example, whose Nat-ural and Political Ob-servations Madeupon the Bills of Mor-tality (1662) standsas the first great ex-ample of modernstatistical data analy-sis, Bernstein alsospends some timetelling us about thegenesis of Lloyd’s ofLondon, the famousinsurance firm. Thisweaving of topicsthat are standard inthe history of proba-bility and statistics

with many that are not is one of the strengths andattractions of the book. Another is the obvious zestwith which Bernstein describes the many intrigu-ing people and tantalizing mysteries so peculiar tothe mathematics of chance.

The subject is certainly not lacking in its mys-teries. The first great puzzle in its history is justwhy it took so long for the mathematics of chanceto develop. The ancient Greeks certainly had boththe necessary mathematical abilities and recre-ational interests, and yet there is not a hint of asystematic mathematical treatment of gambling ineither their literature or that of the Middle Ages.Many explanations—mathematical, conceptual,and economic—have been proposed for this. Thephilosopher Ian Hacking, for example, whose in-fluential and important book The Emergence of

JANUARY 1999 NOTICES OF THE AMS 47

Book Review

Against the Gods:The Remarkable Story of Risk

Reviewed by S. L. Zabell

Against the Gods:The Remarkable Story of RiskPeter L. BernsteinJohn Wiley and Sons, New York, 1996ISBN 0-471-12104-5xi + 383 pp.; Hardcover $27.95

The thesis of this interesting and provocativebook is that modern civilization is largely distin-guished by its successful efforts first to understandand then to control risk. These efforts use, in part,tools of risk management that only became pos-sible after the development of the mathematicalsubjects of probability and statistics. The author’smethod is largely historical and biographical: thefive parts of the book cover the periods before1200, 1200–1700, 1700–1900, 1900–1960, and1960 to the present; the chapters in each part con-tain a series of one or more vignettes of importantcontributors to the subject. Many familiar namescrop up—Cardano, Pascal, Fermat, Graunt, theBernoullis, De Moivre, Bayes, Laplace, Galton,Keynes, and von Neumann—but also a number ofindividuals who will be less familiar: Baumol,Knight, Markowitz, Leland, Rubinstein, and Thaler.

Peter L. Bernstein is the president of a consult-ing firm for institutional investors and the authorof six books on economics and finance. He bringsan unusual and novel perspective to his historicalsurvey of the development of the mathematics ofchance and uncertainty. In the same chapter inwhich he discusses the work of John Graunt, for

S. L. Zabell is professor of mathematics at NorthwesternUniversity. His e-mail address is zabell@math.nwu.edu.

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Probability (1975) created a resurgence of interestin the history of probability, suggested that a nec-essary and special concept of evidence, absent be-fore the seventeenth century, had been a necessaryprecursor for the development of the mathemat-ics of chance. This provocative if not entirely con-vincing thesis was inspired by the ideas of thecontroversial French structuralist philosopherMichel Foucault. Bernstein opts for a much morestandard explanation: that a suitable notion ofchance event was lacking.

The first book on the mathematics of games ofchance in fact dates to the sixteenth century: theLiber de Ludo Alaea of Girolamo Cardano. Car-dano was, by any account, a most curious figure.

In his day a famous physi-cian, he was also a mystic,a polymath, and a prolificauthor, best remembered bymathematicians today asthe author of the Ars Magna(1545), which contains thefirst general discussion ofthe solution of the cubicequation. Mathematiciansoften live relatively un-eventful lives, but Cardanodid not. After persuading areticient Nicolo Tartaglia todivulge his solution of thecubic under a strict pledgeof secrecy, Cardano manyyears later described it inthe Ars Magna. Even thoughCardano acknowledgedTartaglia’s priority, this in-cident sparked an extendedand venomous exchange be-

tween the two. (Cardano’s brilliant student Ferrariwent on to provide a similar solution for the quar-tic.) Perversely, Cardano did not publish his owndiscoveries in the mathematics of gambling; theyfirst appeared in print nearly a century after hisdeath.

The colorful Cardano is by no means unusual,for probability has, by any measure, a history re-plete with curious figures. There is the asceticBlaise Pascal, who abandoned mathematics andthe world to spend his last years as a Jansenistmonk penning the Pensées that would win him im-mortality. There is the secretive James Bernoulli,a member of an enormously gifted but highly com-petitive family, whose proof of the law of largenumbers (the first major limit theorem in math-ematical probability) in his Ars Conjectandi (writ-ten around 1685) languished for nearly twentyyears while he pondered its significance and ulti-mately appeared in 1713, eight years after hisdeath. There is the Huguenot refugee AbrahamDe Moivre, who fled France in 1688 after the re-

vocation of the Edict of Nantes three years earlier,but who, despite his membership in the Royal So-ciety and friendship with Newton, was unable toobtain a university position and ended up spend-ing his days teaching students and consulting atSlaughter’s Coffee House in the Strand. Cardanoand Bernoulli were not the only two who publishedonly after they perished. There is the mysterious(but “ingenious”, as many later described him)Thomas Bayes, whose mathematical works—pub-lished only after his death by his friend RichardPrice—included both a short note pointing out forthe first time the divergent nature of the asymp-totic series for logn, as well as his longer andmore famous “Essay towards solving a problem inthe doctrine of chances”, later to bring him epony-mous fame as the founder of “Bayesian statistics”.

Following de Moivre and Bayes, dominance inmathematical probability (as in so many othermathematical subjects after the death of Newton)passed from England to France. The Marquis deCondorcet, one of the philosophes, enthusiasti-cally argued for the uses of probability in arrivingat a more rational judicial system before he, likethe chemist Lavoisier, perished in the irrational ter-ror of the French Revolution. (The Marquis was thevictim of his aristocractic background and per-haps also of his fondness for omelettes: legend hasit that his lineage was betrayed at a country innby the large number of eggs he requested in anomelette). Condorcet’s younger contemporary,Pierre Simon, the Marquis de Laplace, was at onceboth more politically agile and more mathemati-cally powerful. He survived the Revolution and solived to write his masterpiece, the Théorie Analy-tique des Probabilités, in 1812—a work whose sub-jective view of the nature of probability dominatedthe subject for nearly a century. Then the pendu-lum finally swung back to England and the Englishschool of biometricians—Galton, Edgeworth, Pear-son, Gosset, and Fisher—who, eclectically buildingon these foundations, crafted the beginnings ofmodern statistical science, applying the math-ematics of chance to the collection, summarization,and analysis of data.

This initial part of Bernstein’s book is relatedwith vigor and enthusiasm. It is not the work of aprofessional historian, but this does not seem par-ticularly important; the book is for the most partaccurate in its details and certainly accurate inthe overall picture it conveys. There are someminor errors of fact (for example, Cardano, whowas born in 1501 and died in 1576, is reported tohave been born “about 1500 and died in 1571”).Such errors as exist, however, are neither particu-larly serious or pervasive, and to harp on themwould be churlish and misleading. Bernstein’s pur-pose is to depict the rise of risk management,combining many different strands in a long andcomplex story, and in this he certainly succeeds.

The colorfulCardano is by

no meansunusual, forprobabilityhas, by anymeasure, a

history repletewith curious

figures.

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This is, of course, a very broad canvas to depict,and it is not surprising that some parts of his syn-thesis are more effectively drawn than others. Itis in the second half of the book that Bernsteinseems particularly in his element, as he discussesthe work of Arrow, Keynes, Knight, and von Neu-mann on game theory (Chapter 14); HarryMarkowitz on portfolio selection (Chapter 15);Kahneman and Tversky on prospect theory (Chap-ter 16); Thaler, Shefrin, and Statman on behav-ioral finance (Chapter 17); and the use of deriva-tives and other financial instruments in riskmanagement (Chapter 18). The author tells this partof his history with an attractive verve, and, if any-thing, one finds oneself wishing that he had spenteven more time on this portion than he does.

The book does have some weaknesses. One isthat its presentation of certain topics in statisticsis occasionally confused or confusing. (The dis-cussions of normality and its relationship to in-dependence on pp. 144–150 and of regression tothe mean on pp. 167–170 and 173–174 are two ex-amples.) On the other hand, Bernstein’s discussionof many other topics (for example, the binomial dis-tribution and the work of Kahneman and Tver-sky) are models of clarity, outstanding examplesof popular exposition. The book also resorts on anumber of occasions to an unnecessary hyper-bole. Is it really the case that “Nothing is moresoothing or more persuasive than the computerscreen” (p. 336) or that “Proponents of chaos the-ory … claim to have revealed the hidden source ofinexactitude” (p. 332)? Surely not.

But these are minor blemishes. Bernstein haswritten an interesting, amusing, unpretentious,and attractive book. It discusses a number of ne-glected topics, provides a fresh outlook on someof the more standard ones, and should stimulatemany of its readers to go further into both the fas-cinating literature of the history of probability andstatistics and modern economic applications ofthese subjects.

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