A Philosophical History of Infinitesimals
2026/05/12 by Vladimir Kanovei, Mikhail G. Katz, MIKHAIL KATZ +2 · 1 voice
Arts and Humanities · Mathematics · #Absoluteness #Axiom #Axiom of choice #Calculus (dental) #Historical Philosophy and Science #History and Theory of Mathematics #Infinitesimal #Mathematical and Theoretical Analysis #Predicate (mathematical logic) #Sketch #Transcendental number #Transfinite number #Trichotomy (philosophy) #math.HO
paper · pdf · doi:10.36446/rlf478
openalex publication_date 2026/05/12 · openalex created_date 2026/05/13 · arxiv published 2026/05/13 · arxiv updated 2026/05/13 · openalex updated_date 2026/08/05
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 ; 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.
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