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On the Foundations of the Theory of Probabilities

Published online by Cambridge University Press:  14 March 2022

D. J. Struik
D. J. Struik*
Affiliation:
Mass. Institute of Technology
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Extract

The foundation of the mathematical theory of probabilities is still a controversial subject. There are schools of insufficient reasoning and of cogent reasoning, of a priori determination and of frequency determination, of subjective and of objective probability. Two main difficulties exist. The first is the definition of equally like events. The second difficulty is the relation between the laws of causal natural science (mechanics, electrodynamics, etc.) and the laws of statistical regularity. Is it really necessary to add to the laws of mechanics one or more independent “laws of large numbers” to explain the regularities in dice throwing? In all these cases different answers exist, and the existence of a mathematical theory of probabilities remains a kind of miracle, as esoteric to the further domain of natural science as the resurrection of the dead.

Type
Research Article
Information
Philosophy of Science , Volume 1 , Issue 1 , January 1934 , pp. 50 - 70
Copyright
Copyright © Philosophy of Science Association 1934

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References

1 A clear statement of this aspect of the theory of probability is given by A. Lomnicki, Fundamenta Mathematica (1923), pp. 34–71.

2 Essai philosophique sur les probabilités, 5me ed., Paris, 1825, p. 7.

3 See eg. A. Fisher, The Mathematical Theory of Probabilities, 2nd ed., Macmillan, N. Y. p. 46.

4 Dirichlet, Werke II, p. 60–64. See also E. Bessel- Ha e, Zahlentheorie, in Pascal's Repertorium, 2te Auflage, I, 3 (1929), p. 1518.

5 J. Von Kries, Die Prinzipien der Wahrscheinlichkeitsrechnung. Freiburg, 1886, p. 11. “Die Aufstellung der gleichmöglichen Fälle muss eine in zwingender Weise und ohne jede Willkür sich ergebende sein.”

6 R. von Mises. Vorlesungen aus der Gebiet der Angewandten Mathematik, I. Band. Wahrscheinlichkeitsrechnung, F. Deuticke, Leipzig u. Wien, 1931. See also E. Kamke. Einführung in die Wahrscheinlich keitsrechnung. Hirzel, Leipzig, 1932, and J. L. Coolidge, Mathematical theory of probability, Oxford University Press.

7 Kamke opens his book with an example which shows the weakness of the point of view of the frequency theory. He gives a series of values of the quotient n/N computed from a dice experiment. When computed to one decimal more these values oscillate in such a way that no exact limit seems to be indicated.

8 L. c. p. 82. The kind of function to use for equally likely cases “kann nur durch das Experiment entschieden werden, und hängt, natürlich, von der Art wie die Würfe ausgeführt werden, wesentlich ab.”

9 See eg. P. & T. Ehrenfest, Mecanique Statistique, Encycl. d. sc. mathématiques, V, 2 (1912) p. 211.

10 The theorem is due to Birkhoff, Proc. Nat. Acad. Science, 17 (1931), 656–660. A simple proof by E. Hopf, same Proc. 18 (1932), 93–100. The theorem also holds for more general functions f (Pt). I am indebted to my collegues E. Hopf and N. Wiener for many stimulating remarks on the subject of probability.

11 The necessity of the application of Lebesque integration to a probability problem in physics is already shown by N. Wiener's investigations on the Brownian movement, Proc. Nat. Ac. Sc. 6 (1920) p. 253–260.

12 See eg. Laplace's “Essay” cit. footnote (2).

13 F. Engels, Dialektik der Natur, Marx-Engels Archiv II, 1927, p. 264.

14 H. Poincaré, Calcul des Probabilités, 2e éd. Paris, 1912, pp. 4–20.

15 M. von Smoluchowski, Über den Begriff des Zufalls und den Ursprung der Wahrscheinlichkeitsgesetze in der Physik. “Die Naturwissenschaften” 6 (1918), pp. 253–263.

Both the work of v. Smoluchowski and of Wiener (footnote 11) is related to the Brownian movement.