19th-century real analysis, forward and backward

Jacques Bair; Piotr Blaszczyk; Peter Heinig; Vladimir Kanovei; Mikhail Katz

Antiquitates Mathematicae, Volume 13 (2019), pp. 19-49

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 Archimedean 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 formalization 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.

Article information

DOI: 10.14708/am.v13i1.6440

Classification: 01A55, 01A85
Keywords: butterfly model, continuity, infinitesimals, \emph{limite}, standard part, variable quantity, Cauchy, de Prony