On the Problem of Defining the Present in Special Relativity: A Challenge for Tense Logic

Abstract

It is not easy to make sense of our intuition of an ontological significance of the present in the framework of special relativity. Many philosophers have commented on the issue. Einstein famously expressed a strong feeling of that problem to Carnap, and while Carnap tried to dismiss Einstein's worries (Carnap 1963), many, e.g., Prior (1968), have felt that the problem runs deeper. In the 1990ies, a number of definability and indefinability results settled much of the technical side of the matter. If one takes Minkowski space-time as the formal framework of Robb's axiomatisation or related schemes and requires a relation of ontological presentness (event e is in the present of f) to be an equivalence relation, it turns out that (i) one cannot define a non-trivial relation of presentness, but (ii) one can add such an equivalence relation as a conservative extension of the formalism (Rakić 1997). Thus, there are no technical obstacles that would ban a relation of presentness, but introducing such a relation seems ad hoc, as it amounts to introducing a preferred frame of reference. Now cosmology may or may not provide the resources to single out such a frame—for special relativity it would be nice to be able to give an independent, “stand-alone” definition of presentness by adding structure that is less controversial than a preferred reference frame. In my talk I will propose a radically new definition of “presentness” in a branching version of Minkowski space- time. Put bluntly, I will argue that the problem of defining the present is an artifact of a deterministic prejudice. The punch lines will be that (i) within the confines of a single Minkowski space-time, the notion of presentness does not make much sense, as there is no clear notion of change, (ii) there is a formally rigorous framework for incorporating indeterministic events to yield multiple, branching Minkowski space-times, and (iii) within that formal framework it is possible to give an intuitively satisfactory definition of an ontological notion of presentness. My formalism is based on Belnasp's branching space-times (1992), incorporating some of the ideas behind his new theory of causation (2005).