union but much less strong. Only the manner of the failure has changed. In the 1960s, under Robert S. McNamara, we witnessed the wholesale substitution of civilian mathematical analysis for military expertise. The new breed of the "systems analysts" introduced new standards of intellectual discipline and greatly improved bookkeeping methods, but also a trained incapacity to understand the most important aspects of military power, which happens to be nonmeasurable. Because morale is nonmeasurable it was ignored, in large and small ways, with disastrous effects. We have seen how the pursuit of business-type efficiency in the placement of each soldier destroys the cohesion that makes fighting units effective; we may recall how the Pueblo was left virtually disarmed when it encountered the North Koreans (strong armament was judged as not "cost effective" for ships of that kind). Because tactics, the operational art of war, and strategy itself are not reducible to precise numbers, money was allocated to forces and single weapons according to "firepower" scores, computer simulations, and mathematical studies - all of which maximize efficiency - but often at the expense of combat effectiveness. An even greater defect of the McNamara approach to military decisions was its businesslike "linear" logic, which is right for commerce or engineering but almost always fails in the realm of strategy. Because its essence is the clash of antagonistic and outmaneuvering wills, strategy usually proceeds by paradox rather than conventional "linear" logic. That much is clear even from the most shopworn of Latin tags: si vis pacem, para bellum (if you want peace, prepare for war), whose business equivalent would be orders of "if you want sales, add to your purchasing staff," or some other, equally absurd advice. Where paradox rules, straightforward linear logic is self-defeating, sometimes quite literally. Let a general choose the best path for his advance, the shortest and best-roaded, and it then becomes the worst path of all paths, because the enemy will await him there in greatest strength... Linear logic is all very well in commerce and engineering, where there is lively opposition, to be sure, but no open-ended scope for maneuver; a competitor beaten in the marketplace will not bomb our factory instead, and the river duly bridged will not deliberately carve out a new course. But such reactions are merely normal in strategy. Military men are not trained in paradoxical thinking, but they do no have to be. Unlike the business-school expert, who searches for optimal solutions in the abstract and then presents them will all the authority of charts and computer printouts, even the most ordinary military mind can recall the existence of a maneuvering antagonists now and then, and will therefore seek robust solutions rather than "best" solutions - those, in other words, which are not optimal but can remain adequate even when the enemy reacts to outmaneuver the first approach.

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

Many models are constructed to account for regularly observed phenomena. By design, their direct implications are consistent with reality. But others are built up from first principles, using the profession’s preferred building blocks. They may be mathematically elegant and match up well with the prevailing modeling conventions of the day. However, this does not make them necessarily more useful, especially when their conclusions have a tenuous relationship with reality. Macroeconomists have been particularly prone to this problem. In recent decades they have put considerable effort into developing macro models that require sophisticated mathematical tools, populated by fully rational, infinitely lived individuals solving complicated dynamic optimization problems under uncertainty. These are models that are “microfounded,” in the profession’s parlance: The macro-level implications are derived from the behavior of individuals, rather than simply postulated. This is a good thing, in principle. For example, aggregate saving behavior derives from the optimization problem in which a representative consumer maximizes his consumption while adhering to a lifetime (intertemporal) budget constraint.† Keynesian models, by contrast, take a shortcut, assuming a fixed relationship between saving and national income. However, these models shed limited light on the classical questions of macroeconomics: Why are there economic booms and recessions? What generates unemployment? What roles can fiscal and monetary policy play in stabilizing the economy? In trying to render their models tractable, economists neglected many important aspects of the real world. In particular, they assumed away imperfections and frictions in markets for labor, capital, and goods. The ups and downs of the economy were ascribed to exogenous and vague “shocks” to technology and consumer preferences. The unemployed weren’t looking for jobs they couldn’t find; they represented a worker’s optimal trade-off between leisure and labor. Perhaps unsurprisingly, these models were poor forecasters of major macroeconomic variables such as inflation and growth.8 As long as the economy hummed along at a steady clip and unemployment was low, these shortcomings were not particularly evident. But their failures become more apparent and costly in the aftermath of the financial crisis of 2008–9. These newfangled models simply could not explain the magnitude and duration of the recession that followed. They needed, at the very least, to incorporate more realism about financial-market imperfections. Traditional Keynesian models, despite their lack of microfoundations, could explain how economies can get stuck with high unemployment and seemed more relevant than ever. Yet the advocates of the new models were reluctant to give up on them—not because these models did a better job of tracking reality, but because they were what models were supposed to look like. Their modeling strategy trumped the realism of conclusions. Economists’ attachment to particular modeling conventions—rational, forward-looking individuals, well-functioning markets, and so on—often leads them to overlook obvious conflicts with the world around them.