Typical = random

Abstract

This expository paper advocates an approach to physics in which ``typicality" is identified with a suitable form of algorithmic randomness. To this end various theorems from mathematics and physics are reviewed. Their original versions state that some property F(x) holds for P-almost all x in X, where P is a probability measure on some space X. Their more refined (and typically more recent) formulations show that F(x) holds for all P-random x in X. The computational notion of P-randomness used here generalizes the one introduced by Martin-L"of in 1966 in a way now standard in algorithmic randomness. Examples come from probability theory, analysis, dynamical systems/ergodic theory, statistical mechanics, and quantum mechanics (especially hidden variable theories). An underlying philosophical theme, inherited from von Mises and Kolmogorov, is the interplay between probability and randomness, especially: which comes first?