Author: Hellman, Geoffrey

Source: Journal of Philosophical Logic, Volume 35, Number 6, December 2006 , pp. 621-651(31)

Publisher: Springer

Abstract:

A remarkable development in twentieth-century mathematics is smooth infinitesimal analysis (`SIA'), introducing nilsquare and nilpotent infinitesimals, recovering the bulk of scientifically applicable classical analysis (`CA') without resort to the method of limits. Formally, however, unlike Robinsonian `nonstandard analysis', SIA conflicts with CA, deriving, e.g., `not every quantity is either = 0 or not = 0.' Internally, consistency is maintained by using intuitionistic logic (without the law of excluded middle). This paper examines problems of interpretation resulting from this `change of logic', arguing that standard arguments based on `smoothness' requirements are question-begging. Instead, it is suggested that recent philosophical work on the logic of vagueness is relevant, especially in the context of a Hilbertian structuralist view of mathematical axioms (as implicitly defining structures of interest). The relevance of both topos models for SIA and modal-structuralism as appled to this theory is clarified, sustaining this remarkable instance of mathematical pluralism.

Keywords: infinitesimal analysis; synthetic differential geometry; intuitionistic logic; Kock-Lawrence Axiom; nilsquare infinitesimal; pluralism; structuralism; modal-structuralism; toposes; vagueness

Document Type: Research article

DOI: http://dx.doi.org/10.1007/s10992-006-9028-9

Affiliations: 1: Email: hellm001@umn.edu

Publication date: 2006-12-01

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