The Reason's Proper Study

Abstract

This volume collects together 15 papers in which Hale and Wright develop their neo‐Fregean approach to the philosophy of mathematics. Two key aspects of Frege's philosophy are drawn out and defended: (1) the doctrine that mathematics is a body of knowledge about independently existing objects––Frege's platonism (2) the doctrine that this knowledge is a species of logical knowledge, broadly construed––Frege's logicism. In Wright and Hale's programme, a kind of contextual explanation––an Abstraction Principle––suffices to ground the fundamental concepts of a mathematical domain. A key result has become known as Frege's theorem: the basic axioms of arithmetic can be derived from a single ’abstraction principle’. Though Frege himself considers and rejects this principle as a foundation for arithmetic, Neo‐Fregeans take a more optimistic view. In the papers included in this volume, Wright and Hale combine elements drawn from Frege's own work with contemporary thinking to develop their distinctive approach to the philosophy of mathematics. The ground covered includes articles exploring the metaphysics, epistemology, and philosophy of language that forms the backdrop to the neo‐Fregean project; responses to critics of their programme; detailed exploration and defence of the case for neo‐Fregean logicism about arithmetic; and proposals for extending the neo‐Fregean programme to real analysis and set theory. A substantial introduction gives the framework for the neo‐Fregean programme and outlines how the papers fit together, while a postscript outlines a series of challenges for further research.