Explaining objective probabilities by physical symmetries

Abstract In this paper, I discuss the relationship between the method of arbitrary functions and single-case probabilities, and offer an interpretation of these probabilities by reference to physical symmetries of chance experiments. The method of arbitrary functions attempts to explain outcome frequencies through the underlying deterministic dynamics of chance experiments and the constant proportions of these dynamics that lead to different outcomes in repeated chance experiments. However, the method of arbitrary functions alone appears to provide no comprehensive interpretation of deterministic chance since it relies on distributions of initial conditions that, in turn, require interpretation. Moreover, even without reference to distributions of initial conditions, the method of arbitrary functions may not be able to provide genuine single-case probabilities since the constant proportions appear to become relevant for outcomes only in repeated chance experiments. In this paper, I propose an explanation of the constant proportions of dynamics via the physical properties of single chance experiments. By doing so, I attempt to develop the method of arbitrary functions into a comprehensive interpretation of probability that does not rely on the distribution of initial conditions and covers genuine single-case probabilities. Specifically, I show how the constant proportions result from what I will call the relevant physical symmetries of chance experiments , i.e., from a regular constitution of chance devices with respect to the dynamically realized symmetry transformations in single chance experiments.