Symbol Emergence and Symbolic Mediation in a Background-Independent Relational Ontology: A Preliminary Discussion

This paper constructs a vocabulary for symbols and symbolic mediation on a combinatorial substrate of the kind that background-independent approaches to quantum gravity, group field theory among them, have developed as models of the emergence of a discrete spacetime, in which no arena is given in advance and every structure, extent and order included, is carried by the relations themselves, and applies it to the elementary structure of production. The substrate is taken as a formal template and no physical claim is made of the objects described. On this substrate, time is generated by the system's own records, and the division into system and environment is a choice the describer makes and can vary; a subsystem is a region of a combinatorial complex, and its dynamics is given by an admissibility relation on configurations. Within that setting the paper defines discrimination by equality of induced admissible continuations over a class of contexts, and defines a symbol as a persistent product that substitutes for a discrimination class in the dynamics. A three-step ladder distinguishes traces, products and symbols, and separates a product that participates in a subsystem's continuations from one that stands in for something else. The reverse pass through which a persistent structure conditions the activity that generated it is defined as selection among admissible continuations, and two derived relations are introduced, the set of subsystems a structure mediates and the set of subsystems able to revise it, with definitions that use no magnitude. The paper's contributions are: a repaired discrimination relation that is an equivalence and is indexed to the subsystem's current configuration; a substitution relation with an explicit interaction class and an explicit residue; a state-sufficiency criterion for the decomposition of a subsystem into relational and symbolic sectors; a result that a coupling which has ceased can leave a persistent structure standing where it stood, so that parties who were never party to it are mediated by it without acquiring any means of revising it; and a mapping of the elementary categories of production onto these objects, with the conditions of failure and the open problems stated.