Randomness from initial conditions: The challenge of incompleteness

Abstract

Certain prima facie deterministic theories, notably Bohmian mechanics and kinetic theory, require ``random'' initial conditions in order to achieve their goals. I analyze recent proposals that this idea should be implemented using algorithmic randomness, seen as a mathematical sharpening of the typicality arguments originally used to understand these theories. On the one hand my analysis includes a critique of these arguments and hence a defense of this sharpening. On the other hand, I argue that Chaitin's incompleteness theorems for random sequences make it impossible for a dynamical theory to either process given random initial conditions beyond some finite (but not arbitrary) number of their digits, or else to state such conditions with arbitrary precision without compromising their randomness (these are two sides of the same coin). This challenges the ability of such theories to make any other predictions than those that hold typically in the usual sense of the word (like macroscopic statements), and moves their alleged determinism high up the metaphysical ladder.