Abstract

This paper argues that it is part of the concept of the positive integers that they are for the sake of numbering things (what Benacerraf calls transitive counting). Numbers are necessarily associated with standard, conventionally established counting sets constituted by the Peano axioms; they cannot be specified independently of a paradigm counting stock, any more than lengths can be part of a system of assessment without appeal to some standard object. Scepticism deriving from Kripke and Salmon is countered by acknowledgement that the paradigms themselves are among the means by which the terms of the system get applied and adjudicated