Against the Gods: The Remarkable Story of Risk (13 page)

Recognizing his son's genius, Blaise's father introduced him at the
age of 14 into a select weekly discussion group that met at the home of
a Jesuit priest named Marin Mersenne, located near the Place Royal in
Paris. Abbe Mersenne had made himself the center of the world of science and mathematics during the first half of the 1600s. In addition to
bringing major scholars together at his home each week, he reported by
mail to all and sundry, in his cramped handwriting, on what was new
and significant.'

In the absence of learned societies, professional journals, and other
means for the exchange of ideas and information, Mersenne made a
valuable contribution to the development and dissemination of new
scientific theories. The Academie des Sciences in 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. In 1646, his father
fell on the ice and broke his hip; the bonesetters called in to take care of
M. Pascal happened to be members of a proselytizing Catholic sect called
Jansenists. These people believed that the only path to salvation was
through aceticism, sacrifice, and unwavering attachment to the strait and
narrow. They preached that a person who fails to reach constantly for
ever-higher levels of purity will slip back into immorality. Emotion and
faith were all that mattered; reason blocked the way to redemption.

After repairing the hip of Pascal pere, the Jansenists stayed on for
three months to work on the soul of Pascalfils, who accepted their doctrine with enthusiasm. Now Blaise abandoned both mathematics and
science, along with the pleasures of his earlier life as a man about town. Religion commanded his full attention. All he could offer by way of
explanation was to ask, "Who has placed me here? By whose order and
warrant was this place and this time ordained for me? The eternal
silence of these infinite spaces leaves me in terror."3

The terror became so overwhelming that in 1650, at the age of 27,
Pascal succumbed to partial paralysis, difficulty in swallowing, and
severe headaches. As a cure, his doctors urged him to rouse himself and
resume his pleasure-seeking ways. He lost no time in taking their
advice. When his father died, Pascal said to his sister: "Let us not grieve
like the pagans who have no hope."4 In his renewed activities he
exceeded even his earlier indulgences and became a regular visitor to
the gambling tables of Paris.

Pascal also resumed his researches into mathematics and related subjects. In one of his experiments he proved the existence of vacuums, a
controversial issue ever since Aristotle had declared that nature abhors
a vacuum. In the course of that experiment he demonstrated that barometric pressure could be measured at varying altitudes with the use of
mercury in a tube emptied of all air.

About this time, Pascal became acquainted with the Chevalier de
Mere, who prided himself on his skill at mathematics and on his ability
to figure the odds at the casinos. In a letter to Pascal some time in the
late 1650s, he boasted, "I have discovered in mathematics things so rare
that the most learned of ancient times have never thought of them and
by which the best mathematicians in Europe have been surprised."5

Leibniz himself must have been impressed, for he described the
Chevalier as "a man of penetrating mind who was both a gambler and
a philosopher." But then Leibniz must have had second thoughts, for
he went on to say, "I almost laughed at the airs which the Chevalier de
Mere takes on in his letter to Pascal."6

Pascal agreed with Leibniz. "M. de Mere," he wrote to a colleague,
"has good intelligence but he is not a geometer and this, as you realize, is
a great defect."' Here Pascal sounds like the academic who takes pleasure
in putting down a non-academic. In any case, he underestimated de Mere 8

Yet Pascal himself is our source of information about de Mere's intuitive sense of probabilities. The Chevalier bet repeatedly on outcomes with just a narrow margin in his favor, outcomes that his opponents
regarded as random results. According to Pascal, de Mere knew that the
probability of throwing a 6 with one die rises above 50% with four
throws-to 51.77469136%. The Chevalier's strategy was to win a tiny
amount on a large number of throws in contrast to betting the chateau on
just a few. That strategy also required large amounts of capital, because a
6 might fail to show up for many throws in a row before it appeared in
a cluster that would bring its average appearance to over 50%.9

De Mere tried a variation on his system by betting that sonnez-the
term for double-six-had a better than 50% probability of showing up
on 24 throws of two dice. He lost enough money on these bets to learn
that the probability of double-six was in fact only 49.14% on 24 throws.
Had he bet on 25 throws, where the probability of throwing sonnez
breaks through to 50.55%, he would have ended up a richer man. The
history of risk management is written in red as well as in black.

At the time he first met Pascal, the Chevalier was discussing with a
number of French mathematicians Paccioli's old problem of the pointshow should two players in a game of balla share the stakes when they
leave the game uncompleted? No one had yet come up with an answer.

Although the problem of the points fascinated Pascal, he was reluctant to explore it on his own. In today's world, this would be the topic
for a panel at an annual meeting of one of the learned societies. In
Pascal's world, no such forum was available. A little group of scholars
might discuss the matter in the intimacy of Abbe Mersenne's home, but
the accepted procedure was to start up a private correspondence with
other mathematicians who might be able to contribute 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. Fermat's erudition was awesome.i0 He spoke all the principal European languages and
even wrote poetry in some of them, and he was a busy commentator on
the literature of the Greeks and Romans. Moreover, he was a mathematician of rare power. He was an independent inventor of analytical
geometry, he contributed to the early development of calculus, he did
research on the weight of the earth, and he worked on light refraction
and optics. In the course of what turned out to be an extended corre spondence with Pascal, he made a significant contribution to the theory
of probability.

But Fermat's crowning achievement was in the theory of numbersthe analysis of the structure that underlies the relationships of each individual number to all the others. These relationships present countless
puzzles, not all of which have been resolved to this very day. The
Greeks, for example, discovered what they called perfect numbers,
numbers that are the sum of all their divisors other than themselves, like
6 = 1 + 2 + 3. The next-higher perfect number after 6 is 28 = 1 + 2
+ 4 + 7 + 14. The third perfect number is 496, followed by 8,128. The
fifth perfect number is 33,550,336.

Pythagoras discovered what he called amicable numbers, "One who
is the other I," numbers whose divisors add up to each other. All the
divisors of 284, which are 1, 2, 4, 71, and 142, add up to 220; all the divisors of 220, which are 1, 2, 4, 5, 10, 11, 20, 22, 44, 55, and 110, add up
to 284.

No one has yet devised a rule for finding all the perfect numbers or
all the amicable numbers that exist, nor has anyone been able to explain
all the varying sequences in which they follow one another. Similar difficulties arise with prime numbers, numbers like 1, 3, or 29, that are
divisible only by 1 and by themselves. At one point, Fermat believed he
might have discovered a formula that would always produce a prime
number as its solution, but he warned that he could not prove theoretically that the formula would always do so. His formula produced 5, then
17, then 257, and finally 65,537, all of which were prime numbers; the
next number to result from his formula was 4,294,967,297.

Fermat is perhaps most famous for propounding what has come to
be known as "Fermat's Last Theorem," a note that he scribbled in the
margin of his copy of Diophantus's book Arithmetic. The notion is simple to describe despite the complexity of its proof.

The Greek mathematician Pythagoras first demonstrated that the
square of the longest side of a right triangle, the hypotenuse, is equal to
the sum of the squares of the other two sides. Diophantus, an early
explorer into the wonders of quadratic equations, had written a similar
expression: x4 + y4 + z4 = u2. "Why," asks Fermat, "did not Diophantus
seek two [rather than three] fourth powers such that their sum is square?
The problem is, in fact impossible, as by my method I am able to prove
with all rigor."11 Fermat observes that Pythagoras was correct that a2 + b2 = c2, but a3 + b3 would not be equal to c3, nor would any integer higher than 2 fit the bill: the Pythagorean theorem works only for squaring.

And then Fermat wrote: "I have a truly marvelous demonstration of this proposition which this margin is too narrow to contain."12 With this simple comment he left mathematicians dumbfounded for over 350 years as they struggled to find a theoretical justification for what a great deal of empirical experimentation proved to be true. In 1993, an English mathematician named Andrew Wiles claimed that he had solved this puzzle after seven years of work in a Princeton attic. Wiles's results were published in the Annals of Mathematics in May 1995, but the mathematicians have continued to squabble over exactly what he had achieved.

Fermat's Last Theorem is more of a curiosity than an insight into how the world works. But the solution that Fermat and Pascal worked out to the problem of the points has long since been paying social dividends as the cornerstone of modem insurance and other forms of risk management.

The solution to the problem of the points begins with the recognition that the player who is ahead when the game stops would have the greater probability of winning if the game were to continue. But how much greater are the leading player's chances? How small are the lagging player's chances? How do these riddles ultimately translate into the science of forecasting?

The 1654 correspondence between Pascal and Fermat on this subject signaled an epochal event in the history of mathematics and the theory of probability.*
In response to the Chevalier de Mere's curiosity about the old problem, they constructed a systematic method for analyzing future outcomes. When more things can happen than will happen, Pascal and Fermat give us a procedure for determining the likelihood of each of the possible results-assuming always that the outcomes can be measured mathematically.

They approached the problem from different standpoints. Fermat turned to pure algebra. Pascal was more innovative: he used a geomet ric format to illuminate the underlying algebraic structure. His methodology is simple and is applicable to a wide variety of problems in probability.

The basic mathematical concept behind this geometric algebra had
been recognized long before Fermat and Pascal took it up. Omar
Khayyam had considered it some 450 years earlier. In 1303, a Chinese
mathematician named Chu Shih-chieh, explicitly denying any originality, approached the problem by means of a device that he called the
"Precious Mirror of the Four Elements." Cardano had also mentioned
such a device.13

Chu's precious mirror has since come to be known as Pascal's
Triangle. "Let no one say that I have said nothing new," boasts Pascal
in his autobiography. "The arrangement of the subject is new. When
we play tennis, we both play with the same ball, but one of us places it
better." 14