Mathematical Pluralism: The Case of Smooth Infinitesimal Analysis

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.