Logicism and the problem of infinity: The number of numbers

Abstract

Simple-type theory is widely regarded as inadequate to capture the metaphysics of mathematics. The problem, however, is not that some kinds of structure cannot be studied within simple-type theory. Even structures that violate simple-types are isomorphic to structures that can be studied in simple-type theory. In disputes over the logicist foundations of mathematics, the central issue concerns the problem that simple-type theory fails to assure an infinity of natural numbers as objects. This paper argues that the problem of infinity is based on a metaphysical prejudice in favor of numbers as objects - a prejudice that mathematics can get along without. © The Author [2011]. Published by Oxford University Press. All rights reserved.