arXiv:1808.03345math.HO
Contemporary Infinitesimalist Theories of Continua and their late 19th- and early 20th-century forerunners
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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.
It isn't true for all ordered rings, and the dual numbers are in fact a counterexample to the claim that it is true. Beyond that I'm not sure what to tell you, other than it's fairly easy to see that the dual numbers do satisfy the axioms of an ordered ring that I gave. Here's a…
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