Commuting and noncommuting infinitesimals

TL;DR

This paper reviews the historical development and conceptual differences between commutative hyperreal infinitesimals and noncommutative infinitesimals in geometry, highlighting their roles in calculus and analysis.

Contribution

It provides a comparative analysis of hyperreal and noncommutative infinitesimals, clarifying their mathematical foundations and historical evolution.

Findings

Hyperreal infinitesimals originated in the 1940s with Hewitt.

Noncommutative infinitesimals have been used since the 1990s in geometry.

The paper contrasts commutative and noncommutative approaches to infinitesimals.

Abstract

Infinitesimals are natural products of the human imagination. Their history goes back to the Greek antiquity. Their role in the calculus and analysis has seen dramatic ups and downs. They have stimulated strong opinions and even vitriol. Edwin Hewitt developed hyperreal fields in the 1940s. Abraham Robinson's infinitesimals date from the 1960s. A noncommutative version of infinitesimals, due to Alain Connes, has been in use since the 1990s. We review some of the hyperreal concepts, and compare them with some of the concepts underlying noncommutative geometry.