On the explanation of connectedness of Kolmogorov’s conditions for probabilities
Philosophy
Keywords:
principle of falsification, probability, independence, Cournot’s principle, Bernoulli’s theorem, frequency interpretationAbstract
The methodological significance of Kolmogorov’s conditions for probabilities studied in applications is demonstrated. For instance, on the basis of the first condition about the proximity of theoretical probability and frequencies under a large number of experiments we used many results in probability theory and mathematical statistics. On the basis of the second condition — Cournot’s principle — verification of statistical hypotheses is carried out. The same principle is the foundation of the concept of falsificationism in Popper’s philosophy. The validity of Popper’s falsificationism for formal reasons is directly connected with the correctness and universality of Cournot’s condition. A special attention is paid to studying the explanations of the fact of connectedness of the requirements. It is demonstrated that the explanation of connectedness by the French mathematicians is not empirical in character. The approaches of Shafer and Vovk to the explanation of connectedness of Kolmogorov’s requirements are analyzed. A classification of these explanations according to their feasibility is presented. The best justified explanations are discussed; new approaches to the explanation of connectedness of the requirements are proposed and studied. The basic idea of the new explanation is that the formal dependence of one requirement on another presupposes, in turn, the semantic dependence of the formally independent requirement on the formally dependent one.References
(1). Алимов Ю.И. Альтернатива методу математической статистики. М.: Знание, 1980. 84 с.
(2). Колмогоров А.Н. Основные понятия теории вероятностей. М.: Наука, 1974. 120 с.
(3). Крамер Г. Математические методы статистики. М: Мир, 1975. 648 с.
(4). Курно А. Основы теории шансов и вероятностей. М: Наука, 1970. 384 с.
(5). Проблемы Гильберта. М.: Наука, 1969. 240 с.
(6). Резников В.М. Анализ условий применения теории вероятностей по Колмогорову // Политематический сетевой электронный научный журнал Кубанского государственного аграрного университета. Краснодар: КубГАУ. 2014. № 04(098). URL: http://ej.kubagro.ru/2014/04/pdf/108.pdf (дата обращения: 07.05.2014).
(7). Резников В.М. Вероятностные идеи в философии К. Поппера // Вестник Новосибирского государственного университета. Серия: Философия. Новосибирск, 2013. T. 11, вып. 2. С. 143–148.
(8). Резников В.М. Критический анализ условий применения теории вероятностей по Колмогорову // Вестник Новосибирского государственного университета. Серия: Философия. 2012. T. 10, вып. 1. С. 19–24.
(9). Резников В.М. Об адекватности математической статистики // Знания-Онтологии-Теории: материалы Всеросс. конференции с международным участием, 8–10 октября 2013 г. Новосибирск: Прайс-Курьер, 2013. Т. 2. С. 106–111.
(10). Резников В.М. Об адекватности принципа Курно в математике // Философия математики: актуальные проблемы. Математика и реальность: тез. Третьей Всеросс. науч. конференции. 27–28 сентября 2013 г. М.: Центр стратегической конъюнктуры, 2013. С. 91–95.
(11). Cohen J. The Earth is Round (p < 0,05) // American Psychologist. 1994. Vol. 49, № 12. P. 997–1003.
(12). Fidler F. From statistical significance to effect estimation: Statistical reform in psychology, medicine and ecology. Department of History and Philosophy of Science. The University of Melbourne, 2005. URL: http://www.botany.unimelb.edu.au/envisci/docs/fidler/fidlerphd_aug06.pdf (date of access: 22.05.2014).
(13). Hendricks V., Symons J. Masses of Formal Philosophy. London: Automatic Press, 2006. 264 p. URL: http://www.formalphilosophy.com (date of access: 12.08.2014).
(14). Kolmogoroff A. Grundbegrife der Wahrscheinlichkeitsrechnung. Berlin: Springer, 1933. 62 s.
(15). Popper K. The propensity interpretation of the calculus of probability, and quantum theory // Observation and interpretation: A symposium of philosophers and physicists / ed. by S. Körner. M.H.L. Pryce. L., 1957. P. 65–70.
(16). Shafer G., Vovk V. Probability and Finance It’s Only a Game! N.Y.: A Wiley-Interscience Publication, 2001. 414 p.
(17). Shafer G., Vovk V. The origins and legacy of Kolmogorov’s Grundbegriffe. URL: http://www.probabilityandfinance.com/articles/04.pdf (date of access: 22.05.2014).
(18). Shafer G., Vovk V. The Sources of Kolmogorov’s Grundbegriffe // Statistical Science. 2006. Vol. 21, № 1. P. 70–98.
References
(1). Alimov Yu.I. Al’ternativa metodu matematicheskoj statistiki [An alternative to the method of mathematical statistics]. Moscow, Znanie Publ., 1980, 84 p. (In Russian).
(2). Kolmogorov A.N. Osnovnye ponyatiya teorii veroyatnostej [Basic terms of theory of probability]. Moscow, Nauka Publ., 1974, 120 p. (In Russian).
(3). Cramer G. Matematicheskie metody statistiki [Mathematical methods of statistics]. Moscow, Mir Publ., 1975, 648 p. (In Russian).
(4). Cournot A. Osnovy teorii shnsov i veroyatnostej [Exposition of the theory of chance and probability]. Moscow, Nauka Publ., 1970, 384 p. (In Russian).
(5). Problemy Gil’berta [Problems of Hilbert]. Moscow, Nauka Publ., 1969, 240 p. (In Russian).
(6). Reznikov V.M. [Analysis of application conditions of the theory of probability sensu Kolmogorov]. Politematicheskij setevoj elektronnyj nauchnyj zhurnal Kubanskogo gosudarstvennogo agrarnogo universiteta [Polythematic net online scientific journal of Kuban State Agrarian University]. Krasnodar, Kuban State Agrarian University Publ., 2014, no 4(098). Available at: http://ej.kubagro.ru/2014/04/pdf/108.pdf (accessed: 07.05.2014). (In Russian).
(7). Reznikov V.M. [Probability ideas in K. Popper’s philosophy]. Vestnik Novosibirskogo gosudarstvennogo universiteta. Seriya: Filosofiya [Novosibirsk State University Bulletin. Series “Philosophy”]. 2013, vol. 11, no 2, pp. 143–148. (In Russian).
(8). Reznikov V.M. [Critical analysis of application conditions of the theory of probability sensu Kolmogorov]. Vestnik Novosibirskogo gosudarstvennogo universiteta. Seriya: Filosofiya [Novosibirsk State University Bulletin. Series “Philosophy”]. 2012, vol. 10, no 1, pp. 19–24. (In Russian).
(9). Reznikov V.M. [Concerning adequacy of mathematical statistics]. Znaniya-Ontologii-Teorii: Materialy vserossijskoj konferentsii s mezhdunarodnym uchastiem [Knowledge-Ontologies-Theories: Proceedings of All-Russian Scientific Conference with international participation]. 8-10th October. Novosibirsk, Prais-Kur’er Publ., 2013, vol. 2, pp. 106–111. (In Russian).
(10). Reznikov V.M. [Concerning adequacy of Cournot’s principle]. Filosofiya matematiki: aktual’nye problemy. Matematika i real’nost’: Tezisy tret’ej vserossijskoj nauchnoj konferentsii. [Philosophy of mathematics: current issues. Mathematics and reality: Proceedings of the 3rd All-Russian scientific conference]. 27-28th of September 2013. Moscow, Centre of Strategic Conjuncture Publ., 2013, pp. 91–95. (In Russian).
(11). Cohen J. The Earth is Round (p < 0,05) // American Psychologist. 1994. Vol. 49, № 12. P. 997–1003.
(12). Fidler F. From statistical significance to effect estimation: Statistical reform in psychology, medicine and ecology. Department of History and Philosophy of Science. The University of Melbourne, 2005. Available at: http://www.botany.unimelb.edu.au/envisci/docs/fidler/fidlerphd_aug06.pdf (accessed 22.05.2014).
(13). Hendricks V, Symons J. Masses of Formal Philosophy. London: Automatic Press, 2006. 264 p. Available at: http://www.formalphilosophy.com (accessed 12.08.2014).
(14). Kolmogoroff A. Grundbegrife der Wahrscheinlichkeitsrechnung. Berlin: Springer, 1933. 62 p.
(15). Popper K. The propensity interpretation of the calculus of probability, and quantum theory // Observation and interpretation: A symposium of philosophers and physicists / ed. by S. Körner M.H.L. Pryce. L.1957. P. 65–70.
(16). Shafer G., Vovk V. Probability and Finance It’s Only a Game! N.Y.: A Wiley-Interscience Publication, 2001. 414 p.
(17). Shafer G., Vovk V. The origins and legacy of Kolmogorov’s Grundbegriffe. URL: http://www.probabilityandfinance.com/articles/04.pdf (accessed 22.05.2014).
(18). Shafer G., Vovk V. The Sources of Kolmogorov’s Grundbegriffe // Statistical Science. 2006. Vol. 21, № 1. P. 70–98.
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