probability theory / Blaise Pascal / Pierre de Fermat

19 results back to index

pages: 266 words: 86,324

The Drunkard's Walk: How Randomness Rules Our Lives by Leonard Mlodinow


Albert Einstein, Alfred Russel Wallace, Antoine Gombaud: Chevalier de Méré, Atul Gawande, Brownian motion, butterfly effect, correlation coefficient, Daniel Kahneman / Amos Tversky, Donald Trump, feminist movement, forensic accounting, Gerolamo Cardano, Henri Poincaré, index fund, Isaac Newton, law of one price, pattern recognition, Paul Erdős, probability theory / Blaise Pascal / Pierre de Fermat, RAND corporation, random walk, Richard Feynman, Richard Feynman, Ronald Reagan, Stephen Hawking, Steve Jobs, The Wealth of Nations by Adam Smith, The Wisdom of Crowds, Thomas Bayes, V2 rocket, Watson beat the top human players on Jeopardy!

Though the new ideas would again be developed in the context of gambling, the first of this new breed was more a mathematician turned gambler than, like Cardano, a gambler turned mathematician. His name was Blaise Pascal. Pascal was born in June 1623 in Clermont-Ferrand, a little more than 250 miles south of Paris. Realizing his son’s brilliance, and having moved to Paris, Blaise’s father introduced him at age thirteen to a newly founded discussion group there that insiders called the Académie Mersenne after the black-robed friar who had founded it. Mersenne’s society included the famed philosopher-mathematician René Descartes and the amateur mathematics genius Pierre de Fermat. The strange mix of brilliant thinkers and large egos, with Mersenne present to stir the pot, must have had a great influence on the teenage Blaise, who developed personal ties to both Fermat and Descartes and picked up a deep grounding in the new scientific method.

Henk Tijms, Understanding Probability: Chance Rules in Everyday Life (Cambridge: Cambridge University Press, 2004), p. 16. 4. Ibid., p. 80. 5. David, Gods, Games and Gambling, p. 65. 6. Blaise Pascal, quoted in Jean Steinmann, Pascal, trans. Martin Turnell (New York: Harcourt, Brace & World, 1962), p. 72. 7. Gilberte Pascal, quoted in Morris Bishop, Pascal: The Life of a Genius (1936; repr., New York: Greenwood Press, 1968), p. 47. 8. Ibid., p. 137. 9. Gilberte Pascal, quoted ibid., p. 135. 10. See A.W.F. Edwards, Pascal’s Arithmetical Triangle: The Story of a Mathematical Idea (Baltimore: Johns Hopkins University Press, 2002). 11. Blaise Pascal, quoted in Herbert Westren Turnbull, The Great Mathematicians (New York: New York University Press, 1961), p. 131. 12. Blaise Pascal, quoted in Bishop, Pascal, p. 196. 13. Blaise Pascal, quoted in David, Gods, Games and Gambling, p. 252. 14. Bruce Martin, “Coincidences: Remarkable or Random?”

Marin Mersenne, the great communicator, had died a few years earlier, but Pascal was still wired into the Académie Mersenne network. And so in 1654 began one of the great correspondences in the history of mathematics, between Pascal and Pierre de Fermat. In 1654, Fermat held a high position in the Tournelle, or criminal court, in Toulouse. When the court was in session, a finely robed Fermat might be found condemning errant functionaries to be burned at the stake. But when the court was not in session, he would turn his analytic skills to the gentler pursuit of mathematics. He may have been an amateur, but Pierre de Fermat is usually considered the greatest amateur mathematician of all times. Fermat had not gained his high position through any particular ambition or accomplishment. He achieved it the old-fashioned way, by moving up steadily as his superiors dropped dead of the plague.

pages: 415 words: 125,089

Against the Gods: The Remarkable Story of Risk by Peter L. Bernstein


Albert Einstein, Alvin Roth, Andrew Wiles, Antoine Gombaud: Chevalier de Méré, Bayesian statistics, Big bang: deregulation of the City of London, Bretton Woods, buttonwood tree, capital asset pricing model, cognitive dissonance, computerized trading, Daniel Kahneman / Amos Tversky, diversified portfolio, double entry bookkeeping, Edmond Halley, Edward Lloyd's coffeehouse, endowment effect, experimental economics, fear of failure, Fellow of the Royal Society, Fermat's Last Theorem, financial deregulation, financial innovation, full employment, index fund, invention of movable type, Isaac Newton, John Nash: game theory, John von Neumann, Kenneth Arrow, linear programming, loss aversion, Louis Bachelier, mental accounting, moral hazard, Myron Scholes, Nash equilibrium, Paul Samuelson, Philip Mirowski, probability theory / Blaise Pascal / Pierre de Fermat, random walk, Richard Thaler, Robert Shiller, Robert Shiller, spectrum auction, statistical model, The Bell Curve by Richard Herrnstein and Charles Murray, The Wealth of Nations by Adam Smith, Thomas Bayes, trade route, transaction costs, tulip mania, Vanguard fund, zero-sum game

In 1654, a time when the Renaissance was in full flower, the Chevalier de Mere, a French nobleman with a taste for both gambling and mathematics, challenged the famed French mathematician Blaise Pascal to solve a puzzle. The question was how to divide the stakes of an unfinished game of chance between two players when one of them is ahead. The puzzle had confounded mathematicians since it was posed some two hundred years earlier by the monk Luca Paccioli. This was the man who brought double-entry bookkeeping to the attention of the business managers of his day-and tutored Leonardo da Vinci in the multiplication tables. Pascal turned for help to Pierre de Fermat, a lawyer who was also a brilliant mathematician. The outcome of their collaboration was intellectual dynamite. What might appear to have been a seventeenth-century version of the game of Trivial Pursuit led to the discovery of the theory of probability, the mathematical heart of the concept of risk.

By the late 1660s, Dutch towns that had traditionally financed themselves by selling annuities were able to put these policies on a sound actuarial footing. By 1700, as we mentioned earlier, the English government was financing its budget deficits through the sale of life annuities. The story of the three Frenchmen begins with an unlikely trio who saw beyond the gaming tables and fashioned the systematic and theoretical foundations for measuring probability. The first, Blaise Pascal, was a brilliant young dissolute who subsequently became a religious zealot and ended up rejecting the use of reason. The second, Pierre de Fermat, was a successful lawyer for whom mathematics was a sideline. The third member of the group was a nobleman, the Chevalier de Mere, who combined his taste for mathematics with an irresistible urge to play games of chance; his fame rests simply on his having posed the question that set the other two on the road to discovery.

In addition, Omar Khayyam used technical mathematical observations to reform the calendar and to devise a triangular rearrangement of numbers that facilitated the figuring of squares, cubes, and higher powers of mathematics; this triangle formed the basis of concepts developed by the seventeenth-century French mathematician Blaise Pascal, one of the fathers of the theory of choice, chance, and probability. The impressive achievements of the Arabs suggest once again that an idea can go so far and still stop short of a logical conclusion. Why, given their advanced mathematical ideas, did the Arabs not proceed to probability theory and risk management? The answer, I believe, has to do with their view of life. Who determines our future: the fates, the gods, or ourselves? The idea of risk management emerges only when people believe that they are to some degree free agents. Like the Greeks and the early Christians, the fatalistic Muslims were not yet ready to take the leap.

pages: 233 words: 62,563

Zero: The Biography of a Dangerous Idea by Charles Seife


Albert Einstein, Albert Michelson, Arthur Eddington, Cepheid variable, cosmological constant, dark matter, Edmond Halley, Georg Cantor, Isaac Newton, John Conway, Pierre-Simon Laplace, place-making, probability theory / Blaise Pascal / Pierre de Fermat, retrograde motion, Richard Feynman, Richard Feynman, Solar eclipse in 1919, Stephen Hawking

It was also in zero and the infinite that Pascal, the devout Jansenist, sought to prove God’s existence. He did it in a very profane way. The Divine Wager What is man in nature? Nothing in relation to the infinite, everything in relation to nothing, a mean between nothing and everything. —BLAISE PASCAL, PENSÉES Pascal was a mathematician as well as a scientist. In science Pascal investigated the vacuum—the nature of the void. In mathematics Pascal helped invent a whole new branch of the field: probability theory. When Pascal combined probability theory with zero and with infinity, he found God. Probability theory was invented to help rich aristocrats win more money with their gambling. Pascal’s theory was extremely successful, but his mathematical career was not to last. On November 23, 1654, Pascal had an intense spiritual experience. Perhaps it was the old Jansenist antiscience creed that was building up in him, but for whatever the reason, Pascal’s newfound devotion led him to abandon mathematics and science altogether.

The steepness of the tangent line—its slope—has some important properties in physics: for instance, if you’ve got a curve that represents the position of, say, a bicycle, then the slope of the tangent line to that curve at any given point tells you how fast that bicycle is going when it reaches that spot. Figure 24: Flying off at a tangent For this reason, several seventeenth-century mathematicians—like Evangelista Torricelli, René Descartes, the Frenchman Pierre de Fermat (famous for his last theorem), and the Englishman Isaac Barrow—created different methods for calculating the tangent line to any given point on a curve. However, like Cavalieri, all of them came up against the infinitesimal. To draw a tangent line at any given point, it’s best to make a guess. Choose another point nearby and connect the two. The line you get isn’t exactly the tangent line, but if the curve isn’t too bumpy, the two lines will be pretty close.

It was the first time in history anyone had created a sustained vacuum. No matter the dimensions of the tube that Torricelli used, the mercury would sink down until its highest point was about 30 inches above the dish; or, looking at it another way, mercury could only rise 30 inches to combat the vacuum above it. Nature only abhorred a vacuum as far as 30 inches. It would take an anti-Descartes to explain why. In 1623, Descartes was twenty-seven, and Blaise Pascal, who would become Descartes’s opponent, was zero years old. Pascal’s father, Étienne, was an accomplished scientist and mathematician; the young Blaise was a genius equal to his father. As a young man, Blaise invented a mechanical calculating machine, named the Pascaline, which is similar to some of the mechanical calculators that engineers used before the invention of the electronic calculator.

pages: 289 words: 85,315

Fermat’s Last Theorem by Simon Singh


Albert Einstein, Andrew Wiles, Antoine Gombaud: Chevalier de Méré, Arthur Eddington, Augustin-Louis Cauchy, Fellow of the Royal Society, Georg Cantor, Henri Poincaré, Isaac Newton, John Conway, John von Neumann, kremlinology, probability theory / Blaise Pascal / Pierre de Fermat, RAND corporation, Rubik’s Cube, Simon Singh, Wolfskehl Prize

Once published, proofs would be examined and argued over by everyone and anyone who knew anything about the subject. When Blaise Pascal pressed him to publish some of his work, the recluse replied: ‘Whatever of my work is judged worthy of publication, I do not want my name to appear there.’ Fermat was the secretive genius who sacrificed fame in order not to be distracted by petty questions from his critics. This exchange of letters with Pascal, the only occasion when Fermat discussed ideas with anyone but Mersenne, concerned the creation of an entirely new branch of mathematics – probability theory. The mathematical hermit was introduced to the subject by Pascal, and so, despite his desire for isolation, he felt obliged to maintain a dialogue. Together Fermat and Pascal would discover the first proofs and cast-iron certainties in probability theory, a subject which is inherently uncertain.

Fermat’s Last Theorem has its origins in the mathematics of ancient Greece, two thousand years before Pierre de Fermat constructed the problem in the form we know it today. Hence, it links the foundations of mathematics created by Pythagoras to the most sophisticated ideas in modern mathematics. In writing this book I have chosen a largely chronological structure which begins by describing the revolutionary ethos of the Pythagorean Brotherhood, and ends with Andrew Wiles’s personal story of his struggle to find a solution to Fermat’s conundrum. Chapter 1 tells the story of Pythagoras, and describes how Pythagoras’ theorem is the direct ancestor of the Last Theorem. This chapter also discusses some of the fundamental concepts of mathematics which will recur throughout the book. Chapter 2 takes the story from ancient Greece to seventeenth-century France, where Pierre de Fermat created the most profound riddle in the history of mathematics.

A History of Greek Mathematics, Vols. 1 and 2, by Sir Thomas Heath, 1981, Dover. Mathematical Magic Show, by Martin Gardner, 1977, Knopf. A collection of mathematical puzzles and riddles. River meandering as a self-organization process, by Hans-Henrik Støllum, Science 271 (1996), 1710-1713. Chapter 2 The Mathematical Career of Pierre de Fermat, by Michael Mahoney, 1994, Princeton University Press. A detailed investigation into the life and work of Pierre de Fermat. Archimedes’ Revenge, by Paul Hoffman, 1988, Penguin. Fascinating tales which describe the joys and perils of mathematics. Chapter 3 Men of Mathematics, by E.T. Bell, Simon and Schuster, 1937. Biographies of history’s greatest mathematicians, including Euler, Fermat, Gauss, Cauchy and Kummer. The periodical cicada problem, by Monte Lloyd and Henry S.

pages: 437 words: 132,041

Alex's Adventures in Numberland by Alex Bellos


Andrew Wiles, Antoine Gombaud: Chevalier de Méré, beat the dealer, Black Swan, Black-Scholes formula, Claude Shannon: information theory, computer age, Daniel Kahneman / Amos Tversky, Edward Thorp, family office, forensic accounting, game design, Georg Cantor, Henri Poincaré, Isaac Newton, Myron Scholes, pattern recognition, Paul Erdős, Pierre-Simon Laplace, probability theory / Blaise Pascal / Pierre de Fermat, random walk, Richard Feynman, Richard Feynman, Rubik’s Cube, SETI@home, Steve Jobs, The Bell Curve by Richard Herrnstein and Charles Murray, traveling salesman

He had a couple of questions about gambling, though, that he was unable to answer himself, so in 1654 he approached the distinguished mathematician Blaise Pascal. His chance enquiry was the random event that set in motion the proper study of randomness. Blaise Pascal was only 31 when he received de Méré’s queries, but he had been known in intellectual circles for almost two decades. Pascal had shown such gifts as a young child that at 13 his father had let him attend the scientific salon organized by Marin Mersenne, the friar and prime-number enthusiast, which brought together many famous mathematicians, including René Descartes and Pierre de Fermat. While still a teenager, Pascal proved important theorems in geometry and invented an early mechanical calculation machine, which was called the Pascaline.

Pondering the answers, and feeling the need to discuss them with a fellow genius, Pascal wrote to his old friend from the Mersenne salon, Pierre de Fermat. Fermat lived far from Paris, in Toulouse, an appropriately named city for someone analysing a problem about gambling. He was 22 years older than Pascal and worked as a judge at the local criminal court, dabbling in maths only as an intellectual recreation. Nevertheless, his amateur ruminations had made him one of the most respected mathematicians of the first half of the seventeenth century. The short correspondence between Pascal and Fermat about chance – which they called hasard – was a landmark in the history of science. Between them the men solved both of the literary bon vivant’s problems, and in so doing, set the foundations of modern probability theory. Now for the answers to Chevalier de Méré’s questions.

The rest of science also relies on the language of equations. In 1621, a Latin translation of Diophantus’s masterpiece Arithmetica was published in France. The new edition rekindled interest in ancient problem-solving techniques, which, combined with better numerical and symbolic notation, ushered in a new era of mathematical thought. Less convoluted notation allowed greater clarity in descrig problems. Pierre de Fermat, a civil servant and judge living in Toulouse, was an enthusiastic amateur mathematician who filled his own copy of Arithmetica with numerical musings. Next to a section dealing with Pythagorean triples – any set of natural numbers a, b and c such that a2 + b2 = c2, for example 3, 4 and 5 – Fermat scribbled some notes in the margin. He had noticed that it was impossible to find values for a, b and c such that a3 + b3 = c3.

pages: 262 words: 65,959

The Simpsons and Their Mathematical Secrets by Simon Singh


Albert Einstein, Andrew Wiles, Benoit Mandelbrot, cognitive dissonance, Donald Knuth, Erdős number, Georg Cantor, Grace Hopper, Isaac Newton, John Nash: game theory, mandelbrot fractal, Menlo Park, Norbert Wiener, Norman Mailer, P = NP, Paul Erdős, probability theory / Blaise Pascal / Pierre de Fermat, Richard Feynman, Richard Feynman, Rubik’s Cube, Schrödinger's Cat, Simon Singh, Stephen Hawking, Wolfskehl Prize, women in the workforce

However, that same day I was given a preliminary script for another upcoming episode, “The Saga of Carl,” which contained an entire scene dedicated to the mathematics of probability. “The Saga of Carl” opens with Marge dragging her family away from the television and taking them on an educational trip to the Hall of Probability at Springfield’s Science Museum. There, they watch a video introduced by an actor playing the role of Blaise Pascal (1623–62), the father of probability theory, and they also see an experimental demonstration of probability theory known as the Galton board. This involves marbles rolling down a slope and ricocheting off a series of pins. At each pin, the marbles bounce randomly to the left or right, only to hit the next row of pins and be met by the same random opportunity. The marbles are finally collected in a series of slots and form a humped distribution. The Galton board was named after its English inventor, the polymath Francis Galton (1822–1911).

The second line on Homer’s blackboard is perhaps the most interesting, as it contains the following equation: 3,98712 + 4,36512 = 4,47212 The equation appears to be innocuous at first sight, unless you know something about the history of mathematics, in which case you are about to smash up your slide rule in disgust. For Homer seems to have achieved the impossible and found a solution to the notorious mystery of Fermat’s last theorem! Pierre de Fermat first proposed this theorem in about 1637. Despite being an amateur who only solved problems in his spare time, Fermat was one of the greatest mathematicians in history. Working in isolation at his home in southern France, his only mathematical companion was a book called Arithmetica, written by Diophantus of Alexandria in the third century A.D. While reading this ancient Greek text, Fermat spotted a section on the following equation: x 2 + y 2 = z 2 This equation is closely related to the Pythagorean theorem, but Diophantus was not interested in triangles and the lengths of their sides.

He scribbled a pair of tantalizing sentences in Latin in the margin of his copy of Diophantus’s Arithmetica. He began by stating that there are no whole number solutions for any of the infinite number of equations above, and then he confidently added this second sentence: “Cuius rei demonstrationem mirabilem sane detexi, hanc marginis exiguitas non caperet.” (I have discovered a truly marvelous proof of this, which this margin is too narrow to contain.) Pierre de Fermat had found a proof, but he did not bother to write it down. This is perhaps the most frustrating note in the history of mathematics, particularly as Fermat took his secret to the grave. Fermat’s son Clément-Samuel later found his father’s copy of Arithmetica and noticed this intriguing marginal note. He also spotted many similar marginal jottings, because Fermat had a habit of stating that he could prove something remarkable, but rarely wrote down the proof.

pages: 218 words: 63,471

How We Got Here: A Slightly Irreverent History of Technology and Markets by Andy Kessler


Albert Einstein, Andy Kessler, automated trading system, bank run, Big bang: deregulation of the City of London, Bob Noyce, Bretton Woods, British Empire, buttonwood tree, Claude Shannon: information theory, Corn Laws, Douglas Engelbart, Edward Lloyd's coffeehouse, fiat currency, fixed income, floating exchange rates, Fractional reserve banking, full employment, Grace Hopper, invention of the steam engine, invention of the telephone, invisible hand, Isaac Newton, Jacquard loom, Jacquard loom, James Hargreaves, James Watt: steam engine, John von Neumann, joint-stock company, joint-stock limited liability company, Joseph-Marie Jacquard, Leonard Kleinrock, Marc Andreessen, Maui Hawaii, Menlo Park, Metcalfe's law, Metcalfe’s law, packet switching, price mechanism, probability theory / Blaise Pascal / Pierre de Fermat, profit motive, railway mania, RAND corporation, Robert Metcalfe, Silicon Valley, Small Order Execution System, South Sea Bubble, spice trade, spinning jenny, Steve Jobs, supply-chain management, supply-chain management software, trade route, transatlantic slave trade, transatlantic slave trade, tulip mania, Turing machine, Turing test, William Shockley: the traitorous eight

Sure, they had a gut feel for the likelihood of a caravan being robbed or a squall hitting a ship of Grecian urns, but they were just making it up. Our hero Blaise Pascal did more than invent the calculator for his tax collecting father, lighting a slow fuse on the computer revolution. He also proved that vacuums exist and that pressure could be measured with a tube of inverted mercury – two phenomena that James Watt needed to get his steam engine working. But Pascal’s rapidly firing mathematical mind would go to other, more near-term pursuits. He was a vicious gambler. He had to know when to hold ‘em and when to fold them, as well as calculate the odds on any given roll of the dice. Along with Pierre de Fermat, whose Last Theorem puzzled math-heads, Pascal struggled on one particular gambling puzzle: How two dice rollers would split the stakes or bets if they left before finishing the game.

Did anything besides Velcro and Tang come out of the Space Program? Why does the U.S. have any industrial businesses left? There are too many questions to answer. So instead, I wrote this primer. Enjoy. Send me feedback, ideas and suggestions at with HWGH in the subject. Logic and Memory I hate to admit it, but it was taxes that got it all started. In 1642, 18-year-old Blaise Pascal, the son of a French tax collector, tired of waiting for his dad to come play a game of “le catch”. Blaise’s dad was what is known as a tax farmer, sort of a 17th century version of a loan shark, threat of broken bones and all. Tax farmers advanced tax money to the government and then had a license to collect taxes, hopefully “harvesting” more than they advanced. Elder Pascal was constantly busy calculating and tabulating his potential tax haul.

The computer industry was on its way, albeit at the pace of a woozy escargot. In 1649, King Louis XIV granted Pascal a patent for his odd device but it failed to effect much change over the next 45 years. Pascal, by the way, would contribute more than a mechanical calculator to this tale. He proved that vacuums exist; that one could measure pressure by inverting a tube of mercury; and in figuring out how others could beat the house at gambling, ended up inventing probability theory. 8 HOW WE GOT HERE In 1694, a German, Gottfried Wilhelm von Leibniz, created a box similar to Pascal’s but his could actually multiply. Leibniz used something called a stepped drum, a cylinder with a number of cogs carved into it, and gears that would engage a different number of cogs depending on their position. It was incredibly complex, which is why very few were ever built. Inside a Pascaline or a Leibniz box were two simple elements needed to create the modern computer: Logic and Memory.

pages: 360 words: 85,321

The Perfect Bet: How Science and Math Are Taking the Luck Out of Gambling by Adam Kucharski

Ada Lovelace, Albert Einstein, Antoine Gombaud: Chevalier de Méré, beat the dealer, Benoit Mandelbrot, butterfly effect, call centre, Chance favours the prepared mind, Claude Shannon: information theory, collateralized debt obligation, correlation does not imply causation, diversification, Edward Lorenz: Chaos theory, Edward Thorp, Everything should be made as simple as possible, Flash crash, Gerolamo Cardano, Henri Poincaré, Hibernia Atlantic: Project Express, if you build it, they will come, invention of the telegraph, Isaac Newton, John Nash: game theory, John von Neumann, locking in a profit, Louis Pasteur, Nash equilibrium, Norbert Wiener, p-value, performance metric, Pierre-Simon Laplace, probability theory / Blaise Pascal / Pierre de Fermat, quantitative trading / quantitative finance, random walk, Richard Feynman, Richard Feynman, Ronald Reagan, Rubik’s Cube, statistical model, The Design of Experiments, Watson beat the top human players on Jeopardy!, zero-sum game

The science of chance blossomed in 1654 as the result of a gambling question posed by a French writer named Antoine Gombaud. He had been puzzled by the following dice problem. Which is more likely: throwing a single six in four rolls of a single die, or throwing double sixes in twenty-four rolls of two dice? Gombaud believed the two events would occur equally often but could not prove it. He wrote to his mathematician friend Blaise Pascal, asking if this was indeed the case. To tackle the dice problem, Pascal enlisted the help of Pierre de Fermat, a wealthy lawyer and fellow mathematician. Together, they built on Cardano’s earlier work on randomness, gradually pinning down the basic laws of probability. Many of the new concepts would become central to mathematical theory. Among other things, Pascal and Fermat defined the “expected value” of a game, which measured how profitable it would be on average if played repeatedly.

This would become extremely difficult after even a spin or two. Other options, such as predicting which half of the table the ball lands in, were less sensitive to initial conditions. It would therefore take a lot of spins before the result becomes as good as random. Fortunately for gamblers, a roulette ball does not spin for an extremely long period of time (although there is an oft-repeated myth that mathematician Blaise Pascal invented roulette while trying to build a perpetual motion machine). As a result, gamblers can—in theory—avoid falling into Poincaré’s second degree of ignorance by measuring the initial path of the roulette ball. They just need to work out what measurements to take. THE RITZ WASN’T THE first time a story of roulette-tracking technology emerged. Eight years after Hibbs and Walford had exploited that biased wheel in Reno, Edward Thorp sat in a common room at the University of California, Los Angeles, discussing get-rich-quick schemes with his fellow students.

Los Angeles Times, May 1, 2004. 7Many have told the tale: Ethier, “Testing for Favorable Numbers.” 7When Wilson published his data: Ethier, “Testing for Favorable Numbers.” 9Poincaré had outlined the “butterfly effect: Gleick, James. Chaos: Making a New Science (New York: Open Road, 2011). 9The Zodiac may be regarded: Poincaré, Science and Method. 10Blaise Pascal invented roulette: Bass, Thomas. The Newtonian Casino (London: Penguin, 1990). 10The orbiting roulette ball: The majority of details and quotes in this section are taken from Thorp, Edward. “The Invention of the First Wearable Computer.” Proceedings of the 2nd IEEE International Symposium on Wearable Computers (1998), 4. 13participants were asked to help: Milgram, Stanley. “The Small-World Problem.”

pages: 315 words: 93,628

Is God a Mathematician? by Mario Livio


Albert Einstein, Antoine Gombaud: Chevalier de Méré, Brownian motion, cellular automata, correlation coefficient, correlation does not imply causation, cosmological constant, Dava Sobel, double helix, Edmond Halley, Eratosthenes, Georg Cantor, Gerolamo Cardano, Gödel, Escher, Bach, Henri Poincaré, Isaac Newton, John von Neumann, music of the spheres, Myron Scholes, probability theory / Blaise Pascal / Pierre de Fermat, Russell's paradox, The Design of Experiments, the scientific method, traveling salesman

Games of Chance The serious study of probability started from very modest beginnings—attempts by gamblers to adjust their bets to the odds of success. In particular, in the middle of the seventeenth century, a French nobleman—the Chevalier de Méré—who was also a reputed gamester, addressed a series of questions about gambling to the famous French mathematician and philosopher Blaise Pascal (1623–62). The latter conducted in 1654 an extensive correspondence about these questions with the other great French mathematician of the time, Pierre de Fermat (1601–65). The theory of probability was essentially born in this correspondence. Let’s examine one of the fascinating examples discussed by Pascal in a letter dated July 29, 1654. Imagine two noblemen engaged in a game involving the roll of a single die. Each player has put on the table thirty-two pistoles of gold.

At the age of eight, Descartes entered the Jesuit College of La Flèche, where he studied Latin, mathematics, classics, science, and scholastic philosophy until 1612. Because of his relatively fragile health, Descartes was excused from having to get up at the brutal hour of five a.m., and he was allowed to spend the morning hours in bed. Later in life, he continued to use the early part of the day for contemplation, and he once told the French mathematician Blaise Pascal that the only way for him to stay healthy and be productive was to never get up before he felt comfortable doing so. As we shall soon see, this statement turned out to be tragically prophetic. After La Flèche, Descartes graduated from the University of Poitiers as a lawyer, but he never actually practiced law. Restless and eager to see the world, Descartes decided to join the army of Prince Maurice of Orange, which was then stationed at Breda in the United Provinces (The Netherlands).

The serious study of probability: Recently published, entertaining popular accounts of probability, its history, and its uses include Aczel 2004, Kaplan and Kaplan 2006, Connor 2006, Burger and Starbird 2005, and Tabak 2004. in a letter dated July 29, 1654: Todhunter 1865, Hald 1990. The essence of probability theory: An excellent, popular, brief description of some of the essential principles of probability theory can be found in Kline 1967. Probability theory provides us with accurate information: The relevance of probability theory to many real-life situations is beautifully described in Rosenthal 2006. The person who brought probability: For an excellent biography, see Orel 1996. Mendel published his paper: Mendel 1865. An English translation can be found on the Web page created by R. B. Blumberg at

pages: 790 words: 150,875

Civilization: The West and the Rest by Niall Ferguson


Admiral Zheng, agricultural Revolution, Albert Einstein, Andrei Shleifer, Atahualpa, Ayatollah Khomeini, Berlin Wall, BRICs, British Empire, clean water, collective bargaining, colonial rule, conceptual framework, Copley Medal, corporate governance, creative destruction, credit crunch, David Ricardo: comparative advantage, Dean Kamen, delayed gratification, Deng Xiaoping, discovery of the americas, Dissolution of the Soviet Union, European colonialism, Fall of the Berlin Wall, Francisco Pizarro, full employment, Hans Lippershey, haute couture, Hernando de Soto, income inequality, invention of movable type, invisible hand, Isaac Newton, James Hargreaves, James Watt: steam engine, John Harrison: Longitude, joint-stock company, Joseph Schumpeter, Kitchen Debate, land reform, land tenure, liberal capitalism, Louis Pasteur, Mahatma Gandhi, market bubble, Martin Wolf, mass immigration, means of production, megacity, Mikhail Gorbachev, new economy, Pearl River Delta, Pierre-Simon Laplace, probability theory / Blaise Pascal / Pierre de Fermat, profit maximization, purchasing power parity, quantitative easing, rent-seeking, reserve currency, road to serfdom, Ronald Reagan, savings glut, Scramble for Africa, Silicon Valley, South China Sea, sovereign wealth fund, special economic zone, spice trade, spinning jenny, Steve Jobs, Steven Pinker, The Great Moderation, the market place, the scientific method, The Wealth of Nations by Adam Smith, Thomas Kuhn: the structure of scientific revolutions, Thomas Malthus, Thorstein Veblen, total factor productivity, trade route, transaction costs, transatlantic slave trade, transatlantic slave trade, upwardly mobile, uranium enrichment, wage slave, Washington Consensus, women in the workforce, World Values Survey

.* 1530 Paracelsus pioneers the application of chemistry to physiology and pathology 1543 Nicolaus Copernicus’ De revolutionibus orbium coelestium states the heliocentric theory of the solar system Andreas Vesalius’ De humani corporis fabrica supplants Galen’s anatomical textbook 1546 Agricola’s De natura fossilium classifies minerals and introduces the term ‘fossil’ 1572 Tycho Brahe records the first European observation of a supernova 1589 Galileo’s tests of falling bodies (published in De motu) revolutionize the experimental method 1600 William Gilbert’s De magnete, magnetisque corporibus describes the magnetic properties of the earth and electricity 1604 Galileo discovers that a free-falling body increases its distance as the square of the time 1608 Hans Lippershey and Zacharias Jansen independently invent the telescope 1609 1609 Galileo conducts the first telescopic observations of the night sky 1610 Galileo discovers four of Jupiter’s moons and infers that the earth is not at the centre of the universe 1614 John Napier’s Mirifici logarithmorum canonis descriptio introduces logarithms 1628 William Harvey writes Exercitatio anatomica de motu cordis et sanguinis in animalibus, accurately describing the circulation of blood 1637 René Descartes’ ‘La Géométrie’, an appendix to his Discours de la méthode, founds analytic geometry 1638 Galileo’s Discorsi e dimonstrazioni matematiche founds modern mechanics 1640 Pierre de Fermat founds number theory 1654 Fermat and Blaise Pascal found probability theory 1661 Robert Boyle’s Skeptical Chymist defines elements and chemical analysis 1662 Boyle states Boyle’s Law that the volume occupied by a fixed mass of gas in a container is inversely proportional to the pressure it exerts 1669 Isaac Newton’s De analysi per aequationes numero terminorum infinitas presents the first systematic account of the calculus, independently developed by Gottfried Leibniz 1676 Antoni van Leeuwenhoek discovers micro-organisms 1687 Newton’s Philosophiae naturalis principia mathematica states the law of universal gravitation and the laws of motion 1735 Carolus Linnaeus’ Systema naturae introduces systematic classification of genera and species of organisms 1738 Daniel Bernoulli’s Hydrodynamica states Bernoulli’s Principle and founds the mathematical study of fluid flow and the kinetic theory of gases 1746 Jean-Etienne Guettard prepares the first true geological maps 1755 Joseph Black identifies carbon dioxide 1775 Antoine Lavoisier accurately describes combustion 1785 James Hutton’s ‘Concerning the System of the Earth’ states the uniformitarian view of the earth’s development 1789 Lavoisier’s Traité élémentaire de chimie states the law of conservation of matter By the mid-1600s this kind of scientific knowledge was spreading as rapidly as had the doctrine of the Protestant Reformers a century before.

The great Dutch-Jewish philosopher Baruch or Benedict Spinoza, who hypothesized that there is only a material universe of substance and deterministic causation, and that ‘God’ is that universe’s natural order as we dimly apprehend it and nothing more, died in 1677 at the age of forty-four, probably from the particles of glass he had inhaled doing his day-job as a lens grinder. Blaise Pascal, the pioneer of probability theory and hydrodynamics and the author of the Pensées, the greatest of all apologias for the Christian faith, lived to be just thirty-nine; he would have died even younger had the road accident that reawakened his spiritual side been fatal. Who knows what other great works these geniuses might have brought forth had they been granted the lifespans enjoyed by, for example, the great humanists Erasmus (sixty-nine) and Montaigne (fifty-nine)?

pages: 893 words: 199,542

Structure and interpretation of computer programs by Harold Abelson, Gerald Jay Sussman, Julie Sussman


Andrew Wiles, conceptual framework, Donald Knuth, Douglas Hofstadter, Eratosthenes, Fermat's Last Theorem, Gödel, Escher, Bach, industrial robot, information retrieval, iterative process, loose coupling, probability theory / Blaise Pascal / Pierre de Fermat, Richard Stallman, Turing machine

Tabulation can sometimes be used to transform processes that require an exponential number of steps (such as count-change) into processes whose space and time requirements grow linearly with the input. See exercise 3.27. 35 The elements of Pascal's triangle are called the binomial coefficients, because the nth row consists of the coefficients of the terms in the expansion of (x + y)n. This pattern for computing the coefficients appeared in Blaise Pascal's 1653 seminal work on probability theory, Traité du triangle arithmétique. According to Knuth (1973), the same pattern appears in the Szu-yuen Yü-chien (“The Precious Mirror of the Four Elements”), published by the Chinese mathematician Chu Shih-chieh in 1303, in the works of the twelfth-century Persian poet and mathematician Omar Khayyam, and in the works of the twelfth-century Hindu mathematician Bháscara Áchárya. 36 These statements mask a great deal of oversimplification.

Let (ak+1, bk+1) ⟶ (ak, bk) ⟶ (ak-1, bk-1) be successive pairs in the reduction process. By our induction hypotheses, we have bk-1> Fib(k - 1) and bk> Fib(k). Thus, applying the claim we just proved together with the definition of the Fibonacci numbers gives bk+1> bk + bk-1> Fib(k) + Fib(k - 1) = Fib(k + 1), which completes the proof of Lamé's Theorem. 44 If d is a divisor of n, then so is n/d. But d and n/d cannot both be greater than √n. 45 Pierre de Fermat (1601-1665) is considered to be the founder of modern number theory. He obtained many important number-theoretic results, but he usually announced just the results, without providing his proofs. Fermat's Little Theorem was stated in a letter he wrote in 1640. The first published proof was given by Euler in 1736 (and an earlier, identical proof was discovered in the unpublished manuscripts of Leibniz).

evening star, see Venus event-driven simulation evlis tail recursion exact integer exchange exclamation point in names execute execute-application metacircular nondeterministic execution procedure in analyzing evaluator in nondeterministic evaluator, [2], [3] in register-machine simulator, [2] exp register expand-clauses explicit-control evaluator for Scheme assignments combinations compound procedures conditionals controller data paths definitions derived expressions driver loop error handling, [2] expressions with no subexpressions to evaluate as machine-language program machine model modified for compiled code monitoring performance (stack use) normal-order evaluation operand evaluation operations optimizations (additional) primitive procedures procedure application registers running sequences of expressions special forms (additional), [2] stack usage tail recursion, [2], [3] as universal machine expmod, [2], [3] exponential growth of tree-recursive Fibonacci-number computation exponentiation modulo n expression, see also compound expression; primitive expression algebraic, see algebraic expressions self-evaluating symbolic, see also symbol(s) expression-oriented vs. imperative programming style expt linear iterative version linear recursive version register machine for extend-environment, [2] extend-if-consistent extend-if-possible external-entry extract-labels, [2] #f factorial, see also factorial infinite stream with letrec without letrec or define factorial as an abstract machine compilation of, [2] environment structure in evaluating linear iterative version linear recursive version register machine for (iterative), [2] register machine for (recursive), [2] stack usage, compiled stack usage, interpreted, [2] stack usage, register machine with assignment with higher-order procedures failure continuation (nondeterministic evaluator), [2] constructed by amb constructed by assignment constructed by driver loop failure, in nondeterministic computation bug vs. searching and false false false? fast-expt fast-prime? feedback loop, modeled with streams Feeley, Marc Feigenbaum, Edward Fenichel, Robert Fermat, Pierre de Fermat test for primality variant of Fermat's Little Theorem alternate form proof fermat-test fetch-assertions fetch-rules fib linear iterative version logarithmic version register machine for (tree-recursive), [2] stack usage, compiled stack usage, interpreted tree-recursive version, [2] with memoization with named let Fibonacci numbers, see also fib Euclid's GCD algorithm and infinite stream of, see fibs fibs (infinite stream) implicit definition FIFO buffer filter, [2] filter filtered-accumulate find-assertions find-divisor first-agenda-item, [2] first-class elements in language first-exp first-frame first-operand first-segment first-term, [2] fixed point computing with calculator of cosine cube root as fourth root as golden ratio as as iterative improvement in Newton's method nth root as square root as, [2], [3] of transformed function unification and fixed-length code fixed-point as iterative improvement fixed-point-of-transform flag register flatmap flatten-stream flip-horiz, [2] flip-vert, [2] flipped-pairs, [2], [3] Floyd, Robert fold-left fold-right for-each, [2] for-each-except Forbus, Kenneth D.

pages: 338 words: 106,936

The Physics of Wall Street: A Brief History of Predicting the Unpredictable by James Owen Weatherall


Albert Einstein, algorithmic trading, Antoine Gombaud: Chevalier de Méré, Asian financial crisis, bank run, beat the dealer, Benoit Mandelbrot, Black Swan, Black-Scholes formula, Bonfire of the Vanities, Bretton Woods, Brownian motion, butterfly effect, capital asset pricing model, Carmen Reinhart, Claude Shannon: information theory, collateralized debt obligation, collective bargaining, dark matter, Edward Lorenz: Chaos theory, Edward Thorp, Emanuel Derman, Eugene Fama: efficient market hypothesis, financial innovation, fixed income, George Akerlof, Gerolamo Cardano, Henri Poincaré, invisible hand, Isaac Newton, iterative process, John Nash: game theory, Kenneth Rogoff, Long Term Capital Management, Louis Bachelier, mandelbrot fractal, martingale, Myron Scholes, new economy, Paul Lévy, Paul Samuelson, prediction markets, probability theory / Blaise Pascal / Pierre de Fermat, quantitative trading / quantitative finance, random walk, Renaissance Technologies, risk-adjusted returns, Robert Gordon, Robert Shiller, Robert Shiller, Ronald Coase, Sharpe ratio, short selling, Silicon Valley, South Sea Bubble, statistical arbitrage, statistical model, stochastic process, The Chicago School, The Myth of the Rational Market, tulip mania, V2 rocket, Vilfredo Pareto, volatility smile

And so, de Méré began to ask the mathematicians he met socially about his problem. No one had an answer, or much interest in looking for one, until de Méré tried his problem out on Blaise Pascal. Pascal had been a child prodigy, working out most of classical geometry on his own by drawing pictures as a child. By his late teens he was a regular at the most important salon, run by a Jesuit priest named Marin Mersenne, and it was here that de Méré and Pascal met. Pascal didn’t know the answer, but he was intrigued. In particular, he agreed with de Méré’s appraisal that the problem should have a mathematical solution. Pascal began to work on de Méré’s problem. He enlisted the help of another mathematician, Pierre de Fermat. Fermat was a lawyer and polymath, fluent in a half-dozen languages and one of the most capable mathematicians of his day. Fermat lived about four hundred miles south of Paris, in Toulouse, and so Pascal didn’t know him directly, but he had heard of him through his connections at Mersenne’s salon.

Samuelson’s interest in Bachelier had begun a few days before, when he received a postcard from his friend Leonard “Jimmie” Savage, then a professor of statistics at the University of Chicago. Savage had just finished writing a textbook on probability and statistics and had developed an interest in the history of probability theory along the way. He had been poking around the university library for early-twentieth-century work on probability when he came across a textbook from 1914 that he had never seen before. When he flipped through it, Savage realized that, in addition to some pioneering work on probability, the book had a few chapters dedicated to what the author called “speculation” — literally, probability theory as applied to market speculation. Savage guessed (correctly) that if he had never come across this work before, his friends in economics departments likely hadn’t either, and so he sent out a series of postcards asking if anyone knew of Bachelier.

The halls were filled with men running back and forth executing trades, transferring contracts and bills, bidding on and selling stocks and rentes. Bachelier knew the rudiments of the French financial system, but little more. The Bourse did not seem like the right place for a quiet boy, a mathematician with a scholar’s temperament. But there was no turning back. It’s just a game, he told himself. Bachelier had always been fascinated by probability theory, the mathematics of chance (and, by extension, gambling). If he could just imagine the French financial markets as a glorified casino, a game whose rules he was about to learn, it might not seem so scary. He repeated the mantra — just an elaborate game of chance — as he pushed forward into the throng. “Who is this guy?” Paul Samuelson asked himself, for the second time in as many minutes.

pages: 322 words: 88,197

Wonderland: How Play Made the Modern World by Steven Johnson


Ada Lovelace, Alfred Russel Wallace, Antoine Gombaud: Chevalier de Méré, Berlin Wall, bitcoin, Book of Ingenious Devices, Buckminster Fuller, Claude Shannon: information theory, Clayton Christensen, colonial exploitation, computer age, conceptual framework, crowdsourcing, cuban missile crisis, Drosophila, Edward Thorp, Fellow of the Royal Society, game design, global village, Hedy Lamarr / George Antheil, HyperCard, invention of air conditioning, invention of the printing press, invention of the telegraph, Islamic Golden Age, Jacquard loom, Jacquard loom, Jacques de Vaucanson, James Watt: steam engine, Jane Jacobs, John von Neumann, joint-stock company, Joseph-Marie Jacquard, land value tax, Landlord’s Game, lone genius, mass immigration, megacity, Minecraft, moral panic, Murano, Venice glass, music of the spheres, Necker cube, New Urbanism, Oculus Rift, On the Economy of Machinery and Manufactures, pattern recognition, peer-to-peer,, placebo effect, probability theory / Blaise Pascal / Pierre de Fermat, profit motive, QWERTY keyboard, Ray Oldenburg, spice trade, spinning jenny, statistical model, Steve Jobs, Steven Pinker, Stewart Brand, supply-chain management, talking drums, the built environment, The Great Good Place, the scientific method, The Structural Transformation of the Public Sphere, trade route, Turing machine, Turing test, Upton Sinclair, urban planning, Victor Gruen, Watson beat the top human players on Jeopardy!, white flight, white picket fence, Whole Earth Catalog, working poor, Wunderkammern

He also demonstrated the multiplicative nature of probability when predicting the results of a sequence of dice rolls: the chance of rolling three sixes in a row is one in 216: 1⁄6 x 1⁄6 x 1⁄6. Written in 1564, Cardano’s book wasn’t published for another century. By the time his ideas got into wider circulation, an even more important breakthrough had emerged out of a famous correspondence between Blaise Pascal and Pierre de Fermat in 1654. This, too, was prompted by a compulsive gambler, the French aristocrat Antoine Gombaud, who had written Pascal for advice about the most equitable way to predict the outcome of a dice game that had been interrupted. Their exchange put probability theory on a solid footing and created the platform for the modern science of statistics. Within a few years, Edward Halley (of comet legend) was using these new tools to calculate mortality rates for the average Englishman, and the Dutch scientist Christiaan Huygens and his brother Lodewijk had set about to answer “the question . . . to what age a newly conceived child will naturally live.”

Within a few years, Edward Halley (of comet legend) was using these new tools to calculate mortality rates for the average Englishman, and the Dutch scientist Christiaan Huygens and his brother Lodewijk had set about to answer “the question . . . to what age a newly conceived child will naturally live.” Lodewijk even went so far as to calculate that his brother, then aged forty, was likely to live for another sixteen years. (He beat the odds and lived a decade beyond that, as it turns out.) It was the first time anyone had begun talking, mathematically at least, about what we now call life expectancy. Probability theory served as a kind of conceptual fossil fuel for the modern world. It gave rise to the modern insurance industry, which for the first time could calculate with some predictive power the claims it could expect when insuring individuals or industries. Capital markets—for good and for bad—rely extensively on elaborate statistical models that predict future risk. “The pundits and pollsters who today tell us who is likely to win the next election make direct use of mathematical techniques developed by Pascal and Fermat,” the mathematician Keith Devlin writes.

That regularity may have foiled the swindlers in the short term, but it had a much more profound effect that had never occurred to dice-making guilds: it made the patterns of the dice games visible, which enabled Cardano, Pascal, and Fermat to begin to think systematically about probability. Ironically, making the object of the die itself more uniform ultimately enabled people like Huygens and Halley to analyze the decidedly nonuniform experience of human mortality using the new tools of probability theory. No longer mere playthings, the dice had become, against all odds, tools for thinking. — The logic of games is ethereal. We have no idea how most ancient games were played, either because written rule books did not survive to modern times, or because the rules themselves evolved and then died out before the game’s players adopted the technology of writing. But we know about these games because they were physically embodied in matter: in game pieces, sports equipment, even in rooms or arenas designed to accommodate rules that have long since been lost to history.

pages: 471 words: 124,585

The Ascent of Money: A Financial History of the World by Niall Ferguson


Admiral Zheng, Andrei Shleifer, Asian financial crisis, asset allocation, asset-backed security, Atahualpa, bank run, banking crisis, banks create money, Black Swan, Black-Scholes formula, Bonfire of the Vanities, Bretton Woods, BRICs, British Empire, capital asset pricing model, capital controls, Carmen Reinhart, Cass Sunstein, central bank independence, collateralized debt obligation, colonial exploitation, commoditize, Corn Laws, corporate governance, creative destruction, credit crunch, Credit Default Swap, credit default swaps / collateralized debt obligations, currency manipulation / currency intervention, currency peg, Daniel Kahneman / Amos Tversky, deglobalization, diversification, diversified portfolio, double entry bookkeeping, Edmond Halley, Edward Glaeser, Edward Lloyd's coffeehouse, financial innovation, financial intermediation, fixed income, floating exchange rates, Fractional reserve banking, Francisco Pizarro, full employment, German hyperinflation, Hernando de Soto, high net worth, hindsight bias, Home mortgage interest deduction, Hyman Minsky, income inequality, information asymmetry, interest rate swap, Intergovernmental Panel on Climate Change (IPCC), Isaac Newton, iterative process, John Meriwether, joint-stock company, joint-stock limited liability company, Joseph Schumpeter, Kenneth Arrow, Kenneth Rogoff, knowledge economy, labour mobility, Landlord’s Game, liberal capitalism, London Interbank Offered Rate, Long Term Capital Management, market bubble, market fundamentalism, means of production, Mikhail Gorbachev, money market fund, money: store of value / unit of account / medium of exchange, moral hazard, mortgage debt, mortgage tax deduction, Myron Scholes, Naomi Klein, negative equity, Nick Leeson, Northern Rock, Parag Khanna, pension reform, price anchoring, price stability, principal–agent problem, probability theory / Blaise Pascal / Pierre de Fermat, profit motive, quantitative hedge fund, RAND corporation, random walk, rent control, rent-seeking, reserve currency, Richard Thaler, Robert Shiller, Robert Shiller, Ronald Reagan, savings glut, seigniorage, short selling, Silicon Valley, South Sea Bubble, sovereign wealth fund, spice trade, structural adjustment programs, technology bubble, The Wealth of Nations by Adam Smith, The Wisdom of Crowds, Thomas Bayes, Thomas Malthus, Thorstein Veblen, too big to fail, transaction costs, value at risk, Washington Consensus, Yom Kippur War

There did not yet exist an adequate theoretical basis for evaluating the risks that were being covered. Then, in a remarkable rush of intellectual innovation, beginning in around 1660, that theoretical basis was created. In essence, there were six crucial breakthroughs:1. Probability. It was to a monk at Port-Royal that the French mathematician Blaise Pascal attributed the insight (published in Pascal’s Ars Cogitandi) that ‘fear of harm ought to be proportional not merely to the gravity of the harm, but also to the probability of the event.’ Pascal and his friend Pierre de Fermat had been toying with problems of probability for many years, but for the evolution of insurance, this was to be a critical point. 2. Life expectancy. In the same year that Ars Cogitandi appeared (1662), John Graunt published his ‘Natural and Political Observations . . . Made upon the Bills of Mortality’, which sought to estimate the likelihood of dying from a particular cause on the basis of official London mortality statistics.

Relatively safe bonds, as recommended by Victorian authorities such as A. H. Bailey, head actuary of the London Assurance Corporation? Or riskier but probably higher yielding stocks? Insurance, in other words, is where the risks and uncertainties of daily life meet the risks and uncertainties of finance. To be sure, actuarial science gives insurance companies an in-built advantage over policy-holders. Before the dawn of modern probability theory, insurers were the gamblers; now they are the casino. The case can be made, as it was by Dickie Scruggs before his fall from grace, that the odds are now stacked unjustly against the punters/policy-holders. But as the economist Kenneth Arrow long ago pointed out, most of us prefer a gamble that has a 100 per cent chance of a small loss (our annual premium) and a small chance of a large gain (the insurance payout after disaster) to a gamble that has a 100 per cent chance of a small gain (no premiums) but an uncertain chance of a huge loss (no payout after a disaster).

pages: 532 words: 133,143

To Explain the World: The Discovery of Modern Science by Steven Weinberg


Albert Einstein, Alfred Russel Wallace, Astronomia nova, Brownian motion, Commentariolus, cosmological constant, dark matter, Dava Sobel, double helix, Edmond Halley, Eratosthenes, Ernest Rutherford, fudge factor, invention of movable type, Isaac Newton, James Watt: steam engine, music of the spheres, On the Revolutions of the Heavenly Spheres, Pierre-Simon Laplace, probability theory / Blaise Pascal / Pierre de Fermat, retrograde motion, Thomas Kuhn: the structure of scientific revolutions

In a thoroughly muddled discussion that I cannot understand, Descartes somehow argues that light travels more easily in water than in air, so that for light n is greater than 1. For Descartes’ purposes his failure to explain the value of n didn’t really matter, because he could and did take the value of n from experiment (perhaps from the data in Ptolemy’s Optics), which of course gives n greater than 1. A more convincing derivation of the law of refraction was given by the mathematician Pierre de Fermat (1601–1665), along the lines of the derivation by Hero of Alexandria of the equal-angles rule governing reflection, but now making the assumption that light rays take the path of least time, rather than of least distance. This assumption (as shown in Technical Note 28) leads to the correct formula, that n is the ratio of the speed of light in medium A to its speed in medium B, and is therefore greater than 1 when A is air and B is glass or water.

In 1643 he observed that if a vertical glass tube longer than this and closed at the top end is filled with mercury, then some mercury will flow out until the height of the mercury in the tube is about 30 inches. This leaves empty space on top, now known as a “Torricellian vacuum.” Such a tube can then serve as a barometer, to measure changes in ambient air pressure; the higher the air pressure, the higher the column of mercury that it can support. The French polymath Blaise Pascal is best known for his work of Christian theology, the Pensées, and for his defense of the Jansenist sect against the Jesuit order, but he also contributed to geometry and to the theory of probability, and explored the pneumatic phenomena studied by Torricelli. Pascal reasoned that if the column of mercury in a glass tube open at the bottom is held up by the pressure of the air, then the height of the column should decrease when the tube is carried to high altitude on a mountain, where there is less air overhead and hence lower air pressure.

., 22–23 astronomy and, 19–20, 51, 61, 91, 117, 151, 155, 264 deduction vs. observation and, 132 Francis Bacon and, 202 homework problem, 79–80 Kepler and, 163, 167, 171 magnetism and, 257 mathematics and, 17–20, 203 matter and, 10–13, 111, 274–79 medieval Europe and, 124, 127, 132 religion and, 45, 47, 50 Plato’s Universe (Vlastos), 6n Pleiades, 55, 176 Plotinus, 47 Plutarch, 70, 153–54 Pneumatics (Philo), 35 poetry, 1, 12–14 Pogson, Norman, 88n Polaris (North Star), 75 polygon, 275–76, 294–95 polyhedrons, regular, 10–12, 15, 18, 162–64, 171, 274–79 Pope, Alexander, 252–53 precession of apparent orbit of Sun around Earth, 113–14 of equinoxes, 74–75, 107, 118, 153, 241–42, 244, 248 of perihelia, 241, 244, 250 prediction, 154, 242–43, 265 Priestley, Joseph, 11 Principles of Philosophy (Descartes), 203–4, 212–13 Prior Analytics (Aristotle), 17 prism, 211, 218–20 probability, theory of, 199 Proclus, 51, 97–98 projectiles, thrown objects and, 27, 51, 71, 133, 135, 161, 193–94, 213, 342–46 proportions, theory of, 17 Protestantism, 156–57, 166, 253 protons, 243, 262–64 Prutenic Tables, 158, 166, 172 Ptolemy, Claudius, 48, 51, 79, 330 Arabs and, 105–7, 110, 112–14, 117–18, 141, 160 Aristotelian models vs., 95–99, 106, 112–14, 141–43, 160 chords vs. sines and, 309–11 Copernicus vs., 149–55, 255 Descartes and, 204 Earth’s rotation and, 135 epicycles and, 87, 255, 303–7 equal-angles and equants, 37, 87, 254–55, 324–25 experiment and, 189 Francis Bacon and, 201 Galileo and, 172–73, 179–80, 185 Kepler, 165, 167–68 lunar parallax and, 237n, 307–9 medieval Europe and, 126, 128–29, 141 planetary motion and, 51, 71–74, 87–96, 100, 137, 254–55 refraction and, 37, 79, 137, 330 Tycho and, 159–61, 165 Ptolemy I, 31–32, 35 Ptolemy II, 32 Ptolemy III, 40, 75 Ptolemy IV, 40, 75 Ptolemy XV, 31 pulmonary circulation, 118 Punic War, Second, 39 Pythagoras, 15, 47, 72 Pythagoreans, 15–20, 72, 78–79, 111, 141n, 151, 153–54, 279–82 Pythagorean theorem, 17, 283–84 quadratic equations, 15 quantum chromodynamics, 243 quantum electrodynamics, 180, 263, 268 quantum field theory, 262, 263–64 quantum mechanics, 21, 34, 152, 220, 248–49, 261–65, 268 quarks, 243, 263–65 quaternions, 163 Questiones quandam philosophicae (Newton), 217, 218 Qutb, Sayyid, 123 radiation, 261 radioactivity, 260–62, 264 radio astronomers, 159 radio waves, 259 Ragep, F.

pages: 626 words: 181,434

I Am a Strange Loop by Douglas R. Hofstadter


Albert Einstein, Andrew Wiles, Benoit Mandelbrot, Brownian motion, double helix, Douglas Hofstadter, Georg Cantor, Gödel, Escher, Bach, Isaac Newton, James Watt: steam engine, John Conway, John von Neumann, mandelbrot fractal, pattern recognition, Paul Erdős, place-making, probability theory / Blaise Pascal / Pierre de Fermat, publish or perish, random walk, Ronald Reagan, self-driving car, Silicon Valley, telepresence, Turing machine

The two phenomena involved — integer powers with arbitrarily large exponents, on the one hand, and Fibonacci numbers on the other — simply seemed (like gemstones and the Caspian Sea) to be too conceptually remote from each other to have any deep, systematic, inevitable interrelationship. And then along came a vast team of mathematicians who had set their collective bead on the “big game” of Fermat’s Last Theorem (the notorious claim, originally made by Pierre de Fermat in the middle of the seventeenth century, that no positive integers a, b, c exist such that an + bn equals c n, with the exponent n being an integer greater than 2). This great international relay team, whose final victorious lap was magnificently sprinted by Andrew Wiles (his sprint took him about eight years), was at last able to prove Fermat’s centuries-old claim by using amazing techniques that combined ideas from all over the vast map of contemporary mathematics.

Ted entelechy entrenchedness of “I”: in main brain; in other brains entwinement: of feedback loops; of human souls envelopes in box; not perceivable individually Epi (apparent marble in envelope box); parameters determining reality of; poem about; possible explanatory power of; seeming reality of Epimenides epiphenomena: in brain; in careenium; in envelope box; in minerals; in video feedback episodes as concepts episodic memory: central role of, in “I”-ness; containing precedents for new situations; of dogs; of human beings episodic projectory episodic subjunctory Erdös, Paul Ernst, Tom errors, study of Escher, Maurits Cornelis essence: extraction of, in brain; pinpointing of, as the goal of thought essential incompleteness essentially self-referential quality required to make an “I” esthetic pressures as affecting content études (Chopin) etymologies of words Euclid’s Elements Euclid’s proof of infinitude of primes Eugene Onegin (Pushkin) Euler, Leonhard Everest, Mount everyday concepts defining human reality; blurriness of evolution: of brain complexity; of careenium; and efficiency; of hearts; producing meanings in brains as accidental by-product; producing universality in brains as accidental by-product; throwing consciousness in as a bonus feature existence: blurriness of, of “I” experiences, as co-present with “I”-ness; as determining “I”, not vice versa; “pure”, as unrelated to physics experiencers vs. non-experiencers explanations, proper level of Exploratorium Museum exponential explosions extensible category system of humans; and representational universality; yielding consciousness extra bonus feature of consciousness extra-physical nature of consciousness F F numbers; see also Fibonacci fading afterglow of a soul failures, perception of one’s own faith in one’s own thought processes falafel, savored by two brains Falen, James falsity in mathematics, assumed equivalent to lacking proof Fauconnier, Gilles Fauré, Gabriel fear: of feedback loops; of self-reference; of self-representation in art feedback loops; and central goals of living creatures; content-free; and exponential growth; as germ of consciousness; in growth of human self-symbol; as instinctive taboo; irrelevant to hereness, for SL #642; level-crossing; see also strange loops feeding a formula its own Gödel number feeling one is elsewhere feeling posited to be independent of physics feeling vs. nonfeeling machines feelium, as stuff of experience and sensation Feigen, George Femme du boulanger, La (Pagnol), indirect meanings carried by analogy in Fermat, Pierre de Fermat’s Last Theorem, proof of fetus having no soul Fibonacci (Leonardo di Pisa) Fibonacci numbers; perfect mutual avoidance with powers fidelity of copy of another’s interiority Fields, W. C. fine-grained vs. coarse-grained loops first-person vs. third-person view of “I”; see also The Hard Problem first-person writing style fish, respect for life of flap loop; photo of flexibility, maximal, of machines flip side of Mathematician’s Credo flipping-around of perceptual system flirting with infinity float-ball in flush toilet fluid pointers flush-toilet fill mechanism food: for people in virtual environments; for virtual creatures fooled by the realism of Drawing Hands football: sensation swapped with baseball sensation; on television forces: in the brain; in physics; vs. desires forest vs. trees form–content interplay formula fragment in PM 43 and 49, as uninteresting integers foundations of mathematics, quest for stable fractalic gestalts of video feedback frames as brain structures Frank, Anne Franklin, Benjamin free will: of machine; as opposed to will; overridden by itself Free Willie Frege, Gottlob French people having flipped color qualia French translation: of Bach aria words; of “my leg is asleep” Freud, Sigmund friends: constituting threshold for consciousness; giving rise to “I”; personal identity of; self-inventorying sentences as; sharing of joys and pains of Frucht, William frustration of one’s will funerals, purpose of funneling of complex input into few symbols fusion, psychic: of Chaplin twins; of halves in Twinwirld; of husband and wife; of souls, as inevitable consequence of long-term intimate sharing G g (Gödel number of Gödel’s formula); bypassing of; code number of unpennable line in play “g is not prim”; see also KG galaxies: colliding in cyberspace; colliding in space; emerging from video feedback Galileo Galilei Galois, Évariste, as radical Gandhi, (Mahatma) Mohandas gargantuan integers gases, behavior of gemstones found in the Caspian Sea generosity, as bedrock cause for humans genes: as abstract entities; human, as source of potential strange loop; as patterns copied in different organisms; as physical entities; as remote from real life genetic code Gershwin, George Giant Electronic Brain gleb/knurk fence; see also knurking Glosspan, Aunt (in “Pig”) Glover, Henry glue: on dollops of Consciousness; on envelopes as source of Epi gluons gluttony vs. weight-watching goal-lacking and goal-possessing entities, perceived schism between goal-orientedness of systems, blurriness of goals of living creatures goals, shared God: elusiveness of mathematical truth and; not a player of dice; reality of Göd Gödel, Escher, Bach: central message of; dialogues in; linking author with Bach; typesetting of; video feedback photos in; writing of Gödel, Kurt; arithmetizes PM; believer in PM ’s consistency; birthyear of; as black belt; bypasses indexicality; code for PM symbols and formulas; concocts self-referential statement in PM; discovery of strange loop by; “God” in name of; growing up in Brünn; 1931 article of; as re-analyzer of what meaning is; respect for power of PM; sees analogy between Fibonacci numbers and PM theorems; sees causal potency of meaning of strings; sees representational richness of whole numbers; umlaut in name of; as young mathematician Gödel numbering; freedom in; revealing secondary meaning of Imp Gödel rays Gödelian formulas, infinitude of Gödelian swirl; ease of transportability; isomorphism of any two, at coarse-grained level; pointers in; see also strange loop Gödel’s formula KG: condensability of; described through abbreviations; downward causality and; hiddenness of higher-level view of; high-level and low-level meanings of; higher-level view fantasized as being obvious; inconceivable length of; lack of indexicals in; as repugnant to Bertrand Russell; translated with the indexical pronoun “I”; truth of; unprovability of Gödel’s formula KH, for Super-PM Gödel’s incompleteness theorem: as proven by Chaitin; as proven by Gödel Gödel’s Proof (Nagel and Newman) Gödel’s strange loop: as inevitable as video feedback; as prototype for “I”-ness Gödel–Turing threshold for universality “God’s-eye point of view” Goldbach Conjecture; decided by Göru Golden Rule Goldsmith, John good and bad, sense of, in growing self Göru: falling short of hopes; machine for solving all mathematical problems; machine for telling prims from non-prims; non-realizability of Gott ist unsre Zu versicht (Bach); counterfactual extension of Gould, Glenn gradations of consciousness Graham’s constant graininess: in representations of others; in video loops grandmother cells grand-music neuron grass and souls grasshopper rescued gravitation grazing of paradox Greater Metropolitan You “great soul” as hidden meaning of “magnanimous” and of “Mahatma” green button, fear of pushing Greg and Karen, Twinwirld couple Greg’l and Greg’r griffins, uncertain reality of “grocery store checkout stand”: causality attached to the concept of; as nested concept; vignette involving grocery stores’ meat displays growth rules, recursive, defining sequences of numbers growth rules, typographical, for strings in PM guinea pigs sacrificed for science gulf between truth and provability Gunkel, Pat guns fashioned from sandwiches Guru, machine that tells primes from non-primes Gutman, Hattie Gutman, Kellie H h (Gödel number of Gödel’s formula KH) “h is not a super-prim number” H1 (robot vehicle) halflings in Twinwirld hallucination hallucinated by hallucination halo: counterfactual, around personal identity; of each soul; of national souls halves, left and right, in Twinwirld hammerhead shark, location of hands-in-water experiment hangnails, reality of one’s own hardness and roundness of Epi Hard Problem, The; see also first-person hardware vs. patterns “hard-wiring”: of hemispheric links; hypothetically tweaked; of motor control as source of identity; of perceptual hardware as source of identity Hastorf, Albert H.

This particular ∨-flipping rule happens not to be one of PM’s rules of inference, but it could have been one. The point is just that this rule shows how one can mechanically shunt symbols and ignore their meanings, and yet preserve truth while doing so. This rule is rather trivial, but there are subtler ones that do real work. That, indeed, is the whole idea of symbolic logic, first suggested by Aristotle and then developed piecemeal over many centuries by such thinkers as Blaise Pascal, Gottfried Wilhelm von Leibniz, George Boole, Augustus De Morgan, Gottlob Frege, Giuseppe Peano, David Hilbert, and many others. Russell and Whitehead were simply developing the ancient dream of totally mechanizing reasoning in a more ambitious fashion than any of their predecessors had. Mechanizing the Mathematician’s Credo If you apply PM’s rules of inference to its axioms (the seeds that constitute the “zeroth generation” of theorems), you will produce some “progeny” — theorems of the “first generation”.

The Art of Computer Programming: Fundamental Algorithms by Donald E. Knuth


discrete time, distributed generation, Donald Knuth, fear of failure, Fermat's Last Theorem, Gerard Salton, Isaac Newton, Jacquard loom, Jacquard loom, John von Neumann, linear programming, linked data, Menlo Park, probability theory / Blaise Pascal / Pierre de Fermat, Richard Feynman, sorting algorithm, stochastic process, Turing machine

[For a rigorous study of fundamental concepts about the integers, see the article "On Mathematical Induction" by Leon Henkin, AMM 67 A960), 323-338.] The idea behind mathematical induction is thus intimately related to the concept of number. The first European to apply mathematical induction to rigorous proofs was the Italian scientist Francesco Maurolico, in 1575. Pierre de Fermat made further improvements, in the early 17th century; he called it the "method of infinite descent." The notion also appears clearly in the later writings of Blaise Pascal A653). The phrase "mathematical induction" appar- apparently was coined by A. De Morgan in the early nineteenth century. [See AMM 24 A917), 199-207; 25 A918), 197-201; Arch. Hist. Exact Sci. 9 A972), 1-21.] Further discussion of mathematical induction can be found in G. Polya's book Induction and Analogy in Mathematics (Princeton, N.J.: Princeton University Press, 1954), Chapter 7.

(r - k + 1) r- y-p r + 1 - j 3 ' C) r\ , = 0, integer k < 0. kj In particular cases we have 2— D) Table 1 gives values of the binomial coefficients for small integer values of r and k; the values for 0 < r < 4 should be memorized. Binomial coefficients have a long and interesting history. Table 1 is called "Pascal's triangle" because it appeared in Blaise Pascal's Traite du Triangle Arithmetique in 1653. This treatise was significant because it was one of the first works on probability theory, but Pascal did not invent the binomial coefficients (which were well-known in Europe at that time). Table 1 also appeared in the treatise Szu-yiian Yii-chien ("The Precious Mirror of the Four Elements") by the Chinese mathematician Shih-Chieh Chu in 1303, where they were said to be an old invention. The earliest known detailed discussion of binomial coefficients is in a tenth-century commentary, due to Halayudha, on an ancient Hindu classic, Pihgala's Chandah-sutra.

. | Letting Qn(z) = (z + n- l)/n, we have Q'n{l) = l/n, Qn{l) = 0; hence 1 11 mean(Qn) = -, var(Qn) = ~ • Finally, since Gn(z) = Ilfc=2 Qk{z), it follows that n n 1 mean(Gn) = ^ mean(Qfc) = ^ — = Hn — 1 k=2 k=2 var(Gn) = k=2 Summing up, we have found the desired statistics related to quantity A: A = f min 0, ave Hn — 1, max n — 1, dev JHn — Hn J . A6) The notation used in Eq. A6) will be used to describe the statistical character- characteristics of other probabilistic quantities throughout this book. We have completed the analysis of Algorithm M; the new feature that has appeared in this analysis is the introduction of probability theory. Elementary probability theory is sufficient for most of the applications in this book: The simple counting techniques and the definitions of mean, variance, and standard deviation already given will answer most of the questions we want to ask. More complicated algorithms will help us develop an ability to reason fluently about probabilities. Let us consider some simple probability problems, to get a little more practice using these methods.

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Structure and Interpretation of Computer Programs, Second Edition by Harold Abelson, Gerald Jay Sussman, Julie Sussman


Andrew Wiles, conceptual framework, Donald Knuth, Douglas Hofstadter, Eratosthenes, Gödel, Escher, Bach, industrial robot, information retrieval, iterative process, loose coupling, probability theory / Blaise Pascal / Pierre de Fermat, Richard Stallman, Turing machine, wikimedia commons

Tabulation can sometimes be used to transform processes that require an exponential number of steps (such as count-change) into processes whose space and time requirements grow linearly with the input. See Exercise 3.27. 35 The elements of Pascal’s triangle are called the binomial coefficients, because the row consists of the coefficients of the terms in the expansion of . This pattern for computing the coefficients appeared in Blaise Pascal’s 1653 seminal work on probability theory, Traité du triangle arithmétique. According to Knuth (1973), the same pattern appears in the Szu-yuen Yü-chien (“The Precious Mirror of the Four Elements”), published by the Chinese mathematician Chu Shih-chieh in 1303, in the works of the twelfth-century Persian poet and mathematician Omar Khayyam, and in the works of the twelfth-century Hindu mathematician Bháscara Áchárya. 36 These statements mask a great deal of oversimplification.

Now, assume that the result is true for all integers less than or equal to and establish the result for . Let be successive pairs in the reduction process. By our induction hypotheses, we have and . Thus, applying the claim we just proved together with the definition of the Fibonacci numbers gives , which completes the proof of Lamé’s Theorem. 44 If is a divisor of , then so is . But and cannot both be greater than . 45 Pierre de Fermat (1601-1665) is considered to be the founder of modern number theory. He obtained many important number-theoretic results, but he usually announced just the results, without providing his proofs. Fermat’s Little Theorem was stated in a letter he wrote in 1640. The first published proof was given by Euler in 1736 (and an earlier, identical proof was discovered in the unpublished manuscripts of Leibniz).

The Japanese National Fifth Generation Project: Introduction, survey, and evaluation. In Future Generation Computer Systems, vol. 9, pp. 105-117. –› Feeley, Marc. 1986. Deux approches à l’implantation du language Scheme. Masters thesis, Université de Montréal. –› Feeley, Marc and Guy Lapalme. 1987. Using closures for code generation. Journal of Computer Languages 12(1): 47-66. –› Feller, William. 1957. An Introduction to Probability Theory and Its Applications, volume 1. New York: John Wiley & Sons. Fenichel, R., and J. Yochelson. 1969. A Lisp garbage collector for virtual memory computer systems. Communications of the ACM 12(11): 611-612. –› Floyd, Robert. 1967. Nondeterministic algorithms. JACM, 14(4): 636-644. –› Forbus, Kenneth D., and Johan deKleer. 1993. Building Problem Solvers. Cambridge, MA: MIT Press. Friedman, Daniel P., and David S.

The Art of Computer Programming by Donald Ervin Knuth


Brownian motion, complexity theory, correlation coefficient, Donald Knuth, Eratosthenes, Georg Cantor, information retrieval, Isaac Newton, iterative process, John von Neumann, Louis Pasteur, mandelbrot fractal, Menlo Park, NP-complete, P = NP, Paul Erdős, probability theory / Blaise Pascal / Pierre de Fermat, RAND corporation, random walk, sorting algorithm, Turing machine, Y2K

The value c = — 2 should also be avoided, since the recurrence xm+1 = x2^ — 2 has solutions of the form xm — r2™ + r^2™. Other values of c do not seem to lead to simple relationships mod p, and they should all be satisfactory when used with suitable starting values. Richard Brent used a modification of Algorithm B to discover the prime factor 1238926361552897 of 2256 + 1. [See Math. Comp. 36 A981), 627-630; 38 A982), 253-255.] Fermat's method. Another approach to the factoring problem, which was used by Pierre de Fermat in 1643, is more suited to finding large factors than small .5.4 FACTORING INTO PRIMES 387 nes. [Fermat's original description of his method, translated into English, can e found in L. E. Dickson's monumental History of the Theory of Numbers 1 Carnegie Inst. of Washington, 1919), 357.] Assume that N = uv, where u < v. For practical purposes we may assume hat N is odd; this means that u and v are odd, and we can let x = (u + v)/2, y = {v-u)/2, A2) N = x2-y2, 0 < y < x < N.

Nondecimal number systems were discussed in Europe during the seven- seventeenth century. For many years astronomers had occasionally used sexagesimal 4.1 POSITIONAL NUMBER SYSTEMS 199 arithmetic both for the integer and the fractional parts of numbers, primarily when performing multiplication [see John Wallis, Treatise of Algebra (Oxford: 1685), 18-22, 30]. The fact that any integer greater than 1 could serve as radix was apparently first stated in print by Blaise Pascal in De Numeris Multiplicibus, which was written about 1658 [see Pascal's (Euvres Completes (Paris: Editions de Seuil, 1963), 84-89]. Pascal wrote, "Denaria enim ex instituto hominum, non ex necessitate naturae ut vulgus arbitratur, et sane satis inepte, posita est"; i.e., "The decimal system has been established, somewhat foolishly to be sure, according to man's custom, not from a natural necessity as most people would think."

Each of Chapters 3 and 4 can be used as the basis of a one-semester college course at the junior to graduate level. Although courses on "Random Numbers" and on "Arithmetic" are not presently a part of many college curricula, I be- believe the reader will find that the subject matter of these chapters lends itself nicely to a unified treatment of material that has real educational value. My own experience has been that these courses are a good means of introducing elementary probability theory and number theory to college students. Nearly all of the topics usually treated in such introductory courses arise naturally in connection with applications, and the presence of these applications can be an important motivation that helps the student to learn and to appreciate the theory. Furthermore, each chapter gives a few hints of more advanced topics that will whet the appetite of many students for further mathematical study.