Kindle Notes & Highlights
Read between
May 6, 2016 - March 4, 2021
the classical statement about probability was the definition from Boethius’s On Different Topics; which is based on the definition of endoxa in Aristotle’s Topics and thus has a distinctly subjectivist tone. “Something is readily believable if it seems true to everyone or to most people or the wise. . . . The truth or falsity of the argument makes no difference, if only it has the appearance of truth.”75 But Boethius’s definitions of induction (“discourse by means of which there is a progression from particulars to universals”) and example were likewise standard for the next thousand years.
While the West underwent the simplification of its culture to a semi-tribal level, the torch of knowledge passed instead to the countries of Islam, which proved sympathetic to abstract thought during its early centuries.
Unusually, the Islamic writers regarded Aristotle’s Rhetoric and Poetics as part of his logic.77 His Rhetoric was thus taken rather more seriously than it later was in the European Middle Ages; al-Farabi, the early leader of the Aristotelian school in the ninth century, is said to have read the Rhetoric two hundred times and written seventy books on it.78 (The cavalier attitude to figures evident here is arguably one of the obstacles to the development of a mathematical theory of probability.)
The most influential of this school is Ibn Sina, known in the West as Avicenna. His Treatise on Logic is heavily dependent on Aristotle but has something to add on the distinction between induction and analogy.
But even though analogy can never give us certainty it can provide us with some kind of satisfaction.”
The last noted Islamic thinker is Ibn Rushd, called Averroes in the West.
The revival of learning in the Gregorian reform was as much logic led as law led. At the same time as the rediscovered Digest was having its profound impact, Anselm offered his ontological argument, according to which God could be proved to exist purely by the analysis of concepts. It must represent a valuation of logic than which no greater can be conceived.83 Abelard’s rational questioning of theology is not far behind: “If Adam’s sin was so grievous that it could only be expiated by the death of Christ, what atonement shall be offered for the crime of them who put Christ to death?”
Basic discussions of the distinction between probable and necessary and of Boethius’s definitions are found as far back as 1000.85 Cicero’s De Inventione and the Rhetoric to Herennius were known from about 1080 and stimulated a wide interest in rhetoric.86 Chartres was the center of synthesis of this first version of the ancient heritage with Christian thought. Symbolizing the harmony still perceived, at least by some, between the ancients and Christianity, a statue of Aristotle, representing dialectic, appears on the west front of Chartres Cathedral.87
Abelard refuses to admit such reasoning into logic, arguing that “probability is casual and often adjoined to falsity” and that to admit such reasoning would require the abandonment of established rules of logic such as “whatever implies the antecedent implies the consequent.”
When commenting on Boethius, he is clear, as Boethius is not, on the distinction between arguments from probable premises and arguments that are probable as inferences, that is, where the premises, whether certain or not, provide only partial support for the conclusion; he asserts that both are possible.
At the end of the thirteenth century, Aristotle’s Rhetoric finally began to make some impact in the West.92 Despite Aristotle’s authority by that time, it never became a widely studied work. The first Latin commentary on it explains that the difference between belief (caused by rhetoric) and opinion (caused by dialectic) is not that opinion is accepted more firmly but rather that rhetoric moves the mind to assent by appetite (that is, appeals to emotion), while in dialectic the intellect is moved by its proper object (that is, what is reasonable).93 Some translations were made also of Arabic
...more
In discussing Aristotle’s definition of “probable” in the Topics, Albert distinguishes two senses of the word. “The probable (on which the dialectical syllogism is based) is the likely. Likely is taken in two ways. Either things are likely in themselves, in that the predicate’s being in the subject is itself likely, because the predicate is not in the subject per se nor the subject in the predicate, nor both in both, nor does the predicate necessarily and essentially inhere in the subject, but it is taken as likely not from necessary causes but from signs. Or the inherence is necessary, but is
...more
He goes on to describe dialectical, legal, and medical reasoning as all probabilistic.
For the rhetorical advocate does not persuade completely and every time, being prevented in three ways: by the evil of his cause, the perversity of the judge and the weakness of his suit.
Q
Nor does the doctor always heal, for three reasons: the fallacy of experience, the evil of the disease and the disobedience of the patient, so the desired end does not always follow. Likewise the dialectician is prevented by the weakness of the probable he adduces, by the power of his opponent and by the difficulty of the problem in dispute.”
He says definitely, unlike Aristotle, that induction is a probable argument, since it cannot survey all the instances.
He agrees in theory with the jurists in holding that passing judgment on suspicion is unlawful, but what he means is something different: “According to Cicero a suspicion involves an ill opinion founded on light indications.”99 Suspicion is thus defined as essentially a vice, perhaps better translated as suspiciousness, and is therefore something to be avoided always.
whereas for the jurists no condemnation on suspicion meant that the highest standard of proof was required, for Aquinas it means only that a very low standard is excluded.
“Prudence deals with contingent matters of action. . . . Now to know what is true for the most part one must consider experience. . . . It is from the past that one ought to take a kind of argument to the future; and so memory of the past is necessary for being well advised as to the future.”
In view of Aquinas’s influence, it is significant that he uses the word probable in an unusually free way in ordinary language: “It is probable that parents [in ancient times] addressed certain prayers to God on behalf of their newly-born children”; “It is more probable that [the star that appeared to the Magi] was a newly-created star, not in the heavens, but in the air near the earth.”
The significance is that the century or two after Aquinas was the period when the vernacular languages of Western Europe became capable of supporting abstract discussion and adopted the vocabulary of the Scholastics to do so.
It might be thought that English already had a native word, likely, for probability concepts, but that is not the case. Likely is the English version of the Latin verisimilis, and the first uses of likely and its derivatives are in the Canterbury Tales, of about the same date as Trevisa.118 The early uses of probable and likely in English are nothing like “It will probably rain tomorrow” but more like “What if ever either of the said premyssis concludyng for feith, y have not oonli likli euydencis, whiche ben clepid in scolis of logik ‘probable euydencis’ or ‘probabilitees,’ but y have sure
...more
Most modern European languages have acquired a pair of synonyms corresponding to English probable and likely, French probable and vraisemblable, and so on.120 They are not native to any of these languages but are borrowings from the Scholastic probabilis and versimilis, themselves descended from the Greek pithanon and eikos via Cicero. Although the two Greek terms are not quite synonyms, the two in Latin and its derivatives are in practice perfectly synonymous, despite occasional individual attempts to find small differences between them.
Later humanism did not produce developments important for probability. Where Scholastic logic kept almost wholly to deductive logic, the humanists saw rhetoric as an art of discovering arguments and of embellishing them so as to speak well. There is no serious discussion of the middle ground as there is in Aristotle’s Rhetoric. The frequent mention of probability in humanist rhetoric is to be seen in this context.
Aristotle’s Rhetoric itself, though occasionally read and lectured on, had a low profile compared with the works of Cicero and Quintilian.122 On the other hand, Renaissance literary theory was dominated by Aristotle’s Poetics. It was common to connect what Aristotle said about the probable in the Poetics with his definition in the Topics, and there were some attempts to distinguish between probabilis (what seems so the wise) and verisimilis (what seems so to the vulgar).
The most influential humanist work in the field was Rudolph Agricola’s De Inventione Dialectica. Probability, when it occasionally appears, is a purely rhetorical property.
Henry VIII’s order to the students of Cambridge in 1515 to study Agricola instead of the “frivolous questions” of Scotus is as useful as most government edicts that academic knowledge should be more “relevant.”
This is plainly a collection of the standard examples of Continental law put together implausibly in a single case. Most of the evidence would be inadmissible in modern English law.
It was usual to explain also how opinion (“a probable assent from an intrinsic medium, or some natural connection in things”) differed from faith (“from an extrinsic medium, namely the authority of a speaker”).
The 1638 Logic of the Jesuit Smiglecki, much used as a textbook at Oxford, said that multiplying probable reasons can make an opinion “infinitely more probable” but cannot make it certain, any more than making a body “infinitely more corporeally perfect” can make it into a spirit.
The most popular of the Scholastic logics was that of Burgersdijck, of 1626. It contains what may be one of the rare incursions of reality into pure logic; his example of a universal proposition is “All crows are black,” his countrymen’s recent extraordinary discovery in New Holland having rendered inoperative the previous example, “All swans are white.” Burgersdijck displays no tendency to inductive skepticism as a result of this contretemps. He believes that an induction that does not survey all the instances is adequate provided it is clear that the unsurveyed instances follow the “same
...more
We will see such makers of modern thought as Galileo and Descartes using the Scholastics’ vocabulary to explain the novelty of their own thought.
The big bang theory of the universe is much more probable, on present evidence, than the steady-state theory. But it is a rare scientist who can be found to say exactly how much more probable—or even approximately how much. Physics and astronomy, both ancient and modern, have few explicit probability calculations. Nevertheless, probabilistic reasoning is essential to both sciences, and it is visible at both ends of the theoretical spectrum. At the most theoretical end are overarching theories intended to explain a large and disparate body of observational data. The more complicated the
...more
The widely accepted Duhem thesis, asserting that observation confronts theory only on a holistic basis, was originally asserted only for physics (“an experiment in physics can never condemn an isolated hypothesis but only a whole theoretical group”).1 It is much more plausible there than in, say, anatomy.
Modern methods are essentially a refinement of the older idea that a more accurate measurement of a doubtful quantity can be gained by averaging several inaccurate measurements.
The difference a tradition of research makes to science is obvious in the treatment of the roundness of the earth in early and late ancient science. The basic discovery was made by Parmenides or some other pre-Socratic around 500 B.C. It needed a single flash of inspiration, based largely on its ability to give a coherent account of eclipses.
(The basic thought that experience is to be preferred to theory, and theory accepted only when it agrees with observation, is of course found in that old empiricist, Aristotle.)
from the acknowledged masters of logic, the Greek mathematicians. Of course, their concern was mostly for deductive logic, since their achievement was to organize mathematics as a system of theorems deductively derived from axioms. But one cannot pursue mathematics with deductive logic alone; before a theorem has been proved, and even before the proof has been seriously attempted, the mathematician has reason to think the theorem might be true, so that effort expended on the attempted proof may be worthwhile.8 The ancient Greeks began the tradition, which modern mathematicians have continued,
...more
Some of Aristotle’s discussions of chance reveal a rather acute sense of the subject. While not exactly quantitative, they constitute the closest approach in the ancient world to a mathematical theory of probability.
Aristotle’s science has plenty of observations and plenty of theory. The observations are relevant to the theory, but the theory is often not exactly based on the observations, in the direct sense of a modern (or Galilean, or Copernican) mathematical model, in which numerical data agree with the numerical predictions of the theory.
Aristotle’s interest in physics was mainly in its qualitative aspects, while his purely scientific work was mostly in biology, where classification, structure, and function are more important than numbers. Aristotle is interested in almost everything except numerical measurements. The Aristotelian attitude is not appropriate in astronomy, where numerical observations are the input, and the desired output is numerical predictions of future eclipses and positions of the planets.
Predictions were made purely by tables of numbers with recurrent patterns, which predicted numbers from numbers.
What Hipparchus is doing is similar to modern regression analysis, in which a straight line or other simple curve is made to give as good a fit as possible to a series of points representing observations. The line of best fit does not go through all the points exactly; the deviations of observed points from the true line are regarded as error (due either to observational error or to the intrinsic variability of the subject matter; the latter cause is not relevant in astronomy). It is an exercise in good probabilistic judgment to decide how much to complicate the model to provide a better fit
...more
It is an important fact of probability that a way to get a more accurate value of a quantity that one can only observe inaccurately is to take the average, or mean, of a number of observations. Errors in the individual observations tend to cancel out. Ptolemy
Rather than take means of individual observations as a normal practice, or try to make the theory fit all the observations, as Hipparchus does, he allows theory to confront the set of all observations in a more holistic fashion. This means that the observations to be exhibited are chosen, to some extent, in accordance with the theory. So older observations may be selected, or adjusted, in accordance with what the theory predicts.23 It is a dangerous practice, but arguably necessary within reasonable limits. It is allowed in modern statistics under the name rejection of outliers, to deal with
...more
Some of the best thought of late antiquity comes from the Greek commentators on Aristotle.26 The art of commenting on ancient texts, so central a part of medieval intellectual life, has a tendency to direct attention to something like logical probability. It is in the nature of the exercise that arguments for and against the opinions of the ancients should be collected, sifted, compared, weighed—anything but definitively decided for or against.
The last point is the one emphasized by modern philosophers of science like Duhem and Quine: that one will not easily abandon a good and wide-ranging theory for the sake of a few minor anomalies.
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 world28 and the recovery, translation, and assimilation of all the main ancient scientific texts.
The long debate on why the scientific revolution did not take place before it did is not subject to conclusive resolution, but there is 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
...more
Pecham says, “I have never heard it said before that one can save all phenomena and explain everything that happens. Nevertheless, the thesis of the mathematicians [that is, of Ptolemy] is more probable; to the reasons they give there has never been given, I think, a reasonable response.”
