Against the Gods: The Remarkable Story of Risk

by Peter L. Bernstein

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.

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.

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

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

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.

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.”

Semantics are important here. As Cardano put it, the probability of an outcome is the ratio of favorable outcomes to the total opportunity set. The odds on an outcome are the ratio of favorable outcomes to unfavorable outcomes.

If the odds on a long-shot at the track are 20—to—1, the theoretical probability of that nag’s winning is one out of 21, or 4.8%, not 5%.

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.

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.

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

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.

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 Méré, 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.

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.

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.

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.

“M. de Méré,” he wrote to a colleague, “has good intelligence but he is not a geometer and this, as you realize, is a great defect.”7 Here Pascal sounds like the academic who takes pleasure in putting down a non-academic.

Fermat’s erudition was awesome.10 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.

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.

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.

“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

As Pascal remarked in his correspondence with Fermat, the mathematical laws must dominate the wishes of the players themselves, who are only abstractions of a general principle.

We all have to make decisions on the basis of limited data. One sip, even a sniff, of wine determines whether the whole bottle is drinkable. Courtship with a future spouse is shorter than the lifetime that lies ahead.

Most critical decisions would be impossible without sampling.

The fable about the blind men and the elephant is famous precisely because each man had taken such a tiny sample of the entire animal.

The most interesting early use of sampling was conducted by the King of England, or by his appointed proxies, in a ceremony known as the Trial of the Pyx and was well established by 1279 when Edward I proclaimed the procedure to be followed.1

The members of the Society were not exactly enthusiastic over the prospect of admitting a mere tradesman, but the King advised them that, “if they found any more such Tradesmen, they should be sure to admit them all, without any more ado.” Graunt made the grade.

Wilkins later became Bishop of Chichester, but he is more interesting as an early author of science fiction embellished with references to probability. One of his works carried the entrancing title of The Discovery of a World in the Moone or a discourse tending to prove that ’tis probable there may be another habitable world in that planet, published in 1640.

Information about births and deaths had long been available in parish churches, and the City of London itself had started keeping weekly tallies from 1603 onward.

The word “statistics” is derived from the analysis of quantitative facts about the state.

Hacking points out that so long as taxation was based on land and tillage nobody much cared about how many people there were.

With the Restoration in full sway, the English were finally rid of the intellectual repression that the Puritans had imposed on the nation. The death of absolutism and Republicanism led to a new sense of freedom and progress throughout the country.

Graunt wonders why the records show that so few died of it, as “a great part of men have, at one time or another, had some species of this disease.” He concludes that most of the deaths from ulcers and sores were in fact caused by venereal disease, the recorded diagnoses serving as euphemisms.

In 1674, he reported to the Royal Society that life expectancy at birth was 18; Graunt’s estimate had been 16.15

Although Halley was English, the data he used came from the Silesian town of Breslau—Breslaw, as it was spelled in those days—located in the easternmost part of Germany; since the Second World War the town has been part of Poland and is now known as Wrozlaw.

Halley was then only 35 years old but already one of England’s most distinguished astronomers. Indeed, he was responsible for persuading Isaac Newton in 1684 to publish his Principia, the work in which Newton first set forth the laws of gravity. Halley paid all the costs of publication out of his own modest resources, corrected the page proofs, and put his own work aside until the job was done. The historian James Newman conjectures that the Principia might never have appeared without Halley’s efforts.

Oxford turned him down for a professorship in 1691 because he held “materialistic views” that did not match the religious orthodoxy of Oxford. But the dons relented in 1703 and gave him the job.

Halley

Halley would live to the age of 86. He appears to have been a jolly man, with an “uncommon degree of sprightliness and vivacity,” and had many warm friendships that included Peter the Great of Russia.

For instance, there were 531 people aged 30, and half that number is 265. One could then look through the table for the age group numbering 265, which appeared to be between 57 and 58. Hence, it would be “an even Wager that. . . a Man of 30 may reasonably expect to live between 27 and 28 years.”

At this point Halley launches into a detailed mathematical analysis of the valuation of annuities, including annuities covering two and three lives as well as one. He offers at the same time to provide a table of logarithms to reduce the “Vulgar Arithmetick” imposed by the mass of necessary calculations.

The first record we have of the concept of annuities dates back to 225 AD, when an authoritative set of tables of life expectancies was developed by a leading Roman jurist named Ulpian. Ulpian’s tables were the last word for over 1400 years!

Taking their cue from the Dutch use of annuities as a financing device, the English government had attempted to raise a million pounds by selling annuities that would pay back the original purchase price to the buyer over a period of 14 years—but the contract was the same for everyone, regardless of their age! The result was an extremely costly piece of finance for the government.

After the publication of Halley’s life tables in Transactions in 1693, a century would pass before governments and insurance companies would take probability-based life expectancies into account.

There was even a Royal Academies Company that promised to hire the greatest scholars of the age to teach the 2,000 winners of a huge lottery a subject of their own choosing.

In the absence of mass media, the coffee houses emerged as the primary source of news and rumor.

Then as now, anyone who was seeking insurance would go to a broker, who would then hawk the risk to the individual risk-takers who gathered in the coffee houses or in the precincts of the Royal Exchange.

Underwriters were willing to write insurance policies against almost any kind of risk, including, according to one history, house-breaking, highway robbery, death by gin-drinking, the death of horses, and “assurance of female chastity”—of which all but the last are still insurable.

The Names committed all their worldly possessions and all their financial capital to secure their promise to make good on their customers’ losses. That commitment was one of the principal reasons for the rapid growth of business underwritten at Lloyd’s over the years.

Bottomry was a loan or a mortgage taken out by the owner of a ship to finance the ship’s voyage. No premium as we know it was paid. If the ship was lost, the loan did not have to be repaid.e This early version of marine insurance was still in use up to the Roman era, when underwriting began to make an appearance.