Mathematical approaches to the study of agents

The definition of life remains one of science's most profound challenges, with contemporary approaches usually focusing on two research programmes: Darwinian evolution and self-maintenance in chemical systems. While evolution has been successfully abstracted and mathematically modelled, the concept of a self-sustaining system has so far resisted a comparable level of formalisation. This paper tackles this challenge by reframing the concept of self-sustaining system within a more abstract framework to study `agents': goal-directed systems acting in an environment. We build on an existing conceptual framework comprising three requirements for agents: individuality, normativity (or goal-directedness), and interactional asymmetry. We then provide a systematic analysis, under a unified notation, of several mathematical approaches aiming to formalise these requirements, including the free energy principle, integrated information theory and dynamical systems. Unlike this conceptual framework, which commits to an intrinsic perspective on agency, we focus on a less ontologically committed `as-if' stance. Using this, we discuss links between identity and normativity, and a way to understand actions as if they were produced by causal interventions. Taken together, our systematic analysis clarifies the limitations of current proposals and reveals how they can work synergistically within a unified, mathematical account of agency across natural and artificial domains.