Do Simple Infinitesimal Parts Solve Zeno's Paradox of Measure?

TL;DR

This paper proposes a new infinitesimal atomism view of space using nonstandard analysis to address Zeno's paradox of measure, but it faces significant unresolved issues similar to existing theories.

Contribution

It introduces a novel infinitesimal atomism framework for space based on nonstandard analysis as a response to Zeno's paradox.

Findings

The view satisfies a form of additivity for space regions.

It is more coherent than Skyrms's approach.

It encounters the same problems as standard and finite atomist theories.

Abstract

I develop a new view of the structure of space--called infinitesimal atomism--as a reply to Zeno's paradox of measure. According to this view, space is composed of ultimate parts with infinitesimal size, where infinitesimals are understood within the framework of Robinson's nonstandard analysis. Notably, this view satisfies a version of additivity: for every region that has a size, its size is the sum of the sizes of its disjoint parts. In particular, the size of a finite region is the sum of the sizes of its infinitesimal parts. Although this view is a coherent approach to Zeno's paradox and is preferable to Skyrms's (1983) infinitesimal approach, it faces both the main problem for the standard view (the problem of unmeasurable regions) and the main problem for finite atomism (Weyl's tile argument), leaving it with no clear advantage over these familiar alternatives.