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Downloadable File 53 A. A. Cournot Exposition of the Theory of Chances and Probabilities A. A. Cournot, Exposition de la théorie des chances et des probabilités. Paris, 1843, 1984 Translated by Oscar Sheynin Oscar Sheynin, 2013 Berlin, 2013 NG Verlag The more difficult it seems to establish logically what is variable and obeys chance, the more delightful is the science that determines the results Huygens1 1 Contents Note by Translator Preface 1. Combinations and order 2. Chances and mathematical probability 3. The laws of mathematical probability of the repetition of events 4. Randomness. Physical possibility and impossibility 5. Sale prices of chances and probabilities. The market of chances and games in general 6. Laws of probability. Mean values and medians 7. Variability of chances 8. Posterior probabilities 9. Statistics in general and experimental determination of chances 10. Experimental determination of mean values by observations and formation of tables of probabilities 11. Means of measurements and observations 12. Application to problems in natural philosophy 13. Application to problems in demography and duration of life 14. On insurance 15. Theory of probability of judgements. Applications to judicial statistics of civil cases 16. Theory of probability of judgements, continued. Applications to judicial statistics of criminal cases. Probability of testimonies 17. On the probability of our knowledge and on judgements based on philosophical probabilities. Summary 2 Note by Translator Antoine Augustin Cournot (1801 – 1877) was a mathematician, philosopher, economist and educator (Feller 1961; Etudes 1978). Here, I discuss his contribution of 1843 reprinted in 1984 (Paris, Libraire J. Vrin) as vol. 1 of his Oeuvres Complètes complete with an Introduction and Commentary by B. Bru. In this Introduction, Bru remarked that in 1828 and 1829 Cournot had published two notes on the calculus of chances and combinations. In 1834, Cournot translated J. Herschel’s astronomical treatise and appended a discussion of cometary orbits which constitutes here a large part of Chapter 12. A long paper on the application of the theory of probability to judicial statistics followed in 1838 and was largely reprinted here, in Chapters 15 and 16. Bru’s commentary certainly demanded great efforts; by his permission, I quoted some of them adding his initials (B. B.). Regrettably, many of his comments are too short and for the same reason some of his references are not readily understandable. He repeatedly mentions Condorcet, Lacroix, and certainly Laplace and Poisson as the main authors from whom Cournot had issued and he also notes (see p. 318, comment to p. 108, line 12) that Cournot had seldom indicated them. Cournot reprinted Kramp’s table of the normal distribution appended to his book (1799); it is not included in the translation. There exist translations of Cournot’s book into German (1849) by C. H. Schnuse, who also translated, in 1841, Poisson (1837), and into Russian (Moscow, 1970) by N. S. Chetverikov, the closest student of Chuprov. A few of Cournot’s terms ought to be explained. Element is parameter; philosophical criticism apparently means philosophical discussion; a commensurable number is rational. And the same randomness is the same random variable. See explanation of Bernoulli theorems in § 30. And, finally, I have introduced the then still n unknown notation Cm and n! The Preface begins by an explanation: Cournot wished to make assessable his subject to those unacquainted with the higher chapters of mathematics. In § 123 he even repeated the formulation of the Pythagorean proposition (excluded in the translation) and in § 69 he says that he adduced a table (Kramp’s table) of a certain function (of the exponential function of a negative square) but he did not provide its analytical expression. I doubt that that was good enough, but then, his Chapter 12 was certainly beyond the reach of ordinary readers. As Bru noted in his Introduction, the book was not understood in its entirety either by mathematicians or other scientists [!], frequently quoted […] but rarely read. Cournot’s sentences are long-winded, up to 12 and 13 lines (§§ 117 and 240/4). In many cases, perhaps copying Poisson (1837), he connected the parts of complex sentences by semicolons rather than words which is not easy to understand. Demonstrative pronouns are often lacking, but unnecessary repetitions are plentiful. 3 [1] Cournot was obviously ignorant of precise observations (measurements) and he ignored Gauss. His Chapter 11 is therefore barely useful. [2] Cournot (§ 145) considered himself a pioneer in applying statistics to astronomy, but he forgot to mention William Herschel, and he certainly had no means for studying the starry heaven. [3] He did not study the application of statistics to meteorology although Humboldt, in 1817, had introduced isotherms, cf. Cournot’s definition of the aims of statistics in § 103! [4] While discussing statistics of population (Chapter 13), Cournot had not mentioned Daniel Bernoulli’s classical study of prevention of smallpox published in 1766, and Gavarret’s contribution (1840) escaped his notice. For many decades, in spite of the work of Graunt, Süssmilch and Daniel B., later statisticians had avoided medical statistics. [5] Many elementary calculations (in §§ 13. 70, 165 – 167, 170, 182, 203, 204) are wrong which had not, however, affected Cournot’s general conclusions. [6] Cournot’s description of the Bayes rule (§ 88) is superficial: he did not notice that Bayes had treated an unknown constant as a random variable. True, he (§ 89) remarked that without prior information that rule leads to a subjective result, and he (§ 95) attempted to prove that with a large number of observations that result becomes objective. Cf. Note 9 to Chapter 8. His treatment of the Petersburg game is interesting, but he failed to refer to Condorcet (1784, p. 714) who had remarked that the possibly infinite game nevertheless only provided one trial so that many such games should have been discussed. Freudenthal (1951) expressed the same opinion and provided pertinent recommendations. Poisson’s law of large numbers is ignored; during Poisson’s lifetime, Cournot (1838) had, however, at least twice mentioned it. It is generally known that the reason of that about-face was Bienaymé’s attitude. Cournot’s description of tontines (§ 52) was completely wrong; see also Note 17 to Chapter 14. Poisson (see Preface) indicated that Cournot discern[ed] the difference between chances and probabilities. However, in § 12, see also § 240/3, the latter stated that probability was the ratio of the pertinent chances. And he almost indifferently applied the terms theory, or doctrine, of probability, and of chances. Now, however, I turn to other points whose positive aspect much prevail over the negative sude. [1] Probability. Cournot’s subjective philosophical probabilities (§§ 43, 233 and 240/8) can be related to expert opinions whose study undoubtedly belongs to mathematical statistics. Laplace (1812, Chapter 2) had introduced them, noted their possible application to decisions of tribunals and elections, but did not introduce any special term. In § 18 Cournot offered a definition of probability covering both the discrete and continuous cases (i. e., and geometric probability). He appropriately introduced the ratio of the extents (étendue) of the 4 pertinent chances; nowadays, we would have said of the measures. In § 45 he objected to Laplace’s belief that probability is relative in part to our ignorance and in part to our knowledge. Poisson’s letter to Cournot (see Preface) indicates that the latter discern[ed] the […] difference between the words chance and probability […]. However, in § 12, see also § 240/3, Cournot called probability the ratio of the pertinent chances and thus contradicted Poisson’s inference. And, see above, he almost indifferently applied the terms theory or doctrine of probability; or of chances. [2] Probability density. In 1709 Niklaus Bernoulli introduced a continuous distribution (and its density) and many later scholars, including Laplace and Gauss, applied such distributions and densities. Cournot (§§ 64 – 65, see also § 31) followed suit. His term was curve of probability; it appropriately represented […] the law of probabilities of different values of a variable magnitude. Indeed, in § 73, although after Poisson (1837, § 53), Cournot introduced a grandeur fortuitously taking a series of distinct values. True, in 1756 and 1757 Simpson, in an error-theoretic context, had effectively applied such variables. Cournot (§ 73) also described the determination of the density of a function u of a magnitude x which takes a series of various fortuitous values. Actually, he (§ 74) considered the case of a function of two independent variables and a linear function of many variables. Supposing that u = |x − y|, he concluded that for 0 ≤ a ≤ 1 P(u ≥ a) = (1 − a)2. The probability of the contrary event would have described the once-popular encounter problem (Whitworth 1886 and possibly 1867; Laurent P. H. (1873, pp. 67 – 69): two persons are to meet during a specified time interval but their arrivals are independent and occur at random and the first to arrive only waits a specified time. Before Cournot Bessel (1838, §§ 1 and 2) determined the densities of two functions of a continuous and uniformly distributed variable and Laplace solved such problems even earlier. In § 81 Cournot studied a mixture of densities. Let n1, n2, … observations have densities f1(x), f2(x), …, then the mixture of those observations will have a density equal to the weighted arithmetic mean of f1(x), f2(x), … [3] Median. It was Cournot (§ 34) who introduced this important parameter. [4] Randomness. Cournot (§ 40) defined a random event as an intersection of (two) independent chains of other events and thus revived an ancient idea (Aristotle). In § 45 he mentioned the mathematical theory of randomness acting in the proper field of science and declared that randomness has a notable role […] in governing the world. Lacking was the dialectical link between randomness and necessity which Kant (1781/1911, p.
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