Controversies in the foundations of analysis: Comments on Schubring's Conflicts

TL;DR

This paper defends a historical analysis of infinitesimals and Cauchy's role, addressing criticisms and clarifying misconceptions about the development and interpretation of infinitesimal calculus.

Contribution

It clarifies the historical role of infinitesimals in Cauchy's work and corrects misinterpretations of key figures and texts in the foundations of analysis.

Findings

Cauchy's use of infinitesimals is part of a long tradition.

Schubring misinterprets historical figures on non-Archimedean issues.

The paper clarifies the context of Moigno's critique and Cauchy's scholarship.

Abstract

Foundations of Science recently published a rebuttal to a portion of our essay it published two years ago. The author, G. Schubring, argues that our 2013 text treated unfairly his 2005 book, Conflicts between generalization, rigor, and intuition. He further argues that our attempt to show that Cauchy is part of a long infinitesimalist tradition confuses text with context and thereby misunderstands the significance of Cauchy's use of infinitesimals. Here we defend our original analysis of variousmisconceptions and misinterpretations concerning the history of infinitesimals and, in particular, the role of infinitesimals in Cauchy's mathematics. We show that Schubring misinterprets Proclus, Leibniz, and Klein on non-Archimedean issues, ignores the Jesuit context of Moigno's flawed critique of infinitesimals, and misrepresents, to the point of caricature, the pioneering Cauchy scholarship of D. Laugwitz. Keywords: Archimedean axiom, Cauchy, Felix Klein, hornangle, infinitesimal, Leibniz, ontology, procedure