Mathematics Best Quotes

Anyone who cannot cope with mathematics is not fully human. At best, he is a tolerable subhuman who has learned to wear his shoes, bathe, and not make messes in the house.

The best programs are written so that computing machines can perform them quickly and so that human beings can understand them clearly. A programmer is ideally an essayist who works with traditional aesthetic and literary forms as well as mathematical concepts, to communicate the way that an algorithm works and to convince a reader that the results will be correct.

Women have a passion for mathematics. They divide their age in half, double the price of their clothes, and always add at least five years to the age of their best friend.

I tell them that if they will occupy themselves with the study of mathematics they will find in it the best remedy against the lusts of the flesh.

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

Archimedes will be remembered when Aeschylus is forgotten, because languages die and mathematical ideas do not. “Immortality” may be a silly word, but probably a mathematician has the best chance of whatever it may mean.

Anyone who cannot cope with mathematics is not fully human. At best he is a tolerable subhuman who has learned to wear shoes".

Certainly the best times were when I was alone with mathematics, free of ambition and pretense, and indifferent to the world.

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

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

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

…numerical precision is the very soul of science, and its attainment affords the best, perhaps the only criterion of the truth of theories and the correctness of experiments.

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

One of the best examples of a polymath is Leonardo da Vinci. Born in Italy in 1452, Leonardo was a sculptor, painter, architect, mathematician, musician, engineer, inventor, anatomist, botanist, geologist, cartographer and writer. Although he received an informal education that included geometry, Latin and mathematics, he was essentially an autodidact, or a self-taught individual.

The qualities of character can be arranged in triads, in each of which the first and last qualities will be extremes and vices, and the middle quality a virtue or an excellence. So between cowardice and rashness is courage; between stinginess and extravagance is liberality; between sloth and greed is ambition; between humility and pride is modesty; between secrecy and loquacity, honesty; between moroseness and buffoonery, good humor; between quarrelsomeness and flattery, friendship; between Hamlet’s indecisiveness and Quixote’s impulsiveness is self-control.49 “Right,” then, in ethics or conduct, is not different from “right” in mathematics or engineering; it means correct, fit, what works best to the best result. The

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

Fieldwork is probably always more likely to be holistic than lab work or mathematical modeling because in the field you can’t get away from the whole when a research project starts.

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

... toxic derivatives were underpinned by toxic economics, which, in turn, were no more than motivated delusions in search of theoretical justification; fundamentalist tracts that acknowledged facts only when they could be accommodated to the demands of the lucrative faith. Despite their highly impressive labels and technical appearance, economic models were merely mathematized versions of the touching superstition that markets know best, both at times of tranquility and in periods of tumult.

I know your race and mine are never on the best of terms." There was a cold smile in his voice if not on his face. "But I do only what you force me to. You rationalize, Keeton. You defend. You reject unpalatable truths, and if you can't reject them outright you trivialize them. Incremental evidence is never enough for you. You hear rumors of Holocaust; you dismiss them. You see evidence of genocide; you insist it can't be so bad. Temperatures rise, glaciers melt—species die—and you blame sunspots and volcanoes. Everyone is like this, but you most of all. You and your Chinese Room. You turn incomprehension into mathematics, you reject the truth without even knowing what it is.

Perhaps I could best describe my experience of doing mathematics in terms of entering a dark mansion. You go into the first room and it's dark, completely dark. You stumble around, bumping into the furniture. Gradually, you learn where each piece of furniture is. And finally, after six months or so, you find the light switch and turn it on. Suddenly, it's all illuminated and you can see exactly where you were. Then you enter the next dark room...

The best that Gauss has given us was likewise an exclusive production. If he had not created his geometry of surfaces, which served Riemann as a basis, it is scarcely conceivable that anyone else would have discovered it. I do not hesitate to confess that to a certain extent a similar pleasure may be found by absorbing ourselves in questions of pure geometry.

Our experience teaches us that there are indeed laws of nature, regularities in the way things behave, and that these laws are best expressed using the language of mathematics. This raises the interesting possibility that mathematical consistency might be used to guide us, along with experimental observation, to the laws that describe physical reality, and this has proved to be the case time and again throughout the history of science. We will see this happen during the course of this book, and it is truly one of the wonderful mysteries of our universe that it should be so.

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

The best thing for being sad,” replied Merlyn, beginning to puff and blow, “is to learn something. That is the only thing that never fails. You may grow old and trembling in your anatomies, you may lie awake at night listening to the disorder of your veins, you may miss your only love, you may see the world about you devastated by evil lunatics, or know your honour trampled in the sewers of baser minds. There is only one thing for it then—to learn. Learn why the world wags and what wags it. That is the only thing which the mind can never exhaust, never alienate, never be tortured by, never fear or distrust, and never dream of regretting. Learning is the thing for you. Look at what a lot of things there are to learn—pure science, the only purity there is. You can learn astronomy in a lifetime, natural history in three, literature in six. And then, after you have exhausted a milliard lifetimes in biology and medicine and theocriticism and geography and history and economics—why, you can start to make a cartwheel out of the appropriate wood, or spend fifty years learning to begin to learn to beat your adversary at fencing. After that you can start again on mathematics, until it is time to learn to plough.

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

Solving the value-loading problem is a research challenge worthy of some of the next generation’s best mathematical talent.

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

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

I have heard experimental physicist complain sotto voce that some of the best theoreticians have largely stopped doing physics and started to indulge in what is sometimes described as 'mathematical masturbation'.

Moreover, we look in vain to philosophy for the answer to the great riddle. Despite its noble purpose and history, pure philosophy long ago abandoned the foundational questions about human existence. The question itself is a reputation killer. It has become a Gorgon for philosophers, upon whose visage even the best thinkers fear to gaze. They have good reason for their aversion. Most of the history of philosophy consists of failed models of the mind. The field of discourse is strewn with the wreckage of theories of consciousness. After the decline of logical positivism in the middle of the twentieth century, and the attempt of this movement to blend science and logic into a closed system, professional philosophers dispersed in an intellectual diaspora. They emigrated into the more tractable disciplines not yet colonized by science – intellectual history, semantics, logic, foundational mathematics, ethics, theology, and, most lucratively, problems of personal life adjustment. Philosophers flourish in these various endeavors, but for the time being, at least, and by a process of elimination, the solution of the riddle has been left to science. What science promises, and has already supplied in part, is the following. There is a real creation story of humanity, and one only, and it is not a myth. It is being worked out and tested, and enriched and strengthened, step by step. (9-10)

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

We should not conclude from this that everything depends on waves of irrational psychology. On the contrary, the state of long-term expectation is often steady, and, even when it is not, the other factors exert their compensating effects. We are merely reminding ourselves that human decisions affecting the future, whether personal or political or economic, cannot depend on strict mathematical expectation, since the basis for making such calculations does not exist; and that it is our innate urge to activity which makes the wheels go round, our rational selves choosing between the alternatives as best we are able, calculating where we can, but often falling back for our motive on whim or sentiment or chance.

Yes," I continued, "I discovered this model recently and her style never fails to be mathematically perfect. She seems to come by it naturally. As if she were born resonant. I notice Japanese models tend to do this. Like I said, they seem to have resonance somewhere deep in their culture. But Yuri Nakagawa, she's the best I've ever seen. The best model, with the most powerful resonance. I need her to probe deeper into this profound mathematical instinct, which I call resonance.

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

The numbers were, at best, guesstimates, and all three men knew it. The relevant figure would ultimately be the one that represented the most they could possibly ask from Congress without raising too many questions. Whatever that sum turned out to be, they knew they could count on (Interim Assistant Secretary of the Treasury) Kashkari to perform some sort of mathematical voodoo to justify it:

The best thing for being sad,’ replied Merlyn, beginning to puff and blow, ‘is to learn something. That is the only thing that never fails. You may grow old and trembling in you anatomies, you may lie awake at night listening to the disorder of your veins, you may miss your only love, you may see the world about you devastated by evil lunatics, or know your honour trampled in the sewers of baser minds. There is only one thing for it then – to learn. Learn why the world wags and what wags in it. That is the only thing which the mind can never exhaust, never alienate, never be tortured by, never fear or distrust, and never dream of regretting. Learning is the thing for you. Look at what a lot of things there are to learn – pure science, the only purity there is. You can learn astronomy in a lifetime, natural history in three, literature in six. And then, after you have exhausted a milliard lifetimes in biology and medicine and theocriticism and geography and history and economics – why, you can start to make a cartwheel out of the appropriate wood, or spend fifty years learning to begin to learn to beat your adversary at fencing. After that you can start again on mathematics, until it is time to learn to plough.

The math-powered applications powering the data economy were based on choices made by fallible human beings. Some of these choices were no doubt made with the best intentions. Nevertheless, many of these models encoded human prejudice, misunderstanding, and bias into the software systems that increasingly managed our lives. Like gods, these mathematical models were opaque, their workings invisible to all but the highest priests in their domain: mathematicians and computer scientists. Their verdicts, even when wrong or harmful, were beyond dispute or appeal. And they tended to punish the poor and the oppressed in our society, while making the rich richer.

It is VITAL that they avoid anything which causes their mind to shift into neutral for any length of time. The best thing they can do is memorize scripture from the King James Bible. Mathematical tables can also be helpful — anything that forces their mind to exercise itself. When they find their mind drifting off, they should ask the Lord to help them claim the scriptural promises that we all have the mind of Christ (1 Cor. 2:16, Phil. 2:5, Rom. 12:2, Eph. 4:23, 2 Tim. 1:7, 1 Pet. 1:13). Beyond

Science proceeds by inference, rather than by the deduction of mathematical proof. A series of observations is accumulated, forcing the deeper question: What must be true if we are to explain what is observed? What "big picture" of reality offers the best fit to what is actually observed in our experience? American scientist and philosopher Charles S. Peirce used the term "abduction" to refer to the way in which scientists generate theories that might offer the best explanation of things. The method is now more often referred to as "inference to the best explanation." It is now widely agreed to be the philosophy of investigation of the world characteristic of the natural sciences.

you were talking about physics and if that’s what you’re talking about, then to not know mathematics is a severe limitation in understanding the world.

And inquiries into nature have the best result when they begin with physics and end in mathematics.

I am not qualified to say whether or not God exists. I kind of doubt He does. Nevertheless I'm always saying that the SF( The SF is the supreme Fascist, the Number-One guy up there) has this transfinite book-transfinite being a concept in mathematics that is larger than infinite-that contains the best proofs of all mathematical theorems, proofs that are elegant and perfect.

Aristotle says, “Now what is characteristic of any nature is that which is best for it and gives most joy. Such to man is the life according to reason, since it is that which makes him man.

In almost all textbooks, even the best, this principle is presented so that it is impossible to understand.’ (K. Jacobi, Lectures on Dynamics, 1842-1843). I have not chosen to break with tradition.

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

One of the best advertising people ever was Carl Ally. He said the true creative person wants to be a know-it-all. They want to know about all kinds of things: ancient history, nineteenth-century mathematics, modern manufacturing techniques, flower arranging, and lean hog futures. Because they never know when these ideas might come together to form a new idea. It may happen six minutes later or six years down the road, but they know it will happen.

Some people gain their understanding of the world by symbols and mathematics. Others gain their understanding by pure geometry and space. There are some others that find an acceleration in the muscular effort that is brought to them in understanding, in feeling the force of objects moving through the world. What they want are words of power that stir their souls like the memory of childhood. For the sake of persons of these different types, whether they want the paleness and tenuity of mathematical symbolism, or they want the robust aspects of this muscular engagement, we should present all of these ways. It’s the combination of them that give us our best access to truth

Archimedes will be remembered when Aeschylus is forgotten, because languages die and mathematical ideas do not. ‘Immortality’ may be a silly word, but probably a mathematician has the best chance of whatever it may mean. G.H. Hardy 23

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

Among this bewildering multiplicity of ideals which shall we choose? The answer is that we shall choose none. For it is clear that each one of these contradictory ideals is the fruit of particular social circumstances. To some extent, of course, this is true of every thought and aspiration that has ever been formulated. Some thoughts and aspirations, however, are manifestly less dependent on particular social circumstances than others. And here a significant fact emerges: all the ideals of human behaviour formulated by those who have been most successful in freeing themselves from the prejudices of their time and place are singularly alike. Liberation from prevailing conventions of thought, feeling and behaviour is accomplished most effectively by the practice of disinterested virtues and through direct insight into the real nature of ultimate reality. (Such insight is a gift, inherent in the individual; but, though inherent, it cannot manifest itself completely except where certain conditions are fulfilled. The principal pre-condition of insight is, precisely, the practice of disinterested virtues.) To some extent critical intellect is also a liberating force. But the way in which intellect is used depends upon the will. Where the will is not disinterested, the intellect tends to be used (outside the non-human fields of technology, science or pure mathematics) merely as an instrument for the rationalization of passion and prejudice, the justification of self-interest. That is why so few even of die acutest philosophers have succeeded in liberating themselves completely from the narrow prison of their age and country. It is seldom indeed that they achieve as much freedom as the mystics and the founders of religion. The most nearly free men have always been those who combined virtue with insight. Now, among these freest of human beings there has been, for the last eighty or ninety generations, substantial agreement in regard to the ideal individual. The enslaved have held up for admiration now this model of a man, now that; but at all times and in all places, the free have spoken with only one voice. It is difficult to find a single word that will adequately describe the ideal man of the free philosophers, the mystics, the founders of religions. 'Non-attached* is perhaps the best. The ideal man is the non-attached man. Non-attached to his bodily sensations and lusts. Non-attached to his craving for power and possessions. Non-attached to the objects of these various desires. Non-attached to his anger and hatred; non-attached to his exclusive loves. Non-attached to wealth, fame, social position. Non-attached even to science, art, speculation, philanthropy. Yes, non-attached even to these. For, like patriotism, in Nurse Cavel's phrase, 'they are not enough, Non-attachment to self and to what are called 'the things of this world' has always been associated in the teachings of the philosophers and the founders of religions with attachment to an ultimate reality greater and more significant than the self. Greater and more significant than even the best things that this world has to offer. Of the nature of this ultimate reality I shall speak in the last chapters of this book. All that I need do in this place is to point out that the ethic of non-attachment has always been correlated with cosmologies that affirm the existence of a spiritual reality underlying the phenomenal world and imparting to it whatever value or significance it possesses.

If, redesigning our education system from scratch, it was suggested that we should attempt to teach Swahili to children but carry out those lessons in another foreign tongue, such as Swedish, this would rightly be derided as lunacy. Yet this is not so very far from what we are attempting to do. Take Coyne, for example. He is 14 now. His grasp of English is, at best, tenuous. Despite this, we are trying to teach him to speak French. Equally, his mathematical ability is next to nil; we are trying, in economics lessons, to explain concepts like inflation and money supply to a boy who can’t add..

The best thing for being sad,” replied Merlin, beginning to puff and blow, “is to learn something. That is the only thing that never fails. You may grow old and trembling in your anatomies, you may lie awake at night listening to the disorder of your veins, you may miss your only love, you may see the world about you devastated by evil lunatics, or know your honor trampled in the sewers of baser minds. There is only one thing for it then—to learn. Learn why the world wags and what wags it. That is the only thing which the mind can never exhaust, never alienate, never be tortured by, never fear or distrust, and never dream of regretting. Learning is the thing for you. Look at what a lot of things there are to learn—pure science, the only purity there is. You can learn astronomy in a lifetime, natural history in three, literature in six. And then, after you have exhausted a million lifetimes in biology and medicine and theocriticism and geography and history and economics, why, you can start to make a cartwheel out of the appropriate wood, or spend fifty years learning to begin to learn to beat your adversary at fencing. After that you can start again on mathematics until it is time to learn to plough.”*

I like the term “stretch” for describing what deliberate practice feels like, as it matches my own experience with the activity. When I’m learning a new mathematical technique—a classic case of deliberate practice—the uncomfortable sensation in my head is best approximated as a physical strain, as if my neurons are physically re-forming into new configurations. As any mathematician will admit, this stretching feels much different than applying a technique you’ve already mastered, which can be quite enjoyable. But this stretching, as any mathematician will also admit, is the precondition to getting better.

But Mandelbrot continued to feel oppressed by France’s purist mathematical establishment. “I saw no compatibility between a university position in France and my still-burning wild ambition,” he writes. So, spurred by the return to power in 1958 of Charles de Gaulle (for whom Mandelbrot seems to have had a special loathing), he accepted the offer of a summer job at IBM in Yorktown Heights, north of New York City. There he found his scientific home. As a large and somewhat bureaucratic corporation, IBM would hardly seem a suitable playground for a self-styled maverick. The late 1950s, though, were the beginning of a golden age of pure research at IBM. “We can easily afford a few great scientists doing their own thing,” the director of research told Mandelbrot on his arrival. Best of all, he could use IBM’s computers to make geometric pictures. Programming back then was a laborious business that involved transporting punch cards from one facility to another in the backs of station wagons.

To the average mathematician who merely wants to know his work is securely based, the most appealing choice is to avoid difficulties by means of Hilbert's program. Here one regards mathematics as a formal game and one is only concerned with the question of consistency ... . The Realist position is probably the one which most mathematicians would prefer to take. It is not until he becomes aware of some of the difficulties in set theory that he would even begin to question it. If these difficulties particularly upset him, he will rush to the shelter of Formalism, while his normal position will be somewhere between the two, trying to enjoy the best of two worlds.

To the average mathematician who merely wants to know his work is securely based, the most appealing choice is to avoid difficulties by means of Hilbert's program. Here one regards mathematics as a formal game and one is only concerned with the question of consistency ... . The Realist position is probably the one which most mathematicians would prefer to take. It is not until he becomes aware of some of the difficulties in set theory that he would even begin to question it. If these difficulties particularly upset him, he will rush to the shelter of Formalism, while his normal position will be somewhere between the two, trying to enjoy the best of two worlds.

I advise you to look for a chance to break away, to find a subject you can make your own. That is where the quickest advances are likely to occur, as measured by discoveries per investigator per year. Therein you have the best chance to become a leader and, as time passes, to gain growing freedom to set your own course. If a subject is already receiving a great deal of attention, if it has a glamorous aura, if its practitioners are prizewinners who receive large grants, stay away from that subject. Listen to the news coming from the hubbub, learn how and why the subject became prominent, but in making your own long-term plans be aware it is already crowded with talented people. You would be a newcomer, a private amid bemedaled first sergeants and generals. Take a subject instead that interests you and looks promising, and where established experts are not yet conspicuously competing with one another, where few if any prizes and academy memberships have been given, and where the annals of research are not yet layered with superfluous data and mathematical models.

There is another issue with the largely cognitive approach to management, which we had big-time at Google. Smart, analytical people, especially ones steeped in computer science and mathematics as we were, will tend to assume that data and other empirical evidence can solve all problems. Quants or techies with this worldview tend to see the inherently messy, emotional tension that’s always present in teams of humans as inconvenient and irrational—an irritant that will surely be resolved in the course of a data-driven decision process. Of course, humans don’t always work that way. Things come up, tensions arise, and they don’t naturally go away. People do their best to avoid talking about these situations, because they’re awkward. Which makes it worse.

variety, don't overdo, always have a proper breakfast, have a good daily amount of antioxidants, don't omit or exaggerate with refined sugar and, most of all, don't forget to insert into your diet a good amount of foods containing phosphorous and B-complex vitamins, great substances to let your brain work at its best. Some examples of foods containing these nutritional substances? Cereals, fish, nuts.

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

Lagrange was born in Turin (now Italy), but his family was partly French ancestry on his father's side, who was originally wealthy, managed to squander all the family's fortune in speculations, leaving his son with no inheritance. Later in life, Lagrange described this economic catastrophe as the best thing that had ever happened to him: "Had I inherited a fortune I would probably not have cast my lot with mathematics.

In a seminal 1981 paper, the economist Sherwin Rosen worked out the mathematics behind these “winner-take-all” markets. One of his key insights was to explicitly model talent—labeled, innocuously, with the variable q in his formulas—as a factor with “imperfect substitution,” which Rosen explains as follows: “Hearing a succession of mediocre singers does not add up to a single outstanding performance.” In other words, talent is not a commodity you can buy in bulk and combine to reach the needed levels: There’s a premium to being the best. Therefore, if you’re in a marketplace where the consumer has access to all performers, and everyone’s q value is clear, the consumer will choose the very best. Even if the talent advantage of the best is small compared to the next rung down on the skill ladder, the superstars still win the bulk of the market.

There is an old debate," Erdos liked to say, "about whether you create mathematics or just discover it. In other words, are the truths already there, even if we don't yet know them?" Erdos had a clear answer to this question: Mathematical truths are there among the list of absolute truths, and we just rediscover them. Random graph theory, so elegant and simple, seemed to him to belong to the eternal truths. Yet today we know that random networks played little role in assembling our universe. Instead, nature resorted to a few fundamental laws, which will be revealed in the coming chapters. Erdos himself created mathematical truths and an alternative view of our world by developing random graph theory. Not privy to nature's laws in creating the brain and society, Erdos hazarded his best guess in assuming that God enjoys playing dice. His friend Albert Einstein, at Princeton, was convinced of the opposite: "God does not play dice with the universe.

We have to throw open the Doors of Perception. As William Blake said, “If the doors of perception were cleansed everything would appear to man as it is, infinite.” The infinite is the province of mathematics, not science. Scientists are locked into the finite and the sensory. Ghosts are denizens of the non-sensory, dimensionless, infinite domain, where, currently, our perception fails us. But, one day, we shall all see ghosts. They shall be our best friends … our eternal soul mates!

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

Hobbes's natural philosophy is of the type classically represented by Democritean-Epicurean physics. Yet he regarded, not Epicurus or Democritus, but Plato, as "the best of the ancient philosophers." What he learned from Plato's natural philosophy was not that the universe cannot be understood if it is not ruled by divine intelligence. Whatever may have been Hobbes's private thoughts, his natural philosophy is as atheistic as Epicurean physics. What he learned from Plato's natural philosophy was that mathematics is "the mother of all natural science." By being both mathematical and materialistic-mechanistic, Hobbes's natural philosophy is a combination of Platonic physics and Epicurean physics. From his point of view, premodern philosophy or science as a whole was "rather a dream than science" precisely because it did not think of that combination. His philosophy as a whole may be said to be the classic example of the typically modern combination of political idealism with a materialistic and atheistic view of the whole.

Olo, Remi, Kwuga, Nur, Anajama, Rhoden. Only Olo and Remi were in my group. Everyone else I met in the dining area or the learning room where various lectures were held by professors onboard the ship. They were all girls who grew up in sprawling houses, who’d never walked through the desert, who’d never stepped on a snake in the dry grass. They were girls who could not stand the rays of Earth’s sun unless it was shining through a tinted window. Yet they were girls who knew what I meant when I spoke of “treeing.” We sat in my room (because, having so few travel items, mine was the emptiest) and challenged each other to look out at the stars and imagine the most complex equation and then split it in half and then in half again and again. When you do math fractals long enough, you kick yourself into treeing just enough to get lost in the shallows of the mathematical sea. None of us would have made it into the university if we couldn’t tree, but it’s not easy. We were the best and we pushed each other to get closer to “God.

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

Aristotle, we are invariably told, was "antiquity's most brilliant intellect," and the explanation of this weird assertion, I believe, is best summarized in Anatole France's words: The books that everybody admires are the books that nobody reads. But on taking the trouble to delve in Aristotle's writings, a somewhat different picture emerges. His ignorance of mathematics and physics, compared to the Greeks of his time, far surpasses the ignorance exhibited by this tireless and tiresome writer in the many subjects that he felt himself called upon to discuss.

What’s stopping me is I’m a Dahlite, a heatsinker on Dahl. I don’t have the money to get an education and I can’t get the credits to get an education. A real education, I mean. All they taught me was to read and cipher and use a computer and then I knew enough to be a heatsinker. But I wanted more. So I taught myself.” “In some ways, that’s the best kind of teaching. How did you do that?” “I knew a librarian. She was willing to help me. She was a very nice woman and she showed me how to use computers for learning mathematics. And she set up a software system that would connect me with other libraries.

Here is a typical story about Mr. John Jones. Mr. Jones works in an office. He had hoped for a little raise but his hope, as hopes often are, was disappointed. The salaries of some of his colleagues were raised but not his. Mr. Jones could not take it calmly. He worried and worried and finally suspected that Director Brown was responsible for his failure in getting a raise. We cannot blame Mr. Jones for having conceived such a suspicion. There were indeed some signs pointing to Director Brown. The real mistake was that, after having conceived that suspicion, Mr. Jones became blind to all signs pointing in the opposite direction. He worried himself into firmly believing that Director Brown was his personal enemy and behaved so stupidly that he almost succeeded in making a real enemy of the director. The trouble with Mr. John Jones is that he behaves like most of us. He never changes his major opinions. He changes his minor opinions not infrequently and quite suddenly; but he never doubts any of his opinions, major or minor, as long as he has them. He never doubts them, or questions them, or examines them critically—he would especially hate critical examination, if he understood what that meant. Let us concede that Mr. John Jones is right to a certain extent. He is a busy man; he has his duties at the office and at home. He has little time for doubt or examination. At best, he could examine only a few of his convictions and why should he doubt one if he has no time to examine that doubt? Still, don’t do as Mr. John Jones does. Don’t let your suspicion, or guess, or conjecture, grow without examination till it becomes ineradicable. At any rate, in theoretical matters, the best of ideas is hurt by uncritical acceptance and thrives on critical examination. 2. A mathematical example. Of all quadrilaterals with

The best thing for being sad,’ replied Merlyn, beginning to puff and blow, ‘is to learn something. That is the only thing that never fails. You may grow old and trembling in your anatomies, you may lie awake at night listening to the disorder of your veins, you may miss your only love, you may see the world about you devastated by evil lunatics, or know your honour trampled in the sewers of baser minds. There is only one thing for it then – to learn. Learn why the world wags and what wags it. That is the only thing which the mind can never exhaust, never alienate, never be tortured by, never fear or distrust, and never dream of regretting. Learning is the thing for you. Look at what a lot of things there are to learn – pure science, the only purity there is. You can learn astronomy in a lifetime, natural history in three, literature in six. And then, after you have exhausted a milliard lifetimes in biology and medicine and theo-criticism and geography and history and economics – why, you can start to make a cartwheel out of the appropriate wood, or spend fifty years learning to begin to learn to beat your adversary at fencing. After that you can start again on mathematics, until it is time to learn to plough.

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

In a representative statement from 1963, he claimed, “Man does not know most of the rules on which he acts; and even what we call his intelligence is largely a system of rules which operate on him but which he does not know.”60 This deference to the precognitive or the pre-rational is what separated him from the rational choice and rational expectations models of Chicago School economists, who professed much more faith in the possibility of both formal mathematical modeling and forecasting. As he explained in his Nobel speech, Hayek saw such efforts as not only presumptuous but misleading. The best one could hope for was pattern prediction.

The result would be random little lurches that would result in what is known as a random walk. The best way for us to envision this is to imagine a drunk who starts at a lamppost and lurches one step in a random direction every second. After two such lurches he may have gone back and forth to return to the lamp. Or he may be two steps away in the same direction. Or he may be one step west and one step northeast. A little mathematical plotting and charting reveals an interesting thing about such a random walk: statistically, the drunk’s distance from the lamp will be proportional to the square root of the number of seconds that have elapsed.35 Einstein

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.

Wolfram, one of the most innovative thinkers in scientific computing and in the theory of complex systems, has been best known for the development of Mathematica, a computer program/system that allows a range of calculations not accessible before. After ten years of virtual silence, Wolfram is about to emerge with a provocative book that makes the bold claim that he can replace the basic infrastructure of science. In a world used to more than three hundred years of science being dominated by mathematical equations as the basic building blocks of models for nature, Wolfram proposes simple computer programs instead. He suggests that nature's main secret is the use of simple programs to generate complexity.

Why? This is our world of habituation, where nothing is ever as good as that first time. Unfortunately, things have to work this way because of our range of rewards.86 After all, reward coding must accommodate the rewarding properties of both solving a math problem and having an orgasm. Dopaminergic responses to reward, rather than being absolute, are relative to the reward value of alternative outcomes. In order to accommodate the pleasures of both mathematics and orgasms, the system must constantly rescale to accommodate the range of intensity offered by particular stimuli. The response to any reward must habituate with repetition, so that the system can respond over its full range to the next new thing.

A black man, Benjamin Banneker, who taught himself mathematics and astronomy, predicted accurately a solar eclipse, and was appointed to plan the new city of Washington, wrote to Thomas Jefferson: I suppose it is a truth too well attested to you, to need a proof here, that we are a race of beings, who have long labored under the abuse and censure of the world; that we have long been looked upon with an eye of contempt; and that we have long been considered rather as brutish than human, and scarcely capable of mental endowments I apprehend you will embrace every opportunity to eradicate that train of absurd and false ideas and opinions, which so generally prevails with respect to us; and that your sentiments are concurrent with mine, which are, that one universal Father hath given being to us all; and that he hath not only made us all of one flesh, but that he hath also, without partiality, afforded us all the same sensations and endowed us all with the same facilities. . . . Banneker asked Jefferson “to wean yourselves from those narrow prejudices which you have imbibed.” Jefferson tried his best, as an enlightened, thoughtful individual might. But the structure of American society, the power of the cotton plantation, the slave trade, the politics of unity between northern and southern elites, and the long culture of race prejudice in the colonies, as well as his own weaknesses—that combination of practical need and ideological fixation—kept Jefferson a slaveowner throughout his life.

What can I say? Mathematics is a way not to be wrong, but it isn't a way not to be wrong about everything. (Sorry, no refunds!) Wrongness is like original sin; we are born to it and it remains always with us, and constant vigilance is necessary if we mean to restrict its sphere of influence over our actions. There is real danger that, by strengthening our abilities to analyze some questions mathematically, we acquire a general confidence in our beliefs, which extends unjustifiably to those things we're still wrong about. We become like those pious people who, over time, accumulate a sense of their own virtuousness so powerful as to make them believe the bad things they do are virtuous too. I'll do my best to resist the temptation. But watch me carefully.

Nevertheless, Leibniz remains a great man, and his greatness is more apparent now than it was at any earlier time. Apart from his eminence as a mathematician and as the inventor of the infinitesimal calculus, he was a pioneer in mathematical logic, of which he perceived the importance when no one else did so. And his philosophical hypotheses, though fantastic, are very clear, and capable of precise expression. Even his monads can still be useful as suggesting possible ways of viewing perception, though they cannot be regarded as windowless. What I, for my part, think best in his theory of monads is his two kinds of space, one subjective, in the perceptions of each monad, and one objective, consisting of the assemblage of points of view of the various monads. This, I believe, is still useful in relating perception to physics.

Of Bergson's theory that intellect is a purely practical faculty developed in the struggle for survival, and not a source of true beliefs, we may say, first, that it is only through intellect that we know of the struggle for survival and of the biological ancestry of man: if the intellect is misleading, the whole of this merely inferred history is presumably untrue. If, on the other hand, we agree with M. Bergson in thinking that evolution took place as Darwin believed, then it is not only intellect, but all our faculties, that have been developed under the stress of practical utility. Intuition is seen at its best where it is directly useful—for example, in regard to other people's characters and dispositions. Bergson apparently holds that capacity for this kind of knowledge is less explicable by the struggle for existence than, for example, capacity for pure mathematics. Yet

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

I struggle with words. Never could express myself the way I wanted. My mind fights my mouth, and thoughts get stuck in my throat. Sometimes they stay stuck for seconds or even minutes. Some thoughts stay for years; some have stayed hidden all my life. As a child, I stuttered. What was inside couldn't get out. I'm still not real fluent. I don't know a lot of good words. If I were wrongfully accused of a crime, I'd have a tough time explaining my innocence. I'd stammer and stumble and choke up until the judge would throw me in jail. Words aren't my friends. Music is. Sounds, notes, rhythms. I talk through music. Maybe that's why I became a loner, someone who loves privacy and doesn't reveal himself too easily. My friendliness might fool you. Come into my dressing room and I'll shake your hand, pose for a picture, make polite small talk. I'll be as nice as I can, hoping you'll be nice to me. I'm genuinely happy to meet you and exchange a little warmth. I have pleasant acquaintances with thousands of people the world over. But few, if any, really know me. And that includes my own family. It's not that they don't want to; it's because I keep my feelings to myself. If you hurt me, chances are I won't tell you. I'll just move on. Moving on is my method of healing my hurt and, man, I've been moving on all my life. Now it's time to stop. This book is a place for me to pause and look back at who I was and what I became. As I write, I'm seventy hears old, and all the joy and hurts, small and large, that I've stored up inside me...well, I want to pull 'em out and put 'em on the page. When I've been described on other people's pages, I don't recognize myself. In my mind, no one has painted the real me. Writers have done their best, but writers have missed the nitty-gritty. Maybe because I've hidden myself, maybe because I'm not an easy guy to understand. Either way, I want to open up and leave a true account of who I am. When it comes to my own life, others may know the cold facts better than me. Scholars have told me to my face that I'm mixed up. I smile but don't argue. Truth is, cold facts don't tell the whole story. Reading this, some may accuse me of remembering wrong. That's okay, because I'm not writing a cold-blooded history. I'm writing a memory of my heart. That's the truth I'm after - following my feelings, no matter where they lead. I want to try to understand myself, hoping that you - my family, my friends, my fans - will understand me as well. This is a blues story. The blues are a simple music, and I'm a simple man. But the blues aren't a science; the blues can't be broken down like mathematics. The blues are a mystery, and mysteries are never as simple as they look.

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

Unfortunately, I know that some of you Hushlanders have trouble counting to three. (The Librarian- controlled schools don't want you to be able to manage complex mathematics.) So I've prepared this helpful guide. Definition of "book one": The best place to start a series. You can identify "book one" by the fact that it has a little "1" on the spine. Smedrys do a happy dance when you read book one first. Entropy shakes its angry fist at you for being clever enough to organize the world. Definition of "book two": The book you read after book one. If you start with book two, I will make fun of you. (Okay, so I'll make fun of you either way. But honestly, do you want to give me more ammunition?) Definition of "book three": The worst place, currently, to start a series. If you start here, I will throw things at you. Definition of "book four": And . . . how'd you manage to start with that one? I haven't even written it yet. (You sneaky time travelers.)

Perhaps the most striking illustration of Bayes’s theorem comes from a riddle that a mathematics teacher that I knew would pose to his students on the first day of their class. Suppose, he would ask, you go to a roadside fair and meet a man tossing coins. The first toss lands “heads.” So does the second. And the third, fourth . . . and so forth, for twelve straight tosses. What are the chances that the next toss will land “heads” ? Most of the students in the class, trained in standard statistics and probability, would nod knowingly and say: 50 percent. But even a child knows the real answer: it’s the coin that is rigged. Pure statistical reasoning cannot tell you the answer to the question—but common sense does. The fact that the coin has landed “heads” twelve times tells you more about its future chances of landing “heads” than any abstract formula. If you fail to use prior information, you will inevitably make foolish judgments about the future. This is the way we intuit the world, Bayes argued. There is no absolute knowledge; there is only conditional knowledge. History repeats itself—and so do statistical patterns. The past is the best guide to the future.

Explanation is always incomplete: we can always raise another Why-questions. And the new why-questions may lead to a new theory which not only "explains" the old theory but corrects it. This is why the evolution of Physics is likely to be an endless process of correction and better approximation. And even if one day we should reach a stage where our theories were no longer open to correction, because they are simply true, they would still not be complete - and we should know it. For Godel's famous incompleteness theorem would come into play: in view of the Mathematical background of Physics, at best an infinite sequence of such true theories would be needed in order to answer the problems which any given (formalized) theory would be undecidable. Such considerations do not prove that the objective physical world is incomplete, or undetermined: they only show the essential incompleteness of our efforts. But they also show that it's barely possible (if possible at all) for science to reach a stage in which it can provide genuine support for the view that the physical world is deterministic. Why, the, should we not accept the verdict of common sense- at least until these arguments have been refuted?

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

Perhaps Einstein himself said it best when he said, “I have no special talents.… I am only passionately curious.” In fact, Einstein would confess that he had to struggle with mathematics in his youth. To one group of schoolchildren, he once confided, “No matter what difficulties you may have with mathematics, mine were greater.” So why was Einstein Einstein? First, Einstein spent most of his time thinking via “thought experiments.” He was a theoretical physicist, not an experimental one, so he was continually running sophisticated simulations of the future in his head. In other words, his laboratory was his mind. Second, he was known to spend up to ten years or more on a single thought experiment. From the age of sixteen to twenty-six, he focused on the problem of light and whether it was possible to outrace a light beam. This led to the birth of special relativity, which eventually revealed the secret of the stars and gave us the atomic bomb. From the age of twenty-six to thirty-six, he focused on a theory of gravity, which eventually gave us black holes and the big-bang theory of the universe. And then from the age of thirty-six to the end of his life, he tried to find a theory of everything to unify all of physics. Clearly, the ability to spend ten or more years on a single problem showed the tenacity with which he would simulate experiments in his head.

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

Goodness and Reality being timeless, the best State will be the one which most nearly copies the heavenly model, by having a minimum of change and a maximum of static perfection, and its rulers should be those who best understand the eternal Good. In the second place: Plato, like all mystics, has, in his beliefs, a core of certainty which is essentially incommunicable except by a way of life. The Pythagoreans had endeavoured to set up a rule of the initiate, and this is, at bottom, what Plato desires. If a man is to be a good statesman, he must know the Good; this he can only do by a combination of intellectual and moral discipline. If those who have not gone through this discipline are allowed a share in the government, they will inevitably corrupt it. In the third place: much education is needed to make a good ruler on Plato's principles. It seems to us unwise to have insisted on teaching geometry to the younger Dionysius, tyrant of Syracuse, in order to make him a good king, but from Plato's point of view it was essential. He was sufficiently Pythagorean to think that without mathematics no true wisdom is possible. This view implies an oligarchy. In the fourth place: Plato, in common with most Greek philosophers, took the view that leisure is essential to wisdom, which will therefore not be found among those who have to work for their living, but only among those who have independent means or who are relieved by the State from anxieties as to their subsistence. This point of view is essentially aristocratic.

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.

Nine months later, on September 1, 1939, Oppenheimer and a different collaborator—yet another student, Hartland Snyder—published a paper titled “On Continued Gravitational Contraction.” Historically, of course, the date is best known for Hitler’s invasion of Poland and the start of World War II. But in its quiet way, this publication was also a momentous event. The physicist and science historian Jeremy Bernstein calls it “one of the great papers in twentieth-century physics.” At the time, it attracted little attention. Only decades later would physicists understand that in 1939 Oppenheimer and Snyder had opened the door to twenty-first-century physics. They began their paper by asking what would happen to a massive star that has begun to burn itself out, having exhausted its fuel. Their calculations suggested that instead of collapsing into a white dwarf star, a star with a core beyond a certain mass—now believed to be two to three solar masses—would continue to contract indefinitely under the force of its own gravity. Relying on Einstein’s theory of general relativity, they argued that such a star would be crushed with such “singularity” that not even light waves would be able to escape the pull of its all-encompassing gravity. Seen from afar, such a star would literally disappear, closing itself off from the rest of the universe. “Only its gravitation field persists,” Oppenheimer and Snyder wrote. That is, though they themselves did not use the term, it would become a black hole. It was an intriguing but bizarre notion—and the paper was ignored, with its calculations long regarded as a mere mathematical curiosity.

To claim that mathematics is purely a human invention and is successful in explaining nature only because of evolution and natural selection ignores some important facts in the nature of mathematics and in the history of theoretical models of the universe. First, while the mathematical rules (e.g., the axioms of geometry or of set theory) are indeed creations of the human mind, once those rules are specified, we lose our freedom. The definition of the Golden Ratio emerged originally from the axioms of Euclidean geometry; the definition of the Fibonacci sequence from the axioms of the theory of numbers. Yet the fact that the ratio of successive Fibonacci numbers converges to the Golden Ratio was imposed on us-humans had not choice in the matter. Therefore, mathematical objects, albeit imaginary, do have real properties. Second, the explanation of the unreasonable power of mathematics cannot be based entirely on evolution in the restricted sense. For example, when Newton proposed his theory of gravitation, the data that he was trying to explain were at best accurate to three significant figures. Yet his mathematical model for the force between any two masses in the universe achieved the incredible precision of better than one part in a million. Hence, that particular model was not forced on Newton by existing measurements of the motions of planets, nor did Newton force a natural phenomenon into a preexisting mathematical pattern. Furthermore, natural selection in the common interpretation of that concept does not quite apply either, because it was not the case that five competing theories were proposed, of which one eventually won. Rather, Newton's was the only game in town!

Sometimes you don’t just want to risk making mistakes; you actually want to make them—if only to give you something clear and detailed to fix. Making mistakes is the key to making progress. Of course there are times when it is really important not to make any mistakes—ask any surgeon or airline pilot. But it is less widely appreciated that there are also times when making mistakes is the only way to go. Many of the students who arrive at very competitive universities pride themselves in not making mistakes—after all, that’s how they’ve come so much farther than their classmates, or so they have been led to believe. I often find that I have to encourage them to cultivate the habit of making mistakes, the best learning opportunities of all. They get “writer’s block” and waste hours forlornly wandering back and forth on the starting line. “Blurt it out!” I urge them. Then they have something on the page to work with. We philosophers are mistake specialists. (I know, it sounds like a bad joke, but hear me out.) While other disciplines specialize in getting the right answers to their defining questions, we philosophers specialize in all the ways there are of getting things so mixed up, so deeply wrong, that nobody is even sure what the right questions are, let alone the answers. Asking the wrongs questions risks setting any inquiry off on the wrong foot. Whenever that happens, this is a job for philosophers! Philosophy—in every field of inquiry—is what you have to do until you figure out what questions you should have been asking in the first place. Some people hate it when that happens. They would rather take their questions off the rack, all nicely tailored and pressed and cleaned and ready to answer. Those who feel that way can do physics or mathematics or history or biology. There’s plenty of work for everybody. We philosophers have a taste for working on the questions that need to be straightened out before they can be answered. It’s not for everyone. But try it, you might like it. In

The goal was ambitious. Public interest was high. Experts were eager to contribute. Money was readily available. Armed with every ingredient for success, Samuel Pierpont Langley set out in the early 1900s to be the first man to pilot an airplane. Highly regarded, he was a senior officer at the Smithsonian Institution, a mathematics professor who had also worked at Harvard. His friends included some of the most powerful men in government and business, including Andrew Carnegie and Alexander Graham Bell. Langley was given a $50,000 grant from the War Department to fund his project, a tremendous amount of money for the time. He pulled together the best minds of the day, a veritable dream team of talent and know-how. Langley and his team used the finest materials, and the press followed him everywhere. People all over the country were riveted to the story, waiting to read that he had achieved his goal. With the team he had gathered and ample resources, his success was guaranteed. Or was it? A few hundred miles away, Wilbur and Orville Wright were working on their own flying machine. Their passion to fly was so intense that it inspired the enthusiasm and commitment of a dedicated group in their hometown of Dayton, Ohio. There was no funding for their venture. No government grants. No high-level connections. Not a single person on the team had an advanced degree or even a college education, not even Wilbur or Orville. But the team banded together in a humble bicycle shop and made their vision real. On December 17, 1903, a small group witnessed a man take flight for the first time in history. How did the Wright brothers succeed where a better-equipped, better-funded and better-educated team could not? It wasn’t luck. Both the Wright brothers and Langley were highly motivated. Both had a strong work ethic. Both had keen scientific minds. They were pursuing exactly the same goal, but only the Wright brothers were able to inspire those around them and truly lead their team to develop a technology that would change the world. Only the Wright brothers started with Why. 2.

The textbooks of history prepared for the public schools are marked by a rather naive parochialism and chauvinism. There is no need to dwell on such futilities. But it must be admitted that even for the most conscientious historian abstention from judgments of value may offer certain difficulties. As a man and as a citizen the historian takes sides in many feuds and controversies of his age. It is not easy to combine scientific aloofness in historical studies with partisanship in mundane interests. But that can and has been achieved by outstanding historians. The historian's world view may color his work. His representation of events may be interlarded with remarks that betray his feelings and wishes and divulge his party affiliation. However, the postulate of scientific history's abstention from value judgments is not infringed by occasional remarks expressing the preferences of the historian if the general purport of the study is not affected. If the writer, speaking of an inept commander of the forces of his own nation or party, says "unfortunately" the general was not equal to his task, he has not failed in his duty as a historian. The historian is free to lament the destruction of the masterpieces of Greek art provided his regret does not influence his report of the events that brought about this destruction. The problem of Wertfreíheit must also be clearly distinguished from that of the choice of theories resorted to for the interpretation of facts. In dealing with the data available, the historian needs ali the knowledge provided by the other disciplines, by logic, mathematics, praxeology, and the natural sciences. If what these disciplines teach is insufficient or if the historian chooses an erroneous theory out of several conflicting theories held by the specialists, his effort is misled and his performance is abortive. It may be that he chose an untenable theory because he was biased and this theory best suited his party spirit. But the acceptance of a faulty doctrine may often be merely the outcome of ignorance or of the fact that it enjoys greater popularity than more correct doctrines. The main source of dissent among historians is divergence in regard to the teachings of ali the other branches of knowledge upon which they base their presentation. To a historian of earlier days who believed in witchcraft, magic, and the devil's interference with human affairs, things hàd a different aspect than they have for an agnostic historian. The neomercantilist doctrines of the balance of payments and of the dollar shortage give an image of presentday world conditions very different from that provided by an examination of the situation from the point of view of modern subjectivist economics.

Our political system today does not engage the best minds in our country to help us get the answers and deploy the resources we need to move into the future. Bringing these people in—with their networks of influence, their knowledge, and their resources—is the key to creating the capacity for shared intelligence that we need to solve the problems we face, before it’s too late. Our goal must be to find a new way of unleashing our collective intelligence in the same way that markets have unleashed our collective productivity. “We the people” must reclaim and revitalize the ability we once had to play an integral role in saving our Constitution. The traditional progressive solution to problems that involve a lack of participation by citizens in civic and democratic processes is to redouble their emphasis on education. And education is, in fact, an extremely valuable strategy for solving many of society’s ills. In an age where information has more economic value than ever before, it is obvious that education should have a higher national priority. It is also clear that democracies are more likely to succeed when there is widespread access to high-quality education. Education alone, however, is necessary but insufficient. A well-educated citizenry is more likely to be a well-informed citizenry, but the two concepts are entirely different, one from the other. It is possible to be extremely well educated and, at the same time, ill informed or misinformed. In the 1930s and 1940s, many members of the Nazi Party in Germany were extremely well educated—but their knowledge of literature, music, mathematics, and philosophy simply empowered them to be more effective Nazis. No matter how educated they were, no matter how well they had cultivated their intellect, they were still trapped in a web of totalitarian propaganda that mobilized them for evil purposes. The Enlightenment, for all of its liberating qualities—especially its empowerment of individuals with the ability to use reason as a source of influence and power—has also had a dark side that thoughtful people worried about from its beginning. Abstract thought, when organized into clever, self-contained, logical formulations, can sometimes have its own quasi-hypnotic effect and so completely capture the human mind as to shut out the leavening influences of everyday experience. Time and again, passionate believers in tightly organized philosophies and ideologies have closed their minds to the cries of human suffering that they inflict on others who have not yet pledged their allegiance and surrendered their minds to the same ideology. The freedoms embodied in our First Amendment represented the hard-won wisdom of the eighteenth century: that individuals must be able to fully participate in challenging, questioning, and thereby breathing human values constantly into the prevailing ideologies of their time and sharing with others the wisdom of their own experience.