Mathematics Famous Quotes

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

In the eighteenth century, philosophers considered the whole of human knowledge, including science, to be their field and discussed questions such as: Did the universe have a beginning? However, in the nineteenth and twentieth centuries, science became too technical and mathematical for the philosophers, or anyone else except a few specialists. Philosophers reduced the scope of their inquiries so much that Wittgenstein, the most famous philosopher of this century, said, "The sole remaining task for philosophy is the analysis of language." What a comedown from the great tradition of philosophy from Aristotle to Kant!

He is like the fox, who effaces his tracks in the sand with his tail. {Describing the writing style of famous mathematician Carl Friedrich Gauss}

A famous Japanese Zen master, Hakuun Yasutani Roshi, said that unless you can explain Zen in words that a fisherman will comprehend, you don’t know what you’re talking about. Some fifty years ago a UCLA professor told me the same thing about applied mathematics. We like to hide from the truth behind foreign-sounding words or mathematical lingo. There’s a saying: The truth is always encountered but rarely perceived. If we don’t perceive it, we can’t help ourselves and we can’t much help anyone else.

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

The mathematics of Malthus? A quick Internet search led him to information about a prominent nineteenth-century English mathematician and demographist named Thomas Robert Malthus, who had famously predicted an eventual global collapse due to overpopulation.

It's not for nothing that advanced mathematics tend to be invented in hot countries. It's because of the morphic resonance of all the camels who have that disdainful expression and famous curled lip as a natural result of an ability to do quadratic equations.

As John Adams famously wrote during the American Revolution, “I must study politics and war, that our sons may have liberty to study mathematics and philosophy. Our sons ought to study mathematics and philosophy, geography, natural history and naval architecture, navigation, commerce and agriculture in order to give their children a right to study painting, poetry, music, architecture, statuary, tapestry and porcelain.” So maybe today they’re writing apps rather than studying poetry, but that’s an adjustment for the age.

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.

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

Why does the universe go to all the bother of existing? Is the unified theory so compelling that it brings about its own existence? Or does it need a creator, and, if so, does he have any other effect on the universe? And who created him? Up to now, most scientists have been too occupied with the development of new theories that describe what the universe is to ask the question why. On the other hand, the people whose business it is to ask why, the philosophers, have not been able to keep up with the advance of scientific theories. In the eighteenth century, philosophers considered the whole of human knowledge, including science, to be their field and discussed questions such as: Did the universe have a beginning? However, in the nineteenth and twentieth centuries, science became too technical and mathematical for the philosophers, or anyone else except a few specialists. Philosophers reduced the scope of their inquiries so much that Wittgenstein, the most famous philosopher of this century, said, 'The sole remaining task for philosophy is the analysis of language.' What a comedown from the great tradition of philosophy from Aristotle to Kant! However, if we do discover a complete theory, it should in time be understandable in broad principle by everyone, not just a few scientists. Then we shall all, philosophers, scientists, and just ordinary people, be able to take part in the discussion of the question of why it is that we and the universe exist. If we find the answer to that, it would be the ultimate triumph of human reason--for then we would know the mind of God.

What is real? Is there more to reality than meets the eye? Yes! was Plato’s answer over two millennia ago. In his famous cave analogy, he likened us to people who’d lived their entire lives shackled in a cave, facing a blank wall, watching the shadows cast by things passing behind them, and eventually coming to mistakenly believe that these shadows were the full reality. Plato argued that what we humans call our everyday reality is similarly just a limited and distorted representation of the true reality, and that we must free ourselves from our mental shackles to begin comprehending it.

It was in 1742 that Christian Goldbach put forward his famous conjecture that every even number greater than 2 can be expressed as the sum of two primes.

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

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

Famously, Einstein said that his ‘happiest thought’ occurred here: ‘I was sitting in a chair in the Patent Office at Bern when all of a sudden a thought occurred to me. If a person falls freely he will not feel his own weight. I was startled.’ By thinking of someone falling, for example in a plummeting lift, Einstein had realised that it was impossible to distinguish acceleration and the pull of gravity. And working through the mathematical implications of this made it clear that gravity was an effect that could be produced by a distortion of space and time.

As Bertrand Russell, the famous British mathematical philosopher and Nobel laureate, famously lamented in an essay condemning the rise of Nazi Germany, “the fundamental cause of the trouble is that in the modern world the stupid are cocksure while the intelligent are full of doubt.

No one is alone in this world. No act is without consequences for others. It is a tenet of chaos theory that, in dynamical systems, the outcome of any process is sensitive to its starting point-or, in the famous cliche, the flap of a butterfly's wings in the Amazon can cause a tornado in Texas. I do not assert markets are chaotic, though my fractal geometry is one of the primary mathematical tools of "chaology." But clearly, the global economy is an unfathomably complicated machine. To all the complexity of the physical world of weather, crops, ores, and factories, you add the psychological complexity of men acting on their fleeting expectations of what may or may not happen-sheer phantasms. Companies and stock prices, trade flows and currency rates, crop yields and commodity futures-all are inter-related to one degree or another, in ways we have barely begun to understand. In such a world, it is common sense that events in the distant past continue to echo in the present.

And because I had the latest advanced mathematical training, I was given the job of analyzing the retractable landing gear for Jimmy Doolittle’s Lockheed Orion 9-D, a modification of the basic Orion. That was my first contact with any of the famous early aviators who would frequent the Lockheed plant. Others included Amelia Earhart, Wiley Post, Sir Charles Kingsford-Smith, and Roscoe Turner. Doolittle, of course, was an early record-setting pilot, both military and civilian, with a master’s degree and doctorate in science from M.I.T. Then he was flying for Shell Oil Company, landing in out-of-the-way fields, cow pastures, and other unprepared strips.

One of Lindon's amusing word-unit palindromes reads: "Girl, bathing on Bikini, eyeing boy, finds boy eyeing bikini on bathing girl." Other palindromes are symmetric with respect to back-to-front reading letter by letter-"Able was I ere I saw Elba" (attributed jokingly to Napoleon), or the title of a famous NOVA program: "A Man, a Plan, a Canal, Panama.

In 1931, Kurt Godel proved in his famous second incompleteness theorem that there could be no finitary proof of the consistency of arithmetic. He had killed Hilbert's program with a single stroke. So should you be worried that all of mathematics might collapse tomorrow afternoon? For what it's worth, I'm not. I do believe in infinite sets, and I find the proofs of consistency that use infinite sets to be convincing enough to let me sleep at night.

Like gamblers, baseball fans and television networks, fishermen are enamored of statistics. The adoration of statistics is a trait so deeply embedded in their nature that even those rarefied anglers the disciples of Jesus couldn't resist backing their yarns with arithmetic: when the resurrected Christ appears on the morning shore of the Sea of Galilee and directs his forlorn and skunked disciples to the famous catch of John 21, we learn that the net contained not "a boatload" of fish, nor "about a hundred and a half," nor "over a gross," but precisely "a hundred and fifty three." This is, it seems to me, one of the most remarkable statistics ever computed. Consider the circumstances: this is after the Crucifixion and the Resurrection; Jesus is standing on the beach newly risen from the dead, and it is only the third time the disciples have seen him since the nightmare of Calvary. And yet we learn that in the net there were "great fishes" numbering precisely "a hundred and fifty three." How was this digit discovered? Mustn't it have happened thus: upon hauling the net to shore, the disciples squatted down by that immense, writhing fish pile and started tossing them into a second pile, painstakingly counting "one, two, three, four, five, six, seven... " all the way up to a hundred and fifty three, while the newly risen Lord of Creation, the Sustainer of all their beings, He who died for them and for Whom they would gladly die, stood waiting, ignored, till the heap of fish was quantified. Such is the fisherman's compulsion toward rudimentary mathematics! ....Concerning those disciples huddled over the pile of fish, another possibility occurs to me: perhaps they paid the fish no heed. Perhaps they stood in a circle adoring their Lord while He, the All-Curious Son of His All-Knowing Dad, counted them all Himself!

But why should 1299 CE be considered the founding date of the empire? – there were no famous battles, no declarations of independence or storming of a bastille. The simplest explanations are often the most convincing: that year corresponds to the years 699–700 in the Islamic calendar. By rare mathematical coincidence, the centuries turned at the same time in both the Christian and Islamic calendars. What more auspicious year to mark the founding of an empire that spanned Europe and the Middle East?

When one day Lagrange took out of his pocket a paper which he read at the Académe, and which contained a demonstration of the famous Postulatum of Euclid, relative to the theory of parallels. This demonstration rested on an obvious paralogism, which appeared as such to everybody; and probably Lagrange also recognised it such during his lecture. For, when he had finished, he put the paper back in his pocket, and spoke no more of it. A moment of universal silence followed, and one passed immediately to other concerns.

Distributions can only be based on measurements, but as in the case of measuring intelligence, the nature of measurement is often complicated and troubled by ambiguities. Consider the problem of noise, or what is known as luck in human affairs. Since the rise of the new digital economy, around the turn of the century, there has been a distinct heightening of obsessions with contests like American Idol, or other rituals in which an anointed individual will suddenly become rich and famous. When it comes to winner-take-all contests, onlookers are inevitably fascinated by the role of luck. Yes, the winner of a singing contest is good enough to be the winner, but even the slightest flickering of fate might have changed circumstances to make someone else the winner. Maybe a different shade of makeup would have turned the tables. And yet the rewards of winning and losing are vastly different. While some critics might have aesthetic or ethical objections to winner-take-all outcomes, a mathematical problem with them is that noise is amplified. Therefore, if a societal system depends too much on winner-take-all contests, then the acuity of that system will suffer. It will become less reality-based.

His most famous paradox goes like this. I decide to walk to the ice cream store. Now certainly I can’t get to the ice cream store until I’ve gone halfway there. And once I’ve gone halfway, I can’t get to the store until I’ve gone half the distance that remains. Having done so, I still have to cover half the remaining distance. And so on, and so on. I may get closer and closer to the ice cream store—but no matter how many steps of this process I undergo, I never actually reach the ice cream store. I am always some tiny but nonzero distance away from my two scoops with jimmies. Thus, Zeno concludes, to walk to the ice cream store is impossible.

Other big questions tackled by ancient cultures are at least as radical. What is real? Is there more to reality than meets the eye? Yes! was Plato's answer over two millennia ago. In his famous cave analogy, he likened us to people who'd lived their entire lives shacked ina a cave, facing a blank wall, watching the shadows cast by things passing behind them, and eventually coming to mistakenly believe that these shadows were the full reality. Plato argued that what we humans call our everyday reality is similarly just a limited and distorted representation of the true reality, and that we must free ourselves from our mental shackles to comprehending it.

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

Anatol Rapoport, a mathematical psychologist who was famous for his insights into social interactions: You should attempt to re-express your target’s position so clearly, vividly, and fairly that your target says, ‘Thanks, I wish I’d thought of putting it that way.’ You should list any points of agreement (especially if they are not matters of widespread agreement). You should mention anything that you have learned from your target. Only then are you permitted to say so much as a word of rebuttal or criticism.1 How many times have you heard or participated in a conversation that obeys these rules? Such guidelines have gone out of fashion recently, if they were ever followed.

Even if there is only one possible unified theory, it is just a set of rules and equations. What is it that breathes fire into the equations and makes a universe for them to describe? The usual approach of science of constructing a mathematical model cannot answer the questions of why there should be a universe for the model to describe. Why does the universe go to all the bother of existing? Is the unified theory so compelling that it brings about its own existence? Or does it need a creator, and, if so, does he have any other effect on the universe? And who created him? Up to now, most scientists have been too occupied with the development of new theories that describe what the universe is to ask the question why. On the other hand, the people whose business it is to ask why, the philosophers, have not been able to keep up with the advance of scientific theories. In the eighteenth century, philosophers considered the whole of human knowledge, including science, to be their field and discussed questions such as: did the universe have a beginning? However, in the nineteenth and twentieth centuries, science became too technical and mathematical for the philosophers, or anyone else except a few specialists. Philosophers reduced the scope of their inquiries so much that Wittgenstein, the most famous philosopher of this century, said, “The sole remaining task for philosophy is the analysis of language.” What a comedown from the great tradition of philosophy from Aristotle to Kant! However, if we do discover a complete theory, it should in time be understandable in broad principle by everyone, not just a few scientists. Then we shall all, philosophers, scientists, and just ordinary people, be able to take part in the discussion of the question of why it is that we and the universe exist. If we find the answer to that, it would be the ultimate triumph of human reason – for then we would know the mind of God.

The Pythagoreans were fascinated by the regular solids, symmetrical three-dimensional objects all of whose sides are the same regular polygon. The cube is the simplest example, having six squares as sides. There are an infinite number of regular polygons, but only five regular solids. (The proof of this statement, a famous example of mathematical reasoning, is given in Appendix 2.) For some reason, knowledge of a solid called the dodecahedron having twelve pentagons as sides seemed to them dangerous. It was mystically associated with the Cosmos. The other four regular solids were identified, somehow, with the four “elements” then imagined to constitute the world; earth, fire, air and water. The fifth regular solid must then, they thought, correspond to some fifth element that could only be the substance of the heavenly bodies. (This notion of a fifth essence is the origin of our word quintessence.) Ordinary people were to be kept ignorant of the dodecahedron.

When I was a kid, my mother thought spinach was the healthiest food in the world because it contained so much iron. Getting enough iron was a big deal then because we didn't have 'iron-fortified' bread. Turns out that spinach is an okay source of iron, but no better than pizza, pistachio nuts, cooked lentils, or dried peaches. The spinach-iron myth grew out of a simple mathematical miscalculation: A researcher accidentally moved a decimal point one space, so he thought spinach had 10 times more iron than it did. The press reported it, and I had to eat spinach. Moving the decimal point was an honest mistake--but it's seldom that simple. If it happened today I'd suspect a spinach lobby was behind it. Businesses often twist science to make money. Lawyers do it to win cases. Political activists distort science to fit their agenda, bureaucrats to protect their turf. Reporters keep falling for it. Scientists sometimes go along with it because they like being famous.

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

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?

Everett's approach, which he described as "objectively deterministic" with probability "reappearing at the subjective level," resonated with this strategy. And he was thrilled by the direction. As he noted in the 1956 draft of his dissertation, the framework offered to bridge the position of Einstein (who famously believed that a fundamental theory of physics should not involve probability) and the position of Bohr (who was perfectly happy with a fundamental theory that did). According to Everett, the Many Worlds approach accommodated both positions, the difference between them merely being one of perspective. Einstein's perspective is the mathematical one in which the grand probability wave of all particles relentlessly evolves by the Schrodinger equation, with chance playing absolutely no role. I like to picture Einstein soaring high above the many worlds of Many Worlds, watching as Schrodinger's equation fully dictates how the entire panorama unfolds, and happily concluding that even though quantum mechanics is correct, God doesn't play dice. Bohr's perspective is that of an inhabitant in one of the worlds, also happy, using probabilities to explain, with stupendous precision, those observations to which his limited perspective gives him access.

For almost all astronomical objects, gravitation dominates, and they have the same unexpected behavior. Gravitation reverses the usual relation between energy and temperature. In the domain of astronomy, when heat flows from hotter to cooler objects, the hot objects get hotter and the cool objects get cooler. As a result, temperature differences in the astronomical universe tend to increase rather than decrease as time goes on. There is no final state of uniform temperature, and there is no heat death. Gravitation gives us a universe hospitable to life. Information and order can continue to grow for billions of years in the future, as they have evidently grown in the past. The vision of the future as an infinite playground, with an unending sequence of mysteries to be understood by an unending sequence of players exploring an unending supply of information, is a glorious vision for scientists. Scientists find the vision attractive, since it gives them a purpose for their existence and an unending supply of jobs. The vision is less attractive to artists and writers and ordinary people. Ordinary people are more interested in friends and family than in science. Ordinary people may not welcome a future spent swimming in an unending flood of information. A darker view of the information-dominated universe was described in the famous story “The Library of Babel,” written by Jorge Luis Borges in 1941.§ Borges imagined his library, with an infinite array of books and shelves and mirrors, as a metaphor for the universe. Gleick’s book has an epilogue entitled “The Return of Meaning,” expressing the concerns of people who feel alienated from the prevailing scientific culture. The enormous success of information theory came from Shannon’s decision to separate information from meaning. His central dogma, “Meaning is irrelevant,” declared that information could be handled with greater freedom if it was treated as a mathematical abstraction independent of meaning. The consequence of this freedom is the flood of information in which we are drowning. The immense size of modern databases gives us a feeling of meaninglessness. Information in such quantities reminds us of Borges’s library extending infinitely in all directions. It is our task as humans to bring meaning back into this wasteland. As finite creatures who think and feel, we can create islands of meaning in the sea of information. Gleick ends his book with Borges’s image of the human condition: We walk the corridors, searching the shelves and rearranging them, looking for lines of meaning amid leagues of cacophony and incoherence, reading the history of the past and of the future, collecting our thoughts and collecting the thoughts of others, and every so often glimpsing mirrors, in which we may recognize creatures of the information.

The concept of absolute time—meaning a time that exists in “reality” and tick-tocks along independent of any observations of it—had been a mainstay of physics ever since Newton had made it a premise of his Principia 216 years earlier. The same was true for absolute space and distance. “Absolute, true, and mathematical time, of itself and from its own nature, flows equably without relation to anything external,” he famously wrote in Book 1 of the Principia. “Absolute space, in its own nature, without relation to anything external, remains always similar and immovable.” But even Newton seemed discomforted by the fact that these concepts could not be directly observed. “Absolute time is not an object of perception,” he admitted. He resorted to relying on the presence of God to get him out of the dilemma. “The Deity endures forever and is everywhere present, and by existing always and everywhere, He constitutes duration and space.”45 Ernst Mach, whose books had influenced Einstein and his fellow members of the Olympia Academy, lambasted Newton’s notion of absolute time as a “useless metaphysical concept” that “cannot be produced in experience.” Newton, he charged, “acted contrary to his expressed intention only to investigate actual facts.”46 Henri Poincaré also pointed out the weakness of Newton’s concept of absolute time in his book Science and Hypothesis, another favorite of the Olympia Academy. “Not only do we have no direct intuition of the equality of two times, we do not even have one of the simultaneity of two events occurring in different places,” he wrote.

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

From science, then, if it must be so, let man learn the philosophic truth that there is no material universe; its warp and woof is maya, illusion. Its mirages of reality all break down under analysis. As one by one the reassuring props of a physical cosmos crash beneath him, man dimly perceives his idolatrous reliance, his past transgression of the divine command: “Thou shalt have no other gods before Me.” In his famous equation outlining the equivalence of mass and energy, Einstein proved that the energy in any particle of matter is equal to its mass or weight multiplied by the square of the velocity of light. The release of the atomic energies is brought about through the annihilation of the material particles. The ‘death’ of matter has been the ‘birth’ of an Atomic Age. Light-velocity is a mathematical standard or constant not because there is an absolute value in 186,000 miles a second, but because no material body, whose mass increases with its velocity, can ever attain the velocity of light. Stated another way: only a material body whose mass is infinite could equal the velocity of light. This conception brings us to the law of miracles. The masters who are able to materialise and dematerialise their bodies or any other object and to move with the velocity of light, and to utilise the creative light-rays in bringing into instant visibility any physical manifestation, have fulfilled the necessary Einsteinian condition: their mass is infinite. The consciousness of a perfected yogi is effortlessly identified, not with a narrow body, but with the universal structure. Gravitation, whether the ‘force’ of Newton or the Einsteinian ‘manifestation of inertia’, is powerless to compel a master to exhibit the property of ‘weight’ which is the distinguishing gravitational condition of all material objects. He who knows himself as the omnipresent Spirit is subject no longer to the rigidities of a body in time and space. Their imprisoning ‘rings-pass-not’ have yielded to the solvent: “I am He.

Hypocrisy—in other words, the practice of lying about lying—shields us from seeing ourselves as we are: a collocation of fragments that fit together as a biological unit but not as anything else, not as that ghost which has been called a self, a phantasm whose ecotoplasmic unreality we can never see through. By staying true to the lie of the self, the ego, we can hold onto the illusion that we will be who we are all our lives and not see our selves die a thousand times before our death. While some have dedicated themselves to getting to the bottom of how these parts create the illusion of a whole, this is not how pyramids are built. To get a pyramid off the ground takes a lot of ego—the base material of those stacks of stones that tourists visit while on vacation. Of course, a pyramid is actually a polyhedron, that is, a mathematical conception which pyramids in the physical world resemble . . . at least from a distance. The nearer one gets to a pyramid, the more it reveals itself to be what it is: a roughly pyramidal conglomeration of bricks, a composition of fragments that is not what it seems to be. This is also how it works with humans. The world around us encourages the build up of our egos—those pyramids of self-esteem—as if we needed such encouragement. Although everyone is affected by this pyramid scheme, some participate in it more than others: they are observably more full of themselves and tend to their egos as they would exotic plants in a hothouse. It helps if they can wear down the self-esteem of others, or simply witness this erosion. As the American novelist and essayist Gore Vidal said famously and often: “It is not enough to succeed. Others must fail.” None of this could work without the distance we put between what we are and what we think we are. Then we may appear to exist apart from our constituent elements. Self-esteem would evaporate without a self to esteem. As with pyramids, it is only at a distance that this illusion can be pulled off. Hypocrisy is that distance.

And one of the things that has most obstructed the path of discipleship in our Christian culture today is this idea that it will be a terribly difficult thing that will certainly ruin your life. A typical and often-told story in Christian circles is of those who have refused to surrender their lives to God for fear he would “send them to Africa as missionaries.” And here is the whole point of the much misunderstood teachings of Luke 14. There Jesus famously says one must “hate” all their family members and their own life also, must take their cross, and must forsake all they own, or they “cannot be my disciple” (Luke 14:26–27, 33). The entire point of this passage is that as long as one thinks anything may really be more valuable than fellowship with Jesus in his kingdom, one cannot learn from him. People who have not gotten the basic facts about their life straight will therefore not do the things that make learning from Jesus possible and will never be able to understand the basic points in the lessons to be learned. It is like a mathematics teacher in high school who might say to a student, “Verily, verily I say unto thee, except thou canst do decimals and fractions, thou canst in no wise do algebra.” It is not that the teacher will not allow you to do algebra because you are a bad person; you just won’t be able to do basic algebra if you are not in command of decimals and fractions. So this counting of the cost is not a moaning and groaning session. “Oh how terrible it is that I have to value all of my ‘wonderful’ things (which are probably making life miserable and hopeless anyway) less than I do living in the kingdom! How terrible that I must be prepared to actually surrender them should that be called for!” The counting of the cost is to bring us to the point of clarity and decisiveness. It is to help us to see. Counting the cost is precisely what the man with the pearl and the hidden treasure did. Out of it came their decisiveness and joy. It is decisiveness and joy that are the outcomes of the counting.

On the one hand, they must develop virtuous habits of behavior; and on the other, they must develop their mental powers through the study of such disciplines as mathematics and philosophy. Both of these types of instruction are necessary. To begin with, some people may not have the intellectual capacity to acquire knowledge; they will not be able to understand what the “good life” is, just as others do not have the intellectual power to apprehend higher mathematics. But if they imitate and are guided by those people who have knowledge of the good and who accordingly act virtuously, they, too, will act virtuously even though they do not understand the essential nature of the good life. On the basis of this sort of reasoning, Plato goes on to advocate the necessity of censorship in what he calls an “ideal society”—the society that is portrayed in his most famous book, the Republic. Plato feels that it is necessary to prevent young people from being exposed to certain sorts of experiences if they are to develop virtuous habits and thus lead a good life. ========== Philosophy Made Simple (Richard H. Popkin;Avrum Stroll)

This special was followed one month later by “Bart the Genius.” This was the first genuine episode of The Simpsons , inasmuch as it premiered the famous trademark opening sequence and included the debut of Bart’s notorious catchphrase “Eat my shorts.” Most noteworthy of all, “Bart the Genius” contains a serious dose of mathematics. In many ways, this episode set the tone for what was to follow over the next two decades, namely a relentless series of numerical references and nods to geometry that would earn The Simpsons a special place in the hearts of mathematicians.

In the British statistician R. A. Fisher’s famous formulation, “the ‘one chance in a million’ will undoubtedly occur, with no less and no more than its appropriate frequency, however surprised we may be that it should occur to us.

Light control works; close control leads to overreaction, sometimes causing the machinery to break into pieces. In a famous paper “On Governors,” published in 1867, Maxwell modeled the behavior and showed mathematically that tightly controlling the speed of engines leads to instability.

The distinction between mathematics and science is pretty well settled. It remains mysterious to us why mathematics that is invented for reasons having nothing to do with nature often turns out to be useful in physical theories. In a famous article,8 the physicist Eugene Wigner has written of “the unreasonable effectiveness of mathematics.

A famous thorny issue in philosophy is the so-called infinite regress problem. For example, if we say that the properties of a diamond can be explained by the properties and arrangements of its carbon atoms, that the properties of a carbon atom can be explained by the properties and arrangements of its protons, neutrons and electrons, that the properties of a proton can be explained by the properties and arrangements of its quarks, and so on, then it seems that we're doomed to go on forever trying to explain the properties of the constituent parts. The Mathematical Universe Hypothesis offers a radical solution to this problem: at the bottom level, reality is a mathematical structure, so its parts have no intrinsic properties at all! In other words, the Mathematical Universe Hypothesis implies that we live in a relational reality, in the sense that the properties of the world around us stem not form properties of its ultimate building blocks, but from the relations between these building blocks. The external physical reality is therefore more than the sum of its parts, in the sense that it can have many interesting properties while its parts have no intrinsic properties at all.

Richard Feynman famously said that the first step in discovering a new physical law is to guess it. It

I suppose that this viewpoint-that physical systems are to be regarded as merely computational entities-stems partly from the powerful and increasing role that computational simulations play in modern twentieth-century science, and also partly from a belief that physical objects are themselves merely 'patterns of information', in some sense, that are subject to computational mathematical laws. Most of the material of our bodies and brains, after all, is being continuously replaced, and it is just its pattern that persists. Moreover, matter itself seems to have merely a transient existence since it can be converted from one form into another. Even the mass of a material body, which provides a precise physical measure of the quantity of matter that the body contains, can in appropriate circumstances be converted into pure energy (according to Einstein's famous E=mc^2)-so even material substance seems to be able to convert itself into something with a theoretical mathematical actuality. Furthermore, quantum theory seemst o tell us that material particles are merely 'waves' of information. (We shall examine these issues more thoroughly in Part II.) Thus, matter itself is nebulous and transient; and it is not at all unreasonable to suppose that the persistence of 'self' might have more to do with the preservation of patterns than of actual material particles.

In a famous article,8 the physicist Eugene Wigner has written of “the unreasonable effectiveness of mathematics.

One thing that we conclude from all this is that the 'learning robot' procedure for doing mathematics is not the procedure that actually underlies human understanding of mathematics. In any case, such bottom-up-dominated procedure would appear to be hopelessly bad for any practical proposal for the construction of a mathematics-performing robot, even one having no pretensions whatever for simulating the actual understandings possessed by a human mathematician. As stated earlier, bottom-up learning procedures by themselves are not effective for the unassailable establishing of mathematical truths. If one is to envisage some computational system for producing unassailable mathematical results, it would be far more efficient to have the system constructed according to top-down principles (at least as regards the 'unassailable' aspects of its assertions; for exploratory purposes, bottom-up procedures might well be appropriate). The soundness and effectiveness of these top-down procedures would have to be part of the initial human input, where human understanding an insight provide the necesssary additional ingredients that pure computation is unable to achieve. In fact, computers are not infrequently employed in mathematical arguments, nowadays, in this kind of way. The most famous example was the computer-assisted proof, by Kenneth Appel and Wolfgang Haken, of the four-colour theorem, as referred to above. The role of the computer, in this case, was to carry out a clearly specified computation that ran through a very large but finite number of alternative possibilities, the elimination of which had been shown (by the human mathematicians) to lead to a general proof of the needed result. There are other examples of such computer-assisted proofs and nowadays complicated algebra, in addition to numerical computation, is frequently carried out by computer. Again it is human understanding that has supplied the rules and it is a strictly top-down action that governs the computer's activity.

In this chapter we will look at the entire edifice of QFT. We will see that it is based on three simple principles. We will also list some of its achievements, including some new insights and understandings not previously mentioned. THE FOUNDATION QFT is an axiomatic theory that rests on a few basic assumptions. Everything you have learned so far, from the force of gravity to the spectrum of hydrogen, follows almost inevitably from these three basic principles. (To my knowledge, Julian Schwinger is the only person who has presented QFT in this axiomatic way, at least in the amazing courses he taught at Harvard University in the 1950's.) 1. The field principle. The first pillar is the assumption that nature is made of fields. These fields are embedded in what physicists call flat or Euclidean three-dimensional space-the kind of space that you intuitively believe in. Each field consists of a set of physical properties at every point of space, with equations that describe how these particles or field intensities influence each other and change with time. In QFT there are no particles, no round balls, no sharp edges. You should remember, however, that the idea of fields that permeate space is not intuitive. It eluded Newton, who could not accept action-at-a-distance. It wasn't until 1845 that Faraday, inspired by patterns of iron filings, first conceived of fields. The use of colors is my attempt to make the field picture more palatable. 2. The quantum principle (discetization). The quantum principle is the second pillar, following from Planck's 1900 proposal that EM fields are made up of discrete pieces. In QFT, all physical properties are treated as having discrete values. Even field strengths, whose values are continues, are regarded as the limit of increasingly finer discrete values. The principle of discretization was discovered experimentally in 1922 by Otto Stern and Walther Gerlach. Their experiment (Fig. 7-1) showed that the angular momentum (or spin) of the electron in a given direction can have only two values: +1/2 or -1/2 (Fig. 7-1). The principle of discretization leads to another important difference between quantum and classical fields: the principle of superposition. Because the angular momentum along a certain axis can only have discrete values (Fig. 7-1), this means that atoms whose angular momentum has been determined along a different axis are in a superposition of states defined by the axis of the magnet. This same superposition principle applies to quantum fields: the field intensity at a point can be a superposition of values. And just as interaction of the atom with a magnet "selects" one of the values with corresponding probabilities, so "measurement" of field intensity at a point will select one of the possible values with corresponding probability (see "Field Collapse" in Chapter 8). It is discretization and superposition that lead to Hilbert space as the mathematical language of QFT. 3. The relativity principle. There is one more fundamental assumption-that the field equations must be the same for all uniformly-moving observers. This is known as the Principle of Relativity, famously enunciated by Einstein in 1905 (see Appendix A). Relativistic invariance is built into QFT as the third pillar. QFT is the only theory that combines the relativity and quantum principles.

creation, we see the conscious, ordering mind of God.23 What God has put into the world, a preordained mathematical order, we can trace back to God through that same order. Like Leonardo da Vinci’s famous drawing of the man standing in the square and circle, divine geometric proportion turns out to be written into every feature of our lives and is only waiting to be revealed like a crucial message inscribed in invisible ink. Thanks to Pythagoras’s mystical math, Socrates’s cave suddenly comes alive in the divine order and meaning.

Four Color Conjecture is one of the most famous unsolved problems in mathematics.

With such an illustrious reputation, it would be easy to assume Einstein rarely made mistakes—but that is not the case. To begin with, his development was described as “slow,” and he was considered to be a below-average student.16 It was apparent from an early age that his way of thinking and learning was different from the rest of the students in his class. He liked working out the more complicated problems in math, for example, but wasn’t very good at the “easy” problems.17 Later on in his career, Einstein made simple mathematical mistakes that appeared in some of his most important work. His numerous mistakes include seven major gaffes on each version of his theory of relativity, mistakes in clock synchronization related to his experiments, and many mistakes in the math and physics calculations used to determine the viscosity of liquids.18 Was Einstein considered a failure because of his mistakes? Hardly. Most importantly he didn’t let his mistakes stop him. He kept experimenting and making contributions to his field. He is famously quoted as having said, “A person who never made a mistake never tried anything new.” What’s more, no one remembers him for his mistakes—we only remember him for his contributions.

Weak argument: talk loudly.' Winston Churchill's famous marginal note is a classic example of a meta-equation. More precisely, it is a reminder of the basic principle of total valency: both halves of any equation strive toward self-repetition. In this respect, an equation is the ideal image of any reflection. Mimesis is tautological, and tautology as a universal phenomenon has its equivalent not only in mathematics but also in art; if in mathematics it takes form in an equation, then its ideal genre equivalent lies the riddle (the equation is the rationalization of a riddle, and detective fiction is its dramatization). As it grows into 'higher' genres, the riddle preserves its principle: two equal sides with unknowns, in which the sides demonstrate that they are identical. A riddle is a game. The process of solving it essentially boils down to proving the obvious; one knows from the start that the meanings of the two functions given are equal. This transforms the whole process into a sort of intellectual ostensibility.

glory, at the Science Museum of London. Charles Babbage was a well-known scientist and inventor of the time. He had spent years working on his Difference Engine, a revolutionary mechanical calculator. Babbage was also known for his extravagant parties, which he called “gatherings of the mind” and hosted for the upper class, the well-known, and the very intelligent.4 Many of the most famous people from Victorian England would be there—from Charles Darwin to Florence Nightingale to Charles Dickens. It was at one of these parties in 1833 that Ada glimpsed Babbage’s half-built Difference Engine. The teenager’s mathematical mind buzzed with possibilities, and Babbage recognized her genius immediately. They became fast friends. The US Department of Defense uses a computer language named Ada in her honor. Babbage sent Ada home with thirty of his lab books filled with notes on his next invention: the Analytic Engine. It would be much faster and more accurate than the Difference Engine, and Ada was thrilled to learn of this more advanced calculating machine. She understood that it could solve even harder, more complex problems and could even make decisions by itself. It was a true “thinking machine.”5 It had memory, a processor, and hardware and software just like computers today—but it was made from cogs and levers, and powered by steam. For months, Ada worked furiously creating algorithms (math instructions) for Babbage’s not-yet-built machine. She wrote countless lines of computations that would instruct the machine in how to solve complex math problems. These algorithms were the world’s first computer program. In 1840, Babbage gave a lecture in Italy about the Analytic Engine, which was written up in French. Ada translated the lecture, adding a set of her own notes to explain how the machine worked and including her own computations for it. These notes took Ada nine months to write and were three times longer than the article itself! Ada had some awesome nicknames. She called herself “the Bride of Science” because of her desire to devote her life to science; Babbage called her “the Enchantress of Numbers” because of her seemingly magical math

Introduction: For the past decades, since its invention, solving the Rubik’s Cube (or simply owning one) has been part of many people’s childhood memories and recreational endeavours. Until now, the “cube craze” lives on and more young minds are learning about just how cool it is to work their hands on a Rubik’s Cube and be able solve it as quickly as possible. Even educators are now using Rubik’s Cube in teaching mathematics and engineering subjects because the meticulous process of solving it helps students better understand and apply basic and advanced mathematical concepts. The goal of this book is to further spread the coolness of learning how to solve the Rubik’s Cube by teaching a method that will help you solve the cube in 3 easy ways: Solve the cross (starting point) Solve the edge pieces Solve the corner pieces (end

Across the English Channel, the biggest champion of the new mechanical worldview was René Descartes. Bacon was entirely ignorant of mathematics. Descartes was steeped in it. Reducing the operations of the universe to a series of lines, circles, numbers, and equations suited his reclusive personality. His most famous saying, “I think, therefore I am” (cogito, ergo sum), could be stated less succinctly but more accurately as “Because we are the only beings who do math, we rule.

Newton’s own goal was to demonstrate the dependence of matter on God.10 He did this through his revolutionary concept of force. Nature as described in the Principia is a complex matrix of forces, from centripetal and centrifugal force, to magnetic force and inertial force (as in, “Bodies at rest tend to remain at rest”), to the most famous of all, the force of gravity. These forces, Newton showed, exert a palpable and mathematically predictable influence on the behavior of all physical bodies. Yet they are entirely invisible and beyond any purely physical or mechanical explanation.

It happened because during my first year at Berkeley I arrived late one day at one of [Jerzy] Neyman's classes. On the blackboard there were two problems that I assumed had been assigned for homework. I copied them down. A few days later I apologized to Neyman for taking so long to do the homework — the problems seemed to be a little harder than usual. I asked him if he still wanted it. He told me to throw it on his desk. I did so reluctantly because his desk was covered with such a heap of papers that I feared my homework would be lost there forever. About six weeks later, one Sunday morning about eight o'clock, [my wife] Anne and I were awakened by someone banging on our front door. It was Neyman. He rushed in with papers in hand, all excited: "I've just written an introduction to one of your papers. Read it so I can send it out right away for publication." For a minute I had no idea what he was talking about. To make a long story short, the problems on the blackboard that I had solved thinking they were homework were in fact two famous unsolved problems in statistics. That was the first inkling I had that there was anything special about them. A year later, when I began to worry about a thesis topic, Neyman just shrugged and told me to wrap the two problems in a binder and he would accept them as my thesis. The second of the two problems, however, was not published until after World War II. It happened this way. Around 1950 I received a letter from Abraham Wald enclosing the final galley proofs of a paper of his about to go to press in the Annals of Mathematical Statistics. Someone had just pointed out to him that the main result in his paper was the same as the second "homework" problem solved in my thesis. I wrote back suggesting we publish jointly. He simply inserted my name as coauthor into the galley proof. [interview in the College Mathematics Journal in 1986]

donated skeletal collection; one more skull was just a final drop in the bucket. Megan and Todd Malone, a CT technician in the Radiology Department at UT Medical Center, ran skull 05-01 through the scanner, faceup, in a box that was packed with foam peanuts to hold it steady. Megan FedExed the scans to Quantico, where Diana and Phil Williams ran them through the experimental software. It was with high hopes, shortly after the scan, that I studied the computer screen showing the features ReFace had overlaid, with mathematical precision, atop the CT scan of Maybe-Leoma’s skull. Surely this image, I thought—the fruit of several years of collaboration by computer scientists, forensic artists, and anthropologists—would clearly settle the question of 05-01’s identity: Was she Leoma or was she Not-Leoma? Instead, the image merely amplified the question. The flesh-toned image on the screen—eyes closed, the features impassive—could have been a department-store mannequin, or a sphinx. There was nothing in the image, no matter how I rotated it in three dimensions, that said, “I am Leoma.” Nor was there anything that said, “I am not Leoma.” To borrow Winston Churchill’s famous description of Russia, the masklike face on the screen was “a riddle wrapped in a mystery inside an enigma.” Between the scan, the software, and the tissue-depth data that the software merged with the

Scientists, famously, use a lot of mathematics, at least in many disciplines (including both cognitive science and economics). Much of this is simply statistics, and derives from the importance of exact measurement. But that is not the whole story. Natural languages and the distinctions they draw are evolutionary products shaped by the interaction of cultural and genetic selection. They therefore encode, in deeply embedded and mutually reinforcing levels, distinctions important to folk theories—that is, the nonscientific social-cognitive structures that organize relationships important to special human purposes. Mathematical language is quite different, and this reflects differences between the nature of mathematical and of practical reasoning. Mathematical reasoning begins from sets of rigorously fixed procedural concepts, and these fixed points then have absolute authority, relative to special purposes in application, over what can and cannot be stated in the language. If we suspect there may be some particular structural fact but cannot figure out how to express it mathematically, this does not show that there is no such fact; but it implies that we must do some more work, either logical or empirical or both, before we can say that we are quite sure just what the putative structural fact is that we are trying to claim.

The new empirico-mathematical method seemed to offer a model for analysing everything in secular terms: ethics as well as politics and society, and religion itself. Indeed, religion was first identified (and weakened) in the eighteenth century as yet another human activity, to be examined alongside philosophy and the economy. The European sense of time changed, too: belief in divine providence – ​Second Coming or Final Days – ​gave way to a conviction, also intensely religious, in human progress in the here and now. A youthful Turgot asserted in a famous speech at the Sorbonne in 1750 that: Self-interest, ambition, and vainglory continually change the world scene and inundate the earth with blood; yet in the midst of their ravages manners are softened, the human mind becomes more enlightened . . . and the whole human race, through alternate periods of rest and unrest, of weal and woe, goes on advancing,

This is Lorenz’s famous (and widely misunderstood) butterfly effect: a flap of a butterfly’s wing can cause a hurricane a month later, halfway round the world. If you think that sounds implausible, I don’t blame you. It’s true, but only in a very special sense. The main potential source of misunderstanding is the word ‘cause’. It’s hard to see how the tiny amount of energy in the flap of a wing can create the huge energy in a hurricane. The answer is, it doesn’t. The energy in the hurricane doesn’t come from the flap: it’s redistributed from elsewhere, when the flap interacts with the rest of the otherwise unchanged weather system. After the flap, we don’t get exactly the same weather as before except for an extra hurricane. Instead, the entire pattern of weather changes, worldwide. At first the change is small, but it grows – not in energy, but in difference from what it would otherwise have been. And that difference rapidly becomes large and unpredictable. If the butterfly had flapped its wings two seconds later, it might have ‘caused’ a tornado in the Philippines instead, compensated for by snowstorms over Siberia. Or a month of settled weather in the Sahara, for that matter.

Python is a mainstream programming language that is commonly used to solve cognitive and mathematical problems. Many Python modules and useful Python libraries, such as IPython, Pandas, SciPy, and others, are most commonly used for these tasks. Usage of Business Applications Python is used by many engineers to assemble and maintain their commercial programs or apps. Python is used by many designers to maintain their web-based company sites. An application that runs on the console Python can be used to create help-based software. IPython, for example, can be used to create a variety of support-based applications. Audio or Video-based Application Programming Python is an excellent programming language for a variety of video and audio projects. Python is used by many professionals to create a variety of media applications. You can do this with the help of cplay, another Python compiler. 3D based Computer-Aided Drafting Applications Python is used by many designers to create 3D-based Computer-Aided Drafting systems. Fandango is a very useful Python-based application that allows you to see all of the capabilities of CAD to expand these types of applications. Applications for Business Python is used by many Python experts to create a variety of apps that can be used in a business. Tryton and Picalo are the most famous applications in this regard.

SCALE THE HUMAN MOUNTAIN OF SUMLESS LIES UNTIL YOU LABORIOUSLY REACH THE SUMMIT THEN CAUSE IT TO CRUMBLE BY YOUR EQUALLY SUMLESS BURDEN OF VERITY THAT NO HUMAN MAY FAVOUR YOU WITH A GLANCE ANY MORE AND THOSE WHO DO ARE NO LONGER HUMAN HAVING DIVESTED THEMSELVES OF THEIR HUMANITY AS YOU DID BY VIRTUE OF THE FACT OF * WHAT MAN HAS DONE TO HIMSELF BESIDES , YOU ARE ABLE TO ASCERTAIN HOW MANY '' FRIENDS '' YOU HAVE WHICH IS THE EMPTY SET CONTAINING ONE ELEMENT ONLY : VERITY ! , TO WHICH YOU PERTAIN AS WELL IT IS WHY IT IS THE HARDEST THING TO FIND THE PATH LEADING TO YOURSELF AND IT IS BY THE EMPTY SET THAT ALL OF MATHEMATICS HAS BEEN MADE AN EGREGIOUS LIE TOO IT IS MORE FACILE TO KILL SOMEONE OR , IF YOU ARE UNABLE TO , YOURSELF DO YOU SEE THE POPLAR AND THE ROBIN THAT IS PERCHED ON IT ? ASK THEM ! THEY KNOW HOW TO LIVE YOU DON'T BECAUSE YOU ARE HUMAN AND INTELLIGENT : MAN IS ENDUED WITH HIS SPIRIT OF INVENTION WHICH HAS REDUCED LIFE TO ABSURDITY AS ALL THOSE THEORIES AND TEACHINGS SPRINGING FROM IT HAVE NEVER BENEFITED LIFE , ON THE CONTRARY , DESTROYED IT ! AN APPRECIATION OF THE MAJESTY OF VERITY ALSO ENTAILS THE INEVITABLE CATASTROPHE OF '' BEING '' AND HENCE THE INFELICITY OF YOURSELF WHICH HAS TO BE ASCRIBED TO THOSE PROFOUND TEACHINGS OF MAN AND THE IMPRECATIONS WHICH THEY HEAPED UPON LIFE AND BEHIND WHICH EVERYONE STRIVES TO CONCEAL HIMSELF AS SOMETHING SUBLIME , BROTHERLY , CUNNING , INGENIOUS CONVINCED OF THE '' SUCCESS '' OF SUCH BEING ! INGENUITY AND SUCCESS , DO THOSE TWO WORDS DIFFER ? , AS MAN IS DETREMINED BY THOSE CRITERIA AND HENCE LIFE !... WHAT ALSO COMES TO MIND HERE IS THIS - THERE IS SOMETHING VASTLY ABOMINABLE ABOUT SOCIETY : ITS MEMBERS ARE EVER SO FOND OF ALL THOSE MOVIE STARS AND ALL THOSE OTHER LUMINARIES AND WHAT IS LUMINOUS ABOUT THEM I DO NOT KNOW ! YET THEY ARE IN THE HABIT OF TREATING THOSE VERY SIGNIFICANT PEOPLE DIFFERENTLY FROM ORDINARY PEOPLE SUCH AS A HOUSEMAID OR A GROCER OR A SALESMAN AND SO FORTH , THEREBY CREATING SOMETHING UTTERLY CORRUPT : A FALSE IDEALISM ! THEY NEED THOSE LUMINARIES AS THEY LACK ANY IDEALISM THEMSELVES IN THEIR EVERYDAY REALITY WHICH HAS DEPRAVED THEM OF IT , OVERLOOKING HOWEVER , HOW TRULY ORDINARY IN TRUTH ALL THOSE STARS ARE ! AND ALLOWING THEIR LACK OF IDEALISM TO BE SUPERSEDED BY OTHER PEOPLE'S NONPRESENT IDEALISM ON ACCOUNT OF THEIR PROMINENCE MAKES EVERYTHING LOOK EVEN DARKER IN LIFE , AS THOUGH LIFE CONSISTED IN FAME ! IS THIS WHY IT IS SO DARK IN THE HUMAN WORLD ? AM I THE ONLY PERSON TO APPREHEND DARKNESS IN THEIR LIGHTNESS ? OR WHY IS SO DARK IN THIS WORLD ? SOMETHING LIKE THAT NEEDS TO BE SHRUGGED OFF AS SOMETHING INEXPLICABLY RATIONAL , WHENCE I HAVE ALWAYS THOUGHT MYSELF IRRATIONAL IN NOT GROVELLING BEFORE THOSE WHO ARE EVEN MORE ORDINARY THAN ALL THE OTHER ORDINARY NON-FAMOUS PEOPLE ARE ! IT IS IN PARTICULAR THOSE ALL-IMPORTANT DIGNITARIES WHO TASTE OF METHYLATED SPIRITS IN A MOST ACRID AND NAUSEATING FASHION ! SO MUCH FOR CEANLINESS !...

THIRD EMENDED VERSION , SOME OMISSIONS HAVING BEEN ADDED TO MY LAST '' PUBLICATION '' TO KEEP THE LOGIC MORE LUCID SORRY FOR SETTING EVERYTHING DOWN SO QUICKLY - ''SCALE THE HUMAN MOUNTAIN OF SUMLESS LIES UNTIL YOU LABORIOUSLY REACH THE SUMMIT THEN CAUSE IT TO CRUMBLE BY YOUR EQUALLY SUMLESS BURDEN OF VERITY THAT NO HUMAN MAY FAVOUR YOU WITH A GLANCE ANY MORE AND THOSE WHO DO ARE NO LONGER HUMAN HAVING DIVESTED THEMSELVES OF THEIR HUMANITY AS YOU DID BY VIRTUE OF THE FACT OF WHAT MAN HAS DONE TO HIMSELF BESIDES , YOU ARE ABLE TO ASCERTAIN HOW MANY '' FRIENDS '' YOU HAVE WHICH IS THE EMPTY SET CONTAINING ONE ELEMENT ONLY : VERITY ! , TO WHICH YOU PERTAIN AS WELL IT IS WHY IT IS THE HARDEST THING TO FIND THE PATH LEADING TO YOURSELF AND IT IS BY THE EMPTY SET THAT ALL OF MATHEMATICS HAS BEEN MADE AN EGREGIOUS LIE TOO IT IS MORE FACILE TO KILL SOMEONE OR , IF YOU ARE UNABLE TO , YOURSELF THAN IT IS TO LIVE ! DO YOU SEE THE POPLAR AND THE ROBIN THAT IS PERCHED ON IT ? ASK THEM ! THEY KNOW HOW TO LIVE YOU DON'T BECAUSE YOU ARE HUMAN AND INTELLIGENT : MAN IS ENDUED WITH HIS SPIRIT OF INVENTION WHICH HAS REDUCED LIFE TO ABSURDITY AS ALL THOSE THEORIES AND TEACHINGS SPRINGING FROM IT HAVE NEVER BENEFITED LIFE , ON THE CONTRARY , DESTROYED IT ! AN APPRECIATION OF THE MAJESTY OF VERITY ALSO ENTAILS THE INEVITABLE CATASTROPHE OF '' BEING '' AND HENCE THE INFELICITY OF YOURSELF WHICH HAS TO BE ASCRIBED TO THOSE PROFOUND TEACHINGS OF MAN AND THE IMPRECATIONS WHICH THEY HEAPED UPON LIFE AND BEHIND WHICH EVERYONE STRIVES TO CONCEAL HIMSELF AS SOMETHING SUBLIME , BROTHERLY , CUNNING , INGENIOUS CONVINCED OF THE '' SUCCESS '' OF SUCH BEING ! INGENUITY AND SUCCESS , DO THOSE TWO WORDS DIFFER ? , AS MAN IS DETREMINED BY THOSE CRITERIA AND HENCE LIFE !... WHAT ALSO COMES TO MIND HERE IS THIS - THERE IS SOMETHING VASTLY ABOMINABLE ABOUT SOCIETY : ITS MEMBERS ARE EVER SO FOND OF ALL THOSE MOVIE STARS AND ALL THOSE OTHER LUMINARIES AND WHAT IS LUMINOUS ABOUT THEM I DO NOT KNOW ! YET THEY ARE IN THE HABIT OF TREATING THOSE VERY SIGNIFICANT PEOPLE DIFFERENTLY FROM ORDINARY PEOPLE SUCH AS A HOUSEMAID OR A GROCER OR A SALESMAN AND SO FORTH , THEREBY CREATING SOMETHING UTTERLY CORRUPT : A FALSE IDEALISM ! THEY NEED THOSE LUMINARIES AS THEY LACK ANY IDEALISM THEMSELVES IN THEIR EVERYDAY REALITY WHICH HAS DEPRAVED THEM OF IT , OVERLOOKING HOWEVER , HOW TRULY ORDINARY IN TRUTH ALL THOSE STARS ARE ! AND ALLOWING THEIR LACK OF IDEALISM TO BE SUPERSEDED BY OTHER PEOPLE'S NONPRESENT IDEALISM ON ACCOUNT OF THEIR PROMINENCE MAKES EVERYTHING LOOK EVEN DARKER IN LIFE , AS THOUGH LIFE CONSISTED IN FAME ! IS THIS WHY IT IS SO DARK IN THE HUMAN WORLD ? AM I THE ONLY PERSON TO APPREHEND DARKNESS IN THEIR LIGHTNESS ? OR WHY IS SO DARK IN THIS WORLD ? SOMETHING LIKE THAT NEEDS TO BE SHRUGGED OFF AS SOMETHING INEXPLICABLY RATIONAL , WHENCE I HAVE ALWAYS THOUGHT MYSELF IRRATIONAL IN NOT GROVELLING BEFORE THOSE WHO ARE EVEN MORE ORDINARY THAN ALL THE OTHER ORDINARY NON-FAMOUS PEOPLE ARE ! IT IS IN PARTICULAR THOSE ALL-IMPORTANT DIGNITARIES WHO TASTE OF METHYLATED SPIRITS IN A MOST ACRID AND NAUSEATING FASHION ! SO MUCH FOR CLEANLINESS !... VENERABLE ANCIENT SHADES HOVERING OVER THIS LAKE THAT IS NO MORE AND OF WHICH I AM PART THE WORLD AROUND ME FADES I DISPEL ALL THOSE BLANK AND GRAINED IDEAS MAKING UP HUMAN EXISTENCE I AM NO MORE I DREAM AND HOPEFULLY I WILL NEVER TURN BACK SO AS TO SEE THAT BLANK AND GRAINED HUMAN EXISTENCE AGAIN WHICH CAUSES LIFE TO BLUR SO MUCH THAT I AM NO LONGER IN A POSITION TO SUFFER FOR THIS MUCH GUILT , WHAT IS LIFE ? AMEN !...

But I need you all, my dear students, to speak beauty more than figures, speak phrases of encouragement more than precise mathematical statistics, speak words of innovation more than historical events and you should speak with your soul rather than just for the sake. What good is it to be famous, if you can’t speak well? So, by the end of this year, we shall have many motivational speakers, and all of us will live a motivated life. Speaking is an art, and everyone cannot become an artist. But give in your heart and soul, and nothing is impossible.

Richard Feynman very famously does this in “Six Easy Pieces,” one of his early physics lectures. He basically explains mathematics in three pages. He starts from the number line—counting—and then he goes all the way up to precalculus. He just builds it up through an unbroken chain of logic. He doesn’t rely on any definitions.

Inventor Buckminster Fuller attributed his invention of the geodesic dome to his early rejection of both the standard x, y, z and polar systems in favor of a tetrahedral paradigm. Einstein similarly rejected Euclidean geometry for a non-Euclidian formulation that gives rise to his famous description of space and time curving in relativistic gravitational fields. Both Einstein and Fuller understood explicitly that Euclidean geometry is only one version of the world. Non-Euclidian geometries, spherical geometries, and many other mathematical formulations of space exist, each providing a different set of patterns for the use of inventors, builders, artists, and other innovators. The problem is that we can’t use what we don’t know. Our pattern-recognizing ability benefits from practice with these different versions of space, just as it benefits from familiarity with different forms of hopscotch.

In 1926 Werner Heisenberg developed his now famous uncertainty principle. [The original name used by Heisenberg was the “unsharpness” principle (Unsharfeprinzip). Later the name was mistranslated and popularized as the “uncertainty” principle (Unsicherheisrelation), from Elementary Quantum Chemistry, Second Edition by Frank L. Pilar, page 19.] It's a purely mathematical concept. It applies anywhere that there are waveforms. The Unsharpness Principle originates not from Quantum Mechanics, but rather from Classical Wave mechanics.

It’s not for nothing that advanced mathematics tends to be invented in hot countries. It’s because of the morphic resonance of all the camels, who have that disdainful expression and famous curled lip as a natural result of an ability to do quadratic equations.

But if you want to become famous, the worst possible thing to do is what we did: to pursue mathematics.

E. M. Forster’s famous advice to “Only connect!” is beginning to look superfluous. A theory in which the building blocks of the Universe are mathematical structures—known as graphs—that do nothing but connect has just passed its first experimental test.

What would have happened, I wondered, if Clover and Jotter never ran the river—if they had listened to the critics and doomsayers, or to their own doubts? They brought knowledge, energy, and passion to their botanical work, but also a new perspective. Before them, men had gone down the Colorado to sketch dams, plot railroads, dig gold, and daydream little Swiss chalets stuck up on the cliffs. They saw the river for what it could be, harnessed for human use. Clover and Jotter saw it as it was, a living system made up of flower, leaf, and thorn, lovely in its fierceness, worthy of study for its own sake. They knew every saltbush twig and stickery cactus was, in its own way, as much a marvel as Boulder Dam—shaped to survive against all the odds. In the United States, half of all bachelor’s degrees in science, engineering, and mathematics go to women, yet these women go on to earn only 74 percent of a man’s salary in those fields. A recent study found that it will be another two decades before women and men publish papers at equal rates in the field of botany, a field traditionally welcoming to women. It may take four decades for chemistry, and three centuries for physics. Stereotypes linger of scientists as white-coated, wild-haired men, and they limit the ways in which young people envision their futures. In a famous, oft-replicated study, 70 percent of six-year-old girls, asked to draw a picture of a scientist, draw a woman, but only 25 percent do so at the age of sixteen.

One of these trades could have been right out of the pages of Beat the Market. In 1970 the American Telephone and Telegraph Company (AT&T) sold warrants to purchase thirty-one million shares of common stock at a price of $12.50 per share. Proceeds to the company were some $387.5 million, at the time the most ever for a warrant. Though it was not sufficiently mispriced then, the history of how warrant prices behaved indicated this could happen before it expired in 1975. When it did we bet a significant part of the partnership’s net worth. — We were guided in this trade and thousands of others by a formula that had its beginnings in 1900 in the PhD thesis of French mathematician Louis Bachelier. Bachelier used mathematics to develop a theory for pricing options on the Paris stock exchange (the Bourse). His thesis adviser, the world-famous mathematician Henri Poincaré, didn’t value Bachelier’s effort, and Bachelier spent the rest of his life as an obscure provincial professor.

So what precisely is it that we don’t understand about consciousness? Few have thought harder about this question than David Chalmers, a famous Australian philosopher rarely seen without a playful smile and a black leather jacket—which my wife liked so much that she gave me a similar one for Christmas. He followed his heart into philosophy despite making the finals at the International Mathematics Olympiad—and despite the fact that his only B grade in college, shattering his otherwise straight As, was for an introductory philosophy course. Indeed, he seems utterly undeterred by put-downs or controversy, and I’ve been astonished by his ability to politely listen to uninformed and misguided criticism of his own work without even feeling the need to respond.

In this moment I’m reminded of Thor tutoring me math in the second grade. Thor was a math major in college, and in fact, Thor has an Erdős number of three. This means that Thor studied with someone who studied with someone who studied with Paul Erdős. Erdős was a brilliant mathematician who was as famous for his eccentric lifestyle as his mathematical theorems. I guess this give me an Erdős number of four.

Infinity earned its rightful place as a legitimate mathematical concept towards the end of the nineteenth century. The two big players behind this were German mathematicians Georg Cantor and his champion David Hilbert. No longer just accepting infinity as a general notion but actually investigating it rigorously did not go down well. Contemporary mathematicians described Cantor as a 'corrupter of youth'. This had a detrimental effect on Cantor, who already suffered from depression. Thankfully, Hilbert saw the power of what Cantor had done. He described Cantor's work as 'the finest product of mathematical genius and one of the supreme achievements of purely intellectual human activity', and famously said, 'no one shall expel us from the Paradise that Cantor has created'.

The Historical Setting of Genesis Mesopotamia: Sumer Through Old Babylonia Sumerians. It is not possible at this time to put Ge 1–11 into a specific place in the historical record. Our history of the ancient Near East begins in earnest after writing has been invented, and the earliest civilization known to us in the historical record is that of the Sumerians. This culture dominated southern Mesopotamia for over 500 years during the first half of the third millennium BC (2900–2350 BC), known as the Early Dynastic Period. The Sumerians have become known through the excavation of several of their principal cities, which include Eridu, Uruk and Ur. The Sumerians are credited with many of the important developments in civilization, including the foundations of mathematics, astronomy, law and medicine. Urbanization is also first witnessed among the Sumerians. By the time of Abraham, the Sumerians no longer dominate the ancient Near East politically, but their culture continues to influence the region. Other cultures replace them in the political arena but benefit from the advances they made. Dynasty of Akkad. In the middle of the twenty-fourth century BC, the Sumerian culture was overrun by the formation of an empire under the kingship of Sargon I, who established his capital at Akkad. He ruled all of southern Mesopotamia and ranged eastward into Elam and northwest to the Mediterranean on campaigns of a military and economic nature. The empire lasted for almost 150 years before being apparently overthrown by the Gutians (a barbaric people from the Zagros Mountains east of the Tigris), though other factors, including internal dissent, may have contributed to the downfall. Ur III. Of the next century little is known as more than 20 Gutian kings succeeded one another. Just before 2100 BC, the city of Ur took control of southern Mesopotamia under the kingship of Ur-Nammu, and for the next century there was a Sumerian renaissance in what has been called the Ur III period. It is difficult to ascertain the limits of territorial control of the Ur III kings, though the territory does not seem to have been as extensive as that of the dynasty of Akkad. Under Ur-Nammu’s son Shulgi, the region enjoyed almost a half century of peace. Decline and fall came late in the twenty-first century BC through the infiltration of the Amorites and the increased aggression of the Elamites to the east. The Elamites finally overthrew the city. It is against this backdrop of history that the OT patriarchs emerge. Some have pictured Abraham as leaving the sophisticated Ur that was the center of the powerful Ur III period to settle in the unknown wilderness of Canaan, but that involves both chronological and geographic speculation. By the highest chronology (i.e., the earliest dates attributed to him), Abraham probably would have traveled from Ur to Harran during the reign of Ur-Nammu, but many scholars are inclined to place Abraham in the later Isin-Larsa period or even the Old Babylonian period. From a geographic standpoint it is difficult to be sure that the Ur mentioned in the Bible is the famous city in southern Mesopotamia (see note on 11:28). All this makes it impossible to give a precise background of Abraham. The Ur III period ended in southern Mesopotamia as the last king of Ur, Ibbi-Sin, lost the support of one city after another and was finally overthrown by the Elamites, who lived just east of the Tigris. In the ensuing two centuries (c. 2000–1800 BC), power was again returned to city-states that controlled more local areas. Isin, Larsa, Eshnunna, Lagash, Mari, Assur and Babylon all served as major political centers.

So I went to Case, and the Dean of Case says to us, says, it’s a all men’s school, says, “Men, look at, look to the person on your left, and the person on your right. One of you isn’t going to be here next year; one of you is going to fail.” So I get to Case, and again I’m studying all the time, working really hard on my classes, and so for that I had to be kind of a machine. I, the calculus book that I had, in high school we — in high school, as I said, our math program wasn’t much, and I had never heard of calculus until I got to college. But the calculus book that we had was great, and in the back of the book there were supplementary problems that weren’t, you know, that weren’t assigned by the teacher. The teacher would assign, so this was a famous calculus text by a man named George Thomas, and I mention it especially because it was one of the first books published by Addison-Wesley, and I loved this calculus book so much that later I chose Addison-Wesley to be the publisher of my own book. But Thomas’s Calculus would have the text, then would have problems, and our teacher would assign, say, the even numbered problems, or something like that. I would also do the odd numbered problems. In the back of Thomas’s book he had supplementary problems, the teacher didn’t assign the supplementary problems; I worked the supplementary problems. I was, you know, I was scared I wouldn’t learn calculus, so I worked hard on it, and it turned out that of course it took me longer to solve all these problems than the kids who were only working on what was assigned, at first. But after a year, I could do all of those problems in the same time as my classmates were doing the assigned problems, and after that I could just coast in mathematics, because I’d learned how to solve problems. So it was good that I was scared, in a way that I, you know, that made me start strong, and then I could coast afterwards, rather than always climbing and being on a lower part of the learning curve.

What is at issue here is not the familiar construct of formal mathematics, but a belief in the existence of ω (the set of natural numbers) prior to all mathematical constructions. What is the origin of this belief? The famous saying by Kronecker that God created the numbers, all the rest is the work of Man, presumably was not meant to be taken seriously. Nowhere in the Book of Genesis do we find the passage: And God said, let there be numbers, and there were numbers; odd and even created he them, and he said unto them be fruitful and multiply; and he commanded them to keep the laws of induction. No, the belief in ω stems from the speculations of Greek philosophy on the existence of ideal entities or the speculations of German philosophy on a priori categories of thought.

For a long time I took a purely theological standpoint on the issue, which is actually so fundamental that it can be used as a springboard for any debate – if environment is the operative factor, for example, if man at the outset is both equal and shapeable and the good man can be shaped by engineering his surroundings, hence my parents’ generation’s belief in the state, the education system and politics, hence their desire to reject everything that had been and hence their new truth, which is not found within man’s inner being, in his detached uniqueness, but on the contrary in areas external to his intrinsic self, in the universal and collective, perhaps expressed in its clearest form by Dag Solstad, who has always been the chronicler of his age, in a text from 1969 containing his famous statement “We won’t give the coffee pot wings”: out with spirituality, out with feeling, in with the new materialism, but it never struck them that the same attitude could lie behind the demolition of old parts of town to make way for roads and parking lots, which naturally the intellectual Left opposed, and perhaps it has not been possible to be aware of this until now when the link between the idea of equality and capitalism, the welfare state and liberalism, Marxist materialism and the consumer society is obvious because the biggest equality creator of all is money, it levels all differences, and if your character and your fate are entities that can be shaped, money is the most natural shaper, and this gives rise to the fascinating phenomena whereby crowds of people assert their individuality and originality by shopping in an identical way while those who ushered all this in with their affirmation of equality, their emphasis on material values and belief in change, are now inveighing against their own handiwork, which they believed the enemy created, but like all simple reasoning this is not wholly true either, life is not a mathematical quantity, it has no theory, only practice, and though it is tempting to understand a generation’s radical rethink of society as being based on its view of the relationship between heredity and environment, this temptation is literary and consists more in the pleasure of speculating, that is, of weaving one’s thoughts through the most disparate areas of human activity, than in the pleasure of proclaiming the truth.