Canonical Ensembles, Evolution of Competing Species, and the Arrow of Time

Abstract

At the outset, it may prove helpful to clarify the respective roles of mathematics and philosophy in the study of evolutionary systems. Historically, the two subjects have not always been considered as distinct. (Indeed, when I was a student in Cambridge, applied mathematics was called “natural philosophy”, while philosophy was called “moral science”.) In order to evolve, systems need to be somewhat complex. This complexity forces a certain subtlety on the concepts used to analyze the systems — concepts like “entropy”, “hierarchy”, “fitness” and “species”. Such concepts can easily lead to paradox and confusion if they are not defined and used with considerable care. Mathematics and philosophy share the task of ensuring that the concepts are properly formulated, that they are clearly understood. On the other hand, the distinction between the two subjects may be summarized by the observation that philosophy works with natural language, with words, whereas mathematics works objectively, with symbols. Sciences such as physics, chemistry, biology, economics and sociology begin with verbal formulations, and then move progressively towards symbolic codifications. As this move takes place, mathematics inherits from philosophy the job of validating the science’s concepts and arguments (while philosophy may then assume the job of interpreting the increasingly refined science in “layman’s terms”, in natural language). One special benefit that mathematics has to offer is its universality: a single mathematical model may be capable of describing a wide range of phenomena by means of varying interpretations of the symbols used. The current paper presents a striking example The canonical ensemble model, originally developed in the context of equilibrium statistical mechanics to describe a system that has cooled to an ultimate state of maximal entropy, may be reinterpreted to describe the evolution of a non-equilibrium system away from a state of maximal entropy and towards an ultimate state of certainty. Examples like this are salutary reminders of the need for the precision and clarity that mathematics and philosophy can bring.