Leibniz's contested infinitesimals: Further depictions

TL;DR

This paper defends Leibniz's use of genuine infinitesimals as fictional entities, countering recent claims that non-Archimedean continua contradict Leibnizian concepts, and critiques errors in those claims.

Contribution

It clarifies Leibniz's conception of infinitesimals as fictional entities and refutes recent arguments claiming incompatibility with Leibnizian mathematics.

Findings

Leibniz regarded infinitesimals as fictional mathematical entities.

Recent critiques contain errors and misrepresentations.

Leibniz's infinitesimals are consistent with his broader mathematical framework.

Abstract

We contribute to the lively debate in current scholarship on the Leibnizian calculus. In a recent text, Arthur and Rabouin argue that non-Archimedean continua are incompatible with Leibniz's concepts of number, quantity and magnitude. They allege that Leibniz viewed infinitesimals as contradictory, and claim to deduce such a conclusion from an analysis of the Leibnizian definition of quantity. However, their argument is marred by numerous errors, deliberate omissions, and misrepresentations, stemming in a number of cases from flawed analyses in their earlier publications. We defend the thesis, traceable to the classic study by Henk Bos, that Leibniz used genuine infinitesimals, which he viewed as fictional mathematical entities (and not merely shorthand for talk about more ordinary quantities) on par with negatives and imaginaries.