Where Infinitesimals Come From ..

TL;DR

This paper explores the foundational algebraic structures, like semigroups and magmas, from which infinitesimals originate, highlighting their potential significance in physics and their connection to non-Archimedean properties.

Contribution

It clarifies the basic algebraic structures that give rise to infinitesimals, emphasizing their emergence in non-Archimedean linearly ordered monoids.

Findings

Infinitesimals originate from semigroups and magmas.

Presence of infinitesimals is linked to non-Archimedean properties.

The results have implications for physics and mathematical structures.

Abstract

The presence of infinitesimals is traced back to some of the most general algebraic structures, namely, semigroups, and in fact, magmas, [1], in which none of the structures of linear order, field, or the Archimedean property need to be present. Such a clarification of the basic structures from where infinitesimals can in fact emerge may prove to have a special importance in Physics, as seen in [4-16]. The relevance of the deeper and simpler roots of infinitesimals, as they are given in Definitions 3.1 and 3.2, is shown by the close connection in Theorem 4.1 and Corollary 4.1 between the presence of infinitesimals and the non-Archimedean property, in the particular case of linearly ordered monoids, a case which, however, has a wide applicative interest.