Franklin -- The Science of Conjecture

[About witch trials] I have my ears battered with a thousand such flim-flams as these: "Three persons saw him such a day in the east; three, the next day in the west; at such a hour, in such a place, in such habit"; in earnest, I should not believe myself. How much more natural and likely do I find it that two men should lie, than that one man in twelve hours' time should fly with the wind from east to west? How much more natural that our understanding should be carried from its place by the volubility of our disordered minds, than that one of us should be carried by a strange spirit upon a broomstick, flesh and bones that we are, up the shaft of a chimney? Let us not seek illusions from without and unknown, we who are perpetually agitated with illusions domestic and our own. Methinks one is pardonable in disbelieving a miracle, at least, at all events where one can allude its verification as such, by means not miraculous; and I am of St. Augustine's opinion, that "tis better to lean towards doubt than assurance, in things hard to prove and dangerous to believe." ... It is true, indeed, that the proofs and reasons that are founded upon experience and fact, I do not go to untie, neither have they any end; I often cut them, as Alexander did the Gordian knot. After all, 'tis setting a man's conjectures at a very high price, upon them to cause a man to be roasted alive. -- Montaigne, Essays 3 chp.11.

The story of the decline of Science in the West and its survival in the East is a familiar one. The twelfth century saw the reappearance in Western Europe of the scientific point of view on the world and the recovery, translation, and assimilation of all the main ancient scientific texts. What the later Middle Ages made of their scientific legacy is, by common consent, less impressive than their achievements in such files as philosophy, theology, and law. The long debate of why the scientific revolution did not take place before it did is not subject to conclusive resolution, but there is a wide support for the view that the Scholastic method, relying too much on conceptual and textual analysis, failed to devote enough attention to experiment and measurement and their relation to theory. Nevertheless, there are areas of science in which purely conceptual work is entirely appropriate, namely the more mathematical sciences, and it is there that medieval science is strongest. Optics and astronomy, in particular, were regarded as actually part of mathematics and were central to both teaching and research. (p.140)

It is in [Nicole] Oresme's mathematical works that we must look for a full treatment of his ideas on probability [...] a passage connects relative frequency ideas with symmetric 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 stars is even; the number of 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'... An example of the second way: The number of stars is a cube. Now indeed, we say that it is possible but not, however, probable or credible or likely (probabile aut opinabile aut verisimile), since such numbers are much fewer than others." (pg.141-2)

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. 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 is flat; lords enjoyed droit de seigneur over peasant women; the Scholastic debated how many angels could dance on the head of a pin), raising questions about which age is really the more credulous. (p.181)

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." [...] This symmetry argument is the driving force behind skepticism. (p.196)

[About the book On Signs by the Epicurean Philodemus] The work describes the debate between 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 suitable 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. (p.201)

Earlier Scholastic thought made some attempt to grapple with the problem of why a astronomical theory's agreeing with the observations should be a reason for believing it and whether the belief should amount to certainty or not. Kepler tried to find something in common between the Copernican and Ptolemaic theories, which would account for the appearances and thus explain why they made the same predictions, although one was right and one was wrong about the underlying causes. But Descartes is asserting something over and above a simple recommendation of inference to the best explanation. He asserts that there is no need to find the true underlying model to make the correct prediction. It is the first clear statement of the dream of modern statistical inference: to make true predictions independently of difficult inquiry into inner causes. 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. (p.221)

Ockham is now best known for the principle of economy in reasoning known as Ockham's Razor. This is a misnomer for two reasons. First he did not originate it; there is the inevitable origin in Aristotle. The phrase "We consider it a good principle to explain the phenomena by the simplest hypothesis possible" is in Ptolemy. Formulations like "It is vain to do with more what can be done with fewer" and "A plurality is not to be posited without necessity" are Scholastic commonplaces from the early thirteenth century. Second, though he does repeat these ideas frequently, Ockham's contribution is more to restrict the operation of the principle, in the interests of God's absolute power: "God does many things by means of more which He could have done by means of fewer, simply because He wishes it, and no other cause is to be sought. From the very fact that Ge wishes it, it is done suitably, and not in vain." In the Eucharist, especially, Ockham holds that a plurality of miracles is to be postulated, simply because that pleases God. (p.241)

Serious mathematical thought on the subject [dice], however, needs a sufficiently numerical culture. Such a culture became established in fourteenth-century Italy, which was responsible for a number of the crucial steps in creating a tradition of skill in applied numerical calculation. It was the scene of many of the first mechanical public clocks, for example, of the invention of double-entry bookkeeping and, as we saw, of insurance. Especially notable is the discovery of the correct means of predicting the future numerically, tables of compound interest. As experience was gain with numbers, it was realized that calculation was not an intellectual feat reserved for thinkers of genius but (using Arabic numerals with a zero) could be reduced to simple rules and taught to children. These rules - modern-day primary school arithmetics - were taught (in Italy) to merchant's sons in the commercial arithmetic schools of the Italian city-states. The better teachers in these schools went further, developing much of the modern high school mathematics, in particular algebra, in the sense of problem solving using the manipulation of unknown numbers represented by letters. As with modern textbook authors on algebra, the Italian writers had the custom of illustrating their techniques in a number of unrealistic 'real life' examples to which the techniques 'apply'. Simple number problems arising from dice occasionally make an appearance. (p.292ff)

That the vast majority of probabilistic inferences are unconscious is obvious from observing animals, for it is not just the human environment that is uncertain but the animal one in general. To find a mechanism capable of performing probabilistic inference (as distinct from talking about it), one need look no further than the brain of the rat, which generates behavior acutely sensitive to small changes in the probability of the results of that behavior. Naturally so, since the life on animals is a constant balance between coping adequately with risk or dying. [...] Some further light on what the brain does is cast by simple artificial neural nets, whose behavior after training on noisy data can be interpreted as implicit estimates of probabilities. These animals and machines studies confirm in the most direct way that, to behave probabilistically, it is not necessary to have anything like explicit estimates of probabilities or ways of talking about them. (p.324)

The influence of Aristotle on the development of thought, though widely recognized, is underrated. We are all in his orbit. ("Aristotle's works are full of platitudes in much the same way as Shakespeare's Hamlet is full of quotations") Everything is in Aristotle's somewhere - at least in potency but often in actuality. And the reason we underrate his contributions is, of course, precisely that it has become platitudinous: we forget about it, for the same reason that we forget the air we breathe. Anything that has become background, or context, or tradition is no longer salient, sometimes no longer represented symbolically at all. The Meno theory really is true of what we have learn from Aristotle: we have forgotten that we learned it, but it is still there, waiting to surprise us when we are induced to remember it. But there is another reason that we do not notice our Aristotelianism. Aristotle is a philosopher with more respect than most for 'what seems so to all, or to most, or to the wise'. His philosophy has none of the paradoxes repugnant to common sense that render the thought of other 'great philosophers' so memorable. (p.344)

From around 1770, English law adopted the phrase 'proof beyond reasonable doubt' (originally defined as equivalent to 'moral certainty') for the standard of proof required in a criminal case. But the status of the rule caused confusion. Even when the rule was believed to be understood, no number became attached to it. Instead, attempts to explain it have been purely linguistic, as in the 1947 case of Miller v. Minister for Pensions, in which Lord Denning declared that in a criminal charge, "that degree is well settled. It need not reach a certainty, but it must carry a high degree of probability. Proof beyond a reasonable doubt does not mean proof beyond the shadow of a doubt. The law would fail to protect the community if it admitted fanciful possibilities to deflect the course of justice. If the evidence is so strong against a man as to leave only a remote possibility in his favor, which can be dismissed with the sentence, 'of course it is possible but not in the least probable', the case is proved beyond reasonable doubt, but nothing short of that will suffice." (p.366)