Against the Gods: The Remarkable Story of Risk (10 page)

Paccioli himself sensed the power that the miracle of numbers could
unleash. In the course of the Summa, he poses the following problem:

A and B are playing a fair game of balla. They agree to continue until
one has won six rounds. The game actually stops when A has won
five and B three. How should the stakes be divided?3

This brain-teaser appears repeatedly in the writings of mathematicians during the sixteenth and seventeenth centuries. There are many
variations but the question is always the same: How do we divide the
stakes in an uncompleted game? The answers differed and prompted
heated debates.

The puzzle, which came to be known as the problem of the points,
was more significant than it appears. The resolution of how to divide the
stakes in an uncompleted game marked the beginning of a systematic
analysis of probability-the measure of our confidence that something is
going to happen. It brings us to the threshold of the quantification of risk.

While we can understand that medieval superstitions imposed a
powerful barrier to investigations into the theory of probability, it is
interesting to speculate once again about why the Greeks, or even the
Romans, had no interest in puzzles like Paccioli's.

The Greeks understood that more things might happen in the
future than actually will happen. They recognized that the natural sciences are "the science of the probable," to use Plato's terminology. Aristotle, in De Caelo says, "To succeed in many things, or many times,
is difficult; for instance, to repeat the same throw ten thousand times
with the dice would be impossible, whereas to make it once or twice is
comparatively easy."4

Simple observation would have confirmed these statements. Yet
the Greeks and the Romans played games of chance by rules that make
no sense in our own times. This failure is all the more curious, because
these games were popular throughout antiquity (the Greeks were
already familiar with six-sided dice) and provided a lively laboratory for
studying odds and probabilities.

Consider the games played with astragali, the bones used as dice.
These objects were oblong, with two narrow faces and two wide faces.
The games usually involved throwing four astragali together. The odds
of landing on a wide face are obviously higher than the odds of landing
on a narrow face. So one would expect the score for landing on a narrow face to be higher than the score for landing on a wide face. But the
total scores received for landing on the more difficult narrow faces-1
on one face and 6 on the other-was identical to the scores for the easier wide faces-3 and 4. The "Venus" throw, a play in which each of
the four faces-1, 3, 4, 6-appear, earned the most, although equally
probable throws of 6, 6, 6, 6 or 1, 1, 1, 1 earned less.5

Even though it was common knowledge that long runs of success,
or of failure, were less probable than short runs, as Aristotle had pointed
out, those expectations were qualitative, not quantitative: ". . . to make
it once or twice is comparatively easy."6 Though people played these
games with insatiable enthusiasm, no one appears to have sat down to
figure the odds.

In all likelihood the reason was that the Greeks had little interest in
experimentation; theory and proof were all that mattered to them.
They appear never to have considered the idea of reproducing a certain
phenomenon often enough to prove a hypothesis, presumably because
they admitted no possibility of regularity in earthly events. Precision
was the monopoly of the gods.

By the time of the Renaissance, however, everyone from scientists
to explorers and from painters to architects was caught up in investiga tion, experimentation, and demonstration. Someone who threw a lot of
dice would surely be curious about the regularities that turned up over
time.

A sixteenth-century physician named Girolamo Cardano was just
such a person. Cardano's credentials as a gambling addict alone would
justify his appearance in the history of risk, but he demonstrated extraordinary talents in many other areas as well. The surprise is that
Cardano is so little known. He is the quintessential Renaissance man.7

Cardano was born in Milan about 1500 and died in 1571, a precise
contemporary of Benvenuto Cellini. And like Cellini he was one of the
first people to leave an autobiography. Cardano called his book De Vita
Propria Liber (The Book Of My Life) and what a life it was! Actually,
Cardano's intellectual curiosity was far stronger than his ego. In his
autobiography, for example, he lists the four main achievements of the
times in which he lived: the new era of exploration into the two-thirds
of the world that the ancients never knew, the invention of firearms
and explosives, the invention of the compass, and the invention of
printing from movable type.

Cardano was a skinny man, with a long neck, a heavy lower lip, a
wart over one eye, and a voice so loud that even his friends complained about it. According to his own account, he suffered from diarrhea, ruptures, kidney trouble, palpitations, even the infection of a
nipple. And he boasted, "I was ever hot-tempered, single-minded, and
given to women" as well as "cunning, crafty, sarcastic, diligent, impertinent, sad, treacherous, magician and sorcerer, miserable, hateful, lascivious, obscene, lying, obsequious, fond of the prattle of old men."

Cardano was a gambler's gambler. He confessed to "immoderate
devotion to table games and dice .... During many years .... I have
played not off and on but, as I am ashamed to say, every day." He
played everything from dice and cards to chess. He even went so far as
to recommend gambling as beneficial "in times of great anxiety and grief
.... I found no little solace at playing constantly at dice." He despised
kibitzers and knew all about cheating; he warned in particular against
players who "smear the cards with soap so that they could slide easily
and slip by one another." In his mathematical analysis of the probabilities in dice-throwing, he carefully qualifies his results with "... if the die
is honest." Still, he lost large sums often enough to conclude that "The
greatest advantage from gambling comes from not playing it at all." He was probably the first person in history to write a serious analysis of
games of chance.

Cardano was a lot more than a gambler and part-time mathematician. He was the most famous physician of his age, and the Pope and
Europe's royal and imperial families eagerly sought his counsel. He had
no use for court intrigue, however, and declined their invitations. He
provided the first clinical description of the symptoms of typhus, wrote
about syphilis, and developed a new way to operate on hernias.
Moreover, he recognized that "A man is nothing but his mind; if that
be out of order, all's amiss, and if that be well, the rest is at ease." He
was an early enthusiast for bathing and showering. When he was
invited to Edinburgh in 1552 to treat the Archbishop of Scotland for
asthma, he drew on his knowledge of allergy to recommend bedclothes
of unspun silk instead of feathers, a pillowcase of linen instead of
leather, and the use of an ivory hair comb. Before leaving Milan for
Edinburgh, he had contracted for a daily fee of ten gold crowns for his
services, but when he departed after about forty days his grateful patient
paid him 1,400 crowns and gave him many gifts of great value.

Cardano must have been a busy man. He wrote 131 printed works,
claims to have burned 170 more before publication, and left 111 in
manuscript form at his death. His writings covered an enormous span
of subject matter, including mathematics, astronomy, physics, urine,
teeth, the life of the Virgin Mary, Jesus Christ's horoscope, morality,
immortality, Nero, music, and dreams. His best seller was De Subtilitate
Rerum ("On the Subtlety of Things"), a collection of papers that ran to
six editions; it dealt with science and philosophy as well as with superstition and strange tales.

He had two sons, both of whom brought him misery. In De Vita,
Cardano describes Giambattista, the older and his favorite, as "deaf in his
right ear [with] small, white, restless eyes. He had two toes on his left
foot; the third and fourth counting the great toe, unless I am mistaken,
were joined by one membrane. His back was slightly hunched ...
Giambattista married a disreputable girl who was unfaithful to him;

none of her three children, according to her own admission, had been
fathered by her husband. Desperate after three years of hellish marriage,
Giambattista ordered his servant to bake a cake with arsenic in it and
fed it to his wife, who promptly died. Cardano did everything he could
to save his son, but Giambattista confessed to the murder and was beyond rescue. On the way to his beheading, his guards cut off his left
hand and tortured him. The younger son, Aldo, robbed his father
repeatedly and was in and out of the local jails at least eight times.

Cardano also had a young protege, Lodovico Ferrari, a brilliant
mathematician and for a time Secretary to the Cardinal of Mantua. At the
age of 14, Ferrari came to live with Cardano, devoted himself to the
older man, and referred to himself as "Cardano's Creation." He argued
Cardano's cases in several confrontations with other mathematicians, and
some authorities believe that he was responsible for many of the ideas for
which Cardano has received credit. But Ferrari provided little solace for
the tragedy of Cardano's own sons. A free-spending, free-living man,
Ferrari lost all the fingers of his right hand in a barroom brawl and died
from poisoning-either by his sister or by her lover-at the age of 43.

Cardano's great book on mathematics, Ars Magna (The Great Art),
appeared in 1545, at the same time Copernicus was publishing his discoveries of the planetary system and Vesalius was producing his treatise
on anatomy. The book was published just five years after the first
appearance of the symbols "+" and "-" in Grounde of Artes by an
Englishman named Robert Record. Seventeen years later, an English
book called Whetstone of Witte introduced the symbol "_" because "noe
2 thynges can be more equalle than a pair of paralleles."8

Ars Magna was the first major work of the Renaissance to concentrate
on algebra. In it Cardano marches right into the solutions to cubic and
quadratic equations and even wrestles with the square roots of negative
numbers, unknown concepts before the introduction of the numbering
system and still mysterious to many people.9 Although algebraic notation
was primitive and each author chose his own symbols, Cardano did
introduce the use of a, b, c that is so familiar to algebra students today.
The wonder is that Cardano failed to solve Paccioli's puzzle of the game
of balla. He did try, but, like other distinguished mathematical contemporaries, he failed at the task.

Cardano's treatise on gambling is titled Liber de Ludo Aleae (Book on
Games of Chance). The word aleae refers to games of dice. Aleatorius,
from the same root, refers to games of chance in general. These words
have come down to us in the word aleatory, which describes events whose outcome is uncertain. Thus, the Romans, with their elegant language, have unwittingly linked for us the meanings of gambling and
uncertainty.

Liber de Ludo Aleae appears to have been the first serious effort to
develop the statistical principles of probablity. Yet the word itself does
not appear in the book. Cardano's title and most of his text refer to
"chances." The Latin root of probability is a combination of probare,
which means to test, to prove, or to approve, and ilis, which means able
to be; it was in this sense of provable or worthy of approval that
Cardano might have known the word. The tie between probability and
randomness-which is what games of chance are about-did not come
into common usage for about a hundred years after Liber de Ludo Aleae
was published.

According to the Canadian philosopher Ian Hacking, the Latin root
of probability suggests something like "worthy of approbation. `0 This
was the meaning the word carried for a long time. As an example,
Hacking quotes a passage from Daniel Defoe's novel of 1724, Roxana,
or The Fortunate Mistress. The lady in question, having persuaded a man
of means to take care of her, says, "This was the first view I had of living comfortably indeed, and it was a very probable way." The meaning
here is that she has arrived at a way of life that justifies the esteem of her
betters; she was, as Hacking puts it, "a good leg up from her scruffy
beginnings."11