History of Archimedean and non-Archimedean approaches to uniform processes: Uniformity, symmetry, regularity

Abstract: We apply Nancy Cartwright's distinction between theories and basic models to explore the history of rival approaches to modeling a notion of chance for an ideal uniform physical process known as a fair spinner. This process admits both Archimedean and non-Archimedean models. Advocates of Archimedean models maintain that the fair spinner should satisfy hypotheses such as invariance with respect to rotations by an arbitrary real angle, and assume that the optimal mathematical tool in this context is the Lebesgue measure. Others argue that invariance with respect to all real rotations does not constitute an essential feature of the underlying physical process, and could be relaxed in favor of regularity. We show that, working in ZFC, no subset of the commonly assumed hypotheses determines a unique model, suggesting that physically based intuitions alone are insufficient to pin down a unique mathematical model. We provide a rebuttal of recent criticisms of non-Archimedean models by Parker and Pruss.