Ergodic Theory and the Foundations of Probability

Abstract

In the following, I shall illustrate through examples how useful the notions and results of ergodic theory are for the foundations of probability. Ergodic theory can be discussed from many points of view. It originated with the development of statistical mechanics in classical physics. Later, an abstract formulation of ergodic theory became part of measure theory. The latter treats probabilities as special kinds of measures. Results of ergodic theory can be intuitively explained in probabilistic terms, without special recourse to their physical significance or to the uninterpreted, abstract mathematical formulation. In discussing the relevance of ergodic theory for the foundations of probability, the physical background is significant, however. Most of the notions of the theory are motivated from statistical physics, so that, proceeding in the other direction as it were, an interpretation of the results of ergodic theory can be given in physical terms. It should be noted that there are different possibilities to hand here. Some writers consider the physical application a special case of a priori probabilities.1 Against this view, in which probabilities only relate to ignorance, stand the attempts at justifying probabilistic assumptions from a physical basis.