Who Gave you the Cauchy-Weierstrass Tale? The Dual History of Rigorous Calculus

TL;DR

This paper reevaluates Cauchy's contributions to analysis by exploring a dual historical perspective, emphasizing the infinitesimal approach and contrasting it with the later epsilon-delta formalism, revealing overlooked foundational insights.

Contribution

It introduces a dual historical analysis of Cauchy's work, highlighting the significance of infinitesimal methods alongside the traditional epsilon-delta perspective.

Findings

Cauchy's work aligns with infinitesimal-enriched continuum concepts.

Reinterpretation of Cauchy's foundational stance from a dual perspective.

Identification of early ideas related to growth rates and uniform convergence.

Abstract

Cauchy's contribution to the foundations of analysis is often viewed through the lens of developments that occurred some decades later, namely the formalisation of analysis on the basis of the epsilon-delta doctrine in the context of an Archimedean continuum. What does one see if one refrains from viewing Cauchy as if he had read Weierstrass already? One sees, with Felix Klein, a parallel thread for the development of analysis, in the context of an infinitesimal-enriched continuum. One sees, with Emile Borel, the seeds of the theory of rates of growth of functions as developed by Paul du Bois-Reymond. One sees, with E. G. Bjorling, an infinitesimal definition of the criterion of uniform convergence. Cauchy's foundational stance is hereby reconsidered.