## Arithmetic and Ontology: A Non-realist Philosophy of ArithmeticThis 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." |

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### Contents

35 | |

Notes to Grundlagen | 45 |

Objectivism and Realism in Freges Philosophy | 73 |

The Peano Axioms | 105 |

Existence Number and Realism | 129 |

Arithmetic and Necessity | 159 |

Arithmetic and Rules | 183 |

Thesis One | 215 |

Thesis Three | 254 |

285 | |

Colin Cheyne Numbers Reference and Abstraction | 291 |

Sanford Shieh What Is NonRealism about Arithmetic? | 317 |

Jean Paul Van Bendegem NonRealism Nominalism and Strict | 343 |

Philip Hugly and Charles Sayward Replies to Commentaries | 369 |

About the Contributors | 387 |

Thesis Two | 241 |

### Common terms and phrases

abstract objects actually analytic answer apples argument arithmetical sentences arithmetical signs assertion Bill claim color concept counting definition derivation discourse entity epistemically equations example expressions fiction finite formula Frege function given Grundlagen Hugly and Sayward Hugly's and Sayward's idea identity inference instances involving kind knowledge language logic marbles meaning model theory names natural numbers necessary existence non-realism non-realist non-referential notion number theory number words numbers exist numerical terms occur ontological commitments particular quantification Peano arithmetic Peano axioms perceptual person philosophers philosophy of arithmetic philosophy of mathematics phrases Platonism possible predicate prime is followed proof propositions provable pure arithmetic purports to denote question Quine realist rules seems semantics sense sentences of pure singular reference singular terms someone statements of number substituends substitutional quantification Suppose tautologies theorems thesis things thought tomatoes truth truth-condition truth-value universal quantification variables Wittgenstein zero