## 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 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 |

9 Intuitionism and Philosophy | 318 |

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 |

23 Relevance in Reasoning | 696 |

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### Common terms and phrases

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