Logicism and the Problem of Infinity: The Number of Numbers

Philosophia Mathematica 19 (2):167-212 (2011)
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
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