## The Oxford Handbook of Philosophy of Mathematics and LogicStewart Shapiro, William J. Wainwright Mathematics and logic have been central topics of concern since the dawn of philosophy. Since logic is the study of correct reasoning, it is a fundamental branch of epistemology and a priority in any philosophical system. Philosophers have focused on mathematics as a case study for general philosophical issues and for its role in overall knowledge- gathering. Today, philosophy of mathematics and logic remain central disciplines in contemporary philosophy, as evidenced by the regular appearance of articles on these topics in the best mainstream philosophical journals; in fact, the last decade has seen an explosion of scholarly work in these areas. This volume covers these disciplines in a comprehensive and accessible manner, giving the reader an overview of the major problems, positions, and battle lines. The 26 contributed chapters are by established experts in the field, and their articles contain both exposition and criticism as well as substantial development of their own positions. The essays, which are substantially self-contained, serve both to introduce the reader to the subject and to engage in it at its frontiers. Certain major positions are represented by two chapters--one supportive and one critical. The Oxford Handbook of Philosophy of Math and Logic is a ground-breaking reference like no other in its field. It is a central resource to those wishing to learn about the philosophy of mathematics and the philosophy of logic, or some aspect thereof, and to those who actively engage in the discipline, from advanced undergraduates to professional philosophers, mathematicians, and historians. |

### Contents

Introduction | 3 |

Philosophy of Mathematics in the Modern Period | 29 |

3 Later Empiricism and Logical Positivism | 51 |

4 Wittgenstein on Philosophy of Logic and Mathematics | 75 |

5 The Logicism of Frege Dedekind and Russell | 129 |

6 Logicism in the Twentyfirst Century | 166 |

7 Logicism Reconsidered | 203 |

8 Formalism | 236 |

15 Nominalism | 483 |

16 Nominalism Reconsidered | 515 |

17 Structuralism | 536 |

18 Structuralism Reconsidered | 563 |

19 Predicativity | 590 |

20 MathematicsApplication and Applicability | 625 |

21 Logical Consequence Proof Theory and Model Theory | 651 |

22 Logical Consequence From a Constructivist Point of View | 671 |

9 Intuitionism and Philosophy | 318 |

10 Intuitionism in Mathematics | 356 |

11 Intuitionism Reconsidered | 387 |

12 Quine and the Web of Belief | 412 |

13 Three Forms of Naturalism | 437 |

14 Naturalism Reconsidered | 460 |

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### Common terms and phrases

abstract accept according algebra analysis application argument arithmetic assumptions axioms believe called Cambridge claim classical completeness concepts concerning conclusion consequence consider consistent construction contains course deductive defined definition determine developed discussion distinct domain empirical example existence expressed fact finite first-order follows formal formula Foundations Frege function geometry given holds idea identity inference infinite interpretation introduced intuition intuitionistic intuitionistic logic kind knowledge language laws least logic mathematical objects mathematicians mathematics matter meaning methods natural numbers nominalist noted notion objects ontology operations Oxford particular philosophy physical position possible predicative principle priori problem proof properties propositions pure quantifiers question Quine reasoning reference relation result rules satisfies scientific second-order semantics sense sentence set theory standard statements structure suggested symbols theorem things thought true truth understanding University Press valid variables Wittgenstein