Abstract

The problem of continuity in Zeno's paradoxes can be described as one of incompatibility between accounts emphasizing the phenomenal or the mathematical aspects of continuity, and between these accounts and continuity as we experience it. This incompatibility is first manifested in Zeno's paradoxes on motion, or more properly, his reductio ad absurdum arguments. It arises when we abstract different facets of experienced continuity to create different sorts of accounts . ;An account, being an analysis or abstraction, is a move from an experienced continuum to a conceptual one, and must leave out some aspects of continuity. It either emphasizes, or adds, some of these aspects which may not be emphasized, or even present, in some other account. Attempts to reconcile these accounts without untangling them, or attempts to map them back onto the experiential ground from which they sprang, have generated paradoxes. ;I interpret Zeno's paradoxes as an analysis of continuity deliberately conflating the two types of accounts, with threads of the two accounts intertwined. I will examine the four paradoxes on motion, with their incompatible phenomenal and mathematical elements. I will also lay out the reductio arguments involved here. ;Aristotle provides the prime example of a phenomenal continuum. His analysis involves a finite number of subcontinua which are given one next in succession to the other. It rejects the idea of actually completed infinite division, in favor of potentially infinite divisibility. ;Our example of a mathematical continuum is taken from Cantor. It is based on the notion of an ordered set. The concept of order is abstracted from succession, yet Cantor's ordered set abandons the experiential quality of succession, in favor of mathematical density. His work is as influential for modern mathematics as Aristotle's work was for philosophy: the current mathematical continuum is Cantorian, although this is changing, due to the influence of Abraham Robinson's nonstandard analysis. ;With these two accounts of continuity at hand, I put forward a structured analysis of accounts: phenomenal, mathematical, and otherwise. I use the notion of formalisation set up in this analysis to explain the incompatibilities between the Aristotelian and Cantorean accounts, and the relation between those accounts and experienced continuity. I then apply my notion of formalisation to Zeno's paradoxes, in order to pick out the different threads. ;Even though my account implies that some difference must remain between a continuous phenomenon and any account of continuity, I suggest that we may develop an account which is closer to the phenomenon than either of the above. This new account is based on the mathematical one, and relies on nonArchimedian infinitesimals, as they are presented in the theory of nonstandard analysis, as it has been developed by Robinson