Zeno's paradoxes and the small-scale structure of space and time

Zeno's paradoxes can be resolved either on the assumption that spatial and temporal intervals are infinitely divisible - the approach of present day science - or on the assumption that they are ultimately only finitely divisible - the sort of approach advocated by William James, A.N. Whitehead, Paul Weiss, and others. However, both of these general approaches to the paradoxes do require certain sacrifices in terms of intuitive plausibility. Whether- we resolve Zeno's paradoxes on the assumption that intervals of space and time are infinitely divisible, or on the assumption that they are only finitely divisible, our intuitive, pre-theoretical ideas about what is possible will be offended. This is the sign of a true paradox. The modern mathematical resolution of Zeno's paradoxes is based on the assumption that spatial and temporal intervals are both infinitely divisible and continuous in the mathematical sense. In chapters I to VIII a detailed examination of the modern mathematical resolution of the paradoxes is undertaken. Particular attention is given to Russell and Grunbaum's defences of this approach to the paradoxes because they are the most detailed and sophisticated. The counterintuitive implications of the mathematical resolution of Zeno's paradoxes are drawn out, and an attempt is.made to show that these implications are indeed only counterintuitive, and not paradoxical. In Chapter II Russell and Grunbaum's respective psychological explanations of why we find Zeno's paradoxes persuasive are examined, and are shown to involve certain difficulties. The present day concern with Zeno's paradoxes has given rise to an extensive literature on infinity machines and super-tasks. In Chapter V attention is given to the sort of restrictions which need to be placed on these machines in order to make them kinematically feasible. Chapter IX involves consideration of the finitary approach to resolving the paradoxes - the sort of approach advocated by James, Whitehead, and Weiss. Again, the counterintuitive implications of the idea that intervals of space and time might be only finitely divisible are drawn out, and an attempt is made to show that these implications are only counterintuitive, and do not warrant the rejection of this approach to resolving the paradoxes. Among other things, important problems associated with the nature of geometry in a discrete spacetime are examined in this chapter. If Zeno's paradoxes can be resolved successfully either on the assumption that intervals of space and time are infinitely divisible or on the assumption that they are only finitely divisible, and if both of these approaches lead to counterintuitive conclusions, how is one of these approaches to be singled out as superior or preferable to the other? In Chapter X an attempt is made to answer this question. The modern mathematical resolution of the paradoxes is ultimately favoured, but only after considerable argument. An attempt is made to show why, to what extent, and in exactly what sense there are reasons for preferring the contemporary mathematical resolution of Zeno's paradoxes to its finitary alternatives