The Method of Arbitrary Functions

Abstract

The method of arbitrary functions is a mathematical technique for the determination of probability distributions. It has been used to justify a unique and objectively interpreted concept of probability in games of chance of a mechanical type, and in the theory of dynamical systems of mathematical physics. Relatively recent results in the ergodic theory of classical dynamical systems will be useful for a re-evaluation of the method of arbitrary or random functions. Towards the end of this paper, we will try to show how, and to what degree, these new results justify the hopes of the pioneers of the method. The literature on probability in dynamical systems, which goes back almost one hundred years, has been unjustly forgotten. It forms an important chapter in the foundations of probability, and includes technical and interpretative insights of great relevance. It may be surprising to find out that probabilities in games of chance which are supposed to be mechanical systems are given an objective interpretation. Nowadays many a philosopher of probability advocates the view that a theory of genuine chance phenomena has to be based on a quantum-mechanically motivated notion of indeterminism. The possibility of 'a causal theory of probability' shows that this propensity point of view, whenever it is applied on a macroscopic level, can be rivalled by a view in which a non-classical notion of indeterminism is not needed. The literature on the method of arbitrary functions and ergodic theory contains studies of specific systems with given equations of motion. Beginning with the work of Ya. Sinai in 1963, the existence of dynamical systems which are measure-theoretically isomorphic to certain stochastic processes has been established. These rigorous results of mathematical physics, and their relevance for the interpretation of probability, have so far been left largely unnoticed by philosophers. Now