Bernstein’s Against the Gods: A Remarkable Story of Risk gives insight into historical and current manifestations of risk. In his unique analysis of the role of risk in our society, Bernstein argues that the notion of bringing risk under control is one of the central ideas that distinguishes modern times from the semi-remote past. I say “semi-remote” because he links the distant past and the bridges similarities with a more current past. Against the Gods chronicles a significant intellectual adventure that liberated humanity from oracles and basic notions; by means of the powerful tools of risk management that are available to us today and been enhanced. In this paper, I will be evaluating and discussing a specific chapter in Against the Gods: A Remarkable Story of Risk: The French Connection. A story of three French men who made major contributions to risk management. I will be providing an analysis, suggestions, and take-away from this section of the book and risk management altogether.
The definition of risk varies by situation. A basic definition of risk is, “the probability of the occurrence with some accuracy” (Hofmann, Risk and Risk Perception). It is essentially the possibility of loss or gain and the degree of probability of each of these possible outcomes. A situation in which a loss is possible is referred to as a loss exposure. This is risk as we know it today. Thousands of years before Internet users could play online poker, people in different ancient civilizations played games with dice and bones. Also, people played games that evolved into chess and checkers over two thousand years ago. Some historical evidence that gaming gave rise to probability theory, important to risk management, comes from writings by Dante and Galileo. The famous mathematicians which will be discussed in this paper, Blaise Pascal and Pierre de Fermat, wrote each other about games of chance in the 1600s, a correspondence that is believed to have given rise to modern probability theory used today.
Twelve years after Galileo Galilei’s death, three French men: Blaise Pascal, Pierre de Fermat, and the Chevalier de Méré commenced their exploration of probability analysis. This development is the subject of this chapter. These men saw beyond the traditional methods and approaches to gambling, a common practice in their day, and devised a foundation for estimating probability. Probability is the likelihood that an event will occur. Bernstein states that, “what had been just a rudimentary idea became a fully developed theory that opened the way to significant practical applications” (Bernstein, 1996 p.57). The theory provided challenging complex numbers to make decisions rather than merely taking educated guesses on beliefs.
The first contributor, Blaise Pascal, was a distinguished mathematician and “occasional philosopher”. He was often torn between pursuing either mathematics or religion. The conflict between these stems from the matter that math and science inadvertently force the reinterpretation of religion, where its teachings conflicts with apparent truth. This gap Pascal could never bridge becomes relevant later. Pascal’s interest in mathematics might have been bred by his father who was also a mathematician as well as a tax collector. In his early years, Pascal invented what would today be a calculator for his father’s accounts. This was an important contribution because, as any calculator would, it facilitated addition, subtraction, multiplying, and dividing. I think the fact that he took it upon himself to create this calculator says a lot about his interest in mathematics. Inventing such device, especially in his time considering the lack of tools, highlights his qualities of patience, innovation, and erudition. Sometime later Pascal’s father had an accident and injured his hip, those called to take care of him were Catholic preachers called Jansenists. Their ideologies emphasized human depravity, original sin, averting mortality, and the necessity of divine grace. According to the Jansenists, “emotion and faith were all that mattered: reason blocked away the way to redemption” (Bernstein, 1996 p.59). This encounter was important because Pascal accepted their teachings and abandoned math, science, and his usual hobbies; now religion predominated. Overwhelmed with questions and a lack of answers, he endured a partial paralysis of difficulty swallowing and unbearable headaches. Doctors advised him to return to his old ways. As a result, Pascal continued his research of mathematics and science and began gambling.
There on after, Pascal familiarized himself with our second French man, another adept mathematician, the Chevalier de Méré. De Méré’s ability was figuring out the odds at casinos. France has a rich history associated with gambling. Thanks to the dynamic development of gambling in France, casinos were created in the eighteenth century. His strategy was to “win a tiny amount on a large number of throws in contrast to betting the house on just a few” (Bernstein, 1996 p.61). This strategy also required large amounts of capital. When Pascal met de Méré, de Méré was discussing the problem of the points, a classical problem in probability theory, with other French men. Pascal wanted to analyze this on his own and was put in touch with Pierre de Fermat to aid him. At this point our third French mathematician whose talent was analytical geometry is introduced, Pierre de Fermat. The correspondence that followed between Fermat and Pascal, was fundamental in the development of probability.
The question that surrounds the problem of the points is, how should the total stake be equitably divided when a game is terminated prematurely. “It’s clear that the player who is closer to winning should get a larger share of the stakes. Since the player who had won more points is closer to winning, the player with more points should receive a larger share of the stakes. Bernstein poses the questions, “how much greater are the leading players chances? How small are the lagging players chances? How do these riddles ultimately translate into the science of forecasting?” (Bernstein, 1996 p.63). So, how do we quantify the differential? Both Pascal and Fermat solved the problem but used different reasoning. While Fermat employs an algebraic approach, Pascal used a geometric format to illustrate the fundamental algebraic structure. Pascal’s developed the Pascal’s Triangle. Pascal’s Triangle is an infinite equilateral triangle of numbers that follows a rule of adding the two numbers above to get the number below. Two of the sides are all 1’s, and because the triangle is infinite there is no bottom side. [Pascal’s triangle will be found at the end of this paper].
For purposes of simplification, Bernstein discusses an example of the baseball World Series. An event we are familiar with. What is the probability that team A will win the World Series after it has lost the first game? The game will now be determined by who wins four out of six games instead of four out of seven. Now we must figure out how many combinations of wins and losses there are by using Pascal’s Triangle. The question Bernstein challenges is, “How many different sequences of six games are possible, and how many of those victories and losses would result in your team winning the four games it needs for victory?” (Bernstein, 1996 p.65). The relevant row of the Triangle is the seventh row 1 6 15 20 15 6 1. Therefore, there are 22 possible combinations were team A will win, 1+6+15=22, and 42 combinations where team B would win the Series. In conclusion, the probability is 22/64.
Another example of Pascal’s approach is presented by the Richard J. Larsen in An Introduction to Probability and Its Applications:
Suppose A and B are playing a series of games where the winner of the overall match is the first player to win a total of k games. However, the contest is interrupted before either competitor has achieved that number. If A needs to take m more games to win the match, while B is n short of the necessary k, how should the stakes be divided? (Larsen 147).
Pascal asserts that if one player requires n more victories to win the stakes, and the other lacks m victories, then the amount of the stakes given to the former should be proportional to sum of the first m entries in the (n+m)th row of Pascal's triangle, divided by the sum of the entries in that row. Similarly, the second player should receive the proportion of the sum of the last n entries of that row, divided by the sum of that same row. Here is Pascal's triangle for row (1+3) = 4, arranged in the matrix format used by Pascal, so that the "rows" involved are diagonal rows, read from lower left to upper right:
1 1 1 1
1 2 3
1 3
1
The first player would receive 7/8 of the share, while the second would receive 1/8th. Pascal and Fermat conclude that the share of the stakes that is received by a player should be proportional to their probability of winning if the game were to continue at the point it was stopped.
Comparatively, Fermat also came up with his mathematical discovery. Other than his help in founding probability, Fermat is most known for “Fermats Last Theorem. It should be noted that this work was based off initial findings of Pythagoras during the 6th century. Fermats Last Theorem states that there are no natural numbers (1, 2, 3,…) x, y, and z such that xn + yn = zn, in which n is a natural number greater than 2. For example, if n = 3, Fermat’s last theorem states that no natural numbers x, y, and z exist such that x3 + y3 = z3. The equation is now to the power 3, rather than the power 2. Fermat concluded that among the infinity of numbers there were none that fitted this new equation. Whereas Pythagoras’ equation had many possible solutions, Fermat claimed that his equation was insoluble. Fermat went even further, believing that if the power of the equation were increased further, then these equations would also have no solutions:
x3 + y3 = z3,
x4 + y4 = z4,
x5 + y5 = z5,
x6 + y6 = z6,
:
:
The mathematical short-hand for this family of insoluble equations is:
xn + yn = zn, where n is any number greater than 2.
According to Fermat, none of these equations could be solved and he noted this in the margin of his Arithmetica. To back up his theorem he had developed an argument or mathematical proof, and following the first marginal note he comments:
It is impossible for a cube to be the sum of two cubes, a fourth power to be the sum of two fourth powers, or in general for any number that is a power greater than the second to be the sum of two like powers. I have discovered a truly marvelous demonstration of this proposition that this margin is too narrow to contain."
Fermat believed he could prove his theorem, but he never committed his proof to paper. After his death, mathematicians across Europe tried to rediscover the proof of what became and is now known as Fermat’s Last Theorem.
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After interpreting the theorems and correspondences, Bernstein introduces Ian Hacking, a Canadian philosopher and leading world scholar. He specializes in fields of philosophy and the history of science. Hacking’s early research focused on the concept of probability and statistical inference. In this, he shows how, during the second half of the nineteenth and the first part of the twentieth century, probability theory and statistics came to change both science and our everyday life. Pascal combined his thoughts about life and religion in Pensées. Pascal’s Wager questions, “God is, or he is not. Which way should we incline? Reason cannot answer” (Bernstein, 1996 p.69).
“Pascal’s line of analysis to answer this question is the beginning of the theory of decision making,” says Hacking (Bernstein, 1996 p. 69). Considered as a descriptive account of human choice, decision theory, has come to serve as the foundation for modern economics. For our purposes, however, it is unnecessary to claim that people do, for the most part, act as the theory says they should. For us it is enough to treat the theory as a normative account: an account of how a rational agent "ought" to act in any given circumstance. People usually make decisions based on past experiences and perceived outcomes. Although, this doesn’t apply to Pascal’s notion of Gods existence. In this case, you either believe and act accordingly, or don’t. According to Bernstein, the insight that directs Pascal to a decision is, “the only way to choose between a bet that God exists and a bet that there is no God down that infinite distance of Pascal’s coin-tossing game is to decide whether an outcome in which God’s exists is preferable—more valuable in some sense- than an outcome in which God does not exist, even though the probability may be 50-50" (Bernstein, 1996 p.70). Pascal argues that argues that if we do not know whether God exists then we should play it safe rather than risk being sorry. The winner of the bet that God does exist has the possibility of salvation, and logically salvation is better than eternal damnation. The conclusion that Pascal’s Wager draws from this is that belief in the Christian God is the rational course of action, even if there is no evidence that he exists. If the Christian God does not exist, then it is of little importance whether we believe or disbelieve in him. If the Christian God does exist, then it is of great importance that we do believe in him. Pascal’s Wager is an attempt to justify belief in God not with an appeal to evidence for his existence but rather with an appeal to self-interest. It is in our interests to believe in the God of Christianity, the argument suggests, and it is therefore rational for us to do so.
In the fifth part of the book, Bernstein touches upon a book by Pascal’s associates, whom primary authorship is attributed to Antoine Arnauld. The final parts of the book cover the process of developing a hypothesis from a limited set of facts, best known as statistical inference. According to the encyclopedia of philosophy, said book is divided into four parts: conceiving, judging, reasoning and ordering. Conceiving is “the simple view we have of things that present themselves to the mind” and is what we do when we represent things to the mind in the form of ideas before making judgments about them. Judging is the act of bringing together different ideas in the mind and affirming or denying one or the other. Reasoning is the “action of the mind in which it forms a judgment from several others.” Ordering is the mental action of arranging “ideas, judgments and reasoning’s” in such a way that the arrangement is “best suited for knowing the subject”. The final chapter of the book recounts a game where there are ten players who risk one coin in hopes of winning the 9 coins of their opponents. According to Bernstein, the author points out that there are “nine degrees of probability of losing a coin for only one of gaining nine” (Bernstein, 1996 p. 71). And, according to Hacking this is the first time probability has been measured.
In conclusion, Blaise Pascal introduced the first systematic mathematical method for calculating probability of future events. The problem of the points is often referred to as a catalyst for the start of the probability theory. Reveals how real-world situations prompt the creation of new mathematics. Pascal and Fermat provided different mathematical solutions to the problem of how two players split the stakes in a game when they leave a game uncompleted. Blaise Pascal, the “father of the modern theory of decision-making”, constructed a systematic method for analyzing the probability of future outcomes using a simple triangle, Pascal’s triangle. While Pierre de Fermat’s Last Theorem states that there are no natural numbers (1, 2, 3,…) x, y, and z such that xn + yn = zn, in which n is a natural number greater than 2. As a final point, understanding and redefining the history and risk of probability can help us make decisions in our own life as well as in the finance, insurance, management and health care industries.