# Cauchy's infinitesimals, his sum theorem, and foundational paradigms

## TL;DR

This paper reinterprets Cauchy's sum theorem, analyzing his original proof and contrasting it with modern frameworks, highlighting epistemological differences and the role of infinitesimals in foundational analysis.

## Contribution

It offers a new perspective on Cauchy's proof by aligning it with a different modern interpretive paradigm rather than the traditional Weierstrassian framework.

## Key findings

- Cauchy's proof aligns more closely with a non-Weierstrassian framework.
- The analysis clarifies the role of infinitesimals in Cauchy's original reasoning.
- Epistemological discussion on interpretive paradigms in mathematical analysis.

## Abstract

Cauchy's sum theorem is a prototype of what is today a basic result on the convergence of a series of functions in undergraduate analysis. We seek to interpret Cauchy's proof, and discuss the related epistemological questions involved in comparing distinct interpretive paradigms. Cauchy's proof is often interpreted in the modern framework of a Weierstrassian paradigm. We analyze Cauchy's proof closely and show that it finds closer proxies in a different modern framework. Keywords: Cauchy's infinitesimal; sum theorem; quantifier alternation; uniform convergence; foundational paradigms.