Revisiting the Sources of Borel's Interest in Probability: Continued Fractions, Social Involvement, Volterra's Prolusione

Abstract

In this paper, we revisit the origins of Emile Borel's developing interest in probability around 1905. This resulted from new findings in his research on continued fraction, but it also cannot be separated from the discovery of new applications for the mathematics of randomness (such as biology or economics) and of their importance as a life‐changing tool for the citizen. In particular, we underline the role of a paper published by Vito Volterra in Borel's Revue du Mois.

Henri Poincaré, 1912–2012

“…He now doubted that diffraction could explain long-distance transmission and he noted that the alternative explanation by ionization of the upper atmosphere might perhaps account for the superior quality of nighttime transmissions. 32 In 1911, an American student of Sommerfeld, Herman William March, filed a dissertation on the propagation of telegraphic waves around the earth and obtained results contradicting Poincaré's. In a letter to Sommerfeld and in a note to the Compte rendus, Poincaré identified a fatal error in March's calculation.…”

“…32 Because of the symmetries, they are the same for the Hamiltonian and for its quadratic part Q 0 .…”

“…Nothing seemed to predispose him to take the plunge and to devote important efforts to study, refine and popularize the calculus of probability whose dubious reputation in the mathematical community -on which we commented above -might have led to rebuke. The context of Borel's turn since 1905, the date of publication of his first work in this domain, has been studied in detail several times, for instance in the papers [32] and [59]. The main difference which can be found between the discovery of probability by Poincaré and by Borel is that, for the latter, it arose from reflections within the mathematical field and more specifically by considerations on the status of mathematical objects -in particular about real numbers.…”

“…From his very first paper, Borel announced that he adopted the conventionalism of Poincaré ([13], p. 123 ). But his aim was to illustrate the role that the (then novel) Lebesgue integral and measure theory could play, after he discovered with amazement their use in [95] by the Swedish mathematician Anders Wiman (on this subject, see [32]). 'The methods adopted by Mr. Lebesgue allow us to examine [.…”

Poincaré and probability

“…One significant challenge in dealing with this subject is that probability penetrated Poincaré's work almost by force, forcing his hand several times, his main achievement consisting in building dykes so that the mathematician might venture with a dry foot onto this rather marshy terrain. His successor Borel had a somewhat different attitude toward choosing to apply probability theory in many domains, reflecting the fact that Borel had encountered probability in a more spectacular way than Poincaré (see [11]). …”