On the Persistence of Particles

This paper is about the metaphysical debate whether objects persist over time by the selfsame object existing at different times (nowadays called `endurance' by metaphysicians), or by different temporal parts, or stages, existing at different times (called ` perdurance'). I aim to illuminate the debate by using some elementary kinematics and real analysis: resources which metaphysicians have, surprisingly, not availed themselves of. There are two main results, which are of interest to both endurantists and perdurantists. (1): I describe a precise formal equivalence between the way that the two metaphysical positions represent the motion of the objects of classical mechanics (both point-particles and continua). (2): I make precise, and prove a result about, the idea that the persistence of objects moving in a void is to be analysed in terms of tracking the continuous curves in spacetime that connect points occupied by matter. The result is entirely elementary: it is a corollary of the Heine-Borel theorem.