Leibniz's Principles and Topological Extensions

TL;DR

This paper formalizes Leibniz's philosophical principles within topological extensions, revealing that they can be simultaneously satisfied only in pairs, not all three at once.

Contribution

It provides a precise topological framework linking Leibniz's principles to separation, compactness, and analyticity, and demonstrates their mutual compatibility limitations.

Findings

Leibniz's principles correspond to topological properties

Only two of the three principles can be fulfilled simultaneously

The framework unifies philosophical ideas with topological structures

Abstract

Three philosophical principles are often quoted in connection with Leibniz: "objects sharing the same properties are the same object", "everything can possibly exist, unless it yields contradiction", "the ideal elements correctly determine the real things". Here we give a precise formulation of these principles within the framework of the Topological Extensions of [8], structures that generalize at once compactifications, completions, and nonstandard extensions. In this topological context, the above Leibniz's principles appear as a property of separation, a property of compactness, and a property of analyticity, respectively. Abiding by this interpretation, we obtain the somehow surprising conclusion that these Leibnz's principles can be fulfilled in pairs, but not all three together.