A philosophical history of infinitesimals

TL;DR

This paper examines Leibnizian infinitesimals through modern set theory, proposing a new formal framework that challenges traditional philosophical views and incorporates nonstandard analysis concepts.

Contribution

It introduces a novel formalization of Leibnizian infinitesimals using ringinals and nonstandard analysis, avoiding ultrafilters and the axiom of choice.

Findings

Infinitesimals can be formalized without ultrafilters.

Analysis with unlimited numbers is possible in conservative set-theoretic extensions.

A recent theory formalizes Leibnizian principles while challenging traditional views.

Abstract

We explore the issue of providing a foundational framework for Leibnizian infinitesimals in the light of modern standard and nonstandard approaches. We outline a trichotomy of ordinals, cardinals and ringinals as a historiographic tool. A ringinal is a concept of infinite number, arithmetic in nature, different from Cantor's transfinite ordinals and cardinals. The continuum is not necessarily identifiable with R; even if one seeks such an identification, infinitesimals are not ruled out. Analysis with unlimited numbers (via the predicate standard) is possible in a conservative extension of Zermelo-Fraenkel set theory and in this sense is epistemologically 'safe'. We sketch a recent theory of infinitesimal analysis that formalizes Leibnizian definitions and heuristic principles while eschewing both the axiom of choice and ultrafilters, thus challenging received philosophical views on the nature of infinitesimals.