A philosophical history of infinitesimals
Fuente:
arXiv
Gespeichert in:
| Hauptverfasser: | , , , |
|---|---|
| Format: | Preprint |
| Veröffentlicht: |
2026
|
| Schlagworte: | |
| Online-Zugang: | |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| _version_ | 1866910215576748032 |
|---|---|
| author | Kanovei, Vladimir Katz, Mikhail G. Kudryk, Taras Kuhlemann, Karl |
| author_facet | Kanovei, Vladimir Katz, Mikhail G. Kudryk, Taras Kuhlemann, Karl |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_13102 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A philosophical history of infinitesimals Kanovei, Vladimir Katz, Mikhail G. Kudryk, Taras Kuhlemann, Karl History and Overview 01A45, 01A61, 01A85, 01A90, 26E35 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. |
| title | A philosophical history of infinitesimals |
| topic | History and Overview 01A45, 01A61, 01A85, 01A90, 26E35 |
| url | https://arxiv.org/abs/2605.13102 |