A topological interpretation of three Leibnizian principles within the functional extensions

Three philosophical principles are often quoted in connection with Leibniz: “objects sharing the same properties are the same object” (Identity of indiscernibles), “everything can possibly exist, unless it yields contradiction” (Possibility as consistency), and “the ideal elements correctly determine the real things” (Transfer). Here we give a precise logico-mathematical formulation of these principles within the framework of the Functional Extensions, mathematical structures that generalize at once compactifications, completions, and elementary extensions of models. In this context, the above Leibnizian principles appear as topological or algebraic properties, namely: a property of separation, a property of compactness, and a property of directeness, respectively. Abiding by this interpretation, we obtain the somehow surprising conclusion that these Leibnizian principles may be fulfilled in pairs, but not all three together.