This volume documents a lively exchange between five philosophers of mathematics.  It also introduces a new voice in one central debate in the philosophy of mathematics.  Non-realism, i.e., the view supported by Hugly and Sayward in their monograph, is an original position distinct from the widely known realism and anti-realism.  Non-realism is characterized by the rejection of a central assumption shared by many realists and anti-realists, i.e., the assumption that mathematical statements purport to refer to objects.  The defense of their main argument for the thesis that arithmetic lacks ontology brings the authors to discuss also the controversial contrast between pure and empirical arithmetical discourse.  Colin Cheyne, Sanford Shieh, and Jean Paul Van Bendegem, each coming from a different perspective, test the genuine originality of non-realism and raise objections to it.  Novel interpretations of well-known arguments, e.g., the indispensability argument, and historical views, e.g. Frege, are interwoven with the development of the authors’ account.  The discussion of the often neglected views of Wittgenstein and Prior provide an interesting and much needed contribution to the current debate in the philosophy of mathematics.    Contents  Acknowledgments  Editor’s Introduction   Philip HUGLY and Charles SAYWARD: Arithmetic and Ontology a Non-Realist Philosophy of Arithmetic  Preface  Analytical Table of Contents  Chapter 1. Introduction  Part One: Beginning with Frege  Chapter 2. Notes to Grundlagen  Chapter 3. Objectivism and Realism in Frege’s Philosophy of Arithmetic  Part Two: Arithmetic and Non-Realism  Chapter 4. The Peano Axioms  Chapter 5. Existence, Number, and Realism   Part Three: Necessity and Rules  Chapter 6. Arithmetic and Necessity   Chapter 7. Arithmetic and Rules  Part Four: The Three Theses  Chapter 8. Thesis One   Chapter 9. Thesis Two   Chapter 10. Thesis Three  References   Commentaries  Colin Cheyne, Numbers, Reference, and Abstraction   Sanford Shieh, What Is Non-Realism about Arithmetic?   Jean Paul Van Bendegem, Non-Realism, Nominalism and Strict Fi-nitism. The Sheer Complexity of It All     Replies to Commentaries  Philip Hugly and Charles Sayward, Replies to Commentaries   About the Contributors   Index

Arithmetic and Ontology: A Non-realist Philosophy of Arithmetic (Poznan Studies in the Philosophy of the Sciences & the Humanities) PDF

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