19th century real analysis, forward and backward

TL;DR

This paper reevaluates 19th century Cauchy's work in real analysis, highlighting his use of infinitesimals and limits, and challenges misconceptions about his approach and the evolution of analysis.

Contribution

It offers a new formalization of Cauchy's procedures using modern infinitesimal analysis and clarifies his role as a pioneer of infinitesimal techniques.

Findings

Cauchy's use of infinitesimals aligns with modern infinitesimal analysis.

The debate at Ecole Polytechnique was about rigor, not infinitesimals.

Cauchy was a pioneer of infinitesimal techniques in analysis.

Abstract

19th century real analysis received a major impetus from Cauchy's work. Cauchy mentions variable quantities, limits, and infinitesimals, but the meaning he attached to these terms is not identical to their modern meaning. Some Cauchy historians work in a conceptual scheme dominated by an assumption of a teleological nature of the evolution of real analysis toward a preordained outcome. Thus, Gilain and Siegmund-Schultze assume that references to limite in Cauchy's work necessarily imply that Cauchy was working with an Archi-medean continuum, whereas infinitesimals were merely a convenient figure of speech, for which Cauchy had in mind a complete justification in terms of Archimedean limits. However, there is another formalisation of Cauchy's procedures exploiting his limite, more consistent with Cauchy's ubiquitous use of infinitesimals, in terms of the standard part principle of modern infinitesimal analysis. We challenge a misconception according to which Cauchy was allegedly forced to teach infinitesimals at the Ecole Polytechnique. We show that the debate there concerned mainly the issue of rigor, a separate one from infinitesimals. A critique of Cauchy's approach by his contemporary de Prony sheds light on the meaning of rigor to Cauchy and his contemporaries. An attentive reading of Cauchy's work challenges received views on Cauchy's role in the history of analysis, and indicates that he was a pioneer of infinitesimal techniques as much as a harbinger of the Epsilontik.