The Science of Conjecture: Evidence and Probability Before Pascal

by James Franklin

Of course Nicholas admits that some things seem to come into and go out of existence, but he asserts that this cannot really be so. They must consist of invisible parts that combine and disperse; the parts are eternal, while combination and dispersion are only changes of relation, and relations do not exist. So the next corollary is a thoroughgoing atomism. “Thus I have here sufficiently probable ways for concluding that the conclusion of the eternity of things is probable, but because I cannot show those tiny whitenesses coming and going like grains, some will perhaps not believe in them; but it is not to be denied on this account.

Atomism in turn implies the doctrine of the eternal return; the atoms eventually reconstitute themselves just as the planets eventually return to the same positions. This includes mental entities: “We can however say that it can be sustained probably, and it is probable, that the acts of our mind are eternal by being recapitulated.”

The conclusions, stated baldly (as they are by Nicholas), are unusual. A little meditation on them will show that principles very like them are assumed time and time again, without being explicitly stated. Of course, it is true that no modern philosopher (or almost none)76 would state anything so optimistic as “everything is well-disposed.” But at a lower level of generality, things are otherwise. There are Ockham’s Razor, for instance, the currently fashionable “inference to the best explanation,” and various older “principles of least action.” Certain modern philosophers have been happy, in unguarded moments, to admit to a preference for desert landscapes, as if this were an intellectual virtue that all would commend. It appears to be, as William James says in another context, “a sort of philosophic faith, bred like most faiths from an aesthetic demand.”77 And the popularity of the theory of evolution in philosophical circles is surely partly attributable to its licensing of speculations about the supposed advantages of anything whatsoever in the living world. What are all these but particular versions of Nicholas of Autrecourt’s “all entities of the universe are rightly disposed”? At least he states this explicitly as a probable principle, and at least his theory contains a God, whose tastes may be conjectured to lean toward a classical simplicity. The rational support for the modern versions of his principle is not so clear.

It will have been observed that whenever Nicholas puts forward a new outrageous conclusion, he is careful to say it is only probable. Scattered remarks explain his understanding of probability, the largest being this: “If these reasons are not found absolutely conclusive, nevertheless positing them is probable and more probable than the reasons for the opposite conclusion. To anyone not predisposed more to one side than the other, the degree of probability will appear to predominate (gradus probabilitatis excedens) in these reasons. I speak thus because in the books of others I have seen few reasons for their obscure conclusions that I cannot give probable answers to, and if they say that I deny principles known per se, it is amazing how they express notorious falsehoods. . . . Was it not sufficiently proved above that to say that [the proposition] when blackness succeeds, whiteness no longer exists, is not a principle known by the meaning of its terms, nor through our experience? It is remarkable too how they consider as known per se principles the opposites of which were agreed to by almost everyone before Aristotle, at least the most serious.”

The professionals in that field decided that faith indeed was in opposition. Nicholas was summoned to the Papal Curia in Avignon around 1340, and after some years of investigation thirty-two propositions from his works were chosen for condemnation. They involve the skeptical theses of the Letters to Bernard. The condemned propositions include: (2)

Nicholas insisted to the commission of inquiry that all his assertions were made disputatively rather than definitively and that he always meant that one should adhere strictly to the faith. The commission called this a “foxy excuse” and sentenced him to burn his own writings in Paris. He did so in 1347 and is last heard of a few years later as dean of Metz. A few contemporary references show he was known and understood; one of his readers was Nicole Oresme, certainly a man capable of appreciating Nicholas’s achievement, though Oresme’s own genius was more mathematical than philosophical.82 Nicholas’s arguments were still known some fifty years later, but he was thereafter unknown until modern times.

Nicholas of Autrecourt and Nicole Oresme belong to the last generation that grew up before the Black Death. Of those mentioned in this story, Nicholas and William of Ockham were probably plague victims, as were those, unknown to us, who would have been their successors. The plague’s impact on population and economic life, though catastrophic, was not quite an isolated event—there had been decline before, and recurrences of the disease prevented recovery later. But in matters of the mind, there was a distinct break in the intellectual tradition. The urban monasteries, the centers of intellectual training, were hit by deaths much more than the general population, and elementary education suffered severely. “Few could be found in the good towns and castles who knew how or were willing to instruct children in the rudiments of grammar.” Oxford, become “like a worthless fig tree without fruit,” was estimated to have lost 80 percent of its students.83

What passes for philosophy around 1400 combines either originality with stupidity, as with Wyclif, or learning with complete unoriginality, as with Pierre d’Ailly. Ailly repeats the phrases of the Ockhamists, among them, “The probable taken more widely is what is true, either necessarily or contingently, but is not evident” and “More probable in natural light than its opposite.”84 But the life has gone out of the exercise. The spirit of the age is represented better by Petrarch. His work of 1370, On His Own Ignorance and that of Many Others, was composed after he left Venice, stung by the remark of some young Averroists that he was a “good man, but a scholar of little merit.” Its tone may be gathered from its remarks on skepticism: “Socrates says: ‘This one thing I know, that I know nothing.’ This most humble confession of ignorance Arce-silas blames as still too bold. . . . The Academy is disproved and rebutted long since, and it is established that something can be known when God reveals it. Therefore, it may be sufficient to know as much as is necessary for salvation.”

Bacon and Descartes: Certainty? or Moral Certainty? It was seen in chapter 6 that in Galileo, for example, there was an insistence on certainty in science, combined with some use of probability in practice. The demand for certainty is not confined to science, of course. It is characteristic of the period 1630–60, especially, also in philosophy, moral theory, and political theory.

In Bacon there are a few hints that there are degrees of certainty in science,87 but on the whole Bacon is the most insistent on certainty of all the founders of the scientific revolution. He is against the simple inductive enumeration of the logicians, simply on the grounds of its fallibility:

At first sight, this is rather surprising, in view of Bacon’s championing of experience and his professional knowledge of law. But it appears that his conception of induction was formed in an area of law other than the law of evidence. His rivalry with Coke concerned matters of principle as well as personality and included a fundamentally different view of the relation of legal principles to cases: while Coke kept very closely to precedents, Bacon favored the more axiomatic, rule-based approach of Continental law.89 To decide what law is applicable, one collects as many relevant cases as possible, both positive and negative, and infers the rules that will cover the positive cases, while not being contradicted by the negative ones. The cases do not “support” the rules probabilistically but instead impose deductive constraints on what the correct rule could be.90 In the scientific context, this raises the problem of the reliability of the reports of the positive and negative cases themselves, since Bacon proposes to have investigators using many reports, not only those of their own experiments.

Bacon’s actual legal experience was not so much in pleading factual cases in court as in conducting examinations on behalf of the government. His most original remarks on natural philosophy are those that urge the examination of nature “under constraint and vexed; that is to say, when by art and the hand of man she is forced out of her natural state, and squeezed and moulded.”92 So certainty is attained because nature will be made to yield up her secrets by torture.

that could, ideally, be checked. Descartes too is on the side of certainty as the only worthwhile ideal. Rule 2 of his Rules for the Direction of the Mind is, “We should attend only to those objects of which our minds seem capable of having certain and indubitable cognition.

He attributes various evils to the failure to separate probabilities and certainties. “We would be well-advised not to mix any conjectures into the judgements we make about the truth of things. It is most important to bear this point in mind. The main reason why we can find nothing in ordinary philosophy which is so evident and certain as to be beyond dispute is that students of the subject first of all are not content to acknowledge what is clear and certain, but on the basis of merely probable conjectures venture also to make assertions on obscure matters about which nothing is known.”

But what certainty is available outside mathematics and pure philosophy? Can Descartes maintain that the truths of empirical science, including his own discoveries, have the certainty of demonstrations? In the exuberance of youth, his answer was yes. But even an intellectual ego as robust as the Cartesian proved subject to the erosion of certainties from long experience of life. The source of trouble was argument “from effects to causes”; very frequently in Descartes’ science he infers the truth of a theory from its ability to explain phenomena. But surely this cannot be a demonstrative argument, since different arrays of causes might produce the same observable effects?

The passage continues with an idea like Bacon’s: crucial experiments can be devised to decide between competing hypotheses with certainty.

The modern economic modeling that attempts to forecast unemployment, interest rates, and so on without any commitment to grand economic theories is a continuation of Descartes’ project.108 More relevant to Descartes’ own time is the problem of the truth of the atomic hypothesis: it could, more or less, explain appearances, but is that a good reason for believing it true, in the absence of any more direct evidence?

Descartes also admits that, “since in everyday life we must often act without delay, it is a most certain truth that when it is not in our power to discern the truest opinions, we must follow the most probable. Even when no opinions appear more probable than any others, we must still adopt some; and having done so we must then regard them not as doubtful, from a practical point of view, but as most true and certain, on the grounds that the reason which made us adopt them is itself true and certain.”

Gassendi’s strategy is the same as the one Nicholas of Autrecourt deployed against the Aristotelians: to put forward hypotheses, or possibilities contrary to what Descartes claims to have proved, “suggestions which do not concern the actual results which you have undertaken to prove, but merely the method and validity of the proof.”

Certain Jesuit Scholastics of the seventeenth century pursued some interesting ideas on the problem of induction. In theory, they take up the problem where Scotus and Ockham left it and take no notice of purveyors of novelties like Descartes, but in fact their problem is very like Descartes’ one of predicting the behavior of clocks whose internal mechanisms one does not know but whose past behavior one has observed.

pupil of Lugo’s argues that it is a matter of probability, founded on relative frequencies: “Every reasonable presumption must be founded on the probability of the thing presumed being greater than that of its opposite . . . we presume the natures of things are not disturbed from the fact that through long induction we can see that God ordinarily keeps to this mode of governing the visible world.”

Hobbes, though inclined to follow Bacon and Descartes in exalting deductive reasoning, does insist on the fallibility of induction, while allowing that it is a good bet. He even assigns a number to the degree of reasonableness of an inductive argument, “for the signs are but conjectural, and according as they have often or seldom failed, so their assurance is more or less; but never full and evident: for though a man have always seen the day and night to follow one another hitherto; yet can he not thence conclude they shall do so, or that they have done so eternally: experience concludeth nothing universally. If the signs hit twenty times for one missing, a man may lay a wager of twenty to one on the event; but may not conclude it for a truth.”

such a rule prescribes infinite odds, that is, a judgment of certainty, when the previous cases have been all of one kind. That it has been unable to discover what the correct odds are has been a notorious source of embarrassment for the theory of probability ever since. The problem has stayed so comprehensively unsolved that it has been more or less agreed to forget about it.

Pascal’s most successful experimental work in science was his investigation of pressure in fluids. His debate on his results with the Scholastic Père Noël provided the occasion also for considerations on the philosophy of science, which raised many of the issues still current in that subject. They include the role of deduction versus probability in the evaluation of alternative hypotheses, the significance of crucial experiments, the place of metaphysics in science, and the degree to which hypotheses can survive adjustments to fit the experimental facts. At issue was the explanation of experiments in which inverted glass tubes of water or mercury of sufficient height were observed to have spaces under the glass at the top of the tube. Were these essentially vacuums, as Pascal thought, or had some subtle matter passed through the glass to fill the space? And was the correct explanation of the phenomenon that of Pascal, according to which the pressure of the outside air could support a column of a certain height but no higher, or the Scholastics’ maxim that “nature abhors a vacuum”?

In discussing the “decisiveness” of his experiment, he subscribes to a fully deductivist ideal of science: Only what is clear and evident to the senses or reason, or logically deducible from such principles, is certain; “and all that is based on neither of them passes for doubtful and uncertain.

Religion too is an old enemy of probability—and for much the same reason as philosophy. It has certainty to offer, and acceptance of that certainty is not only a privilege but a duty. A strong though not always dominant strand in religion has been the condemnation of half measures, epitomized in the Book of Revelation’s “you are neither hot nor cold. I wish you were one or the other, but since you are neither, but only lukewarm, I will spit you out of my mouth.”1 When St. Paul mentions the distinction between probable and necessary reasons, it is to claim that he has the latter: “My speech and my message were not in the plausible words (pithanologia, Vulgate: persuasibilibus verbis) of wisdom, but in demonstration of the Spirit and of power.”2

In deciding on the reasonableness of faiths that assert some historical facts, like Christianity, Islam, and Judaism, there is room to debate the reliability of historical knowledge. And one reaction to conflicting dogmatisms has been a moderate position that denies the possibility, and necessity, of knowing some of the matters in dispute between the warring sects.

Of all arguments for the existence of God, the most durable is the design argument. It comes in two flavors, deductive and nondeductive. The deductive type, of which the most famous example is Thomas Aquinas’s Fifth Way, argues that teleology, or directedness, in nature necessarily implies an Orderer, in much the same way as a street sign’s having a meaning implies that someone wrote it to have that meaning. This is offered as a philosophical intuition into the nature of things. Take it or leave it. The other, more popular, kind of design argument is quite different. Exemplified by such modern versions as “What is the chance of a tornado blowing through a junkyard and leaving a 747?” it is a nondeductive argument and hence subject to infinite discussions of its plausibility. Is there an alternative explanation of apparent order? Can nature self-organize? Can order come to be by chance? Can the analogy between the world and an artifact be defeated by some parallel argument, like an argument from disorder?

Sextus Empiricus reports ancient design arguments. “Just as the man who is familiar with ships, as soon as he sees in the distance a ship with a favouring wind behind it and with all its sails well set, concludes that there is somebody who directs its course and brings it into its appointed havens, so too those who first looked up to heaven and beheld the sun running its courses from east to west and the orderly processions of the stars sought for the Artificer of this most beautiful array, conjecturing that it had not come about spontaneously but by the agency of some superior and imperishable nature, which is God.”

(To which Sextus mentions the reply: “You possess also a small portion of the great quantity of gall existing in the Universe, and phlegm and blood. It will follow, therefore, that the Universe is gall-making, and productive of blood.” Analogies are always in danger of being outflanked by parallel arguments.)

These arguments were taken up by the revealed religions, when their apologists saw the need to take seriously the problems of the relation of faith and reason. Christians were encouraged by St. Paul’s saying that “ever since the creation of the world his invisible nature, namely his eternal power and deity, have been clearly perceived in the things that have been made.”

According to the church fathers, the relation of theology to probability consists mainly in accusing other disciplines of being restricted to “mere probability,” the implication being that theology could of course do better.