The Sources of Kolmogorov’s Grundbegriffe
Glenn Shafer and Vladimir Vovk
Source: Statist. Sci. Volume 21, Number 1
(2006), 70-98.
Abstract
Andrei Kolmogorov’s Grundbegriffe der Wahrscheinlichkeits-rechnung put probability’s modern mathematical formalism in place. It also provided a philosophy of probability—an explanation of how the formalism can be connected to the world of experience. In this article, we examine the sources of these two aspects of the Grundbegriffe—the work of the earlier scholars whose ideas Kolmogorov synthesized.
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Keywords: Axioms for probability; Borel; classical probability; Cournot’s principle; frequentism; Grundbegriffe der Wahrscheinlichkeits-rechnung; history of probability; Kolmogorov; measure theory
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References
Andersen, E. S. and Jessen, B. (1948). On the introduction of measures in infinite product sets. Det Kongelige Danske Videnskabernes Selskab, Matematisk-Fysiske Meddelelser 25 (4), 8 pp.
Mathematical Reviews (MathSciNet): MR27041
Zentralblatt MATH: 0031.01305
Bachelier, L. (1900). Théorie de la spéculation. Ann. Sci. École Norm. Supér. (3) 17 21--86. This was Bachelier's doctoral dissertation. Reprinted in facsimile in 1995 by Éditions Jacques Gabay, Paris. An English translation, by A. J. Boness, appears in P. H. Cootner, ed. (1964). The Random Character of Stock Market Prices 17--78. MIT Press.
Mathematical Reviews (MathSciNet): MR1508978
Bachelier, L. (1910). Les probabilités à plusieurs variables. Ann. Sci. École Norm. Supér. (3) 27 339--360.
Mathematical Reviews (MathSciNet): MR1509125
Bachelier, L. (1912). Calcul des probabilités. Gauthier-Villars, Paris.
Barone, J. and Novikoff, A. (1978). A history of the axiomatic formulation of probability from Borel to Kolmogorov. I. Arch. Hist. Exact Sci. 18 123--190.
Mathematical Reviews (MathSciNet): MR482894
Bartlett, M. S. (1949). Probability in logic, mathematics and science. Dialectica 3 104--113.
Mathematical Reviews (MathSciNet): MR30919
Digital Object Identifier: doi:10.1111/j.1746-8361.1949.tb00858.x
Zentralblatt MATH: 0034.07003
Bayer, R., ed. (1951). Congrès international de philosophie des sciences, Paris, 1949 4. Calcul des probabilités. Hermann, Paris.
Bernoulli, J. (1713). Ars Conjectandi. Thurnisius, Basel. This pathbreaking work appeared eight years after Bernoulli's death. A facsimile reprinting of the original Latin text is sold by Éditions Jacques Gabay, Paris. A German translation appeared in 1899 (Wahrscheinlichkeitsrechnung von Jakob Bernoulli. Anmerkungen von R. Haussner, Ostwald's Klassiker, Nr. 107--108, Engelmann, Leipzig), with a second edition (Deutsch, Frankfurt) in 1999. A Russian translation of Part IV, which contains Bernoulli's law of large numbers, appeared in 1986:\textcyrYa. Bernulli, O zakone bolp1xih qisel. Nauka, Moscow. It includes a preface by Kolmogorov, dated October 1985, and commentaries by other Russian authors. B. Sung's English translation of Part IV, dated 1966, remains unpublished but is available in several university libraries in the United States. O. Sheynin's English translation of Part IV, dated 2005, can be downloaded from www.sheynin.de.
Bernstein, F. (1912). Über eine Anwendung der Mengenlehre auf ein aus der Theorie der säkularen Störungen herrührendes Problem. Math. Ann. 71 417--439.
Mathematical Reviews (MathSciNet): MR1511668
Digital Object Identifier: doi:10.1007/BF01456856
Bernstein, S. N. (1917).\textcyrOpyt aksiomaticheskogo obosnovaniya teorii veroyatnosteĭ (On the axiomatic foundation of the theory of probability).\textcyrSoobshcheniya Harp1kovskogo Matematicheskogo Obshchestva (Communications of the Kharkiv Mathematical Society) 15 209--274. Reprinted in S. N. Bernstein (1964).\textcyrSobranie Sochineniĭ 10--60. Nauka, Moscow.
Bernstein, S. N. (1927).\textcyrTeoriya veroyatnosteĭ (Theory of Probability).\textcyrGosudarstvennoe Izdatelp1stvo (State Publishing House), Moscow and Leningrad. Second edition 1934, fourth 1946. This work was included in the Grundbegriffe's bibliography.
Bertrand, J. (1889). Calcul des probabilités. Gauthier-Villars, Paris. Some copies of the first edition are dated 1888. Second edition 1907. Reprinted by Chelsea, New York, 1972.
Mathematical Reviews (MathSciNet): MR392452
Blackwell, D. (1956). On a class of probability spaces. Proc. Third Berkeley Symp. Math. Statist. Probab. 2 1--6. Univ. California Press, Berkeley.
Mathematical Reviews (MathSciNet): MR84882
Zentralblatt MATH: 0073.12301
Blum, A. and Mespoulet, M. (2003). L'Anarchie bureaucratique. Statistique et pouvoir sous Staline. Découverte, Paris.
Bohlmann, G. (1901). Lebensversicherungs-Mathematik. In Encyklopädie der Mathematischen Wissenschaften 1(2) 852--917. Teubner, Leipzig.
Borel, E. (1895). Sur quelques points de la théorie des fonctions. Ann. Sci. École Norm. Supér. (3) 12 9--55.
Mathematical Reviews (MathSciNet): MR1508908
Borel, E. (1897). Sur les séries de Taylor. Acta Math. 20 243--247. Reprinted in Borel (? 2 661--665.
Mathematical Reviews (MathSciNet): MR1554891
Digital Object Identifier: doi:10.1007/BF02417979
Borel, E. (1898). Leçons sur la théorie des fonctions. Gauthier-Villars, Paris.
Borel, E. (1905). Remarques sur certaines questions de probabilité. Bull. Soc. Math. France 33 123--128. Reprinted in Borel (? 2 985--990.
Mathematical Reviews (MathSciNet): MR1504512
Borel, E. (1906). La valeur pratique du calcul des probabilités. La revue du mois 1 424--437. Reprinted in Borel (? 2 991--1004.
Borel, E. (1909a). Les probabilités dénombrables et leurs applications arithmétiques. Rend. Circ. Mat. Palermo 27 247--270. Reprinted in Borel (? 2 1055--1079.
Borel, E. (1909b). Éléments de la théorie des probabilités. Gauthier-Villars, Paris. Third edition 1924. The 1950 edition was translated into English by J. E. Freund and published as Elements of the Theory of Probability by Prentice-Hall in 1965.
Mathematical Reviews (MathSciNet): MR33328
Borel, E. (1912). Notice sur les travaux scientifiques. Gauthier-Villars, Paris. Prepared by Borel to support his candidacy to the Académie des Sciences. Reprinted in Borel (? 1 119--190.
Zentralblatt MATH: 43.0061.07
Borel, E. (1914). Le Hasard. Alcan, Paris. The first and second editions both appeared in 1914, with later editions in 1920, 1928, 1932, 1938 and 1948.
Zentralblatt MATH: 45.1271.08
Borel, E. (1930). Sur les probabilités universellement négligeables. C. R. Acad. Sci. Paris 190 537--540. Reprinted as Note IV of Borel (1939).
Borel, E. (1939). Valeur pratique et philosophie des probabilités. Gauthier-Villars, Paris. Reprinted in 1991 by Éditions Jacques Gabay, Paris.
Borel, E. (1972). Œuvres de Émile Borel. Centre National de la Recherche Scientifique, Paris. Four volumes.
Bourbaki, N. (pseudonym) (1994). Elements of the History of Mathematics. Springer, Berlin. Translated from the 1984 French edition by J. Meldrum.
Mathematical Reviews (MathSciNet): MR1290116
Zentralblatt MATH: 0803.01002
Broggi, U. (1907). Die Axiome der Wahrscheinlichkeitsrechnung. Ph.D. thesis, Universität Göttingen. Excerpts reprinted in Schneider (1988) 359--366.
Bru, B. (2001). Émile Borel. In Statisticians of the Centuries (C. C. Heyde and E. Seneta, eds.) 287--291. Springer, New York.
Mathematical Reviews (MathSciNet): MR1863206
Bru, B. (2003). Souvenirs de Bologne. Journal de la Société Française de Statistique 144 134--226. Special volume on history.
Bruns, H. (1906). Wahrscheinlichkeitsrechnung und Kollektivmasslehre. Teubner, Leipzig and Berlin. Available at historical.library.cornell.edu.
Buffon, G.-L. (1777). Essai d'arithmétique morale. In Supplément à l'Histoire naturelle 4 46--148. Imprimerie Royale, Paris.
Cantelli, F. P. (1916a). La tendenza ad un limite nel senso del calcolo delle probabilità. Rend. Circ. Mat. Palermo 41 191--201. Reprinted in Cantelli (? 175--188.
Cantelli, F. P. (1916b). Sulla legge dei grandi numeri. Atti Reale Accademia Nazionale Lincei, Memorie Cl. Sc. Fis. 11 329--349. Reprinted in Cantelli (? 189--213.
Cantelli, F. P. (1917). Sulla probabilità come limite della frequenza. Atti Reale Accademia Nazionale Lincei 26 39--45. Reprinted in Cantelli (? 214--221.
Cantelli, F. P. (1932). Una teoria astratta del calcolo delle probabilità. Giornale dell'Istituto Italiano degli Attuari 3 257--265. Reprinted in Cantelli (? 289--297.
Cantelli, F. P. (1935). Considérations sur la convergence dans le calcul des probabilités. Ann. Inst. H. Poincaré 5 3--50. Reprinted in Cantelli (? 322--372.
Mathematical Reviews (MathSciNet): MR1508022
Cantelli, F. P. (1958). Alcune memorie matematiche. Giuffrè, Milan.
Carathéodory, C. (1914). Über das lineare Mass von Punktmengen---eine Verallgemeinerung des Längenbegriffs. Nachr. Akad. Wiss. Göttingen Math.-Phys. II Kl. 4 404--426.
Carathéodory, C. (1918). Vorlesungen über reelle Funktionen. Teubner, Leipzig and Berlin. Second edition 1927.
Castelnuovo, G. (1919). Calcolo delle probabilitá. Albrighi e Segati, Milan, Rome, and Naples. Second edition in two volumes, 1926 and 1928. Third edition 1948.
Chuprov, A. A. (1910).\textcyrOcherki po teorii statistiki (Essays on the Theory of Statistics), 2nd ed. Sabashnikov, St. Petersburg. The first edition appeared in 1909. The second edition was reprinted by the State Publishing House, Moscow, in 1959.
Church, A. (1940). On the concept of a random sequence. Bull. Amer. Math. Soc. 46 130--135.
Mathematical Reviews (MathSciNet): MR911
Digital Object Identifier: doi:10.1090/S0002-9904-1940-07154-X
Cifarelli, D. M. and Regazzini, E. (1996). de Finetti's contribution to probability and statistics. Statist. Sci. 11 253--282.
Mathematical Reviews (MathSciNet): MR1445983
Digital Object Identifier: doi:10.1214/ss/1032280303
Project Euclid: euclid.ss/1032280303
Zentralblatt MATH: 0955.01552
Copeland, A. H., Sr. (1932). The theory of probability from the point of view of admissible numbers. Ann. Math. Statist. 3 143--156.
Cournot, A.-A. (1843). Exposition de la théorie des chances et des probabilités. Hachette, Paris. Reprinted in 1984 as Volume I (B. Bru, ed.) of Cournot (1973--1984).
Mathematical Reviews (MathSciNet): MR768180
Cournot, A.-A. (1973--1984). Œuvres complètes. Vrin, Paris. Ten volumes, with an eleventh to appear.
Czuber, E. (1903). Wahrscheinlichkeitsrechnung und ihre Anwendung auf Fehlerausgleichung, Statistik und Lebensversicherung. Teubner, Leipzig. Second edition 1910, third 1914.
d'Alembert, J. (1761). Réflexions sur le calcul des probabilités. Opuscules mathématiques 2 1--25.
d'Alembert, J. (1767). Doutes et questions sur le calcul des probabilités. Mélanges de littérature, d'histoire, et de philosophie 5 275--304.
Daniell, P. J. (1918). A general form of integral. Ann. of Math. (2) 19 279--294.
Mathematical Reviews (MathSciNet): MR1502538
Digital Object Identifier: doi:10.2307/1967495
JSTOR: links.jstor.org
Daniell, P. J. (1919a). Integrals in an infinite number of dimensions. Ann. of Math. (2) 20 281--288.
Mathematical Reviews (MathSciNet): MR1502563
Digital Object Identifier: doi:10.2307/1967122
JSTOR: links.jstor.org
Daniell, P. J. (1919b). Functions of limited variation in an infinite number of dimensions. Ann. of Math. (2) 21 30--38.
Mathematical Reviews (MathSciNet): MR1503526
Digital Object Identifier: doi:10.2307/2007133
JSTOR: links.jstor.org
Daniell, P. J. (1920). Further properties of the general integral. Ann. of Math. (2) 21 203--220.
Mathematical Reviews (MathSciNet): MR1503617
Digital Object Identifier: doi:10.2307/2007258
JSTOR: links.jstor.org
Daniell, P. J. (1921). Integral products and probability. Amer. J. Math. (2) 43 143--162.
Mathematical Reviews (MathSciNet): MR1506439
Digital Object Identifier: doi:10.2307/2370323
JSTOR: links.jstor.org
Daston, L. (1979). d'Alembert's critique of probability theory. Historia Math. 6 259--279.
Mathematical Reviews (MathSciNet): MR540919
Digital Object Identifier: doi:10.1016/0315-0860(79)90125-3
Zentralblatt MATH: 0422.01010
Daston, L. (1994). How probabilities came to be objective and subjective. Historia Math. 21 330--344.
Mathematical Reviews (MathSciNet): MR1295723
Digital Object Identifier: doi:10.1006/hmat.1994.1028
Zentralblatt MATH: 0805.01009
de Finetti, B. (1930). A proposito dell'estensione del teorema delle probabilità totali alle classi numerabili. Rend. Reale Instituto Lombardo Sci. Lettere 63 901--905, 1063--1069.
de Finetti, B. (1939). Compte rendu critique du colloque de Genève sur la théorie des probabilités. Actualités Scientifiques et Industrielles 766. Hermann, Paris. Number 766 is the eighth fascicle of Wavre (1938--1939).
de Moivre, A. (1718). The Doctrine of Chances: Or, A Method of Calculating the Probability of Events in Play. Pearson, London. Second edition 1738, third 1756.
Zentralblatt MATH: 0153.30801
Dieudonné, J. (1948). Sur le théorème de Lebesgue--Nikodym. III. Ann. Univ. Grenoble 23 25--53.
Mathematical Reviews (MathSciNet): MR28924
Doob, J. L. (1953). Stochastic Processes. Wiley, New York.
Mathematical Reviews (MathSciNet): MR58896
Zentralblatt MATH: 0053.26802
Doob, J. L. (1989). Kolmogorov's early work on convergence theory and foundations. Ann. Probab. 17 815--821.
Mathematical Reviews (MathSciNet): MR1009435
Digital Object Identifier: doi:10.1214/aop/1176991247
Project Euclid: euclid.aop/1176991247
Zentralblatt MATH: 0687.01008
Doob, J. L. (1994). The development of rigor in mathematical probability, 1900--1950. In Pier (1994b) 157--170. Reprinted in Amer. Math. Monthly 103 (1996) 586--595.
Mathematical Reviews (MathSciNet): MR1404084
Digital Object Identifier: doi:10.2307/2974673
JSTOR: links.jstor.org
Dörge, K. (1930). Zu der von R. von Mises gegebenen Begründung der Wahrscheinlichkeitsrechnung. Math. Z. 32 232--258.
Mathematical Reviews (MathSciNet): MR1545164
Digital Object Identifier: doi:10.1007/BF01194632
Ellis, R. L. (1849). On the foundations of the theory of probabilities. Trans. Cambridge Philos. Soc. 8 1--6. The paper was read on February 14, 1842. Part 1 of Volume 8 was published in 1843 or 1844, but Volume 8 was not completed until 1849.
Faber, G. (1910). Über stetige Funktionen. II. Math. Ann. 69 372--443.
Mathematical Reviews (MathSciNet): MR1511593
Digital Object Identifier: doi:10.1007/BF01456327
Fechner, G. T. (1897). Kollektivmasslehre. Engelmann, Leipzig. Edited by G. F. Lipps.
Fréchet, M. (1915a). Définition de l'intégrale sur un ensemble abstrait. C. R. Acad. Sci. Paris 160 839--840.
Fréchet, M. (1915b). Sur l'intégrale d'une fonctionnelle étendue à un ensemble abstrait. Bull. Soc. Math. France 43 248--265.
Mathematical Reviews (MathSciNet): MR1504744
Fréchet, M. (1930). Sur l'extension du théorème des probabilités totales au cas d'une suite infinie d'événements. Rend. Reale Instituto Lombardo Sci. Lettere 63 899--900, 1059--1062.
Fréchet, M. (1937--1938). Recherches théoriques modernes sur la théorie des probabilités. Gauthier-Villars, Paris. This work was listed in the Grundbegriffe's bibliography as in preparation. It consists of two books, Fréchet (? and Fréchet (? The two books together constitute Fascicle 3 of Volume 1 of Émile Borel's Traité du calcul des probabilités et ses applications.
Fréchet, M. (1937). Généralités sur les probabilités. Variables aléatoires. Gauthier-Villars, Paris. Second edition 1950. This is Book 1 of Fréchet (?
Mathematical Reviews (MathSciNet): MR38581
Fréchet, M. (1938a). Méthode des fonctions arbitraires. Théorie des événements en chaîne dans le cas d'un nombre fini d'états possibles. Gauthier-Villars, Paris. Second edition 1952. This is Book 2 of Fréchet (1937--1938).
Fréchet, M. (1938b). Exposé et discussion de quelques recherches récentes sur les fondements du calcul des probabilités. Actualités Scientifiques et Industrielles 735 23--55. Hermann, Paris. In Wavre (1938--1939), second fascicle, entitled Les fondements du calcul des probabilités.
Fréchet, M. (1951). Rapport général sur les travaux du Colloque de Calcul des Probabilités. In Bayer (? 3--21.
Fréchet, M. and Halbwachs, M. (1924). Le calcul des probabilités à la portée de tous. Dunod, Paris.
Gnedenko, B. V. and Kolmogorov, A. N. (1948).\textcyrTeoriya veroyatnosteĭ (Probability theory). In\textcyrMatematika v SSSR za tridcatp1 let 1917--1947 (Thirty Years of Soviet Mathematics 1917--1947) 701--727.\textcyrGostehizdat, Moscow and Leningrad. English translation in Sheynin (1998) 131--158.
Gnedenko, B. V. and Kolmogorov, A. N. (1949).\textcyrPredelp1nye raspredeleniya dlya summ nezavisimyh sluchaĭnyh velichin. State Publishing House, Moscow. Translated into English by K. L. Chung and published in 1954 as Limit Distributions for Sums of Independent Random Variables, Addison--Wesley, Cambridge, MA, with an appendix by J. L. Doob.
Mathematical Reviews (MathSciNet): MR62975
Zentralblatt MATH: 0056.36001
Haberman, S. J. (1996). Advanced Statistics 1. Description of Populations. Springer, New York.
Mathematical Reviews (MathSciNet): MR1409815
Hadamard, J. (1922). Les principes du calcul des probabilités. Revue de métaphysique et de morale 39 289--293. A slightly longer version of this note, with the title ``Les axiomes du calcul des probabilités,'' was included in Oeuvres de Jacques Hadamard 4 2161--2162. Centre National de la Recherche Scientifique, Paris, 1968.
Hausdorff, F. (1901). Beiträge zur Wahrscheinlichkeitsrechnung. Sitzungsber. Königlich Sächs. Gesellschaft Wiss. Leipz. Math.-Phys. Kl. 53 152--178.
Hausdorff, F. (1914). Grundzüge der Mengenlehre. von Veit, Leipzig.
Hawkins, T. (1975). Lebesgue's Theory of Integration: Its Origins and Development, 2nd ed. Chelsea, New York. First edition 1970, Univ. Wisconsin Press, Madison. The second edition differs only slightly from the first, but it corrects a consequential error on p. 104. Second edition reprinted in 1979 by Chelsea, New York, and then in 2001 by the American Mathematical Society, Providence, RI.
Mathematical Reviews (MathSciNet): MR1857634
Zentralblatt MATH: 0986.26002
Helm, G. (1902). Die Wahrscheinlichkeitslehre als Theorie der Kollektivbegriffe. Annalen der Naturphilosophie 1 364--384.
Hilbert, D. (1902). Mathematical problems. Bull. Amer. Math. Soc. 8 437--479. Hilbert's famous address to the International Congress of Mathematicians in Paris in 1900, in which he listed twenty-three open problems central to mathematics. Translated from the German by M. W. Newson.
Mathematical Reviews (MathSciNet): MR1557926
Digital Object Identifier: doi:10.1090/S0002-9904-1902-00923-3
Zentralblatt MATH: 0979.01028
Hochkirchen, T. (1999). Die Axiomatisierung der Wahrscheinlichkeitsrechnung und ihre Kontexte: Von Hilberts sechstem Problem zu Kolmogoroffs Grundbegriffen. Vandenhoeck and Ruprecht, Göttingen.
Mathematical Reviews (MathSciNet): MR1738288
Zentralblatt MATH: 0944.01007
Holgate, P. (1997). Independent functions: Probability and analysis in Poland between the wars. Biometrika 84 161--173.
Mathematical Reviews (MathSciNet): MR1766039
Digital Object Identifier: doi:10.1093/biomet/84.1.159
JSTOR: links.jstor.org
Jeffreys, H. (1931). Scientific Inference. Cambridge Univ. Press. Second edition 1957, third 1973.
Jessen, B. (1930). Über eine Lebesguesche Integrationstheorie für Funktionen unendlich vieler Veränderlichen. In Den Syvende Skandinaviske Mathatikerkongress I Oslo 19--22 August 1929 127--138. A. W. Brøggers Boktrykkeri, Oslo.
Jessen, B. (1935). Some analytical problems relating to probability. J. Math. Phys. Mass. Inst. Tech. 14 24--27.
Johnson, N. L. and Kotz, S., eds. (1997). Leading Personalities in Statistical Sciences. Wiley, New York.
Mathematical Reviews (MathSciNet): MR1469759
Zentralblatt MATH: 0878.01018
Kahane, J.-P. (1994). Des séries de Taylor au mouvement brownien, avec un aperçu sur le retour. In Pier (1994b) 415--429.
Mathematical Reviews (MathSciNet): MR1298638
Kamlah, A. (1983). Probability as a quasi-theoretical concept--- J. V. Kries' sophisticated account after a century. Erkenntnis 19 239--251.
Keynes, J. M. (1921). A Treatise on Probability. Macmillan, London.
Zentralblatt MATH: 48.0615.08
Khinchin, A. Ya. (1929).\textcyrUchenie Mizesa o veroyatnostyakh i printsipy fizicheskoĭ statistiki (Mises' work on probability and the principles of statistical physics).\textcyrUspekhi Fizicheskikh Nauk 9 141--166.
Khinchin, A. Ya. (1961). On the Mises frequentist theory.\textcyrVoprosy filosofii (Questions of Philosophy) 15 (1, 2) 91--102, 77--89. Published after Khinchin's death by B. Gnedenko. English translation in Sheynin (1998) 99--137, reproduced with footnotes by R. Siegmund-Schultze in Science in Context 17 (2004) 391--422. We have seen only this English translation, not the original.
Mathematical Reviews (MathSciNet): MR2106024
Khinchin, A. Ya. and Kolmogorov, A. N. (1925). Über Konvergenz von Reihen, deren Glieder durch den Zufall bestimmt werden.\textcyrMatematicheskiĭ Sbornik (Sbornik: Mathematics) 32 668--677. Translated into Russian in Kolmogorov (1986) 7--16 and thence into English in Kolmogorov (1992) 1--10.
Knobloch, E. (2001). Emile Borel's view of probability theory. In Probability Theory: Philosophy, Recent History and Relations to Science (V. F. Hendricks, S. A. Pedersen and K. F. Jørgensen, eds.) 71--95. Kluwer, Dordrecht.
Mathematical Reviews (MathSciNet): MR1894709
Kolmogorov, A. N. (1928). Über die Summen durch den Zufall bestimmter unabhängiger Grössen. Math. Ann. 99 309--319. An addendum appears in 1930: 102 484--488. The article and the addendum are translated into Russian in Kolmogorov (1986) 20--34 and thence into English in Kolmogorov (1992) 15--31.
Mathematical Reviews (MathSciNet): MR1512451
Digital Object Identifier: doi:10.1007/BF01459098
Kolmogorov, A. N. (1929).\textcyrObshchaya teoriya mery i isqislenie veroyatnosteĭ (The General Theory of Measure and the calculus of probability). In\textcyrSbornik rabot Matematicheskogo Razdela, Kommunisticheskaya Akademiya, Sekciya Estestvennyh i Tochnyh Nauk (Collected Works of the Mathematical Section, Communist Academy, Section for Natural and Exact Sciences) 1 8--21. The Socialist Academy was founded in Moscow in 1918 and was renamed The Communist Academy in 1923 (Vucinich, ? The date 8 January 1927, which appears at the end of the article in the journal, was omitted when the article was reproduced in the second volume of Kolmogorov's collected works (Kolmogorov, 1986, 48--58). The English translation (Kolmogorov, 1992, 48--59) modernizes the article's terminology somewhat: $M$ becomes a ``measure'' instead of a ``measure specification.''
Mathematical Reviews (MathSciNet): MR861120
Kolmogorov, A. N. (1931). Über die analytischen Methoden in der Wahrscheinlichkeitsrechnung. Math. Ann. 104 415--458. Dated July 26, 1930. Translated into Russian in Kolmogorov (1986) 60--105 and thence into English in Kolmogorov (1992) 62--108.
Mathematical Reviews (MathSciNet): MR1512678
Digital Object Identifier: doi:10.1007/BF01457949
Kolmogorov, A. N. (1933). Grundbegriffe der Wahrscheinlichkeitsrechnung. Springer, Berlin. A Russian translation by G. M. Bavli, appeared under the title\textcyrOsnovnye ponyatiya teorii veroyatnosteĭ (Nauka, Moscow) in 1936, with a second edition, slightly expanded by Kolmogorov with the assistance of A. N. Shiryaev, in 1974, and a third edition ($\!\!\!$\textcyrFAZIS, Moscow) in 1998. An English translation by N. Morrison appeared under the title Foundations of the Theory of Probability (Chelsea, New York) in 1950, with a second edition in 1956.
Mathematical Reviews (MathSciNet): MR494348
Kolmogorov, A. N. (1935).\textcyrO nekotoryh novyh techeniyah v teorii veroyatnosteĭ (On some modern currents in the theory of probability). In\textcyrTrudy 2-go Vsesoyuznogo Matematicheskogo Sp2ezda, Leningrad, 24--30 Iyunya 1934 g. (Proceedings of the 2nd All-Union Mathematical Congress, Leningrad, 24--30 June 1934) 1 (Plenary Sessions and Review Talks) 349--358. \textcyrIzdatelp1stvo AN SSSR, Leningrad and Moscow. English translation in Sheynin (? 165--173.
Kolmogorov, A. N. (1939). Letter to Maurice Fréchet. Fonds Fréchet, Archives de l'Académie des Sciences, Paris.
Kolmogorov, A. N. (1948).\textcyrEvgeniĭ Evgenievich Slutskiĭ: Nekrolog (Obituary for Evgeny Evgenievich Slutsky).\textcyrUspekhi Matematicheskikh Nauk (Russian Mathematical Surveys) 3(4) 142--151. English translation in Sheynin (? 77--88, reprinted in Math. Sci. 27 67--74 (2002).
Mathematical Reviews (MathSciNet): MR1950284
Kolmogorov, A. N. (1956).\textcyrTeoriya veroyatnosteĭ (Probability theory). In\textcyrMatematika, ee soderzhanie, metody i znachenie (A. D. Aleksandrov, A. N. Kolmogorov and M. A. Lavrent'ev, eds.) 2 252--284. Nauka, Moscow. The Russian edition had three volumes. The English translation, Mathematics, Its Content, Methods, and Meaning, was first published in 1962 and 1963 in six volumes by the American Mathematical Society, Providence, RI, and then republished in 1965 in three volumes by the MIT Press, Cambridge, MA. Reprinted by Dover, New York, 1999. Kolmogorov's chapter occupies pp. 33--71 of Part 4 in the 1963 English edition and pp. 229--264 of Volume 2 in the 1965 English edition.
Kolmogorov, A. N. (1986).\textcyrIzbrannye trudy. Teoriya veroyatnosteĭ i matematicheskaya statistika. Nauka, Moscow.
Mathematical Reviews (MathSciNet): MR861120
Kolmogorov, A. N. (1992). Selected Works of A. N. Kolmogorov 2. Probability Theory and Mathematical Statistics. Kluwer, Dordrecht. Translation by G. Lindquist of Kolmogorov (1986).
Mathematical Reviews (MathSciNet): MR1153022
Laemmel, R. (1904). Untersuchungen über die Ermittlung von Wahrscheinlichkeiten. Ph.D. thesis, Universität Zürich. Excerpts reprinted in Schneider (1988) 367--377.
Lebesgue, H. (1901). Sur une généralisation de l'intégrale définie. C. R. Acad. Sci. Paris 132 1025--1028.
Lebesgue, H. (1904). Leçons sur l'intégration et la recherche des fonctions primitives. Gauthier-Villars, Paris. Second edition 1928.
Lévy, P. (1925). Calcul des probabilités. Gauthier-Villars, Paris.
Lévy, P. (1937). Théorie de l'addition des variables aléatoires. Gauthier-Villars, Paris. Second edition 1954.
Lévy, P. (1959). Un paradoxe de la théorie des ensembles aléatoires. C. R. Acad. Sci. Paris 248 181--184. Reprinted in Levy (? 6 67--69.
Mathematical Reviews (MathSciNet): MR102852
Lévy, P. (1973--1980). Œuvres de Paul Lévy. Gauthier-Villars, Paris. In six volumes. Edited by D. Dugué.
Łomnicki, A. (1923). Nouveaux fondements du calcul des probabilités (Définition de la probabilité fondée sur la théorie des ensembles). Fund. Math. 4 34--71.
Łomnicki, Z. and Ulam, S. (1934). Sur la théorie de la mesure dans les espaces combinatoires et son application au calcul des probabilités. I. Variables indépendantes. Fund. Math. 23 237--278.
Lorentz, G. G. (2002). Mathematics and politics in the Soviet Union from 1928 to 1953. J. Approx. Theory 116 169--223.
Mathematical Reviews (MathSciNet): MR1911079
Digital Object Identifier: doi:10.1006/jath.2002.3670
Zentralblatt MATH: 1006.01009
Loveland, J. (2001). Buffon, the certainty of sunrise, and the probabilistic reductio ad absurdum. Arch. Hist. Exact Sci. 55 465--477.
Mathematical Reviews (MathSciNet): MR1827868
Digital Object Identifier: doi:10.1007/s004070000035
Zentralblatt MATH: 0978.01022
Mac Lane, S. (1995). Mathematics at Göttingen under the Nazis. Notices Amer. Math. Soc. 42 1134--1138.
Mathematical Reviews (MathSciNet): MR1350011
Maistrov, L. E. (1974). Probability Theory: A Historical Sketch. Academic Press, New York. Translated and edited by S. Kotz.
Mathematical Reviews (MathSciNet): MR378008
Zentralblatt MATH: 0308.01001
Markov, A. A. (1900).\textcyrIschislenie veroyatnosteĭ (Calculus of Probability).\textcyrTipografiya Imperatorskoĭ Akademii Nauk, St. Petersburg. Second edition 1908, fourth 1924.
Markov, A. A. (1912). Wahrscheinlichkeitsrechnung. Teubner, Leipzig. Translation of second edition of Markov (1900). Available at historical.library.cornell.edu.
Zentralblatt MATH: 43.0291.01
Martin, T. (1996). Probabilités et critique philosophique selon Cournot. Vrin, Paris.
Mathematical Reviews (MathSciNet): MR1652541
Martin, T. (1998). Bibliographie cournotienne. Annales littéraires de l'Université de Franche-Comté, Besançon.
Martin, T. (2003). Probabilité et certitude. In Probabilités subjectives et rationalité de l'action (T. Martin, ed.) 119--134. CNRS Éditions, Paris.
Masani, P. R. (1990). Norbert Wiener, 1894--1964. Birkhäuser, Basel.
Mathematical Reviews (MathSciNet): MR1032520
Zentralblatt MATH: 0681.01016
Mazliak, L. (2003). Andrei Nikolaevitch Kolmogorov (1903--1987). Un aperçu de l'homme et de l'œuvre probabiliste. Prépublication PMA-785, Univ. Paris VI. Available at www.proba.jussieu.fr.
Meinong, A. (1915). Über Möglichkeit und Wahrscheinlichkeit: Beiträge zur Gegenstandstheorie und Erkenntnistheorie. Barth, Leipzig.
Nikodym, O. (1930). Sur une généralisation des intégrales de M. J. Radon. Fund. Math. 15 131--179.
Ondar, Kh. O., ed. (1981). The Correspondence Between A. A. Markov and A. A. Chuprov on the Theory of Probability and Mathematical Statistics. Springer, New York. Translated from the Russian by C. M. and M. D. Stein.
Mathematical Reviews (MathSciNet): MR667526
Zentralblatt MATH: 0476.62002
Onicescu, O. (1967). Le livre de G. Castelnuovo Calcolo della probabilità e applicazioni comme aboutissant de la suite des grands livres sur les probabilités. In Simposio Internazionale di Geometria Algebrica (Roma, 30 Settembre--5 Ottobre 1965) xxxvii--liii. Edizioni Cremonese, Rome.
Mathematical Reviews (MathSciNet): MR488197
Pier, J.-P. (1994a). Intégration et mesure 1900--1950. In Pier (? 517--564.
Mathematical Reviews (MathSciNet): MR1298642
Pier, J.-P., ed. (1994b). Development of Mathematics 1900--1950. Birkhäuser, Basel.
Mathematical Reviews (MathSciNet): MR1298630
Zentralblatt MATH: 0796.00016
Poincaré, H. (1890). Sur le problème des trois corps et les équations de la dynamique. Acta Math. 13 1--271.
Poincaré, H. (1896). Calcul des probabilités. Leçons professées pendant le deuxième semestre 1893--1894. Carré, Paris. Available at historical.library.cornell.edu.
Poincaré, H. (1912). Calcul des probabilités. Gauthier-Villars, Paris. Second edition of Poincaré (1896).
Porter, T. (1986). The Rise of Statistical Thinking, 1820--1900. Princeton University Press, Princeton, NJ.
Rademacher, H. (1922). Einige Sätze über Reihen von allgemeinen Orthogonalfunktionen. Math. Ann. 87 112--138.
Mathematical Reviews (MathSciNet): MR1512104
Digital Object Identifier: doi:10.1007/BF01458040
Radon, J. (1913). Theorie und Anwendungen der absolut additiven Mengenfunktionen. Akad. Wiss. Sitzungsber. Kaiserl. Math.-Nat. Kl. 122 1295--1438. Reprinted in his Gesammelte Abhandlungen 1 45--188. Birkhäuser, Basel, 1987.
Reichenbach, H. (1916). Der Begriff der Wahrscheinlichkeit für die mathematische Darstellung der Wirklichkeit. Barth, Leipzig.
Reichenbach, H. (1932). Axiomatik der Wahrscheinlichkeitsrechnung. Math. Z. 34 568--619.
Mathematical Reviews (MathSciNet): MR1545274
Digital Object Identifier: doi:10.1007/BF01180610
Rogers, L. C. G. and Williams, D. (2000). Diffusions, Markov Processes, and Martingales. 1. Foundations, reprinted 2nd ed. Cambridge Univ. Press.
Mathematical Reviews (MathSciNet): MR1796539
Zentralblatt MATH: 0949.60003
Schneider, I., ed. (1988). Die Entwicklung der Wahrscheinlichkeitstheorie von den Anfängen bis 1933: Einführungen und Texte. Wissenschaftliche Buchgesellschaft, Darmstadt.
Mathematical Reviews (MathSciNet): MR998777
Segal, I. E. (1992). Norbert Wiener. November 26, 1894--March 18, 1964. Biographical Memoirs 61 388--436. National Academy of Sciences, Washington.
Seneta, E. (1997). Boltzmann, Ludwig Edward. In Johnson and Kotz (? 353--354.
Seneta, E. (2004). Mathematics, religion and Marxism in the Soviet Union in the 1930s. Historia Math. 31 337--367.
Mathematical Reviews (MathSciNet): MR2079595
Digital Object Identifier: doi:10.1016/S0315-0860(03)00046-6
Zentralblatt MATH: 1085.01017
Shafer, G. and Vovk, V. (2001). Probability and Finance: It's Only a Game! Wiley, New York.
Mathematical Reviews (MathSciNet): MR1852450
Zentralblatt MATH: 0985.91024
Shafer, G. and Vovk, V. (2005). The origins and legacy of Kolmogorov's Grundbegriffe. Working Paper No. 4. Available at www.probabilityandfinance.com.
Zentralblatt MATH: 05191852
Sheynin, O. (1996). Aleksandr A. Chuprov: Life, Work, Correspondence. The Making of Mathematical Statistics. Vandenhoeck and Ruprecht, Göttingen.
Mathematical Reviews (MathSciNet): MR1384301
Zentralblatt MATH: 0845.01011
Sheynin, O., ed. (1998). From Markov to Kolmogorov. Russian papers on probability and statistics. Containing essays of S. N. Bernstein, A. A. Chuprov, B. V. Gnedenko, \mboxA. Ya. Khinchin, A. N. Kolmogorov, A. M. Liapunov, A. A. Markov and V. V. Paevsky. Hänsel-Hohenhausen, Egelsbach, Germany. Translations from Russian into English by the editor. Deutsche Hochschulschriften No. 2514. In microfiche.
Sheynin, O., ed. (2000). From Daniel Bernoulli to Urlanis. Still more Russian Papers on Probability and Statistics. Hänsel-Hohenhausen, Egelsbach, Germany. Translations from Russian into English by the editor. Deutsche Hochschulschriften No. 2696. In microfiche.
Zentralblatt MATH: 0943.01009
Siegmund-Schultze, R. (2004). Mathematicians forced to philosophize: An introduction to Khinchin's paper on von Mises' theory of probability. Sci. Context 17 373--390.
Mathematical Reviews (MathSciNet): MR2106023
Digital Object Identifier: doi:10.1017/S0269889704000171
Zentralblatt MATH: 1181.01049
Sierpiński, W. (1918). Sur une définition axiomatique des ensembles mesurables (L). Bull. Internat. Acad. Sci. Cracovie A 173--178. Reprinted in W. Sierpiński (1975). Oeuvres choisies 2 256--260. PWN (Polish Scientific Publishers), Warsaw.
Slutsky, E. (1922).\textcyrK voprosu o logicheskih osnovah teorii veroyatnosti (On the question of the logical foundation of the theory of probability).\textcyrVestnik Statistiki (Bulletin of Statistics) 12 13--21.
Slutsky, E. (1925). Über stochastische Asymptoten und Grenzwerte. Metron 5 3--89.
Steinhaus, H. (1923). Les probabilités dénombrables et leur rapport à la théorie de la mesure. Fund. Math. 4 286--310.
Steinhaus, H. (1930a). Über die Wahrscheinlichkeit dafür, daß der Konvergenzkreis einer Potenzreihe ihre natürliche Grenze ist. Math. Z. 31 408--416. Received by the editors 5 August 1929.
Mathematical Reviews (MathSciNet): MR1545123
Digital Object Identifier: doi:10.1007/BF01246422
Steinhaus, H. (1930b). Sur la probabilité de la convergence de séries. Première communication. Studia Math. 2 21--39. Received by the editors 24 October 1929.
Stigler, S. M. (1973). Simon Newcomb, Percy Daniell, and the history of robust estimation 1885--1920. J. Amer. Statist. Assoc. 68 872--879.
Mathematical Reviews (MathSciNet): MR362575
Digital Object Identifier: doi:10.2307/2284515
JSTOR: links.jstor.org
Zentralblatt MATH: 0286.01011
Tornier, E. (1933). Grundlagen der Wahrscheinlichkeitsrechnung. Acta Math. 60 239--380.
Mathematical Reviews (MathSciNet): MR1555367
Digital Object Identifier: doi:10.1007/BF02398272
Ulam, S. (1932). Zum Massbegriffe in Produkträumen. In Verhandlung des Internationalen Mathematiker-Kongress Zürich 2 118--119.
Venn, J. (1888). The Logic of Chance, 3rd ed. Macmillan, London and New York. First edition 1866, second 1876.
Zentralblatt MATH: 20.0208.06
Ville, J. (1939). Étude critique de la notion de collectif. Gauthier-Villars, Paris. This differs from Ville's dissertation, which was defended in March 1939, only in that a 17-page introductory chapter replaces the dissertation's one-page introduction.
von Bortkiewicz, L. (1901). Anwendungen der Wahrscheinlichkeitsrechnung auf Statistik. In Encyklopädie der Mathematischen Wissenschaften 1 821--851. Teubner, Leipzig.
von Kries, J. (1886). Die Principien der Wahrscheinlichkeitsrechnung. Eine logische Untersuchung. Mohr, Freiburg. The second edition, which appeared in 1927, reproduced the first without change and added a new 12-page foreword.
von Mises, R. (1919). Grundlagen der Wahrscheinlichkeitsrechnung. Math. Z. 5 52--99.
Mathematical Reviews (MathSciNet): MR1544374
Digital Object Identifier: doi:10.1007/BF01203155
von Mises, R. (1928). Wahrscheinlichkeit, Statistik und Wahrheit. Springer, Vienna. Second edition 1936, third 1951. A posthumous fourth edition, edited by his widow Hilda Geiringer, appeared in 1972. English editions, under the title Probability, Statistics and Truth, appeared in 1939 and 1957.
Mathematical Reviews (MathSciNet): MR350796
Zentralblatt MATH: 0242.60001
von Mises, R. (1931). Wahrscheinlichkeitsrechnung und ihre Anwendung in der Statistik und theoretischen Physik. Deuticke, Leipzig and Vienna.
von Plato, J. (1994). Creating Modern Probability: Its Mathematics, Physics, and Philosophy in Historical Perspective. Cambridge Univ. Press.
Mathematical Reviews (MathSciNet): MR1265718
Vucinich, A. (2000). Soviet mathematics and dialectics in the Stalin era. Historia Math. 27 54--76.
Mathematical Reviews (MathSciNet): MR1745515
Digital Object Identifier: doi:10.1006/hmat.1999.2261
Wald, A. (1938). Die Widerspruchfreiheit des Kollectivbegriffes. In Actualités Scientifiques et Industrielles 735 79--99. Hermann, Paris. Titled Les fondements du calcul des probabilités, Number 735 is the second fascicle of Wavre \mbox{(?
Wavre, R. (1938--1939). Colloque consacré à la théorie des probabilités. Hermann, Paris. This celebrated colloquium, chaired by Maurice Fréchet, was held in October 1937 at the University of Geneva. The proceedings were published by Hermann in eight fascicles in their series Actualités Scientifiques et Industrielles. The first seven fascicles appeared in 1938 as numbers 734 through 740; the eighth, de Finetti's summary of the colloquium, appeared in 1939 as number 766 (de Finetti, ?
Whittle, P. (2000). Probability via Expectation, 4th ed. Springer, New York. The first two editions (Penguin, 1970; Wiley, 1976) were titled Probability. The third edition, also by Springer, appeared in 1992.
Mathematical Reviews (MathSciNet): MR1163374
Zentralblatt MATH: 0755.60001
Wiener, N. (1920). The mean of a functional of arbitrary elements. Ann. of Math. (2) 22 66--72.
Mathematical Reviews (MathSciNet): MR1502573
Digital Object Identifier: doi:10.2307/1967855
JSTOR: links.jstor.org
Wiener, N. (1921a). The average of an analytical functional. Proc. Natl. Acad. Sci. U.S.A. 7 253--260.
Wiener, N. (1921b). The average of an analytical functional and the Brownian movement. Proc. Natl. Acad. Sci. U.S.A. 7 294--298.
Wiener, N. (1923). Differential-space. J. Math. Phys. Mass. Inst. Tech. 2 131--174.
Wiener, N. (1924). The average value of a functional. Proc. London Math. Soc. 22 454--467.
Wiener, N. (1956). I am a Mathematician. The Later Life of a Prodigy. Doubleday, Garden City, NY.
Mathematical Reviews (MathSciNet): MR77455
Wiener, N. (1976--1985). Collected Works with Commentaries. MIT Press, Cambridge, MA. Four volumes. Edited by P. Masani. Volume 1 includes Wiener's early papers on Brownian motion (Wiener, ? Wiener, ? Wiener, ? Wiener, ? Wiener, ?, with a commentary by K. Itô.
Wiman, A. (1900). Über eine Wahrscheinlichkeitsaufgabe bei Kettenbruchentwicklungen. Öfversigt af Kongliga Svenska Vetenskaps-Akademiens Förhandlingar. Femtiondesjunde Årgången 57 829--841.
Wiman, A. (1901). Bemerkung über eine von Gyldén aufgeworfene Wahrscheinlichkeitsfrage. Håkan Ohlssons boktrykeri, Lund.
