Instants and Instantaneous Velocity

1 The Problem of Instantaneous Rates of Change The concept of the instantaneous rate of change of some physical quantity, position most strikingly, poses a significant philosophical dilemma. Notoriously, Zeno argued, in his arrow paradox, that the very concept of an instantaneous rate of change in position is both unavoidable in the explanation of motion and incoherent. Recently worries about instantaneous rates of change have received an additional boost from concerns about the definition and reality of instantaneous velocity in classical physics(cf. Carroll, 2002; Meyer, 2003; Smith, 2003). The clearest recent treatment of the range of possible solutions to the problems posed by instantaneous velocity is by Frank Arntzenius(2000). Although Arntzenius argues that none of the traditional treatments of motion really serve to resolve the tension between Zeno’s argument and the unavoidable role of instantaneous velocity in classical physics, Marc Lange(2005) has argued that we can resolve this dilemma with a dispositional analysis of instantaneous velocity. This paper will argue that the puzzles about instantaneous velocity, and rates of change more generally, are the result of a failure to recognize an ambiguity in the concept of an instant, and therefore of an instantaneous state. We will conclude that there are two distinct conceptions of a temporal instant: (i) instants conceived as fundamentally distinct zero-duration temporal atoms and (ii) instants conceived as the boundary of, or between, temporally extended durations. Since the concept of classical instantaneous velocity is welldefined only on the second conception of instants, we will conclude that this distinction allows us to avoid the above dilemma. If instantaneous velocity is well-defined then the states of a system at various instants are not logically distinct and thus we cannot generate Zeno’s paradox. However, if we assume that the instants are metaphysically distinct, then instantaneous velocity is not well-defined and thus the second horn of the dilemma about the causal-explanatory role of instantaneous velocity cannot be generated.