On the Duality of Diachrony and Synchrony: A Mathematical Theory of Identity

摘要

In this paper we provide an analysis of the relationship between diachrony and synchrony, which entails a dynamic extensional mereological theory of identity. By 'extensional mereological' we mean that it is assumed that an individual, at any given time, is the sum of the properties and relations which he or she instantiates; by 'dynamic' we mean that change is incorporated as a fundamental part of the theory of identity which is proposed. In what follows, the theory of hypergraphs, particularly the theory of transverse hypergraphs, is used to prove that the relationship between diachrony and synchrony is one of 'weak duality'. What is meant by 'duality' here (strong duality), stems from the duality between the logical operators 'or' and 'and' of propositional logic. Furthermore, by invoking a quasi-Leibnizian identity principle, viz., that no two distinct properties of the same kind are possessed by an individual at any specific time, an intermediate duality between diachronic and synchronic perspectives is represented. It is also proved that there is a subclass of individuals whose diachronic and synchronic perspectives are strongly dual. Then, by exploiting the harmonic theory of hypergraphs, it is shown to how to devise well-defined measures of inter-and intra-personal resemblance. Lastly, using the colouring theory of hypergraphs, we consider an application to gender identity. It is proved that given a certain richness in our taxonomy of properties, if the collection of an individual's behaviours in his or her respective social settings is sufficiently cohesive, and there is an appropriate relation of duality between his or her synchronic and diachronic representations, then it is possible that some social role in which he or she engages will witness him or her behaving in both of two differently gendered ways-at the same time, no less.