DETERMINISTIC VERSUS INDETERMINISTIC DESCRIPTIONS: NOT THAT DIFFERENT AFTER ALL?

Abstract

The general theme of this paper is the elimination, or replacement, of deterministic descriptions by stochastic ones and of stochastic descriptions being replaced by deterministic ones. In particular, I discuss how far this replacement can be pushed. I tackle these issues for discrete-time measure-theoretic dynamical systems, which widely occur in the sciences, e.g., in meteorology, population dynamics, and generally Newtonian and statistical mechanics. I start by showing that all stochastic descriptions can be replaced by deterministic ones and, conversely, that most deterministic descriptions can be replaced by stochastic ones. I argue that often there are no clear, general principles that call for either the deterministic or stochastic description. Given all this, it might still be hypothesised that the deterministic descriptions needed to replace the stochastic ones are very different from the usual deterministic systems. I provide examples showing that this is not the case. Also, it might be conjectured that the stochastic descriptions needed to replace the deterministic ones at every level of observational accuracy are very different from, and much less random than, the paradigmatic stochastic systems, e.g., Bernoulli or Markov processes. By adapting and extending recent results in ergodic theory, I show that also that conjecture is misguided: (aperiodic and irreducible) Markov processes are the most random stochastic descriptions by which deterministic descriptions can be replaced at every level of accuracy. They model a wide and important class of deterministic systems. All this illustrates that deterministic and indeterministic descriptions are interconvertible in a strong way.