The Science of Conjecture: Evidence and Probability Before Pascal

by James Franklin

By far the most original of the successes of medieval mathematical science were those relating to the conceptual analysis of motion, and of continuous variation more generally, associated with the fourteenth-century Merton school at Oxford and its French counterparts. The most impressive of these thinkers was Nicole Oresme, who was active in Paris in the decades just after the Black Death of 1350; he later became bishop of Lisieux.

As financial adviser to the king of France, he recommended against debasement of the coinage, a primitive Keynesian technique whereby the king spent more than he had.

Oresme is plainly no straightforward monarchist; his commentary on Aristotle’s Politics makes him one of the last chief theorists of medieval limited monarchy, before political thought was diverted into absolute monarchy during the Renaissance.

The last step involves a transition from relative frequencies in finite sets to comparisons in infinite sets, and Oresme’s intuition, though sound, would need explication in the mathematical machinery of 1900, not that of 1350. The important thing is that the probability argument is perfectly clear, and perfectly correct; it is an instance of the probable argument schema: “The vast majority of As are Bs. This is an A (about which there is no more relevant information). Therefore, this is a B.”

A related passage connects these relative frequency ideas with symmetry notions of insufficient reason. He divides the possible into three. “Either it is equally possible, or it is improbable, or it is probable. An example of the first way: The number of the stars is even; the number of the stars is odd. One is necessary, the other impossible. However, we have doubts as to which is necessary, so that we say of each that it is possible . . . sometimes in such cases we have no reason for one part; and sometimes we do have a reason, and then it is called a ‘problem.’ . . .

The criticism Oresme is advancing is one that would now be described by saying that astrology is not structurally stable. If the genius of ideas is measured by the time taken for them to become common currency, then this one is a winner. It is almost totally original; although a few thirteenth-century writers had realized that there would be some problem if the ratios of the orbits were irrational,41 only Oresme understands what the problem really is. On the way to his result, Oresme, in 1350, has come up with the ideas of fractional powers and probability based on relative frequency, ideas that took three hundred years to reappear; structural stability did not appear again until Poincaré’s work of the 1890s.42 The idea is that a theory dealing with physical things must make predictions based on measurements, which do not have infinite accuracy. Therefore the theory must, at least usually, make similar predictions for sufficiently close initial conditions.

the Newtonian theory makes predictions about the orbits of planets; if one puts into the equations a slightly wrong position, the theory’s prediction of the orbit will be wrong, but only slightly. Similarly, Euclid’s results on perfect spheres apply to real spheres, which are imperfect, not because of some magical idealization but because of the purely geometrical fact that the volume of, for example, an approximate sphere is close to the volume of a perfect sphere that it approximates. Physical systems that are not like this are unpredictable, even though deterministic; they are now much studied under the name of “chaos”43 (another word possibly introduced into French by Oresme).44 Oresme is saying that astrology is not structurally stable and so cannot make any predictions.

The idea that diversity and variety are necessary to beauty has had a long history since then, although as modern architecture and town planning show, it is far from being completely accepted.

The relevance of diversity and variety to matters of explanation is one of the themes of another of Oresme’s works, On the Marvels of Nature. The general thesis is that marvels can be explained naturally: “The natural causes there assigned and the manner of finding them are possible and much more likely than that demons or unknown influences are the causes of the aforementioned effects.” There is simply too much credulity among people, Oresme thinks. “It seems to me that to believe easily is and has been the cause of the destruction of natural philosophy; and also in faith it makes and will make great dangers, and it will be the cause of receiving Antichrist and the introduction of a new law.”

One of his lines of argument is that certain things commonly thought to require explanation do not need any at all. Variety as such needs no cause; if some men are sodomites and others only attracted to black women, it can be put down simply to the variety of things. Similarly there is no need to posit demonic or astral influences, or direct intervention by God, to explain a run of coincidences. Oresme gives the same kind of naturalistic explanation of prophecies and their apparent success: “I also say that, if someone says many things, it is difficult for him not to spe...

looks backward to the Scholastic discussions of Ockham’s Razor (discussed in chapter 8) and forward to the ideas of Copernicus, Kepler, and Galileo on the choice of scientific theories according to criteria of simplicity:

The detectable influence of Oresme’s opinions concerning probability and incommensurability is almost zero. One of the few to know anything about them is Gerson (noted in chapter 4 as the inventor of the phrase “moral certainty”), who mentions that Oresme shows that questions of the commensurability of the heavenly motions and the influence of the stars are completely uncertain and can be discussed only in terms of “rhetorical probability.”

The Renaissance took place along an axis stretching from Rome to London. Actual additions to knowledge in the period are associated with regions far from that axis: Salamanca, for example, and the ports from which the Indies were discovered. At the other end of Europe it was still possible to receive a solid medieval education in cities such as Cracow.

Much has been written on the status of scientific theories in the century of the scientific revolution. The most prominent fact is that the leading writers—Bacon, Galileo, Descartes, and Newton—stated emphatically that their results were certain. Their statements were undoubtedly sincere, but one should not be too quick to conclude that the science of the time was universally thought to be a search for certainty. It is proverbial that very successful people tend to possess, besides unlimited energy, an utter conviction that their own opinions are right. There is less talk about certainty in the lesser, but still exceptionally brilliant, scientists of the seventeenth century. There is also less insistence on certainty in some of the works of the great men themselves, in places where they stopped advertising their projects and dealt with the detailed business of explaining their results.

Kepler and Galileo share the same project, the mathematization of science, support the same unorthodox thesis, heliocentrism, and often propose the same arguments. They both interpret astronomy realistically—Kepler was the first to call attention to the fact that the preface to De Revolutionibus was not written by Copernicus, and Galileo rejected the suggestion that he regard Copernicanism as fictitious to reconcile it with theology. They distinguish between probable and demonstrative arguments for that thesis in the same way. Their genius is equally exceptional. It has therefore often been wondered why they took so little notice of each other; Galileo, for example, never took seriously Kepler’s principal discovery, that the planetary orbits are elliptical. Of course, geniuses are very busy people and often have no time to read one another’s works. But in this case, their styles of thinking and writing show a fundamental incompatibility sufficient to make them almost incomprehensible to each other.

It has been common to see, in consequence of the (true) ellipse theory and the (false) Platonic solids theory, two Keplers, or at least two opposing aspects of Kepler’s thought—one, rationalist and looking forward to modern scientific method, with its curve fitting and hypothesis testing; the other backward looking and “mystical,” misled by all manner of unlikely Pythagorean speculations. Kepler’s works reveal no such split. The difference between the two theories is simply that the ellipse theory turned out to be right, the solids theory wrong. Kepler’s manner of thinking, and of argument, is the same in both cases and is the one that lies at the bottom of the success of modern mathematical science. It involves finding the harmonies of the world, to use Kepler’s words, or in the language of a more pedestrian century, argument from concomitant variation.

Kepler differed from Galileo, and indeed from everyone else up to his time, in demanding an extremely high standard of agreement.

Kepler sees the divide between necessary and probable reasons for his theory as a result of introducing physics into astronomy. “As is customary in the physical sciences, I mingle the probable and the necessary and draw a plausible (probabilem) conclusion from the mixture. For since I have mingled celestial physics with astronomy in this work, no one should be surprised at a certain amount of conjecture. This is the nature of physics, of medicine, and of all the sciences which make use of other axioms besides the most certain evidence of the eyes. . . . Thus the physical difference is now immediately apparent [among the three opinions of Ptolemy, Copernicus, and Brahe]: by way of conjecture indeed, but yielding nothing in certainty to conjectures of doctors on physiology, or to any other natural science.”

Since it is demonstrable that there are only five regular solids—the tetrahedron, the cube, the octahedron, the dodecahedron, and the icosahedron—the correspondence would explain why there were as many planets as there were.

Understanding Galileo’s opinions on scientific theories is much complicated by the fact that his main works were written with a view to avoiding problems with the Inquisition.

The reasoning behind the condemnation takes us back to the end of the previous century, when the Jesuits, by then leaders in the world of academic science, applied the Scholastic method to astronomical questions, treating them as if they were problems of canon law or moral theology. The following passage, written about 1590, is typical of the style of argument; it is the same style as found in, for example, the debates on probabilism, with its mixture of considerations of plausibility and appeals to authority. It is worth quoting at some length, to convey the exact consistency of the intellectual bog from which modern science had to extricate itself.

As evidence of the forms of thought that modern science had to overcome, this is particularly convincing in that the author of this rubbish is the young Galileo himself. It is from unpublished notes, which have been found to be a collage of the lecture notes of a number of Jesuit professors at the Collegio Romano, dating from the late 1570s and the 1580s. These professors made considerable play of probabilities and their comparisons, in the same way as the passage just quoted.82 Galileo later wrote, “If what we are discussing were a point of law or of the humanities, in which neither true nor false exists, one might trust in subtlety of mind and readiness of tongue and in the greater experience of the writers, and expect him who excelled in those things to make his reasoning more plausible, and one might judge it to be the best. But in natural sciences whose conclusions are true and necessary and have nothing to do with human will, one must take care not to place oneself in the defense of error; for here a thousand Demostheneses and a thousand Aristotles would be left in the lurch by every mediocre wit who happened to hit upon the truth for himself.”

Galileo’s reference to law and the humanities is exactly right. Men like his opponents, and his earlier self, do not believe anything is established in science, or ever will be. They expect science to continue like law, the disputes of the learned stretching out in commentaries and glosses till Judgment Day. In their favor it must be admitted that very little had been established in science for centuries. They were not to know that the telescope was about to be invented and Galileo about to exist. What is most blameworthy in their activities is the same tendency as in moral theology, to exaggerate the worth of arguments from tradition or “extrinsic probability.”

This is supported by a very true maxim of Aristotle’s which teaches that, ‘It is pointless to use many to accomplish what can be done with fewer.’”

It is certainly true that it is useful to consider the comparison of hypotheses here. As with modern creationism, the defense of improbable hypotheses discredits religion.

Discussion of the simplicity of hypotheses has advanced little since Mersenne. Everyone agrees that the simplicity of a scientific theory is a reason to believe it (or at least to “accept” it, in the jargon of the century of noncommitment). But why should it be? Does a theory’s simplicity make it more probable? If so, why? If not, why does it constitute a reason for preferring the theory? Newton’s “Nature is pleased with simplicity, and affects not the pomp of superfluous causes” is magisterial but unsupported by reasons.96 God may have the potential to be “pleased” with simplicity, but why Nature?

a comparatively minor technical dispute on evidence for the Copernican hypothesis occasioned a different contribution by Galileo on the relation of observation to theory, one that is the ancestor of all modern statistical techniques of fitting theories to data. It concerns methods for combining discordant and error-prone observations. This matter had been treated to some extent by Ptolemy and Copernicus, but Galileo’s version is considerably more precise.

He puts forward the principles that observations are “equally prone to err in one direction and the other” and that careful observations are “more likely to err little than much.”

Astronomy has many difficulties with errors in measurement but very few with variability in the subject matter itself. The more mundane sciences may be able to measure more directly and accurately, but the variability in what is being studied makes the extraction of laws and the prediction of new cases very hard. Superficially, it is easy to know whether an herb cures a certain disease, by testing it on many cases and seeing if a cure results. Just occasionally, it is as easy as that, but almost always there are so many variables, spontaneous cures, extenuating circumstances, and possible excuses that it is almost impossible to extract a truth from any reasonable amount of data. It was only from the late nineteenth century that the sophisticated statistical techniques of biometry (modern statistics) were developed in agriculture and genetics and were applied in psychology, sociology, and drug trials.

Lacking any idea of such techniques, ancient study of the “soft” sciences could not make much sense of the mass of experience and observation, however assiduously collected. The situation was made worse by the fact that some of the “sciences” in which experience was most attended to were ones like physiognomics and divination, in which the data are, on a modern understanding, all noise and no laws.

The allowable kinds of sign are therefore various and must be put together: “In general it is foolish to rely on a single sign: you will more likely have confidence in your conclusions when you find several signs pointing the same way.”4 This is indeed the main reason that a nondeductive science will look so different from a deductive one. Euclid does not collect different proofs for the same theorem, since the theorems are given the highest possible degree of belief by a single proof. But when conclusions are not susceptible of deductive proof, it will always be worthwhile to collect more “signs pointing the same way.” This means that wide experience is an advantage in sciences like physiognomy. One will need long experience to discern the difference between pallor due to terror and that due to fatigue and, generally, the congruity between different subtle shades of expression and conditions of mind.5 In such a science there must also arise problems when the signs do not point the same way, that is, when evidence conflicts: “It is safest, however, to refrain from all positive assertion when you find that your signs are inconsistent and contrary to one another in detail, unless they belong to classes, some of which you have determined to be more trustworthy than others.”

Many ancient authors extolled the virtues of experiment and observation.9 But the Alexandrians, for all their research grants, found it a good deal harder to learn from experience than to praise doing so. The fields in which the careful recording of observational data perhaps went furthest in antiquity were divination and astrology. Cicero reports the Stoic arguments for divination and their replies to the Skeptics. It is an important passage both for its suggestion that in sciences without certainty a reasonable level of mistakes can be tolerated without that making the science worthless, and also for its use of random phenomena similar to dice throwing as a model of uncertainty:

But his main excuse is the one that still serves so well for sciences with poor predictive power, like economics and meteorology, the excuse from complexity: “The ancient configurations of the planets, upon the basis of which we attach to similar aspects of our own day the effects observed by the ancients in theirs, can be more or less similar to the modern aspects, and that, too, at long intervals, but not identical, since the exact return of all the heavenly bodies and the earth to the same positions, unless one holds vain opinions of his ability to comprehend and know the incomprehensible, either takes place not at all or at least not at all in the period of time that falls within the experience of man; so that for this reason predictions sometimes fail, because of the disparity of the examples on which they are based.”

It is in medicine that there is most pressure to find correct methods of learning from experience. And there is always money available for it. The subject matter, unfortunately, is recalcitrant; though there are true generalizations to be found, in contrast to divination or astrology, it is much harder than in astronomy to discover simple hypotheses true universally. That is what gives credibility to Cicero’s comparison of medicine with divination. As in divination, so in medicine, and especially in pharmacology, one must generalize from experience, without knowledge of underlying causes: “The power of scammony root for purging . . . I see, which is enough; why they have the power I do not know.”14

A rationalist approach, combined with experiment, was pursued in early Alexandria by doctors like Erasistratus, whose neurophysiological experiments on criminals show a zeal for finding causes equaled only by the worst of the Nazi doctors.

In opposition to these and other Dogmatist schools seeking the causes of medical phenomena, there arose the Empiric school, which denied there was any knowledge of causes to be found and restricted itself to describing the succession of phenomena. Such an attitude also had some basis in the works of Hippocrates.

The Aphorisms of Hippocrates open with the famous saying, “Life is short, art long, opportunity fleeting, experiment...

The significance of the passage is its choice of a middle path in finding the correct reference class for a generalization, still one of the hardest problems to solve in probabilistic reasoning.

The Empiric school itself did not arise until there was a sufficient number of Dogmatist schools to combat. Like any skepticism, it throve best in the presence of dogmatisms in collision. In general, its lines of argument were taken from the Skeptics of the Academy and were a straightforward application of the skeptical method to medicine.

Though Sextus Empiricus was a member of the school shortly after Menodotus, he tells us little about any aspects of the Empirics’ views other than the purely negative. Our knowledge of the positive side of their doctrines, on how to learn from experience, comes from two obscure books of Galen, An Outline of Empiricism, which exists only in a Latin translation of 1341, and On Medical Experience, most of which survives only in Arabic.27 Both works were almost unknown until modern times.

The Empirics rejected inference to unseen causes for various reasons, among others that a cause could in principle be inferred from an effect that had been observed only once. Dogmatists were not, therefore, in a position to object to the Empirics’ use of inference from what has been observed “very many times.” To admit that in doubtful cases a second case is useful, as the Dogmatists do, opens the door to considering the whole question of inductive skepticism and the worth of large numbers of cases.

The Dogmatists naturally demanded to know how many times counted as “very many” and pointed to the absurdity of taking, for example, forty-nine times as not enough to support an inference but fifty times as enough. They asked again how experience worthy of inference could be composed of experiences each of which was not worthy of inference.31 A natural reply would have been that the probability of the induction increases gradually with the number of observed instances, but the Empirics were diverted from making this discovery by having on hand another intellectual structure into which the case would fit. They simply answered that the situation is the same as a heap: it is impossible to say exactly how many grains of wheat are needed to make a heap; nevertheless it remains true that very many grains form a heap.

As to the certainty or otherwise of their methods, the Empirics, though apparently not admitting the fallibility in principle of induction from what has been seen “very many times,” did admit to making mistakes because their experience was limited. They hastened to add, of course, that the Dogmatists did too, without admitting it. Dogmatism, they said, could at best only reach the level of plausibility and likelihood.

Besides direct experience, the Empiric will acquire knowledge through “history”—what has been written down by others. Here the Empirics make a start on discussion of the credibility of testimony, or what was called by later historians the “signs of true histories.” Agreement among authorities is important, but not everything repeated by many writers is to be believed (for example, things written under the influence of dogmatic misconceptions):

These considerations are in principle equally applicable to works of history in the usual sense, but historians in ancient times do not seem to have developed any theory of evaluation of written evidence as self-conscious as that of the medical writers.

It is significant that the Empirics ask why agreement should be taken to be the sign of truth and answer simply, “It is known from experience.” They give the same reason for believing in “transition to the similar,” that is, argument from analogy. “Empirical transition . . . relies on what is known naturally, not because it is plausible (suasibile, probably translating pithanon) that something similar should have similar effects, lack similar things, or be similarly affected; it is not because of this or because of anything else of this sort that one insists on transition, but only because we know from experience that similar things are like this.”38 So there is no concession to Dogmatic a priori reasoning; all conclusions are to depend on experience. Because of this, one can learn things about analogy that would not be suggested by pure reason.

Jewish law discusses a number of cases that involve proportions in populations. The style of reasoning resembles that of the Greek medical writers, but the examples are quite different, though a few of them are also medical. Jewish law deals with far more than what other societies regard as legal. It lays down an order for all activities of life. There is immense pressure on the interpreters of the law to produce answers for all doubtful cases and methods that can be thought to solve in principle all future doubts. While a decision procedure for doubtful cases need not be one that aims at producing the answer that is most probably true, it can be expected that there will be some connection between what the law lays down and what is most likely true.

Maimonides comes close to evaluating “doubts” by counting cases that could be regarded (though he does not say so) as equiprobable.

Avicenna was aware of one of the major problems in experimentation in the low sciences: that the variability of the subject matter makes it difficult to sort out the causes of any given effect.

Biology was especially subject to domination by unhelpful arguments from analogy, whereby superficial aspects of plants and animals caused them to be taken as symbols of something else. One of the things most notably wrong with Renaissance thought was the immense proliferation of signs. Though allegory was certainly popular earlier (medieval bestiaries spend a good deal of space on what virtues animals symbolize), it did not so completely create a veil of signs between man and the world as it did in the Renaissance.