Leibniz's Principle of the Identity of Indiscernibles: Symmetry and the Relativity of Identity

Abstract

This thesis examines the relationship between Leibniz’s Principle of Identity of Indiscernibles and symmetry. In his 1717 correspondence with Samuel Clarke, Leibniz argued that “There is no such thing as a pair of individuals that are indiscernible from each other” (Leibniz 16). In other words, any objects sharing all their properties are in fact one and the same object. This is Leibniz’s Principle of Identity of Indiscernibles (the “PII”). The principle and its converse Leibniz’s Law express a conditional relationship between the identity of an object and its properties. Our investigation will use applications of Leibniz’s principles from the history of philosophy to examine this relationship and we’ll find that imperfect applications result in either perfect qualitative identity between multiple objects (“multiple indiscernibles”), imperfect qualitative identity between multiple objects (“incongruent counterparts”), or, finally, a relative identity between two facets of one object (a “singular discernible”). My project will also trace the historical thread leading from Leibniz to the development of symmetry groups in mathematics. Leibniz’s principles are embedded in science’s ability to distinguish the objective from the subjective, owing to their usefulness discerning an object’s intrinsic properties (properties belonging to the object itself) from extrinsic properties (properties based in relations the object is in with other objects). Symmetry is the relativity of identity, and the PII is an exploratory instrument illuminating this relationship: it injects structure into investigations of identity, but also affords the opportunity to capture pre-existing convictions about identity a thinker brings to the application.