What Numbers Really Cannot Be and What They Plausibly Are

Abstract We show that structuralism has the very serious defect of having no satisfactory notion of identity which can be associated with its central notion: structure. We also refute the structural thesis about the nature of the natural numbers by showing that there are at least two completely different structures that are entitled to be taken as ‘the structure of the natural numbers’, and any choice between them would arbitrarily favor one of them over the equally legitimate other. Finally, we argue for the high plausibility of the identification of the natural numbers with the finite von Neumann ordinals.