Toward a history of mathematics focused on procedures

TL;DR

This paper uses Abraham Robinson's modern infinitesimal framework to reinterpret historical mathematical procedures, emphasizing their problem-solving techniques over foundational issues, and showing its usefulness in understanding pioneers' methods.

Contribution

It introduces a new approach to analyze historical analysis procedures using modern infinitesimals, bypassing traditional foundational interpretations.

Findings

Robinson's framework helps interpret historical techniques.

Historical procedures align with modern infinitesimal methods.

The approach clarifies the problem-solving focus of early mathematicians.

Abstract

Abraham Robinson's framework for modern infinitesimals was developed half a century ago. It enables a re-evaluation of the procedures of the pioneers of mathematical analysis. Their procedures have been often viewed through the lens of the success of the Weierstrassian foundations. We propose a view without passing through the lens, by means of proxies for such procedures in the modern theory of infinitesimals. The real accomplishments of calculus and analysis had been based primarily on the elaboration of novel techniques for solving problems rather than a quest for ultimate foundations. It may be hopeless to interpret historical foundations in terms of a punctiform continuum, but arguably it is possible to interpret historical techniques and procedures in terms of modern ones. Our proposed formalisations do not mean that Fermat, Gregory, Leibniz, Euler, and Cauchy were pre-Robinsonians, but rather indicate that Robinson's framework is more helpful in understanding their procedures than a Weierstrassian framework.