March 20, 2023

Von Mises, Popper, and the Cournot Principle

Tessa Murthy, Carnegie Mellon University

Abstract: Adherents to non-frequentist metaphysics of probability have often claimed that the axioms of frequentist theories can be derived as theorems of alternate characterizations of probability by means of statistical convergence theorems. One representative such attempt is Karl Popper’s “bridge” between propensities and frequencies, discussed in his books Logic of Scientific Discovery and Realism and the Aim of Science. What makes Popper’s argument particularly interesting is that he seemed much more sympathetic to the frequentists than the other theorists who promoted similar claims. In particular, while Richard von Mises criticized the use of the Law of Large Numbers in interderivability arguments, Popper was clearly aware of this worry—he even joined von Mises in criticizing his contemporary Frechet’s more heavy-handed application of it. What Popper thought set his version of the bridge argument apart was his use of the almost-sure strong law in place of the weak law. SLLN has a measure zero exclusion clause, which Popper claimed could be unproblematically interpreted as probability zero. While in other contexts he agreed with von Mises that taking low propensity to entail low frequency requires frequentist assumptions, the zero case, according to Popper, was special. To defend this claim, he relied on a contextually odd application of the Cournot principle.

In this project I investigate two related, understudied elements of the Popper/von Mises dispute. First, I provide an explanation of the mutual misunderstanding between von Mises and Popper about the admissibility of the claim that measure zero sets are probability zero sets. Popper takes von Mises as levying the criticism that claims along the lines that “low propensity means common in a long sequence of trials” are inaccurate (a claim von Mises elsewhere makes) when in fact von Mises is instead concerned that such claims are circular or fundamentally frequentist. This explains, but does not entirely justify, Popper’s appeal to the Cournot principle. Second, I relay the worry that the use of convergence theorems in the context of propensities requires more auxiliary information than similar uses in frequentist theories. The SLLN, for example, requires that subsequent trials satisfy independence conditions (usually i.i.d.). I provide a charitable interpretation of Popper’s project that better justifies that experimental iterations satisfy the antecedent of the SLLN, and also makes sense of Popper’s references to Doob’s impossibility theorem. I conclude by reflecting that though this historical investigation paints Popper’s approach as more statistically informed than is commonly thought, it does not get him entirely out of trouble.