The Science of Conjecture: Evidence and Probability Before Pascal

by James Franklin

Though mostly too busy writing voluminously in praise of experiments to actually perform any, Francis Bacon does report an interesting experiment on the germination of seeds that involves the use of a control.

With the low sciences one at least has the opportunity of confirming or disconfirming a hypothesis by collecting further evidence. The peculiar difficulty of historiography is that this is rarely possible, as the sum of evidence on a given question about the past is usually fixed.

The social sciences in general were in a poor state of development in the Middle Ages and Renaissance. While legal science made some progress on the problem of evaluating the testimony of witnesses, history did not. As is well known, medieval histories, chronicles, travelers’ tales, bestiaries, and so on generally combine the perfectly true with the totally impossible, with little attempt to sort out the two. In the absence of any generally recognized concept of probability, readers’ attitudes to the contents of such works could only oscillate unstably between credulity and skepticism.

The frequently repeated tale that the fragments of the True Cross would have added up to a whole forest appears to be a modern myth.74 Indeed, it is only one of a cluster of widely credited myths about the Middle Ages (in the Middle Ages it was believed the earth was flat; lords enjoyed droit de seigneur over peasant women; the Scholastics debated how many angels could dance on the head of a pin),75 raising questions about which age is really the more credulous. Is the level of common “knowledge” about the Middle Ages actually negative?

Even so, one can often see in medieval historians that a critical skill is not absent; it is just that it is not their practice to remold the stories they are preserving in the image of their own critical opinion. They will, for example, retail contradictory stories and perhaps note that one is “scarcely credible,” rather than excising one of them. Compared to modern historians, they prefer to present evidence rather than conclusions and leave evaluation to the reader.77 Their practice is perhaps more comparable to that of a present-day archivist than a historian.

One of the more acute medieval historians is Bernard Gui, whose skills in evidence evaluation in history are close to those in his work as inquisitor. Occasionally even the actual evidence is the same; he has inquisitorial records sent to him under seal to assist his writing on history.83 His extensive researches in a wide variety of oral and written sources leave no doubt of the sincerity of his statement, “I have inquired (inquisivi) into the truth of what has gone before.”84 He is of course concerned with the lack of harmony in the sources, which he attributes to the errors of copyists and (being one of the first to do so) to the “diversity of authors’ opinions and positions.” In cases in which the authorities are equally trustworthy and the issue is minor, he is prepared to simply “present the contrary arguments and leave it to the reader to judge and choose.”

Some progress was made on evaluating the authenticity of documents, which were often as deserving of skepticism as relics. Men of law possessed the relevant skills, being trained both in the evaluation of evidence and in close attention to documents.88 Since Roman law gave a high evidentiary value to documents, it had encountered the problems of their forgery and loss. Both the Digest and the Code have sections “on faith in documents” that raise the problem of determining authenticity but say little about how to resolve it beyond a vague mention of “comparison of handwriting,” and some doubts about the value of “private documents.

The humanists, for all their criticism of science, logic, law, and so on, were not incompetent at everything. Two arts in which they particularly excelled were philology and invective. Petrarch was strong on both, as in a letter to the emperor in which he mocks a document allegedly written by Julius Caesar for anachronistically using the royal plural.100 But these skills, and their combination, reached a high point in the writings of Lorenzo Valla. Counterintuitive theses maintained by him include: that all Scholastic disputes on metaphysics are caused by misuse of words; that the Latin of medieval law is barbarous (“to the highest degree gothic”101—the persistence of this word to describe things medieval is a sign of Valla’s success); that historians and rhetoricians are superior to philosophers; that logic is a subdivision of rhetoric, not vice versa;102 that the lay state is better than the religious; and that Quintilian is a better stylist than Cicero.

His project was to replace Scholastic logic and metaphysics with dialectic and rhetoric, as understood by Cicero, Quintilian, and Boethius. This involved a revival of the arguing on both sides of a question, the urbane skepticism and topical argument of classical rhetoric, in opposition to Scholastic “dogmatism.”106 As in Cicero, an emphasis on the likely followed. After dividing premises of arguments into the necessary and the likely, Valla draws distinctions among the latter. “But as long as the reason is not plainly true but half-true and half-certain, then the conclusion is not necessary but half-necessary, which when it has much force is called likely and credible, that is, exceedingly possible, and when it has scant force is called possible, that is, scarcely likely or credible.”

There is more along the same lines, with appropriate details and references to the original documents. It is customary to say that arguments from silence are weak, but enough of them together, as in this case, can be strong enough.

A different kind of humanistic project in history is that of Machiavelli’s Discourses. He attempts to make history a science by extracting maxims from history: generalizations that are summaries of and lessons taught by the many individual events of the past. A typical example is “Dictatorship is advantageous in times of emergency.” Naturally, there are continual problems with the wealth of counterexamples to the maxims; Machiavelli attempts to explain them away individually, rather than saying that the maxims only hold true for the most part.124 His advice on what action to take in case of doubt is unhelpful: “In all discussions one should consider which alternative involves fewer inconveniences and should adopt this as the better course; for one never finds any issue that is clear cut and not open to question.”125 To consider only the payoff, and not the probability of occurrence, will not lead to a satisfactory decision theory.

The pose of skepticism affected by a section of the humanist movement sometimes included skepticism about historical evidence.126 The rhetorical temperament of certain humanists was impressed by simplistic arguments such as the dilemma posed by one of them: an observer is either privy to the counsels of the Prince or he is not. If not, he cannot tell the truth. If so, he is a partisan of the prince and will not tell the truth.127

“Serious and trustworthy historians, of which sort some undoubtedly exist both on ecclesiastical and secular affairs, provide the theologian with a probable argument either for corroborating propositions of his own, or for refuting false opinions of his adversaries.”

Philosophy and religion are old enemies of probability. Philosophers from the earliest times have wished to distinguish themselves from the spinners of mere rhetoric by offering certainty. Parmenides distinguished sharply between truth, associated with Being, and the opinion of men, called “likely” and associated with non-Being.1 Logical reasoning is intended, by Parmenides, Plato, Aristotle, and their followers, to establish the foundations of knowledge beyond all doubt, and correspondingly likelihoods are banished as other people’s business.

The word “likely” is clearly intended to situate the theory somewhere between the completely worthless and the certainly true, but beyond that it seems to have no meaning. It confirms rather than undermines the contrast between likelihood and truth.

Those religions, too, that offer doctrines (as opposed to practices and wisdom) usually believe those doctrines are certain, doubt being often regarded as an insult to the God who has deigned to reveal the truth. But as history has progressed, it has been found impossible to ignore the fact that the more certainties that are offered by philosophy and religion, the more those supposed certainties conflict. In philosophy, one reaction has been moderate skepticism or fallibilism, which emphasizes the merely probable character of all theory and the sufficiency of probability as a guide to life. That reaction is rare before the twentieth century.

The Stoic school inherited most of the opinions of Aristotle. The Stoics did, however, concentrate strongly on the certainty of knowledge, to the neglect of the beginnings Aristotle had made on nondeductive inference.

The Stoics had a place for the plausible (pithanon) and the reasonable (eulogon), but their use of these words made little of their probabilistic aspects. They are reported as offering these definitions: “A plausible (pithanon) judgement is one that induces to assent, for example, ‘Whoever gave birth to anything, is that thing’s mother.’ This, however, is not necessarily true, since the hen is not mother of the egg. . . . A reasonable (eulogon) proposition is one that has many things in its favour, for example, ‘I shall be alive tomorrow.’”5 But these ideas are not expanded. The Stoics’ exaltation of the reasonable as the highest good in action led to a number of jokes at their expense, to the effect that the reasonable did not seem to be reasonable for anything.

Since the time of Democritus, one of the chief irritants producing the pearl of philosophy has been the challenge of skepticism, the worry that false sense impressions are sometimes indistinguishable from true ones. The experience of illusions of perception is familiar—the oar that appears bent in water, the tower that appears round from a distance but square nearby. As Aristotle explains, “Which, then, of these impressions are true and which are false are not obvious, for the one set is no more true than the other, but both are alike. And this is why Democritus, at any rate, says that either there is no truth or to us at least it is not evident.”9This symmetry argument is the driving force behind skepticism: “There is no mark to distinguish the true and the false,” as Cicero puts it.10

The skeptical position is an easy one to propose but has always been dogged by the objection that genuine skepticism would result in paralysis.12 Carneades’ mitigated skepticism is an answer to this objection.

But though Cicero continually writes about probability, his account of what probability is supposed to be is curiously content free. Sextus Empiricus is a follower of a more extreme skeptical school and does not believe in probability, but his account of what Carneades said is much clearer. The difference is significant for the later history of probability, because Cicero was one of the best-known ancient authors, while Sextus Empiricus was almost unknown until nearly 1600.

The key phrase in Carneades’ theory, according to Sextus, is “probable presentation” (pithane phantasia). The discussion is in the context of perceptual errors, such as a coiled rope that one may at first take to be a snake. The presentation, or what appears to the observer, is either true or false, but which it is, according to earlier skeptical arguments he has given, cannot be known. Distinctions can be drawn, however, in the way the presentation appears. A first division is between presentations that appear true and those that appear false. The latter are not of interest, since the observer will not tend to base action on them—even if, by chance, they are true.14 More important are the divisions among apparently true presentations.

Carneades then discusses an even higher grade of presentation, that which is irreversible and also tested; that is, the possible sources of error in the organ of perception and in the environment have been checked and found unproblematic. How much inquiry is necessary in any particular case depends on the importance of getting the right answer. “Just as in ordinary life when we are investigating a small matter we question a single witness, but in a greater matter several, and when the matter investigated is still more important we cross-question each of the witnesses on the testimony of the others—so likewise, says Carneades, in trivial matters we employ as criterion only the probable presentation, but in greater matters the irreversible, and in matters which contribute to happiness the tested presentation.”15

There is much here that accords with later theories of probability: the various grades of probability, the high degree of probability that provides a good criterion for action, the increase in probability through the coherence of different pieces of evidence. What is unique about Carneades’ theory is that his probability is a property of perception rather than of propositions or beliefs and is genuinely equated with appearance of truth or likeness to truth.

The difference between Carneades’ theory and other theories should not be exaggerated, since the Skeptics were capable of distinguishing between a presentation and the assent to it and saying that their skepticism was really about assent.

In expounding his own philosophy in other works, Cicero again approves of advancing doctrines as only probable, but one is left none the wiser as to what this means. In addition to this he inserts into his long works on argument a few remarks of more or less Aristotelian tenor on probability as a property of arguments.

One of the pleasures of studying the ancient world is that a large proportion of what one has to read is of good quality, since the better works have enjoyed a higher survival rate over the centuries than the rubbish. Scholars of ancient philosophy have been less than overjoyed, therefore, at the recovery of part of a large library of Epicurean philosophy in the ruins of Herculaneum, the city destroyed with Pompeii in the eruption of Vesuvius in 79 A.D. The library is a small sample of the vast production in late antiquity of books about words. Among the least uninteresting of the papyri so far deciphered is the end of a book On Signs by the Epicurean Philodemus. It

Philodemus led a school of Epicurean philosophy in the area of Naples in the mid-first century B.C., with which Cicero had some connection. The work describes the debate between the Stoics and Epicureans over, essentially, the problem of induction, or the inference to general facts from observations. To infer “All men are mortal” from “All observed men are mortal” requires, according to the Stoics, following Aristotle, rational insight into the nature of man. The Epicureans maintain that there are no such rational insights into natures and that one can only make the inference from suitably repeated and suitably checked experiences. The debate is in principle the same as the modern one about whether inductive argument needs laws of nature and a uniformity of nature principle, or whether it is a purely statistical procedure like arguing from sample to population.26 In the ancient debate, the logical details on both sides never become completely clear.

Sextus Empiricus shows the fallibility of induction continuing to be a live issue in philosophy. It would be a mistake, he says, to suppose that because most animals eat by moving the lower jaw, all do, since the crocodile moves the upper jaw.

Sextus goes further, and poses the problem of induction as a general skeptical philosophical challenge: “For, when they [Stoic dogmatists] propose to establish the universal from particulars by means of induction, they will effect this by a review either of all or of some of the particular instances. But if they review some, the induction will be insecure, since some of the particulars omitted in the induction may contravene the universal; while if they are to review all, they will be toiling at the impossible, since the particulars are infinite and indefinite.”33

Avicenna became known in the West chiefly as a medical writer; he and Galen were the most respected authorities until well into the sixteenth century. In this capacity he transmitted something of the ancient medical writers’ emphasis on reasoning from what happens “for the most part.” He notes the fallibility of induction, referring to Sextus’s example of the crocodile.34 But he suggests an answer that has considerable appeal: he expands a little the connection briefly found in Aristotle between reasoning from what happens “for the most part” and the exclusion of the hypothesis of chance: “Indeed, when we have the repeated sensation of [connection] between one thing and another, such as between purging and scammony, and between the observed movements and things in the heavens, this sensation repeats itself in our memory; following which, we acquire experience of it, the cause of it being a concomitant syllogism or memory: If this thing, for example purging relative to scammony, were due to chance, accidental, with no necessity in nature, it would not be possible in the majority of cases for the same to happen—so that, when it is not thus, the mind finds that fact extraordinary and looks for a reason . . . we have experienced this often and then reasoned that if it were not owing to the nature of scammony but only by chance, this would happen only on certain occasions.”35

Avicenna has tried to use the Aristotelian not-by-chance argument to show that inductive reasoning is justified. His reasoning her...

Aquinas admits that a good deal of common knowledge concerns things about which one can make conclusions true conjecturally or true only in most cases. This includes knowledge even of such a crucial and intimate matter as whether one is in a state of grace: “something can be known conjecturally by signs; and in this way one knows that one has grace, in that he sees he delights in God, and despises worldly things.”

Inferring the intentions of others from their actions is of course also probabilistic.

As Aquinas recognizes, this is a substantial qualification of the Aristotelian thesis that “there is no science of the contingent”: “There can be no science of future contingents considered per se. But there can be science of them considered in their causes, in that some sciences can know there are certain inclinations to such and such effects.”

One could, in principle, use this reasoning to justify a science of dice throwing, which certainly deals with inclinations to future contingents.41 But Aquinas himself uses it to reconcile astrology and free will. Because the human will is free, the stars cannot determine it. But they can “dispositively incline” it, so that, since most men follow their passions, astrologers often predict the truth, “especially in public events, which depend on the multitude.”42 This last remark, with its distinction between the individual and the multitude, introduces Aquinas’s most important contribution to the theory of probability. He understands that in cases of variable outcomes, one can predict what a group will do more confidently than what the individuals that make it up will do. It is the idea behind the modern “law of large numbers,” the assertion that random effects in individuals even out as one takes a large sample.

If Aquinas had had a somewhat more favorable estimate of the natural inclinations of the majority, he might have been tempted to favor democracy. This course was taken two centuries later by Nicholas of Cusa, who held that the majority of citizens could be expected to make the right decision. “Otherwise it would happen that a natural appetite would be frustrated in many cases, which is considered most unfitting by the philosophers.”

Scotus and Ockham on Induction Aquinas does not give the impression of being genuinely concerned by the problem of induction or by skepticism in general. His successors in the fourteenth century did reply to the newly revived ancient skeptical arguments. Duns Scotus advances a not-by-chance argument combined with a realism about causes, similar to Avicenna’s: “Even though a person does not experience every single individual, but only a great many, nor does he experience them at all times, but only frequently, still he knows infallibly that it is always this way and holds for all instances.”

So in principle, “experimental knowledge, however frequent, does not imply necessarily that it is so in all cases, but only probably.”50

William of Ockham presents the same combination as Scotus, asserting that causal connections are in some way necessary, while undermining the supposed absoluteness of the necessity. Causal connection is such that it can in principle be inferred from a single observation, “since there is no reason why one heat should be more calefactive than another.” But in practice, if we observe that administering a certain kind of herb results in a reduction of fever, we cannot be sure what it is in the herb that effects the cure.51

In the early fourteenth century, philosophy came to be pursued by an established community of academics. There is much pushing of intellectual positions to their “logical” conclusions, irrespective of whether those conclusions contradicted a balanced and reasonable view of the world.54 More and more, writing was devoted to critiques of one another’s works, finer logical distinctions were made, and a wave of modesty swept through philosophical style. Such a culture naturally supported an attention to finer classifications of propositions than to the gross categories of true and false, and the grades of probability that could attach to propositions became something of an obsession. The beginnings are found in Scotus:

Although it is erroneous to say that God is not three and one and that the world had no beginning, it is nevertheless not erroneous to assert that, faith aside, it is more probable that God is not three and one or that the world never began than to assert their opposites. For nothing prevents some false propositions from being more probable than true ones. For I understand “probable” so widely (however erroneous this may seem to many) that, faith aside, it is in a way probable or even provable that everything happens of necessity.

One theoretical issue about the notion of probability was taken up by these writers. If probability can increase, and one can speak of “equally probable,” “more probable,” and “much more probable,” can probability if sufficiently increased become certainty? Or are probability and certainty of their nature different? Opinion was divided.

The language used here suggests that probability, if not exactly a quantity to which numbers can be attached, is at least capable of continuous variation. The early fourteenth century is chiefly known in the history of science for its analysis of the notion of continuous variation, under the name of “the intension and remission of forms.” Buridan and Oresme in Paris and the Merton school in Oxford studied such continuously varying quantities as speed and acceleration (which they were the first to distinguish), heat and the lightness and darkness of surfaces.63 Their discoveries were the necessary preliminaries to the calculus of the seventeenth century. Potentially, probability too is capable of being seen as a continuously varying quantity.

These ideas date to the 1330s and 1340s. Time for the development of ideas was running out. Nicholas of Autrecourt Before the disaster of the Black Death in 1348–49, there appeared two men of immense genius, Nicholas of Autrecourt and Nicole Oresme. The plague deprived them of their audience, and they were not influential figures in the history of thought. Their writings were simply too profound to be understood by the intellectual pygmies by whom they had the misfortune to be succeeded.

Nicholas of Autrecourt has been called “the medieval Hume” for his criticisms of the received views on sense knowledge and causality. The parallel with Hume is indeed very close. His opinions appear first in his Letters to Bernard of Arezzo, written in the 1330s. Bernard is an orthodox Ockhamist Aristotelian and so believes that, for example, although there could be the appearance of a star without there being a star, if God created a vision, that does not happen in the normal course of nature. He believes also that philosophy consists of demonstrated certainties. Nicholas argues, in his First Letter to Bernard, that no propositions of this sort can be known with “evidence reducible to that of the law of contradiction.” His criticism, like Hume’s, keeps to a purely logical plane:

Nicholas goes on to show without difficulty that if certainty means the certainty of logical demonstration, almost nothing is certain: not the existence of the objects of the senses, nor whether any proposition exists.

The Second Letter to Bernard is a more elaborate treatise on “evident certainty.” Nicholas argues that every certitude is reducible to the “first principle” that contradictories cannot be simultaneously true. It follows that “the certitude of evidence has no degrees” and that “in every consequence reducible to the first principle by as many intermediates as you please, the consequent is really identical with the antecedent or with part of what is signified by the antecedent.” The domain of what can really be demonstrated is thus very narrow: From

The passage is interesting for the suggestion that if inductive arguments are the only probable arguments, there is a problem with the knowledge of their premises: if the premises are not absolutely certain (as Nicholas has argued), whence do they derive their probability? It is hard to point to any progress on this question since Nicholas’s time.

There is another treatise by Nicholas, the Exigit, which expounds an anti-Aristotelian, and apparently totally original, picture of the world. The details of Nicholas’s worldview are remarkable. His idea is to oppose to the Aristotelian world picture, whose necessity he has denied, his own view, which he asserts is “probable.” Of the infinitely many things that could possibly exist, or ways things could be, he needs some probable principle to determine which ones do in fact exist. His central probable principle is that a thing exists if it is good that it should exist. His notion of good seems to be as much aesthetic as moral. “It may be taken that all entities of the universe are rightly disposed, that is, it may be taken that those things exist if it is good that they exist, and those things do not exist if it is bad that they exist. This proposition comes to the understanding from inspecting what happens in things of nature and things of art. In things of art the artificer is the measure of the good . . . the fore teeth of man are sharper for cutting food and the back teeth flatter for chewing, an example Aristotle uses.”69 It is to be noted that he does not derive this principle from anything about God, whose existence he regards as not demonstrable.70