Leibniz vs Ishiguro: Closing a quarter-century of syncategoremania

TL;DR

This paper examines Leibniz's infinitesimals, arguing they are best understood as pure fictions rather than logical fictions, challenging Ishiguro's interpretation and providing textual and historical evidence.

Contribution

It offers a new interpretation of Leibniz's infinitesimals as pure fictions, countering Ishiguro's logical fiction view, supported by textual and historical analysis.

Findings

Leibniz describes infinitesimals as useful fictions.

Textual evidence shows Leibniz's infinitesimals violate the Archimedean property.

Infinitesimals are better understood as pure fictions, not logical fictions.

Abstract

Did Leibniz exploit infinitesimals and infinities `a la rigueur, or only as shorthand for quantified propositions that refer to ordinary Archimedean magnitudes? Chapter 5 in (Ishiguro 1990) is a defense of the latter position, which she reformulates in terms of Russellian logical fictions. Ishiguro does not explain how to reconcile this interpretation with Leibniz's repeated assertions that infinitesimals violate the Archimedean property, viz., Euclid's Elements, V.4. We present textual evidence from Leibniz, as well as historical evidence from the early decades of the calculus, to undermine Ishiguro's interpretation. Leibniz frequently writes that his infinitesimals are useful fictions, and we agree; but we shall show that it is best not to understand them as logical fictions; instead, they are better understood as pure fictions. Keywords: Archimedean property; infinitesimal; logical fiction; pure fiction; quantified paraphrase; law of homogeneity