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.