## 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. |

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### Contents

Philosophy of Mathematics and Its Logic Introduction | 4 |

Apriority and Application Philosophy of Mathematics in the Modern Period | 30 |

Latter Empiricism and Logical Positivism | 52 |

Wittgenstein on Philosophy of Logic and Mathematics | 76 |

The Logicism of Frege Debekind and Russell | 130 |

Logicism in the Twentyfirst Century | 167 |

Logicism Reconsidered | 204 |

Formalism | 237 |

Nominalism | 484 |

Nominalism Reconsidered | 516 |

Structuralism | 537 |

Structuralism Reconsidered | 564 |

Predicativity | 591 |

MathematicsApplication and Applicability | 626 |

Logical Consequence Proof Theory and Model Theory | 652 |

Logical Consequence From a Constructivist Point of View | 672 |

Intuitionism and Philosophy | 319 |

Intuitionism in Mathematics | 357 |

Intuitionism Reconsidered | 388 |

Quine and the Web of Belief | 413 |

Three Forms of Naturalism | 438 |

Naturalism Reconsidered | 461 |

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### Common terms and phrases

abstract algebra analysis application argument arithmetic assumptions axioms Boolos Brouwer Burgess Cambridge canonical cardinal claim classical logic concepts construction deductive system deﬁned deﬁnition Descartes difﬁculty Disjunctive Syllogism distinct domain Dummett empirical entities epistemic epistemological example existence expressed ﬁgure ﬁnd ﬁnite ﬁrst ﬁrst-order formal formula Frege Fregean function geometry given Go¨del higher-order higher-order logic Hilbert holism Hume’s Principle idea impredicative inference inﬁnite interpretation intuition intuitionism intuitionistic logic justiﬁcation Kant Kant’s knowledge language logical consequence math mathematical objects mathematicians meaning model theory model-theoretic natural numbers neo-Fregean nominalist notion ontology Oxford University Press paradox philosophy of mathematics predicative premises priori problem proof properties propositions pure quantiﬁers question Quine Quine’s real numbers reasoning reﬂection relation relevant Resnik rules Russell Russell’s satisﬁes scientiﬁc second-order logic semantics sense sentence sequence set theory Shapiro signiﬁcant speciﬁc Stewart Shapiro structure subset sufﬁcient symbols theorem true valid variables Wittgenstein