April 3, 2023

The Curious Historiography of the Foundations of Probability

Michael Acree, Senior Statistician (Retired), University of California, San Francisco

Abstract: It is apparent to reflection that the modern, dualistic concept of probability comprises two aspects: an aleatory, mathematical concept based on games of chance and an epistemic aspect pertaining to knowledge, evidence, and belief. The early historiography of probability, such as the work by David, focused on the history of gambling rather than the concept of probability per se. The first major work to tackle the conceptual foundations of probability was Ian Hacking’s The Emergence of Probability. Hacking tried to uncover the epistemological ground for the merging of the two concepts, but didn’t really address the tensions that occurred when Bernoulli formally brought them together in 1713. The revolution in historiography of probability occurred in 1978 with a paper by Shafer. Virtually all attention to Bernoulli’s Ars Conjectandi had focused only on the closing pages, where he famously proved the weak law of large numbers; Shafer was the first to notice, at least since the 18th century, Bernoulli’s struggle to integrate the two concepts.  He also attributes the success of Bernoulli’s dualistic concept to Bernoulli’s widow and son having withheld publication for 8 years following his death, and to eulogies having created the impression that Bernoulli had succeeded in his ambition of applying the calculus of chances to “civil, moral, and economic matters.” Lambert improved Bernoulli’s formula for the combination of probabilities 50 years later, but did not address the question of a metric for epistemic probability, or the meaningfulness of combining them with chances.  I suggest that no such meaningful combination is possible. The most important consequence of the dualistic concept of probability is the concept of statistical inference, which stands without warrant. So, ironically, Bernoulli is celebrated for what he failed to achieve, whereas Shafer, in pointing that out, is uncelebrated for his—both presumably due to theorists wanting to realize the promise of the union of these two incompatible concepts.