Bertrand's Paradox is a Paradox of Infinity. - Ryan, Patrick J. (Synthese, 2025)
Since its formulation in the nineteenth century, Bertrand's paradox has played a prominent role in discussions of probability theory and has received a great deal of attention from philosophers, mathematicians, and physicists alike. This is because the paradox is often interpreted as refuting the Principle of Indifference, a central pillar of classical probability theory. In this paper, I articulate a novel interpretation of Bertrand's paradox according to which it is ultimately a paradox of infinity produced by a conflict between our intuitions and techniques for measuring infinite sets and the properties we wish to preserve when doing so. I proceed by first analyzing Bertrand's original formulation of the paradox and show that one can make sense of this discussion entirely in terms how we deal with the infinite sets in question. I then discuss two possible approaches for dealing with this construal and argue that the paradox can be avoided by jettisoning our Cantorian intuitions for measuring infinite sets. Next, I analyze recent mathematical frameworks developed to resolve the paradox and argue that disagreement between these accounts indicates that a "meta-level" Bertrand's paradox is lurking in the wings. This recurrence of the paradox leads us into deep questions about the possible indeterminacy latent in all attempts to measure infinite sets. Finally, I take stock of what we have learned from my analysis, delineate possible approaches for dealing with the meta-level paradox, and suggest directions for future research.