probability theory / Blaise Pascal / Pierre de Fermat Concepts

Against the Gods: The Remarkable Story of Risk

by Peter L. Bernstein · 23 Aug 1996 · 415pp · 125,089 words

complex, and breakdowns can be catastrophic, with farreaching consequences. We must be constantly aware of the likelihood of malfunctions and errors. Without a command of probability theory and other instruments of risk management, engineers could never have designed the great bridges that span our widest rivers, homes would still be heated by

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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

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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

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worked out the theory of probability, even if the Chevalier de Mere had never confronted Pascal with his brainteaser. As the years passed, mathematicians transformed probability theory from a gamblers' toy into a powerful instrument for organizing, interpreting, and applying information. As one ingenious idea was piled on top of another, quantitative

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rolled dice for the universe, with Zeus winning the heavens, Poseidon the seas, and Hades, the loser, going to hell as master of the underworld. Probability theory seems a subject made to order for the Greeks, given their zest for gambling, their skill as mathematicians, their mastery of logic, and their obsession

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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

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's work became available for other mathematicians to build on, his generalizations about probabilities in gambling would have significantly accelerated the advance of mathematics and probability theory. He defined, for the first time, what is now the conventional format for expressing probability as a fraction: the number of favorable outcomes divided by

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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

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Paris and the Royal Society in London, which were founded about twenty years after Mersenne's death, were direct descendants of Mersenne's activities. Although Blaise Pascal's early papers in advanced geometry and algebra impressed the high-powered mathematicians he met at Abbe Mersenne's, he soon acquired a competing interest

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something to the investigation. In 1654, Pascal turned to Pierre de Carcavi, a member of Abbe Mersenne's group, who put him in touch with Pierre de Fermat, a lawyer in Toulouse. Pascal could not have approached anyone more competent to help him work out a solution to the problem of the points

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be proportional to that which they had the right to expect from fortune .... [T]his just distribution is known as the division." The principles of probability theory determine the division, because they determine the just distribution of the stakes. Seen in these terms, the Pascal-Fermat solution is clearly colored by the

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for his religious books, and, a short while later, took up residence in the monastery of Port-Royal in Paris. Yet traces of the old Blaise Pascal lingered on. He established the first commercial bus line in Paris, with all the profits going to the monastery of Port-Royal. In July 1660

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outstanding scholars in person. And he made a number of contributions to mathematics on his own, including an analysis of the use of conjecture and probability theory in applications of the law. To complicate matters further, Daniel Bernoulli had a brother five years older than he, also named Nicolaus; by convention, this

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the dice, but Bernoulli introduces us to the risk-taker-the player who chooses how much to bet or whether to bet at all. While probability theory sets up the choices, Bernoulli defines the motivations of the person who does the choosing. This is an entirely new area of study and body

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to fashion his primary case around the desire for wealth and opportunity. His emphasis was on decision-making rather than on the mathematical intricacies of probability theory. He announces at the outset that his aim is to establish "rules [that] would be set up whereby anyone could estimate his prospects from any

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and incorporate it into degrees of belief developed from prior information? Is the theory of probability a mathematical toy or a serious instrument for forecasting? Probability theory is a serious instrument for forecasting, but the devil, as they say, is in the details-in the quality of information that forms the basis

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chapter describes a sequence of giant steps over the course of the eighteenth century that revolutionized the uses of information and the manner in which probability theory can be applied to decisions and choices in the modern world. The first person to consider the linkages between probability and the quality of information

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of conjecture begins. In a sense, conjecture is the process of estimating the whole from the parts. Jacob's analysis begins with the observation that probability theory had reached the point where, to arrive at a hypothesis about the likelihood of an event, "it is necessary only to calculate exactly the number

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a proper academic position. He supported himself by tutoring in mathematics and by acting as a consultant to gamblers and insurance brokers on applications of probability theory. For that purpose, he maintained an informal office at Slaughter's Coffee House in St. Martin's Lane, where he went most afternoons after his

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personality who deserves a brief digression here; we shall encounter him again in Chapter 12. Gauss had been exploring some of the same areas of probability theory that had occupied Laplace's attention for many years. Like Gauss, Laplace had been a child prodigy in mathematics and had been fascinated by astronomy

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fraud in order to accommodate draft-dodgers. Cournot's remark that the average man would be some sort of monstrosity reflected his misgivings about applying probability theory to social as opposed to natural data. Human beings, he argued, lend themselves to a bewildering variety of classifications. Quetelet believed that a normally distributed

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book with an attack on traditional views of probability; many of our old friends are victims, including Gauss, Pascal, Quetelet, and Laplace. He declares that probability theory has little relevance to real-life situations, especially when applied with the "incautious methods and exaggerated claims of the school of Laplace."22 An objective

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nature is not troubled by difficulties of analysis, nor should humanity be so troubled. Keynes rejects the term "events" as used by his predecessors in probability theory, because it implies that forecasts must depend on the mathematical frequencies of past occurrences. He preferred the term "proposition," which reflects degrees of belief about

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is a synthesis of the ideas of Pascal, de Moivre, Bayes, Laplace, Gauss, Galton, Daniel Bernoulli, Jevons, and von Neumann and Morgenstern. It draws on probability theory, on sampling, on the bell curve and dispersion around the mean, on regression to the mean, and on utility theory. Markowitz has told me that

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black ball from two different urns, each holding 100 balls. Urn 1 held 50 balls of each color; the breakdown in Urn 2 was unknown. Probability theory would suggest that Urn 2 was also split 50-50, for there was no basis for any other distribution. Yet the overwhelming preponderance of the

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example of what happens when "oldfashioned human hunches" take over, had occurred only twenty years before Pascal and Fermat first laid out the principles of probability theory; the memory of it must still have been vivid when they began their historic deliberations. Perhaps they ignored the challenge of valuing an option because

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, forecasting tools based on nonlinear models or on computer gymnastics are subject to many of the same hurdles that stand in the way of conventional probability theory: the raw material of the model is the data of the past. The past seldom obliges by revealing to us when wildness will break out

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York: Carol Publishing Group, A Lyle Stuart Book. David, Florence Nightingale, 1962. Games, Gods, and Gambling. New York: Hafner Publishing Company.* Davidson, Paul, 1991. "Is Probability Theory Relevant for Uncertainty? A Post Keynesian Perspective." Journal of Economic Perspectives, Vol. 5, No. 1 (Winter), pp. 129-143. Davidson, Paul, 1996. "Reality and Economic

The Drunkard's Walk: How Randomness Rules Our Lives

by Leonard Mlodinow · 12 May 2008 · 266pp · 86,324 words

, which according to what we calculated above, would come to 55 percent. These three laws, simple as they are, form much of the basis of probability theory. Properly applied, they can give us much insight into the workings of nature and the everyday world. We employ them in our everyday decision making

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it “a wonderfully confusing little problem” and noted that “in no other branch of mathematics is it so easy for experts to blunder as in probability theory.” Of course, to a mathematician a blunder is an issue of embarrassment, but to a gambler it is an issue of livelihood. And so it

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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

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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

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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

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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

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1050, published by another Chinese mathematician, Zhu Shijie, in 1303, discussed in a work by Cardano in 1570, and plugged into the greater whole of probability theory by Pascal, who ended up getting most of the credit.10 But the prior work didn’t bother Pascal. “Let no one say I have

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to cash. How perfectly, he wondered, can the roulette wheels in Monte Carlo really work? The roulette wheel—invented, at least according to legend, by Blaise Pascal as he was tinkering with an idea for a perpetual-motion machine—is basically a large bowl with partitions (called frets) that are shaped like

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(New York: Warner Books, 1983), p. 41. 10. See Arthur De Vany, Hollywood Economics (Abington, U.K.: Routledge, 2004). 11. William Feller, An Introduction to Probability Theory and Its Applications, 2nd ed. (New York: John Wiley and Sons, 1957), p. 68. Note that for simplicity’s sake, when the opponents are tied

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: 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

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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?” Skeptical Inquirer 22, no. 5 (September/October 1998). 15.

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Daniel Kahneman, “Belief in the Law of Small Numbers,” Psychological Bulletin 76, no. 2 (1971): 105–10. 19. Jakob Bernoulli, quoted in L. E. Maistrov, Probability Theory: A Historical Sketch, trans. Samuel Kotz (New York: Academic Press, 1974), p. 68. 20. Stigler, The History of Statistics, p. 77. 21. E. T. Bell

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(2004): 5–8. 10. John Batt, Stolen Innocence (London: Ebury Press, 2005). 11. Stephen J. Watkins, “Conviction by Mathematical Error? Doctors and Lawyers Should Get Probability Theory Right,” BMJ 320 (January 1, 2000): 2–3. 12. “Royal Statistical Society Concerned by Issues Raised in Sally Clark Case,” Royal Statistical Society, London, news

Shape: The Hidden Geometry of Information, Biology, Strategy, Democracy, and Everything Else

by Jordan Ellenberg · 14 May 2021 · 665pp · 159,350 words

report on Bachelier’s thesis, emphasizing the modesty of his student’s goals: “[O]ne might fear that the author has exaggerated the applicability of Probability Theory as has often been done. Fortunately, this is not the case . . . he strives to set limits within which one can legitimately apply this type of

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the purely abstract space of the English language itself. PONDENOME OF DEMONSTURES OF THE REPTAGIN Markov’s original work was a purely abstract exercise in probability theory. Were there applications? “I am concerned only with questions of pure analysis,” Markov wrote in a letter. “I refer to the question of the applicability

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of probability theory with indifference.” According to Markov, Karl Pearson, the eminent statistician and biometrician, had “not done anything worthy of note.” Apprised some years later of the

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things. Our observation about primes isn’t just a fact, it’s a fact with a name: it’s called Fermat’s Little Theorem, after Pierre de Fermat, the first person to write it down.* No matter which prime number n you take, however large it may be, 2 raised to the nth

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one, he told Frénicle, which he would definitely have included in the letter “if he did not fear being too long.” This move is classic Pierre de Fermat. If you’ve heard his name at all, it’s not because of Fermat’s Little Theorem, but the other FLT, Fermat’s Last Theorem

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in games of chance, where the walk through the tree is always random, at least in part. Pierre de Fermat, when he wasn’t writing letters about prime numbers, was corresponding with the mathematician and mystic Blaise Pascal about the problem of the gambler’s ruin. In this game, Akbar and Jeff go head to

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exactly as quickly as it mounted. In fact it is none other than the normal distribution, or bell curve, which plays a central role in probability theory. The bell curve is treated, by people who know a little bit of math, with a kind of fetishistic reverence. And it does describe an

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and a music teacher who ran away from home at fourteen to be a magician in New York, then went to City College to study probability theory after a fellow practitioner told him it would make him better at cards. He met Martin Gardner, a fellow enthusiast of both math and magic

Radical Uncertainty: Decision-Making for an Unknowable Future

by Mervyn King and John Kay · 5 Mar 2020 · 807pp · 154,435 words

Solow, occupied adjoining offices at MIT for over half a century. As Samuelson relates, ‘When young he [Solow] would say, if you don’t regard probability theory as the most interesting subject in the world, then I feel sorry for you. I always agreed with that.’ 9 The appeal of

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probability theory is understandable. But we suspect the reason that such mathematics was, as we shall see, not developed until the seventeenth century is that few real-

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human reasoning reportedly began when the Chevalier de Méré, an inveterate gambler, sought the advice of the mathematician and philosopher Blaise Pascal. Pascal in turn consulted an even more distinguished French polymath, Pierre de Fermat. The resulting exchange of letters between Pascal and Fermat in the winter of 1653–4 represents the first formal analysis

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scientific reasoning in the seventeenth century; such reasoning was a prerequisite for the Industrial Revolution and the unprecedented economic growth it generated. The advance of probability theory would contribute to that economic development through the creation of markets in risk. The first venues for managing risk were the coffee shops of London

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is hardly surprising that subsequent generations have been prone to exaggerate the scope of these powerful ideas. By the early twentieth century, the value of probability theory was well established in understanding games of chance, and in the analysis of data which are generated by a stationary process. The achievements of the

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established the existence of God from the premiss of total ignorance.’ 19 Keynes certainly had in mind the famous ‘wager’ of Pascal, the founder of probability theory: ‘God is, or He is not. Reason can decide nothing here . . . you must wager. It is not optional . . . Let us weigh the gain and the

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only darkness everywhere. Nature presents to me nothing which is not matter of doubt and concern . . . The true course is not to wager at all. —BLAISE PASCAL , Pensées 1 B y the early twentieth century, the uses of probability were well established in understanding games of chance, such as cards, roulette, or

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had the upper hand. In his 1843 System of Logic , the British philosopher John Stuart Mill criticised the French mathematician Pierre-Simon Laplace for applying probability theory ‘to things of which we are completely ignorant’. 2 Another French mathematician, Joseph Bertrand, went further. 3 He lambasted his countrymen for making absurd assumptions

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a strong sense of the uncertainty of the logical basis upon which it seems to rest. —John Maynard Keynes 1 F rom the beginnings of probability theory, mathematicians realised that a further logical step was necessary to translate that theory into advice as to when to gamble and when to keep your

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), chapter 11.3.1. 13 Bayes is buried in Bunhill Fields Cemetery in the City of London. 14 An excellent discussion of the development of probability theory may be found in Daston (1995). 15 The original problem is from Selvin et al. (1975) p. 67 who named the host Monte (sic ) Hall

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and his sister, who was a member of the economics faculty at Oxford, wrote a moving memoir which gave special emphasis to his contributions to probability theory (Paul 2012). 9 Ramsey (1926). 10 de Finetti (1989) p. 219 – although do not expect to understand his reasoning! 11 A few economists continued to

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be lognormal . 5 Table 205, Statistical Abstract of the United States: 2011 , p. 135. 6 The distribution was first published by Poisson, together with his probability theory, in 1837 in his work Recherches sur la Probabilité des Jugements en Matière Criminelle et en Matière Civile . 7 Zipf (1935 and 1949). 8 Technically

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Hole, 317–18 economics: adverse selection process, 250–1 , 327 ; aggregate output and GDP, 95 ; ambiguity of variables/concepts, 95–6 , 99–100 ; appeal of probability theory, 42–3 ; ‘bubbles’, 315–16 ; business cycles, 45–6 , 347 ; Chicago School, 36 , 72–4 , 86 , 92 , 111–14 , 133–7 , 158 , 257–8 , 307

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, 53 , 56 , 57 , 59–60 , 106 ; Pascal’s wager, 64 , 80 ; posterior distribution, 100 ; ‘probabilistic turn’ in human thought, 20 , 49 , 53–4 , 55–68 ; probability theory, 42–3 , 55 , 58 , 59–68 , 69–70 , 71–2 , 105 ; the problem of points, 59–60 , 61 , 64–5 , 106 , 113 ; puzzle-mystery distinction

Alex's Adventures in Numberland

by Alex Bellos · 3 Apr 2011 · 437pp · 132,041 words

, 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

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the erotic effect of eatg something labelled with the number 284, while a partner was eating something labelled 220. It was only in 1636 that Pierre de Fermat discovered the second set of amicable numbers: 17,296 and 18,416. Because of the advent of computer processing, more than 11 million amicable pairs

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to Reno to meet the mathematician who sets the odds for more than half of the world’s slot machines. His job has historical pedigree – probability theory was first conceived in the sixteenth century by the gambler Girolamo Cardano, our Italian friend we met earlier when discussing cubic equations. Rarely, however, has

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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

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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. The first question

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up? 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

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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. How many times do you need to throw a pair of dice so that it

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bandits; they are also illustrative in explaining how the insurance industry works. Insurance is very much like playing the slots. Both are systems built on probability theory in which the losses of almost everyone pay for the winnings of a few. And both can be fantastically profitable for those controlling the payback

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] occurs for millions of trials in succession, and yet this is what a good coin will do rather regularly,’ he wrote in An Introduction to Probability Theory and Its Applications. ‘If a modern educator or psychologist were to describe the long-run case histories of individual coin-tossing games, he would classify

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wider discipline, one that was less about statecraft and more about the mathematics of collective behaviour. He could not have done this without advances in probability theory, which provided techniques to analyse the randomness in data. In Brussels in 1853 Quételet hosted the first international conference on statistics. Quételet’s insights on

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is the sum of the two numbers above it. Pascal’s triangle with only squares divisible by 2 in white. The triangle is named after Blaise Pascal, even though he was a latecomer to its charms. Indian, Chinese and Persian mathematicians were all aware of the pattern centuries before he was. Unlike

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its first appearance in mathematics literature in de Moivre’s 1718 book on gaming called The Doctrine of Chances. This was the first textbook on probability theory, and another example of how scientific knowledge flourished thanks to gambling. I’ve been treating the bell curve as if it is one curve, when

Zero: The Biography of a Dangerous Idea

by Charles Seife · 31 Aug 2000 · 233pp · 62,563 words

, Science, Wired UK, The Sciences, and numerous other publications. He holds an M.S. in mathematics from Yale University and his areas of research include probability theory and artificial intelligence. His most recent book is Alpha & Omega. He lives in Washington, D.C. zero The Biography of a Dangerous Idea [CHARLES SEIFE

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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

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the time of the Pascals’ conversion, a friend of Étienne’s—a military engineer—came to visit and repeated Torricelli’s experiment for the Pascals. Blaise Pascal was enthralled, and started performing experiments of his own, using water, wine, and other fluids. The result was New experiments concerning the vacuum, published in

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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

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God. Probability theory was invented to help rich aristocrats win more money with their gambling. Pascal’s theory was extremely successful, but

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,000,000. Which envelope should you choose? B, obviously! Its value is much greater. It is not difficult to show this using a tool from probability theory called an expectation, which is a measure of how much we expect each envelope to be worth. Envelope A might or might not have a

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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

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number of important new theorems, but Desargues’s colleagues couldn’t understand his terminology and concluded that Desargues was nuts. Though a few mathematicians, like Blaise Pascal, picked up on Desargues’s work, it was forgotten. None of this mattered to Jean-Victor Poncelet. As Monge’s student, Poncelet had learned the

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. Carol Volk. Chicago: University of Chicago Press, 1991. The Epic of Gilgamesh. Trans. N. K. Sanders. London: Penguin Books, 1960. Feller, William. An Introduction to Probability Theory and Its Applications. New York: John Wiley and Sons, 1950. Feynman, Richard. QED. Princeton, N.J.: Princeton University Press, 1985. Feynman, Richard, et al. The

Human Compatible: Artificial Intelligence and the Problem of Control

by Stuart Russell · 7 Oct 2019 · 416pp · 112,268 words

dice games as his main example. (Unfortunately, his work was not published until 1663.13) In the seventeenth century, French thinkers including Antoine Arnauld and Blaise Pascal began—for assuredly mathematical reasons—to study the question of rational decisions in gambling.14 Consider the following two bets: A: 20 percent chance of

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: he actually made paper wheels inscribed with symbols, by means of which he could generate logical combinations of assertions. The great seventeenth-century French mathematician Blaise Pascal was the first to develop a real and practical mechanical calculator. Although it could only add and subtract and was used mainly in his father

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, who went on to win the 2011 Turing Award, had been working on methods for uncertain reasoning based in probability theory.44 AI researchers gradually accepted Pearl’s ideas; they adopted the tools of probability theory and utility theory and thereby connected AI to other fields such as statistics, control theory, economics, and operations

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truth in any particular situation can be checked. Perhaps the most famous theorem is Fermat’s Last Theorem, which was conjectured by the French mathematician Pierre de Fermat in 1637 and finally proved by Andrew Wiles in 1994 after 357 years of effort (not all of it by Wiles).1 The theorem can

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, or into some still-to-be-invented hybrid design. Appendix C UNCERTAINTY AND PROBABILITY Whereas logic provides a general basis for reasoning with definite knowledge, probability theory encompasses reasoning with uncertain information (of which definite knowledge is a special case). Uncertainty is the normal epistemic situation of an agent in the real

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the seventeenth century, only recently has it become possible to represent and reason with large probability models in a formal way. The basics of probability Probability theory shares with logic the idea that there are possible worlds. One usually starts out by defining what they are—for example, if I am rolling

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(sometimes called outcomes): 1, 2, 3, 4, 5, 6. Exactly one of them will be the case, but a priori I don’t know which. Probability theory assumes that it is possible to attach a probability to each world; for my die roll, I’ll attach 1/6 to each world. (These

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usually have little license plates that uniquely identify them. Identity is something our minds sometimes attach to objects for our own purposes. The combination of probability theory with an expressive formal language is a fairly new subfield of AI, often called probabilistic programming.4 Several dozen probabilistic programming languages, or PPLs, have

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, initially published anonymously, is often called The Port-Royal Logic: Antoine Arnauld, La logique, ou l’art de penser (Chez Charles Savreux, 1662). See also Blaise Pascal, Pensées (Chez Guillaume Desprez, 1670). 15. The concept of utility: Daniel Bernoulli, “Specimen theoriae novae de mensura sortis,” Proceedings of the St. Petersburg Imperial Academy

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, 241 preferences. See human preferences preference utilitarianism, 220 Price, Richard, 54 pride, 230–31 Primitive Expounder, 133 prisoner’s dilemma, 30–31 privacy, 70–71 probability theory, 21–22, 273–84 Bayesian networks and, 275–77 first-order probabilistic languages, 277–80 independence and, 274 keeping track of not directly observable phenomena

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UBI (universal basic income), 121 uncertainty AI uncertainty as to human preferences, principle of, 53, 175–76 human uncertainty as to own preferences, 235–37 probability theory and, 273–84 United Nations (UN), 250 universal basic income (UBI), 121 Universal Declaration of Human Rights (1948), 107 universality, 32–33 universal Turing machine

When Einstein Walked With Gödel: Excursions to the Edge of Thought

by Jim Holt · 14 May 2018 · 436pp · 127,642 words

qualms, we are given to believe, was psychological as well as mathematical. Borel retreated from the abstractions of set theory to the safer ground of probability theory. “Je vais pantoufler dans les probabilités,” as he charmingly put it (“I’m going to dally with probability”; pantoufler literally means “play around in my

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: How could the real infinite, which was supposed to be an attribute of God alone, be present in the finite world he created? It was Blaise Pascal who was most agitated by this question. None of his contemporaries embraced the idea of the infinite more passionately than did Pascal. And no one

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had frustrated other mathematicians. Nonstandard analysis has since found many adherents in the international mathematical community, especially in France, and has been fruitfully applied to probability theory, physics, and economics, where it is well suited to model, say, the infinitesimal impact that a single trader has on prices. Beyond his achievement as

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which it must travel. But this can’t be right, can it? Our explanation for the route taken by the light beam—first formulated by Pierre de Fermat in the seventeenth century as the “principle of least time”—assumes that the light somehow knows where it is going in advance and that it

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”?) Paul Erdős (1913–1996) was one of the supreme mathematicians of the last century. He was also one of the world’s leading experts on probability theory; indeed, something he invented called the probabilistic method is often simply referred to as the Erdős method—thus making his name synonymous with probability. In

Artificial Intelligence: A Modern Approach

by Stuart Russell and Peter Norvig · 14 Jul 2019 · 2,466pp · 668,761 words

’s history. In the early decades, rational agents were built on logical foundations and formed definite plans to achieve specific goals. Later, methods based on probability theory and machine learning allowed the creation of agents that could make decisions under uncertainty to attain the best expected outcome. In a nutshell, AI has

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and Economic Behavior (1944). Economics is no longer the study of money; rather it is the study of desires and preferences. Decision theory, which combines probability theory with utility theory, provides a formal and complete framework for individual decisions (economic or otherwise) made under uncertainty—that is, in cases where probabilistic descriptions

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of the environment that are of current interest. Most work on this problem has been done for stochastic, continuous-state environments with the tools of probability theory, as explained in Chapter 14. In this section we will show an example in a discrete environment with deterministic sensors and nondeterministic actions. The example

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black king even if the latter tries to avoid this fate, since Black cannot keep guessing the right evasive moves indefinitely. In the terminology of probability theory, detection occurs with probability 1. The KBNK endgame—king, bishop and knight versus king—is won in this sense; White presents Black with an infinite

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how detailed they want to be in specifying their model, and what details they want to leave out. We will see in Chapter 12 that probability theory allows us to summarize all the exceptions without explicitly naming them. 7.7.2A hybrid agent The ability to deduce various aspects of the state

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be true, believes it to be false, or has no opinion. These logics therefore have three possible states of knowledge regarding any sentence. Systems using probability theory, on the other hand, can have any degree of belief, or subjective likelihood, ranging from 0 (total disbelief) to 1 (total belief). It is important

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not to confuse the degree of belief in probability theory with the degree of truth in fuzzy logic. Indeed, some fuzzy systems allow uncertainty (degree of belief) about degrees of truth. For example, a probabilistic

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sees a car parked on the street, one is normally willing to believe that it has four wheels even though only three are visible. Now, probability theory can certainly provide a conclusion that the fourth wheel exists with high probability; yet, for most people, the possibility that the car does not have

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becomes inappropriate, even though there is no new evidence of faulty brakes. These considerations have led researchers to consider how to embed default reasoning within probability theory or utility theory. 10.6.2Truth maintenance systems We have seen that many of the inferences drawn by a knowledge representation system will have only

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with methods for the specific tasks of representing and updating the belief state over time and predicting outcomes. Chapter 18 looks at ways of combining probability theory with expressive formal languages such as firstorder logic and general-purpose programming languages. Chapter 15 covers utility theory in more depth, and Chapter 16 develops

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possible worlds are strictly ruled out (all those in which the assertion is false), probabilistic assertions talk about how probable the various worlds are. In probability theory, the set of all possible worlds is called the sample space. The possible worlds are mutually exclusive and exhaustive—two possible worlds cannot both be

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, that is, particular possible worlds. A fully specified probability model associates a numerical probability P(ω) with each possible world.1 The basic axioms of probability theory say that every possible world has a probability between 0 and 1 and that the total probability of the set of possible worlds is 1

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example, we might ask for the probability that the two dice add up to 11, the probability that doubles are rolled, and so on. In probability theory, these sets are called events—a term already used extensively in Chapter 10 for a different concept. In logic, a set of worlds corresponds to

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: For example, when rolling fair dice, we have P(Total = 11) = P((5, 6))+ P((6, 5)) = 1/36 + 1/36 = 1/18. Note that probability theory does not require complete knowledge of the probabilities of each possible world. For example, if we believe the dice conspire to produce the same number

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world is represented by a set of variable/value pairs. A more expressive structured representation is also possible, as shown in Chapter 18. Variables in probability theory are called random variables, and their names begin with an uppercase letter. Thus, in the dice example, Total and Die1 are random variables. Every random

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) and (12.5) are often called Kolmogorov’s axioms in honor of the mathematician Andrei Kolmogorov, who showed how to build up the rest of probability theory from this simple foundation and how to handle the difficulties caused by continuous variables.4 While Equation (12.2) has a definitional flavor, Equation (12

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with choosing values for the probabilities of logically related propositions: If Agent 1 expresses a set of degrees of belief that violate the axioms of probability theory then there is a combination of bets by Agent 2 that guarantees that Agent 1 will lose money every time. For example, suppose that Agent

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satisfies these axioms. The world being the way it is, however, practical demonstrations sometimes speak louder than proofs. The success of reasoning systems based on probability theory has been much more effective than philosophical arguments in making converts. We now look at how the axioms can be deployed to make inferences. 12

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need probability to tell us how likely it is. What this section has shown is that even seemingly complicated problems can be formulated precisely in probability theory and solved with simple algorithms. To get efficient solutions, independence and conditional independence relationships can be used to simplify the summations required. These relationships often

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develop formal representations for such relationships as well as algorithms that operate on those representations to perform probabilistic inference efficiently. Summary This chapter has suggested probability theory as a suitable foundation for uncertain reasoning and provided a gentle introduction to its use. •Uncertainty arises because of both laziness and ignorance. It is

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unobserved aspects of the world, thereby improving on the decisions of a purely logical agent. Conditional independence makes these calculations tractable. Bibliographical and Historical Notes Probability theory was invented as a way of analyzing games of chance. In about 850 CE the Indian mathematician Mahaviracarya described how to arrange a set of

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was posthumous (1663). By that time, probability had been established as a mathematical discipline due to a series of results from a famous correspondence between Blaise Pascal and Pierre de Fermat in 1654. The first published textbook on probability was De Ratiociniis in Ludo Aleae (On Reasoning in a Game of Chance) by Huygens (1657

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nothing but the want of art.” The connection between probability and reasoning dates back at least to the nineteenth century: in 1819, Pierre Laplace said, “Probability theory is nothing but common sense reduced to calculation.” In 1850, James Maxwell said, “The true logic for this world is the calculus of Probabilities, which

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inherent. More recent work by Bacchus, Grove, Halpern, and Koller (1992) extends Carnap’s methods to first-order theories. The first rigorously axiomatic framework for probability theory was proposed by Kolmogorov (1950, first published in German in 1933). Renyi (1970) later gave an axiomatic presentation that took conditional probability, rather than absolute

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arguments for the validity of the axioms, Cox (1946) showed that any system for uncertain reasoning that meets his set of assumptions is equivalent to probability theory. This gave renewed confidence to probability fans, but others were not convinced, objecting to the assumption that belief must be represented by a single number

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for the surprising success of naive Bayesian reasoning even in domains where the independence assumptions are clearly violated. There are many good introductory textbooks on probability theory, including those by Bertsekas and Tsitsiklis (2008), Ross (2015), and Grinstead and Snell (1997). DeGroot and Schervish (2001) offer a combined introduction to probability and

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(Pearl and McKenzie, 2018), provides a less mathematical but more readable and wide-ranging introduction. Uncertain reasoning in AI has not always been based on probability theory. As noted in Chapter 12, early probabilistic systems fell out of favor in the early 1970s, leaving a partial vacuum to be filled by alternative

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as a competing approach to probability. Pearl (1988) and Ruspini et al. (1992) analyze the relationship between the Dempster-Shafer theory and standard probability theory. In many cases, probability theory does not require probabilities to be specified exactly: we can express uncertainty about probability values as (second-order) probability distributions, as explained in Chapter

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to fuzzy set theory; papers on fuzzy applications are collected in Zimmermann (1999). Fuzzy logic has often been perceived incorrectly as a direct competitor to probability theory, whereas in fact it addresses a different set of issues: rather than considering uncertainty about the truth of well-defined propositions, fuzzy logic handles vagueness

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systems and has much in common with probability (Dubois and Prade, 1994). Many AI researchers in the 1970s rejected probability because the numerical calculations that probability theory was thought to require were not apparent to introspection and presumed an unrealistic level of precision in our uncertain knowledge. The development of qualitative probabilistic

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values. Goldszmidt and Pearl (1996) take a similar approach. Work by Darwiche and Ginsberg (1992) extracts the basic properties of conditioning and evidence combination from probability theory and shows that they can also be applied in logical and default reasoning. Several excellent texts (Jensen, 2007; Darwiche, 2009; Koller and Friedman, 2009; Korb

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states in terms of which world states were possible, but could say nothing about which states were likely or unlikely. In this chapter, we use probability theory to quantify the degree of belief in elements of the belief state. As we show in Section 14.1, time itself is handled in the

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uncertain world—at least as much as possible and on average. In this chapter, we fill in the details of how utility theory combines with probability theory to yield a decision-theoretic agent—an agent that can make rational decisions based on what it believes and what it wants. Such an agent

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highway. Summary This chapter shows how to combine utility theory with probability to enable an agent to select actions that will maximize its expected performance. •Probability theory describes what an agent should believe on the basis of evidence, utility theory describes what an agent wants, and decision theory puts the two together

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general-purpose inference algorithms, roughly analogous to sound and complete logical inference algorithms such as resolution. There are two routes to introducing expressive power into probability theory. The first is via logic: to devise a language that defines probabilities over first-order possible worlds, rather than the propositional possible worlds of Bayes

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the same sense as “temporal logic”—a logical system specialized for probabilistic reasoning. To apply probability logic to tasks such as proving interesting theorems in probability theory, a more expressive language was needed. Gaifman (1964a) proposed a first-order probability logic, with possible worlds being first-order model structures and with probabilities

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(1990) and Bacchus (1990) also built on Gaifman’s approach, exploring some of the basic knowledge representation issues from the perspective of AI rather than probability theory and mathematical logic. The subfield of probabilistic databases also has logical sentences labeled with probabilities (Dalvi et al., 2009)—but in this case probabilities are

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). The Most Human Human. Doubleday. Christin, A., Rosenblat, A., and Boyd, D. (2015). Courts and predictive algorithms. Data & Civil Rights. Chung, K. L. (1979). Elementary Probability Theory with Stochastic Processes (3rd edition). SpringerVerlag. Church, A. (1936). A note on the Entscheidungsproblem. JSL, 1, 40–41 and 101–102. Church, A. (1956). Introduction

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. Mathematical model. Econometrica, 17, 200–211. Darwiche, A. (2001). Recursive conditioning. AIJ, 126, 541. Darwiche, A. and Ginsberg, M. L. (1992). A symbolic generalization of probability theory. In AAAI-92. Darwiche, A. (2009). Modeling and reasoning with Bayesian networks. Cambridge University Press. Darwin, C. (1859). On The Origin of Species by Means

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., Kavukcuoglu, K., Ranzato, M., and LeCun, Y. (2009). What is the best multi-stage architecture for object recognition? In ICCV-09. Jaynes, E. T. (2003). Probability Theory: The Logic of Science. Cambridge Univ. Press. Jeffrey, R. C. (1983). The Logic of Decision (2nd edition). University of Chicago Press. Jeffreys, H. (1948). Theory

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Implementing Dynamical Systems. MIT Press. Renner, G. and Ekart, A. (2003). Genetic algorithms in computer aided design. ComputerAided Design, 35, 709–726. Rényi, A. (1970). Probability Theory. Elsevier. Resnick, P. and Varian, H. R. (1997). Recommender systems. CACM, 40, 56–58. Rezende, D. J., Mohamed, S., and Wierstra, D. (2014). Stochastic backpropagation

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NeurIPS 25. Solomonoff, R. J. (1964). A formal theory of inductive inference. Information and Control, 7, 1–22, 224–254. Solomonoff, R. J. (2009). Algorithmic probability–theory and applications. In Emmert-Streib, F. and Dehmer, M. (Eds.), Information Theory and Statitical Learning. Springer. Sondik, E. J. (1971). The Optimal Control of Partially

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. N. (1998). Statistical Learning Theory. Wiley. Vapnik, V. N. and Chervonenkis, A. Y. (1971). On the uniform convergence of relative frequencies of events to their probabilities. Theory of Probability and Its Applications, 16, 264–280. Vardi, M. Y. (1996). An automata-theoretic approach to linear temporal logic. In Moller, F. and Birtwistle

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model (abstract description of reality), 84 small-scale, 31 model (in logic), 232, 364, 272, 295, 345 model (in machine learning), 669, 671 model (in probability theory), 407 model-based reflex agents, 78 reinforcement learning, 841, 966 vision, 988 MODEL-BASED-REFLEX-AGENT, 71 model-free agent, 74 Model-free reinforcement learning

Fermat’s Last Theorem

by Simon Singh · 1 Jan 1997 · 289pp · 85,315 words

in the fact that the problem itself is supremely simple to understand. It is a puzzle that is stated in terms familiar to every schoolchild. Pierre de Fermat was a man in the Renaissance tradition, who was at the centre of the rediscovery of ancient Greek knowledge, but he asked a question that

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tragedy, involving all the greatest heroes of mathematics. 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

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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. To convey the extraordinary character of Fermat and his contribution to mathematics, which goes far

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that I have been able to convey the creativity and heroism that was required during Wiles’s ten-year ordeal. In telling the tale of Pierre de Fermat and his baffling riddle I have tried to describe the mathematical concepts without resorting to equations, but inevitably x, y and z do occasionally rear

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which has its roots in ancient Greece, but which only reached full maturity in the seventeenth century. It was then that the great French mathematician Pierre de Fermat inadvertently set it as a challenge for the rest of the world. One great mathematician after another had been humbled by Fermat’s legacy and

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to any higher number, finding whole number solutions turns from being relatively simple to being mind-bogglingly difficult. In fact, the great seventeenth-century Frenchman Pierre de Fermat made the astonishing claim that the reason why nobody could find any solutions was that no solutions existed. Fermat was one of the most brilliant

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. No clues were ever found as to what Fermat’s proof might have been. In Chapter 2 we shall find out more about the mysterious Pierre de Fermat and how his theorem came to be lost, but for the time being it is enough to know that Fermat’s Last Theorem, a problem

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something like a mushroom on stilts – who solves partial differential equations mentally; and even he’s given up.’ Arthur Porges, ‘The Devil and Simon Flagg’ Pierre de Fermat was born on 20 August 1601 in the town of Beaumont-de-Lomagne in south-west France. Fermat’s father, Dominique Fermat, was a wealthy

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. Mersenne travelled throughout France and further afield, spreading news of the latest discoveries. In his travels he would make a point of meeting up with Pierre de Fermat and, indeed, seems to have been Fermat’s only regular contact with other mathematicians. Mersenne’s influence on this Prince of Amateurs must have been

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to suffer jealous nit-picking. 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

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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

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Fermat and Pascal would discover the first proofs and cast-iron certainties in probability theory, a subject which is inherently uncertain. Pascal’s interest in the subject had been sparked by a professional Parisian gambler, Antoine Gombaud, the Chevalier de

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infinite excitement and one worth playing, because multiplying an infinite prize by a finite probability results in infinity. As well as sharing the parentage of probability theory, Fermat was deeply involved in the founding of another area of mathematics, calculus. Calculus is the ability to calculate the rate of change, known as

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used to describe Newton’s law of gravity and his laws of mechanics, which depend on distance, velocity and acceleration. The discovery of calculus and probability theory would have been more than enough to earn Fermat a place in the mathematicians’ hall of fame, but his greatest achievement was in yet another

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branch of mathematics. While calculus has since been used to send rockets to the moon, and while probability theory has been used for risk assessment by insurance companies, Fermat’s greatest love was for a subject which is largely useless – the theory of numbers

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books which made up the Arithmetica, only six would survive the turmoils of the Dark Ages and go on to inspire the Renaissance mathematicians, including Pierre de Fermat. The remaining seven books would be lost during a series of tragic events which would send mathematics back to the age of the Babylonians. During

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Omar and now the Turks, a few precious volumes of the Arithmetica made their way back to Europe. Diophantus was destined for the desk of Pierre de Fermat. Birth of a Riddle Fermat’s judicial responsibilities occupied a great deal of his time, but what little leisure he had was devoted entirely to

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might have seemed like a foolhardy dream but young Andrew was right in thinking that he, a twentieth-century schoolboy, knew as much mathematics as Pierre de Fermat, a genius of the seventeenth century. Perhaps in his naïvety he would stumble upon a proof which other more sophisticated minds had missed. Despite his

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simple trick which had eluded the great professors of the past. The keen twentieth-century amateur was to a large extent on a par with Pierre de Fermat when it came to knowledge of mathematical techniques. The challenge was to match the creativity with which Fermat used his techniques. Within a few weeks

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mathematicians, professional and amateur, who were persisting in their attempts to prove Fermat’s Last Theorem – perhaps Fermat’s Last Theorem was undecidable! What if Pierre de Fermat had made a mistake when he claimed to have found a proof? If so, then there was the possibility that the Last Theorem was undecidable

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could be undecidable. In conclusion, Fermat’s Last Theorem might be true, but there may be no way of proving it. The Compulsion of Curiosity Pierre de Fermat’s casual jotting in the margin of Diophantus’ Arithmetica had led to the most infuriating riddle in history. Despite three centuries of glorious failure and

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solutions, namely Proving that this elliptic equation has only one set of whole number solutions is an immensely difficult task, and in fact it was Pierre de Fermat who discovered the proof. You might remember that in Chapter 2 it was Fermat who proved that 26 is the only number in the universe

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the most brilliant mathematician in the world, and after 350 years of frustration number theorists believed that they had at last got the better of Pierre de Fermat. Now Wiles was faced with the humiliation of having to admit that he had made a mistake. Before confessing to the error he decided to

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-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

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, 60–61 death 67 education 37 and elliptic equations 184 and Father Mersenne 41–2 ill with plague 38–9 observations and theorems 70–73 probability theory 43–4, 45–6 reluctant to reveal proofs 42 Fermat’s Last Theorem challenge of 72–4 computers unable to prove 177–8 Miyaoka’s