A Note on Two Indeterminacies: Free IMTL and Empirical Enquiry

L.E.J. Brouwer's intuitionistic mathematics introduces an element of indeterminacy into mathematics; an indeterminacy stemming from the role of infinity in mathematics and reflected in the rejection of the law of excluded middle by formal intuitionistic logic. There has been recent interest in applying some of the insights of intuitionistic mathematics and logic in order to inject an element of indeterminacy into physical considerations. In this note, I point out that in fact this emerging physical application involves two distinct notions of indeterminacy that interact delicately. In what follows I will set out this distinction, provide the outlines of a formal framework in which we can represent the separate notions, and show how they can interact consistently. The framework, in brief puts a metric temporal logic (a logic with indexed temporal operators) on the base of a free intuitionistic logic. That is a logic whose propositional fragment is intuitionistic and whose quantificational portion is free. Thus the title: Free Intuitionistic Metric Temporal Logic (Free IMTL). I speak of this technical combination as a "framework" for two reasons. First, I mean the machinery to be sufficiently flexible to accommodate different choices for the temporal axioms. Indeed I'll mention a number of options. Secondly, I leave the work of providing an adequate semantics for another occasion. I will also show how this combination elucidates a pair of distinct philosophical approaches to temporal indeterminacy. I call them the "Aristotelian approach" and the "broadly Kantian approach." ("Broadly" Kantian, because this approach itself has two versions: the strictly Kantian view and the Brouwerian view.)