Is mathematical history written by the victors?

TL;DR

This paper reevaluates the history of infinitesimal mathematics by analyzing historical procedures through modern infinitesimal theories, highlighting overlooked contributions and clarifying their formalization.

Contribution

It offers a new perspective on historical infinitesimals by connecting their procedures to modern frameworks like nonstandard analysis, challenging traditional narratives.

Findings

Historical infinitesimal procedures align with modern nonstandard analysis concepts.

Many procedures were only formalized with modern infinitesimals.

The paper clarifies the consistency issues of historical infinitesimals.

Abstract

We examine prevailing philosophical and historical views about the origin of infinitesimal mathematics in light of modern infinitesimal theories, and show the works of Fermat, Leibniz, Euler, Cauchy and other giants of infinitesimal mathematics in a new light. We also detail several procedures of the historical infinitesimal calculus that were only clarified and formalized with the advent of modern infinitesimals. These procedures include Fermat's adequality; Leibniz's law of continuity and the transcendental law of homogeneity; Euler's principle of cancellation and infinite integers with the associated infinite products; Cauchy's infinitesimal-based definition of continuity and "Dirac" delta function. Such procedures were interpreted and formalized in Robinson's framework in terms of concepts like microcontinuity (S-continuity), the standard part principle, the transfer principle, and hyperfinite products. We evaluate the critiques of historical and modern infinitesimals by their foes from Berkeley and Cantor to Bishop and Connes. We analyze the issue of the consistency, as distinct from the issue of the rigor, of historical infinitesimals, and contrast the methodologies of Leibniz and Nieuwentijt in this connection.