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