The Science of Conjecture: Evidence and Probability Before Pascal

by James Franklin

The first edition of The Science of Conjecture reconfigured the field of the history of probability in two ways. The first concerned a wider than usual conception of the nature of probability. The second involved a Renaissance-free view of the history of Western ideas.

The book’s conception of the subject matter stemmed from an objective Bayesian (or logical probabilist) theory of probability.

as developed by Keynes in his Treatise on Probability and by later authors,1 the main notion of probability is of an objective logical relation holding between a body of evidence and a conclusion.

Logical probability may or may not be numerical; even if it is, qualitative or approximate judgments are often of most importance.

on the law of evidence, which, over many centuries of thought, especially in medieval Roman law, had developed evidential concepts like the modern proof beyond reasonable doubt. Moral theory and business were also familiar with concepts of probabilities and risks, mostly quantified only loosely.

During the late twentieth century, debate on “interpretations of probability” largely took the form of pitched battles between frequentists and related schools (who held that probability dealt with relative frequencies or objective propensities) and subjective Bayesians (who took probability to be about degrees of belief, subject to some constraints). But in recent years, a more objective and logical Bayesian interpretation has gradually come to the fore.

Nassim Nicholas Taleb’s The Black Swan was the most successful of several works that explained an inherent difficulty in predicting rare events. Because they are rare, there is little relevant evidence, so purely inductive methods are unreliable and must be supplemented with expert opinion to situate the event in a context; unfortunately, expert opinion is chronically unreliable, especially with very small probabilities.10

Since these abilities are acquired even before infants have learned to speak, it is clear that humans have pre-linguistic abilities to respond to and reason about probabilities, confirming the view of The Science of Conjecture that the story of probability is one of bringing to consciousness existing but implicit probabilistic knowledge.

second main idea behind The Science of Conjecture was to take a pro-medieval, anti-Renaissance approach to the early modern history of ideas. The scholastics and legal writers of the late middle ages developed the main concepts of the evaluation of uncertain evidence as well as of commercial risk.

science strictly speaking was not the natural bent of the scholastics, who were much more at home in disciplines that rely on conceptual analysis, such as economic theory and law. That has not yet been generally accepted.

On the subject matter of The Science of Conjecture itself, the development of ideas about evidence and probability up to 1650, there have not been any major new works.

One of the main findings of The Science of Conjecture was that late medieval canon lawyers, reflecting on business practice with contracts like insurance, annuities, and options that depend on chance outcomes, made great strides in understanding the pricing of risk.

The chapter on probability in religious argument showed the beginnings of a divergence between English and continental thought related to “reasonableness,” with English works following Hooker’s Laws of Ecclesiastical Polity, emphasizing the need for reasonableness and moderation in matters of religion and politics.

At a deeper cultural level, Anna Wierzbicka’s English: Meaning and Culture describes how John Locke’s writings on probability, reasonableness, and moderation became ingrained in the English language. The words “reasonable” and “probably” appear in modern English with a frequency and range of application very much higher than their cognates in other European languages, as do words indicating hedges relating to degrees of evidence such as “presumably,” “apparently,” “clearly.” These phenomena indicate an Anglophone mindset that particularly values attention to the uncertainties of evidence.

some of the most conceptually tangled but crucial concepts of modern life: risk, evidence, and uncertainty.

J. Franklin, What Science Knows: And How It Knows It (New York, 2009).

S. B. McGrayne, The Theory That Would Not Die: How Bayes’ Rule Cracked the Enigma Code, Hunted Down Russian Submarines, and Emerged Triumphant from Two Centuries of Controversy (New Haven, 2011).

Hannam, God’s Philosophers: How the Medieval World Laid the Foundations of Modern Science (London, 2009), published in the United States as The Genesis of Science: How the Christian Middle Ages Launched the Scientific Revolution (Washington, DC, 2011).

A. Wierzbicka, English: Meaning and Culture (Oxford, 2006),

“Probability is the very guide of life,” in Bishop Butler’s famous phrase.1 He does not mean, of course, that calculations about dice are the guide of life but that real decision making involves an essential element of reasoning with uncertainty.

This is a history of rational methods of dealing with uncertainty. It treats, therefore, methods devised in law, science, commerce, philosophy, and logic to get at the truth in all cases in which certainty is not attainable.

Three levels of probabilistic reasoning are distinguishable: 1. Unconscious inference, the reactions to uncertain situations that the brain delivers automatically at a subsymbolic level.

Ordinary language reasoning about probabilities. It is this middle level that is the main subject of this book. It may avoid numbers entirely, as in “proof beyond reasonable doubt” in law or the nonnumerical judgments of plausibility that scientists and detectives make in evaluating their hypotheses. Or it may involve rough numerical estimates of probabilities, as in racecourse odds and guesses about the risks of rare events. 3. Formal mathematical reasoning of the kind found in textbooks of probability and statistics. The higher levels may be more noble and perfect, but they are so at a cost: they are less widely applicable.

The theme of this book, then, must be the coming to consciousness of uncertain inference.

There is one further essential distinction to be made. Probability is of two kinds.2 There is factual or stochastic or aleatory probability, dealing with chance setups such as dice throwing and coin tossing, which produce characteristic random or patternless sequences.

there is logical or epistemic probability, or non-deductive logic, concerned with the relation of partial support or confirmation, short of strict entailment, between one proposition and another.

How probable a hypothesis is, on given evidence, determines the degree of belief it is rational to have in that hypothesis, if that is all the evidence one has that is relevant to it.

In the period covered by the present study, logical probability was the main focus of interest, and the word probability was reserved solely for this case. The little study of factual probability there was—concerned with dice and insurance—was not seen as connected with logical probability.

Consequently, the book opens with three chapters on the law of evidence, in which there has been the most consistent tradition of dealing explicitly with evidence that falls short of certainty.

reader with an average familiarity with received ideas on intellectual history is asked to make a small number of reorientations, at least provisionally. The first concerns probability specifically. Two points should be made, to avoid perceptions that early writers are indulging merely in confused “anticipations” of later mathematical discoveries. The first is that the process of discovering the principles of uncertain reasoning is far from over. It can sometimes appear that, beginning with Fermat and Pascal’s success with dice in 1654, there has been a successful colonization of all areas of uncertain reasoning by the mathematical theory of probability.

there is no agreement on, for example, how to combine evidence for conclusions in computerized expert systems for medical diagnosis. The disagreements are fundamental and are about quite simple issues that have occupied thinkers about uncertain inference for two thousand years: how to decide the strength with which evidence supports a conclusion, how to combine pieces of evidence that support each other, and what to do when pieces of evidence conflict.

second point is that while the probability of outcomes of dice throws is essentially numerical, and advances in understanding are measured by the ability to calculate the right answers, it is otherwise with logical probability.

Even now, the degree to which evidence supports hypotheses in law or science is not usually quantified, and it is debatable whether ...

One should not give in to the easy assumption that numbers are good, words bad.

The template antiquity > medieval decline > renaissance > scientific revolution does not fit the history of probability; certainly not the history of logical probability. In particular, it is not possible to read the story with the medieval Scholastics as “them” and the men of the seventeenth century as “us.” The Scholastics made many advances in the clarification and deepening of concepts necessary to understand probability. And contrary to the myths put about by their many enemies, they explained themselves perfectly clearly.

the reader is asked to regard it as normal to find many ideas developing in legal contexts. Like the Scholastics, lawyers are often thought of as pursuing esoteric interests of little consequence for the outside world and as, by and large, enemies of scientific progress. It is argued that the prominence of both Scholastics and lawyers is not unique to probability but that their contributions to the development of modern ideas generally have been substantially underrated.

Mathematical modeling is always difficult, as is evident in contemporary parallel cases such as the mathematization of continuity that led to the calculus of Newton and Leibniz. The very idea of a “geometry of chance,” as Pascal put it, is revolutionary.

The purpose of history may not be to teach us lessons, but the story told here does have a certain contemporary relevance, even though it ends in 1660.

In the philosophy of science, Popper, Kuhn, and their schools denied that observational evidence could make scientific theories more probable, and attention in the field moved to sociological and other nonevidential influences.4 Postmodernism, presuming rather than arguing for the absence of objective methods of evaluating theories, offered a number of other reasons—or rather causes—of actual beliefs, such as the demands of “power.”5 The situation is not so bad in law, which has largely retained a commitment to the objectivity of evidence, but even there, theory is not as robust as practice.

The past is a counterweight to these febrile inanities of pygmies who stand on the shoulders of giants only to mock their size. Just as one who feels battered by the relentless enfant terriblisme of “modern” art or music can revive his spirit by communion with Vermeer or Mozart, so the friend of reason can draw comfort from the achievements left by the like-minded of the past. The story of the discovery of rational methods of evaluating evidence can serve as a point of reference and can supply material for the defenses of rationality that will have to be undertaken.

THE LAW OF EVIDENCE is the central thread in the history of probability. The reason is that the law must decide disputes. To do so, it must often decide who or what to believe. What witnesses are credible? Which documents? What is to be done about conflicting testimony? Is evidence of what a witness heard another person say to be admitted? And on what basis should one make the final decision on whose evidence is to be believed? It is possible to leave all these questions to the discretion of the judge, but a developed system of law will tend to introduce rules of evidence, so that this part of the law, like others, will be aboveboard and seen to be so.

law keeps its conceptual achievements, and they remain accessible.

“Civilization is the art of living in towns of such size that everyone does not know everyone else.”1 With anonymity comes litigation, and with litigation comes conflict of evidence.

As they do not (at least avowedly) rely on any evaluation of evidence by a rational agent, it is correct to say that “procedure was not essentially deliberation, but action.”

Greek thought of the classical period of course had a great deal to say about reasoning in legal cases—but more from the point of view of rhetoric than in a strictly legal context.

Late antiquity is normally regarded as “decadent” in such intellectual fields as science and mathematics, literature, perhaps philosophy. By contrast, law is one of the few areas of thought in which the legacy of antiquity consists almost exclusively of the productions of the last centuries of the ancient world. It consists, in particular, of the two great compilations of ancient law, the Jewish Talmud and the Roman Corpus of Civil Law.

The classical period of Roman law is that of Ulpian and other jurists of the period around 150–250 A.D. Excerpts from their works were collected, and sometimes adjusted, by a commission set up by the Emperor Justinian. This collection, the Digest or Pandects, was declared law in 533; all other writings of the jurists were declared to be no longer law and have almost entirely disappeared. The Digest forms the largest part of Justinian’s complete codification of Roman law, the Corpus of Civil Law. The Talmud and the Corpus are almost independent, except that a few of the older precepts of Jewish law were inserted into Roman law by the Christian emperors.

On matters of evidence, the Talmud and the Digest share recognizably similar ideas on three matters: rules concerning witnesses, the high standard of evidence required in criminal cases, and reasoning from “presumptions.”

The contact between Jewish and Greek thought produced the first treatises on the theme so central to medieval philosophy, the harmony of faith and reason. Of the many themes of Aquinas and others that can be found in the works of Jesus’ contemporary Philo of Alexandria, one of the most characteristic is the “reasons for the laws”: what seems in the Old Testament to be purely an expression of God’s will is often justified rationally by Philo and later philosophers.

For why should the statement of a witness made in accusation of another be accepted in preference to the words of the accused in his own defense? Where there is neither deficiency nor excess it is clearly best to suspend judgment.”13 In the Talmud itself, the demand for a high standard of evidence in criminal cases developed into a prohibition of any uncertainty in evidence: