Abstract
We study the origins of the axiomatization of subjective probabilities. Starting with the problem of how to measure subjective probabilities, our main goal was to search for the first explicit uses of the definition of subjective probability using betting odds or ratios, i.e., using the Dutch book argument, as it is presently known. We have found two authors prior to Ramsey (The foundations of mathematics and other logical essays. Routledge & Kegan Paul, 1931, [43]) and de Finetti (Fund Math 17:298–329, 1931, [20]) that used the mentioned definition: Émile Borel, in an article of 1924, and Jean-Baptiste Estienne, a French army officer, in a series of four articles published in 1903 and 1904. We tried to identify, in the references given by Borel and Estienne, inspirations common to Ramsey and de Finetti in order to determine, in the literature on the probability of the beginning of the last century, at least some elements that point to specific events that lead to the referred axiomatization. To the best of our knowledge, the genesis of the axiomatic approach in the subjective school was not traced yet, and this untold history can give us a better understanding of recent developments and help us, as applied scientists, in future works.
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- 1.
We are using the term “axiomatization” as the setting of unproved “basic statements about the concept (such as the geometry of the plane) to be studied, using certain undefined technical terms as well as the terms of classical logic.” [52, p. 9].
- 2.
- 3.
[34, p. 4].
- 4.
- 5.
Probability was not regarded as an interesting research topic by pure mathematicians. As an example of this feeling, see below the remarks made by Camille Jordan about the probability lessons he had to teach at the Polytechnique. The only exception was, perhaps, the Russian school in St. Petersburg led by Markov and Tchebychev.
- 6.
See [50] for an English translation.
- 7.
In a footnote to §2, “The Relation to Experimental Data,” of his book, [33, p. 3], mentioned that “In establishing the premises necessary for the applicability of the theory of probability to the world of actual events, the author has used, in large measure, the work of R. v. Mises, pp. 21–27.” See [49]. Therefore, although Kolmogorov’s approach is strictly mathematical, i.e., can be adopted regardless of the interpretation given to the axioms, the frequentist school rapidly embraced it.
- 8.
[34, p. 15].
- 9.
[43, p. 31] says: “The subject of our inquiry is the logic of partial belief, and I do not think we can carry it far unless we have at least an approximate notion of what partial belief is, and how, if at all, it can be measured.” On his side, [20, pp. 302–303] says “Now it is a question of measuring subjective probability, that is, to translate in the determination of a number, our degree of uncertainty about a given sentences; this is the first problem that presents when one wants to establish the calculation of probabilities according to the subjectivistic conception.”
- 10.
See [40, p. 202]: “So there is one general rule, namely, that we should consider the whole circuit [the sample space], and the number of those casts which represents in how many ways the favorable result can occur, and compare that number to the remainder of the circuit, and according to that proportion should the mutual wagers be laid so that one may contend on equal terms.”
- 11.
Academic competition held every year between senior high school students.
- 12.
See [22]. Bertrand and Jordan were the chairs of the session when he presented the work.
- 13.
Distant 120 km from Montpellier.
- 14.
The same year he won the first prize for mathematics in the concours général, the same prize won by Estienne in 1882.
- 15.
Preparatory school of teachers for the high school level.
- 16.
Poincaré and Karl Pearson, for instance, wrote contributions about probability and statistics.
- 17.
This allows one to conjecture that Borel and Estienne eventually met.
- 18.
Five of them written by Borel himself.
- 19.
- 20.
[16, p. 61].
- 21.
[16, p. 63].
- 22.
[39, pp. 10–11].
- 23.
In fact, Jordan taught in 1888 — see “Camille JORDAN: Leçons de Probabilités à l’Ecole Polytechnique (1888),"available at http://www.jehps.net/decembre2009.html — which gives us evidence that Estienne was, indeed, taught by Bertrand.
- 24.
That is why we have class notes of Jordan’s 1888 lecture notes.
- 25.
See [16, p. 63].
- 26.
According to [19, p. 205], Jordan wrote in 1894: “I would like to see [...] disappear without regret [from the course of analysis] the three lessons that I devote to the calculus of probabilities”.
- 27.
- 28.
- 29.
[29, p. 10].
- 30.
He goes on bringing an example: “When the dimensions of a rectangular field are given the values 200m and 300m, we are forced to admit that the field has [an area of] 6 hectares; it would be a mistake to claim that the numeral 6 is imposed by geometry on those who would not have first agreed on the accuracy of the dimensions.” This echoes [2, p. 28]: “If it is alleged that it is impossible to measure in figures the probabilities of which we are speaking, the objection would be as unfounded as if, evaluating the length of a field of rectangular appearance at 300m and the width at 100m, to add, irrespective of any verification, that such measures, however doubtful they may be, and these assessments assign to the field an area of three hectares.”
- 31.
When Estienne took Bertrand’s course, in 1881–1882, the book of his professor has not yet been published. In the preface, [2, p. v] mentions that the book was based on the course he taught at the Collège de France. Bertrand also mentions Jouffret, as “Jauffret” [2, p. xxxvi], citing his colleague’s illustration of the law of large numbers using an example from ballistics.
- 32.
Poincaré was full professor of the chair of probability theory and mathematical physics, being eventually succeeded by Borel [51, p. 36].
- 33.
Like [2], entitled Calcul des Probabilités. A second edition was published in 1912.
- 34.
In [9, pp. 226–227], he wrote that “It is not a difference of nature that separates the objective probability from the subjective probability, but only a difference of degree. A result from probability calculus deserves to be called objective when the probability becomes large enough to be confounded with practical certitude.”
- 35.
In [11], in pages 84 through 86 he explains the method of betting and in the last section (Conclusion and probability of a single trial), he reviews the argument and mentions [21]. In 1928 Borel help was important in the establishment of the Institut Henri Poincaré, a research institution devoted to probability theory and mathematical physics, where several lectures were held in the 1930s, including de Finetti’s, presented in 1935 and published in 1937.
References
Barone, J., Novikoff, A.: A history of the axiomatic formulation of probability from Borel to Kolmogorov: Part I. Arch. Hist. Exact Sci. 18, 123–190 (1978)
Bertrand, J.: Calcul des Probabilités. Gauthier-Villars (1889)
Bohlmann, G.: Lebensversicherungs-mathematik. In: Encyklopädie der Mathematischen Wissenschaften, vol. 1(2), pp. 852–917. Teubner (1901)
Boole, G.: XIII. On the conditions by which the solutions of questions in the theory of probabilities are limited. Philos. Mag. 8, 91–98 (1854)
Borel, É.: Sur quelques points de la théorie des fonctions. Annales Scientifiques de l’École Normale Supérieure 12, 9–5 (1895)
Borel, É.: Remarques sur certaines questions de probabilité. Bulletin de la Société Mathématique de France 33, 123–128 (1905)
Borel, É.: La valeur pratique du calcul des probabilités. Revue du mois 1, 424–437 (1906)
Borel, É.: Les probabilités denombrables et leurs applications arithmétiques. Rendiconti del Circolo Matematico di Palermo 27, 247–270 (1909)
Borel, É.: Le Hasard. Libraire Félix Alcan (1914)
Borel, É.: A propos d’un traité de probabilités. Revue Philosophique de la France et de l’Étranger 98, 321–336 (1924)
Borel, É.: Valeur pratique et philosophie des probabilités. In: Traité du Calcul des Probabilités et de ses applications, vol. IV, fasc. 3, pp. 1–182. Gauthier-Villars (1939)
Bru, B.: Problème de l’efficacité du tir à l’école d’artillerie de Metz. Mathématiques et Sciences Humaines 136, 29–42 (1996)
Bru, B.: Statisticians of the Centuries, Chapter ‘Émile Borel’, pp. 287–291. Springer (2001)
Buffon, G.-L.L.: Essai d’artithmétique morale (1777). In: Ouevres complètes de Buffon, vol. 9, pp. 373–444. Pourrat Frères (1835)
Carnap, R.: Logical Foundations of Probability. University of Chicago Press (1950)
Courtebras, B.: À l’école des probabilités : une histoire de l’enseignement français du calcul des probabilités. Presses universitaires de Franche-Comté (2006)
Cox, R.: Probability, frequency and reasonable expectation. Am. J. Phys. 14, 1–13 (1946)
Crépel, P.: De Condorcet à Arago:l’enseignement des probabilités en France de 1786 à 1830. Bulletin de la Société des Amis de la Bibliothèque de l’École Polytechnique 4, 29–81 (1989)
Crépel, P.: La formation polytechnicienne 1794–1994, chapter ‘Le calcul des probabilités: de l’arithmétique sociale a l’art militaire’, pp. 197–215. Dunod (1994)
de Finetti, B.: Sul significato soggettivo della probabilità. Fund. Math. 17, 298–329 (1931)
de Finetti, B.: La prévision: ses lois logiques, ses sources subjectives, translated in [34]. Annales de l’Institut Henri Poincaré 7, 1–68 (1937)
Estienne, J.B.: Étude sur les erreurs d’observation. Comptes Rendus Hebdomadaires des Séances de l’Académie des Sciences 110, 512 (1890)
Estienne, J.B.: Éssai sur l’art de conjecturer. Revue d’artillerie 61, 405–449, 62:73–117, 64:5–39, 65–97 (1903–1904)
Fréchet, M.: Exposé et discussion de quelques recherches récentes sur les fondements du calcul des probabilités. In: Wavre, R. (ed.) Colloque Consacré a la Théorie des Probabilités: Les fondements du calcul des probabilités, number 735 in Actualités Scientifiques et Industrielles, pp. 23–55. Hermann (1938)
Hacking, I.: The Emergence of Probability: A Philosophical Study of Early Ideas about Probability. Cambridge University Press, Induction and Statistical Inference (1975)
Hilbert, D.: Mathematical problems. Bull. Am. Math. Soc. 8, 437–479 (1902)
Jeffreys, H.: Theory of Probability. Clarendon Press (1939)
Jouffret, E.P.: Étude sur l’établissement et l’usage des tables du tir. Revue d’artillerie 2, 370–389, 3:52–72, 208–227, 324–340, 386–401 (1873a)
Jouffret, E.P.: Sur la probabilité du tir et la méthode des moindres carrés. Revue maritime et coloniale 38, 5–30 (1873b)
Jouffret, E.P.: Sur la probabilité du tir et la méthode des moindres carrés. Revue maritime et coloniale 42, 665–680, 986–1021, 43:131–152 (1874)
Keynes, J.M.: A Treatise on Probability. Macmillan (1921)
Kolmogorov, A.N.: Grundbegriffe der Wahrscheinlichkeitsrechnung. Springer (1933)
Kolmogorov, A.N.: Foundations of the Theory of Probability, Chelsea, 2nd edn. (transl. of [32]) (1956)
Kyburg, H.E., Smokler, H.E.: Studies in Subjective Probability. Wiley (1980)
Lacroix, S.F.: Traité élémentaire du calcul des probabilités. Courcier (1816)
Laplace, P.S.: Théorie Analytique des Probabilités. Courcier (1812)
Laplace, P.S.: Essai Philosophique sur les Probabilités. Bachelier (1814)
Lebesgue, H.: Sur une généralisation de l’intégrale définie. Comptes Rendus Hebdomadaires de Séances de l’Académie des Sciences 132, 1025–1028 (1901)
Meusnier, N.: Sur l’histoire de l’enseignement des probabilités et des statistiques. Eletron. J. Hist. Probab. Stat. 2, 1–20 (2006)
Ore, O.: Cardano. Princeton University Press, The Gambling Scholar (1953)
Poincaré, H.: Calcul des Probabilités. Gauthier-Villars (1896)
Poincaré, H.: La Science et l’hypothèse. Ernest Flammarion (1902)
Ramsey, F.P.: The Foundations of Mathematics and other Logical Essays, Chapter ‘Truth and Probability (1926)’, pp. 156–198. Routledge & Kegan Paul (1931)
Reichenbach, H.: The Theory of Probability. University of California Press (1949)
Savage, L.J.: The Foundations of Statistics. Wiley (1954)
Shafer, G., Vovk, V.: The sources of Kolmogorov’s Grundbegriffe. Stat. Sci. 21, 70–98 (2006)
Venn, J.: The Logic of Chance. McMillan (1866)
von Mises, R.: Wahrscheinlichkeit. Springer, Statistik und Wahrheit (1928)
von Mises, R.: Wahrscheinlichkeitsrechnung und ihre anwendung in der statistik und theoretischen physik. Leipzig (1931)
von Mises, R.: Probability, Statistics and Truth. William Hodge and Co., translation of [48] (1939)
von Plato, J.: Creating Modern Probability. Cambridge University Press (1994)
Wilder, R.L.: Introduction to the Foundations of Mathematics. Wiley (1965)
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Diniz, M.A., Gallo, S. (2018). The Beginnings of Axiomatic Subjective Probability. In: Polpo, A., Stern, J., Louzada, F., Izbicki, R., Takada, H. (eds) Bayesian Inference and Maximum Entropy Methods in Science and Engineering. maxent 2017. Springer Proceedings in Mathematics & Statistics, vol 239. Springer, Cham. https://doi.org/10.1007/978-3-319-91143-4_14
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