A Metric for the Entropic Purpose of a System

Purpose in systems is considered to be beyond the purview of science since it is thought to be intrinsically personal. However, just as Claude Shannon was able to define an impersonal measure of information, so we formally define the (impersonal) 'entropic purpose' of an information system (using the theoretical apparatus of Quantitative Geometrical Thermodynamics) as the line integral of an entropic "purposive" Lagrangian defined in hyperbolic space across the complex temporal plane. We verify that this Lagrangian is well-formed: it has the appropriate variational (Euler-Lagrange) behaviour. We also discuss the teleological characteristics of such variational behaviour (featuring both thermodynamically reversible and irreversible temporal measures), so that a "Principle of Least (entropic) Purpose" can be adduced for any information-producing system. We show that entropic purpose is (approximately) identified with the information created by the system: an empirically measurable quantity. Exploiting the relationship between the entropy production of a system and its energy Hamiltonian, we also show how Landauer's principle also applies to the creation of information; any purposive system that creates information will also dissipate energy. Finally, we discuss how 'entropic purpose' might be applied in artificial intelligence contexts (where degrees of system 'aliveness' need to be assessed), and in cybersecurity (where this metric for 'entropic purpose' might be exploited to help distinguish between people and bots).