Émile Borel Versus John Maynard Keynes: The Two Opposing Views on Probability

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This paper aims to compare French mathematician Émile Borel and British economist John Maynard Keynes with special reference to probability theory. Borel and Keynes were contemporaries who greatly influenced each other across the English channel each other in the twentieth century. In his influential book, Borel (1938) harshly criticized Keynes’s position on probability theory. Borel regarded probability as a measurable object, thus constituting an important branch of mathematics. In contrast, Keynes thought of probability as a non-measurable phenomenon, thereby belonging rather to a branch of logic. Their controversies attracted attention in some parts of academia, generating a considerable output of papers even after their deaths and to the present day. With hindsight, however, it seems that the differences in their views are very deep-rooted and originated in the critical gulf between the abstract-minded French spirit and the empirical-oriented British tradition. In this paper, we wish to offer a series of fresh angles, thus shedding new light on the French-British controversy over probability. The first angle is offered by the rediscovery of Keynes’s romantic poem on probability, which can be found at the very end of Keynes’s 1921 book but had long been neglected until today. The second angle comes from reinvestigation of Keynes’s original yet almost forgotten concept of “interval-valued probability,” the third angle from the new interpretation of Keynes’s strange diagram on non-comparable probabilities. The fourth is prompted by the question of how and to what extent probability is related to non-measurability and ambiguity. Brief reference to Bruno de Finetti will be made in the final section.