What do we know, and how surely do we know it? The general answer was given by Aristotle in the Nicomachean Ethics: certainty can be found only in mathematics, all other knowledge being to some degree doubtful. Much evil has been let loose upon the world by defiance of, or exaggeration of, this simple truth: at the one extreme, by the belief that absolute certainty can be found in non-mathematical dogmas, and at the other, by the vulgar conclusion that since certainty is not possible outside mathematics (nor even inside it, according to a few bold theorists), everything we think we know is really just a set of epiphenomenal delusions arising from our personal and social circumstances. The natural fruit of the first of these errors is obscurantist tyranny; of the second, that mendacious solipsism Americans have come to know so intimately well, according to which, since nothing can be known, language has no content and no purpose but the manipulation of the world for the gratification of our private appetites. Whether that second folly has any worse mischief to unload on us than the indignities we suffered during the 42nd Presidency, we shall eventually find out, since it has colonized a large part of our academic life, and seems still to be increasing its hold on the minds of the intelligent young.

Aristotle’s observation implies that most of what we can hope to know must emerge from the weighing of probabilities. To what branch of human knowledge does this weighing of probabilities, this “science of conjecture,” as James Franklin calls it, itself belong? Without thinking very much, most of us moderns would make a paradox of the whole thing by replying: “to mathematics.” The fact that we can give this answer at all, and be partly right in giving it, is a wonderful thing in itself, almost a miracle. That mathematics, our only stock of certain knowledge, can be used with great precision to tell us useful things about the uncertain majority of human experience, is astounding. It poses, in fact, deep philosophical questions to which convincing answers are in short supply. In mathematical statistics, for example, there is an entity named the Poisson distribution, used for estimating the occurrence of rare events. It was first derived in 1837 by the French mathematician Siméon-Denis Poisson from a study of deaths by horse kicks in the Prussian army. It turns out that if you list the number of cavalry corps in which there were no deaths, one death, two deaths, three deaths, … the numbers you have listed follow an elegant mathematical formula. When I first encountered this in my studies I easily mastered the math but got stuck on the metaphysical question: How did the horses know when to stop kicking? I have still not seen any answer that leaves me entirely satisfied.

The first significant results in the mathematical theory of probability were given to us by Pierre Fermat and Blaise Pascal, who developed the fundamental principles in a correspondence undertaken during the year 1654, in which they discussed two problems posed by the Chevalier de Méré, a professional gambler. (The problems were, first, how to divide the stakes of an unfinished game of chance between two players when one of them is ahead, and second, to quantify the odds in dice-throwing.) In The Science of Conjecture, James Franklin has set out to provide a full account of all non-mathematical approaches to probabilistic reasoning prior to that annus mirabilis and also, in a brief but very useful epilogue, to summarize subsequent non-mathematical developments. This means that he has embraced a very wide field of inquiry indeed, taking in practically all the major intellectual disciplines and pseudo-disciplines, from medicine to moral theology, from rhetoric to astrology. He begins with Bishop Butler’s phrase: “Probability is the very guide of life.” He ends with a stirring, and very timely, defense of rational judgment against “the forces of unreason” that are on the loose in our academies. Franklin teaches mathematics at the University of New South Wales in Sydney, Australia.

The author reminds us that pre-modern thinkers had a keen grasp of the “science of conjecture” long before that science was quantified. Lawyers were specially skilful at weighing, and displaying, probabilities in a convincing way. Charged with having spoken against the supremacy of his King in matters religious, Sir Thomas More was confronted with just one witness, an inveterate liar by reputation. He dealt with that witness’s testimony thus:

Can it therefore seem likely vnto your honorable Lordshipps that I wold, in so weyghty a cause, so unadvisedly overshoote myself as to trust master Rich, a man of me alwaies reputed for one of so litle truth … that I wold vnto him vtter the secreates of my consciens towchinge the Kings supremacye? … A thinge which I neuer wold, after the statute thereof made, reveale either to the kings highnes himself, or to any of his honorable councellours … Can this in your iudgments, my lordes, seeme likely to be true?

Under the circumstances, it was not in the least likely — though, alas, Sir Thomas went to the block anyway.

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The weighing of evidence in courtrooms is, of course, still conducted today — a good illustration of the fact that non-mathematical reasoning about probabilities did not stop abruptly in 1654, and is in fact never likely to stop. Even in our own extremely mathematical age, such methods still form much the larger part of probabilistic thinking and arguing. Even, in fact, in areas where the matters under discussion are of a strictly scientific nature, and in theory quantifiable, the mathematical calculus of probability is often of very little help: think of meteorology. Franklin notes in passing, as many others have done, that mathematics herself, though her truths must always be demonstrated deductively, most often advances by induction and intuition. This function behaves like this here … and here … and here. Perhaps it behaves like this everywhere! Let’s see if I can construct a deductive proof … In an analogy I like very much, the sociologist Erving Goffman speaks of the “front” and the “back” of intellectual work, comparing such work to what goes on in a theater or a restaurant, where the smooth, disciplined, orderly “front” for presentation to the public is supported by a noisy, chaotic “back” where professionals prepare the dishes, or don the costumes, amid much yelling and banging and breakage.

The converse is also true. Just as the post-Pascalian world is rich in unquantified and unquantifiable reasoning about probability, so, it turns out, the ancient and medieval world was by no means innocent of numerical methods for dealing with chance. Gamblers — at any rate, successful gamblers — must always have had some notion of “odds.” We know that they did, for related terms escaped into ordinary language: “vernacular quantification,” Franklin calls it, and quotes passages like Sir Andrew Aguecheek’s “it’s four to one she’ll none of me” in Twelfth Night. More surprising, at any rate to me, is Franklin’s account of the medieval trade in annuities, in which “[m]onasteries were among the principal sellers … and churchmen common among the buyers.” It was all squared with the Church’s prohibitions of usury by dint of some ingenious reasoning, notably in Alexander Lombard’s Treatise on Usury of 1307. Lombard’s main point: the contract is illicit only when one party has notably the better side. If the right price can be found, given the probabilities, then no wrong has been done. Franklin presents these topics in a chapter headed “Aleatory Contracts: Insurance, Annuities and Bets,” the best part of the book, for my money. I was also surprised to learn that the first English state lottery was organized as early as 1566. “The public showed a certain skepticism about the government’s honesty …” Franklin notes drily, and only 34,000 of the 400,000 tickets were sold. Apparently it was not only in their appreciation of drama that the Elizabethan public was more sophisticated than ourselves.

This is not an easy book to read, though it is easier towards the end than at the beginning. I am not sure that Franklin found the best method of organizing his material; however, this is not a very constructive criticism, as I don’t see how a net cast so wide can bring in anything other than an unwieldy mass. The author’s style is at any rate clear and fluent, with an occasional sly Gibbonian aside to make the reader chuckle. Of the Jacobean jurist Sir Edward Coke’s argument that “the Judge ought to be … for the party indifferent,” Franklin observes:

The Jesuits no doubt remained skeptical of the “indifference” of English judges, especially those Jesuits personally tortured by Coke.

Franklin lets all the important sources speak for themselves, in many long quotations — a sensible way to present material of this sort, I think. I learned a lot from The Science of Conjecture. I am glad to have read it, and shall keep it for its reference value. I cannot say I ever picked it up eagerly, though, and I set it aside at last with some relief. This is a dense, quite difficult and often very dry account of a large and important subject.