Abstract

This chapter develops a metaphysics-first view of natural numbers and real numbers. The account gives a philosophical priority to applications: to the application of natural numbers as numbering property-instances, and to the application of real numbers as ratios of extensive magnitudes. Each natural number is individuated by the condition for it to be the number of a property. The account is contrasted with the neo-Fregean approach to natural numbers advocated by Wright; but it does have a natural marriage with the postulationist approach of Fine. This metaphysics of numbers can then be deployed in combination with the principle that, for these ontologies, Individuation Precedes Representation. To be capable of representing numbers of these kinds is to have tacit knowledge of the principles that individuate them. The resulting account has both differences from and affinities with the views of Carnap.