Against the Gods: The Remarkable Story of Risk

by Peter L. Bernstein

  The Renaissance was a time of discovery. Columbus set sail in the year Piero died; not long afterward, Copernicus revolutionized humanity's view of the heavens themselves. Copernicus's achievements required a high level of mathematical skill, and during the sixteenth century advances in mathematics were swift and exciting, especially in Italy. Following the introduction of printing from movable type around 1450, many of the classics in mathematics were translated into Italian and published either in Latin or in the vernacular. Mathematicians engaged in spirited public debates over solutions to complex algebraic equations while the crowds cheered on their favorites.

  The stimulus for much of this interest dates from 1494, with the publication of a remarkable book written by a Franciscan monk named Luca Paccioli.' Paccioli was born about 1445, in Piero della Francesca's hometown of Borgo San Sepulcro. Although Paccioli's family urged the boy to prepare for a career in business, Piero taught him writing, art, and history and urged him to make use of the famous library at the nearby Court of Urbino. There Paccioli's studies laid the foundation for his subsequent fame as a mathematician.

  At the age of twenty, Paccioli obtained a position in Venice as tutor to the sons of a rich merchant. He attended public lectures in philosophy and theology and studied mathematics with a private tutor. An apt student, he wrote his first published work in mathematics while in Venice. His Uncle Benedetto, a military officer stationed in Venice, taught Paccioli about architecture as well as military affairs.

  In 1470, Paccioli moved to Rome to continue his studies and at the age of 27 he became a Franciscan monk. He continued to move about, however. He taught mathematics in Perugia, Rome, Naples, Pisa, and Venice before settling down as professor of mathematics in Milan in 1496. Ten years earlier, he had received the title of magister, equivalent to a doctorate.

  Paccioli's masterwork, Summa de arithmetic, geometria et proportionalita' (most serious academic works were still being written in Latin), appeared in 1494. Written in praise of the "very great abstraction and subtlety of mathematics," the Summa acknowledges Paccioli's debt to Fibonacci's Liber Abaci, written nearly three hundred years earlier. The Summa sets out the basic principles of algebra and contains multiplication tables all the way up to 60 x 60-a useful feature at a time when printing was spreading the use of the new numbering system.

  One of the book's most durable contributions was its presentation of double-entry bookkeeping. This was not Paccioli's invention, though his treatment of it was the most extensive to date. The notion of double-entry bookkeeping was apparent in Fibonacci's Liber Abaci and had shown up in a book published about 1305 by the London branch of an Italian firm. Whatever its source, this revolutionary innovation in accounting methods had significant economic consequences, comparable to the discovery of the steam engine three hundred years later.

  While in Milan, Paccioli met Leonardo da Vinci, who became a close friend. Paccioli was enormously impressed with Leonardo's talents and commented on his "invaluable work on spatial motion, percussion, weight and all forces."2 They must have had much in common, for Paccioli was interested in the interrelationships between mathematics and art. He once observed that "if you say that music satisfies hearing, one of the natural senses ... [perspective] will do so for sight, which is so much more worthy in that it is the first door of the intellect."

  Leonardo had known little about mathematics before meeting Paccioli, though he had an intuitive sense of proportion and geometry. His notebooks are full of drawings made with a straight-edge and a compass, but Paccioli encouraged him to master the concepts he been using intuitively. Martin Kemp, one of Leonardo's biographers, claims that Paccioli "provided the stimulus for a sudden transformation in Leonardo's mathematical ambitions, effecting a reorientation in Leonardo's interest in a way which no other contemporary thinker accomplished." Leonardo in turn supplied complex drawings for Paccioli's other great work, De Divine Proportione, which appeared in two handsome manuscripts in 1498. The printed edition came out in 1509.

  Leonardo owned a copy of the Summa and must have studied it with great care. His notebooks record repeated attempts to understand multiples and fractions as an aid to his use of proportion. At one point, he admonishes himself to "learn the multiplication of the roots from master Luca." Today, Leonardo would barely squeak by in a third-grade arithmetic class.

  The fact that a Renaissance genius like da Vinci had so much difficulty with elementary arithmetic is a revealing commentary on the state of mathematical understanding at the end of the fifteenth century. How did mathematicians find their way from here to the first steps of a system to measuring risk and controlling it?

  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, how
ever, 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

  Hacking cites another example of the changing meaning of probability.12 Galileo, making explicit use of the word probabilitd, referred to Copernicus's theory of the earth revolving around the sun as "improbable," because it contradicted what people could see with their own eyes-the sun revolving around the earth. Such a theory was improbable because it did not meet with approval. Less than a century later, using a new (but not yet the newest) meaning, the German scholar Leibniz described the Copernican hypothesis as "incomparably the most probable." For Leibniz, Hacking writes, "probability is determined by evidence and reason."13 In fact, the German word, wahrscheinlich, captures this sense of the concept well: it translates literally into English as "with the appearance of truth."

  Probability has always carried this double meaning, one looking into the future, the other interpreting the past, one concerned with our opinions, the other concerned with what we actually know. The distinction will appear repeatedly throughout this book.

  In the first sense, probability means the degree of belief or approvability of an opinion-the gut view of probability. Scholars use the term "epistemological" to convey this meaning; epistemological refers to the limits of human knowledge not fully analyzable.