Against the Gods: The Remarkable Story of Risk (12 page)

But the real hero of the story, then, is not Cardano but the times in
which he lived. The opportunity to discover what he discovered had
existed for thousands of years. And the Hindu-Arabic numbering system had arrived in Europe at least three hundred years before Cardano
wrote Liber de Ludo Aleae. The missing ingredients were the freedom of
thought, the passion for experimentation, and the desire to control the
future that were unleashed during the Renaissance.

The last Italian of any importance to wrestle with the matter of
probability was Galileo, who was born in 1564, the same year as
William Shakespeare. By that time Cardano was already an old man.16
Like so many of his contemporaries, Galileo liked to experiment and
kept an eye on everything that went on around him. He even used his
own pulse rate as an aid in measuring time.

One day in 1583, while attending a service in the cathedral in Pisa,
Galileo noticed a lamp swaying from the ceiling above him. As the
breezes blew through the drafty cathedral, the lamp would swing irregularly, alternating between wide arcs and narrow ones. As he watched,
he noted that each swing took precisely the same amount of time, no
matter how wide or narrow the arc. The result of this casual observation was the introduction of the pendulum into the manufacture of
clocks. Within thirty years, the average timing error was cut from fifteen minutes a day to less than ten seconds. Thus was time married to
technology. And that was how Galileo liked to spend his time.

Nearly forty years later, while Galileo was employed as the First and
Extraordinary Mathematician of the University of Pisa and Mathematician to His Serenest Highness, Cosimo II, the Grand Duke of
Tuscany, he wrote a short essay on gambling "in order to oblige him
who has ordered me to produce what occurs to me about the prob lem."17 The title of the essay was Sopra le Scoperte dei Dadi (On Playing
Dice). The use of Italian instead of Latin suggests that Galileo had no
great relish for a topic that he considered unworthy of serious consideration. He appears to have been performing a disagreeable chore in order
to improve the gambling scores of his employer, the Grand Duke.

In the course of the essay, Galileo retraces a good deal of Cardano's
work, though Cardano's treatise on gambling would not be published
for another forty years. Yet Galileo may well have been aware of
Cardano's achievement. Florence Nightingale David, historian and statistician, has suggested that Cardano had thought about these ideas for
so long that he must surely have discussed them with friends. Moreover
he was a popular lecturer. So mathematicians might very well have
been familiar with the contents of Liber de Ludo Aleae, even though they
had never read it.18

Like Cardano, Galileo deals with trials of throwing one or more
dice, drawing general conclusions about the frequency of various combinations and types of outcome. Along the way, he suggests that the
methodology was something that any mathematician could emulate.
Apparently the aleatory concept of probability was so well established
by 1623 that Galileo felt there was little more to be discovered.

Yet a great deal remained to be discovered. Ideas about probability
and risk were emerging at a rapid pace as interest in the subject spread
through France and on to Switzerland, Germany, and England.

France in particular was the scene of a veritable explosion of mathematical innovation during the seventeenth and eighteenth centuries
that went far beyond Cardano's empirical dice-tossing experiments.
Advances in calculus and algebra led to increasingly abstract concepts
that provided the foundation for many practical applications of probability, from insurance and investment to such far-distant subjects as
medicine, heredity, the behavior of molecules, the conduct of war, and
weather forecasting.

The first step was to devise measurement techniques that could be
used to determine what degree of order might be hidden in the uncertain future. Tentative efforts to devise such techniques were under way
early in the seventeenth century. In 1619, for example, a Puritan minister named Thomas Gataker published an influential work, Of the
Nature and Use of Lots, in which he argued that natural law, not divine
law, determined the outcome of games of chance.19 By the end of the seventeenth century, about a hundred years after the death of Cardano
and less than fifty years after the death of Galileo, the major problems
in probability analysis had been resolved. The next step was to tackle
the question of how human beings recognize and respond to the probabilities they confront. This, ultimately, is what risk management and
decision-making are all about and where the balance between measurement and gut becomes the focal point of the whole story.

either Cardano nor Galileo realized that he was on the verge of
articulating the most powerful tool of risk management ever to
be invented: the laws of probability. Cardano had proceeded
from a series of experiments to some important generalizations, but he was
interested only in developing a theory of gambling, not a theory of probability. Galileo was not even interested in developing a theory of gambling.

Galileo died in 1642. Twelve years later, three Frenchmen took a
great leap forward into probability analysis, an event that is the subject
of this chapter. And less than ten years after that, what had been just a
rudimentary idea became a fully developed theory that opened the way
to significant practical applications. A Dutchman named Huygens published a widely read textbook about probability in 1657 (carefully read
and noted by Newton in 1664); at about the same time, Leibniz was
thinking about the possibility of applying probability to legal problems;
and in 1662 the members of a Paris monastery named Port-Royal produced a pioneering work in philosophy and probability to which they
gave the title of Logic. In 1660, an Englishman named John Graunt
published the results of his effort to generalize demographic data from
a statistical sample of mortality records kept by local churches. By the
late 1660s, Dutch towns that had traditionally financed themselves by
selling annuities were able to put these policies on a sound actuarial
footing. By 1700, as we mentioned earlier, the English government was
financing its budget deficits through the sale of life annuities.

The story of the three Frenchmen begins with an unlikely trio who
saw beyond the gaming tables and fashioned the systematic and theoretical foundations for measuring probability. The first, Blaise Pascal, was a
brilliant young dissolute who subsequently became a religious zealot
and ended up rejecting the use of reason. The second, Pierre de Fermat,
was a successful lawyer for whom mathematics was a sideline. The third
member of the group was a nobleman, the Chevalier de Mere, who
combined his taste for mathematics with an irresistible urge to play
games of chance; his fame rests simply on his having posed the question
that set the other two on the road to discovery.

Neither the young dissolute nor the lawyer had any need to experiment in order to confirm their hypotheses. Unlike Cardano, they
worked inductively in creating for the first time a theory of probability.
The theory provided a measure of probability in terms of hard numbers,
a climactic break from making decisions on the basis of degrees of belief.

Pascal, who became a celebrated mathematician and occasional
philosopher, was born in 1623, just about the time Galileo was putting
the finishing touches on Sopra le Scoperte dei Dadi. Born in the wake of
the religious wars of the sixteenth century, Pascal spent half his life torn
between pursuing a career in mathematics and yielding to religious
convictions that were essentially anti-intellectual. Although he was a
brilliant mathematician and proud of his accomplishments as a "geomaster," his religious passion ultimately came to dominate his life.'

Pascal began life as a child prodigy. He was fascinated with shapes
and figures and discovered most of Euclidean geometry on his own by
drawing diagrams on the tiles of his playroom floor. At the age of 16,
he wrote a paper on the mathematics of the cone; the paper was so
advanced that even the great Descartes was impressed with it.

This enthusiasm for mathematics was a convenient asset for Pascal's
father, who was a mathematician in his own right and earned a comfortable living as a tax collector, a functionary known at the time as a
tax farmer. The tax farmer would advance money to the monarch-the
equivalent of planting his seeds-and then go about collecting it from
the citizenry-the equivalent of gathering in a harvest whose ultimate
value, as with all farmers, he hoped would exceed the cost of the seeds.

While Pascal was still in his early teens, he invented and patented a
calculating machine to ease the dreary task of adding up M. Pascal's
daily accounts. This contraption, with gears and wheels that went forward and backward to add and subtract, was similar to the mechanical
calculating machines that served as precursors to today's electronic calculators. The young Pascal managed to multiply and divide on his
machine as well and even started work on a method to extract square
roots. Unfortunately for the clerks and bookkeepers of the next 250
years, he was unable to market his invention commercially because of
prohibitively high production costs.