The Emergence of Mathematical Probability from the Perspective of the Leibniz-Jacob Bernoulli Correspondence

This paper moves out from a defined body of evidence, namely the correspondence between Gottfried Wilhelm Leibniz and Jacob Bernoulli, to see what light it casts on the emergence of mathematical probability in the late seventeenth and early eighteenth centuries. At many times during his life Leibniz repeated his recommendation that serious attention should be paid to developing a theory of probabilistic reasoning. In his Ars Conjectandi (The Art of Conjecturing), Jacob Bernoulli tried to do just what Leibniz at many times recommended, but he did not finish the work before his death in 1705, partly because he lacked information about real-life situations. Although, when they discussed probability and what Bernoulli had accomplished and hoped still to do in letters written between April 1703 and April 1705, Leibniz and Bernoulli did not understand each other perfectly, nevertheless, when one looks more broadly in their writings, one sees that they were in fundamental agreement about their purposes: the new probability logic or the art of conjecturing was to play a role in a practical moral discipline and its problems were to be more like those of law than those of empirical science. Within this practical discipline, the role of mathematics was to supply a rigorous method of reasoning and, in particular, a way of insuring comprehensiveness and consistency when the complexity of the situation might otherwise overwhelm normal reason.

The Correspondence Between Leibniz and Jacob Bernoulli

There are twenty-one extant letters between Leibniz and Jacob Bernoulli (one of which may never have been sent) and evidence that at least five more once existed (see Appendix below for list of letters). The skimpiness of the correspondence was no doubt largely Jacob Bernoulli’s doing—in all there are only approximately 150 extant letters to and from Jacob Bernoulli, whereas there are approximately 2,000 to and from his brother Johann Bernoulli and over 15,000 surviving letters from Leibniz to other people (Nagel in Jacob Bernoulli 1993a, p. 27, n. 2; Hess 1989, pp. 155–56; Ross 1984, p. 9). The correspondence between Jacob Bernoulli and Leibniz got off to a slow start because Leibniz was not in Hannover when Bernoulli first wrote him in 1687. Once Leibniz replied in 1690, Bernoulli allowed more than five years to pass before writing again in October 1695. Meanwhile, however, between 1693 and Jacob’s October 1695 letter to Leibniz, Johann Bernoulli, then living in Basel like his brother, exchanged 17 letters with Leibniz, often sending and receiving greetings between Leibniz and Jacob, but also reporting a growing conflict between the brothers. In April of 1695, Johann reported to Leibniz that he had shown Leibniz’s letters to his brother in the hopes they would convince Jacob of his culpability in his conflict with Johann. He said that Jacob’s failure to write to Leibniz himself was due not only to Jacob’s character, but also because Jacob had been sick (GM III, p. 173). Johann was sure that Jacob would write to Leibniz in the near future, but he was afraid that he would have nothing good to say about Johann. When Jacob then wrote to Leibniz on 9/19 October 1695, it was soon after Johann had left Basel to take up his new post in Groningen—Johann’s first letter to Leibniz from Holland was written from Amsterdam on 8/18 October 1695, just the day before his brother’s letter.

In the period after Johann left Basel for Groningen, there followed a more regular if not copious exchange of letters between Jacob Bernoulli and Leibniz. Leibniz replied to Jacob in December 1695. Each man wrote to the other at least once in 1696 and at least once in 1697. Then Jacob let another five years pass before writing to Leibniz again on 15 November 1702, his letter then being occasioned by receipt of documents making him a member of the Berlin Society of Sciences, which had been established at Leibniz’s instigation. Such a happy occasion was hardly one for complaining, but Jacob went on to explain...