Contemporary Infinitesimalist Theories of Continua and their late 19th- and early 20th-century forerunners

TL;DR

This paper reviews modern infinitesimalist theories of continua, such as nonstandard analysis and surreal numbers, tracing their historical roots to late 19th and early 20th-century approaches.

Contribution

It provides a comprehensive historical overview linking contemporary infinitesimalist theories to their late 19th and early 20th-century forerunners.

Findings

Modern theories emerge from historical algebraic, geometric, and analytic infinitesimalist approaches.

Connections between contemporary and historical theories are elucidated.

The paper highlights the evolution of infinitesimalist ideas over time.

Abstract

The purpose of this paper is to provide a historical overview of some of the contemporary infinitesimalist alternatives to the Cantor-Dedekind theory of continua. Among the theories we will consider are those that emerge from nonstandard analysis, nilpotent infinitesimalist approaches to portions of differential geometry and the theory of surreal numbers. Since these theories have roots in the algebraic, geometric and analytic infinitesimalist theories of the late nineteenth and early twentieth centuries, we will also provide overviews of the latter theories and some of their relations to the contemporary ones.