## Proof and Other Dilemmas: Mathematics and PhilosophyFor the majority of the twentieth century, philosophers of mathematics focused their attention on foundational questions. However, in the last quarter of the century they began to return to basics, and two new schools of thought were created: social constructivism and structuralism. The advent of the computer also led to proofs and development of mathematics assisted by computer, and to questions concerning the role of the computer in mathematics. This book of sixteen original essays is the first to explore this range of new developments in the philosophy of mathematics, in a language accessible to mathematicians. Approximately half the essays were written by mathematicians, and consider questions that philosophers have not yet discussed. The other half, written by philosophers of mathematics, summarise the discussion in that community during the last 35 years. A connection is made in each case to issues relevant to the teaching of mathematics. |

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User Review - fpagan - LibraryThingA collection of essays by mathematicians (Ms) and philosophers of math (PMs) on proof and how it is changing, social constructivist views of math, the nature of math objects and math knowledge, and ... Read full review

### Contents

Proof and How it is Changing | 1 |

Its Nature and Significance Michael Detlefsen | 3 |

Implications of Experimental Mathematics for the Philosophy of Mathematics Jonathan Borwein | 33 |

On the Roles of Proof in Mathematics Joseph Auslander | 61 |

U SocialConstructivist Views of Mathematics | 79 |

When Is a Problem Solved? Philip J Davis | 81 |

Mathematical Practice as a Scientific Problem Reuben Hersh | 95 |

Social Constructs? Julian Cole | 109 |

The Nature of Mathematical Objects 0ystein Linnebo | 205 |

When is One Thing Equal to Some Other Thing? Barry Mazur | 221 |

TV The Nature of Mathematics and its Applications | 243 |

Mathematics as the Science of Relations as Such R S D Thomas | 245 |

What is Mathematics? A Pedagogical Answer to a Philosophical Question Guershon Harel | 265 |

What Will Count as Mathematics in 2100? Keith Devlin | 291 |

The Case of Addition Mark Steiner | 313 |

ProbabilityA Philosophical Overview Alan Hdjek | 323 |

The Nature of Mathematical Objects and Mathematical Knowledge | 129 |

The Existence of Mathematical Objects Charles Chihara | 131 |

Mathematical Objects Stewart Shapiro | 157 |

Mathematical Platonism Mark Balaguer | 179 |

Glossary of Common Philosophical Terms | 341 |

About the Editors | 345 |

### Common terms and phrases

abstract objects algebraic analysis answer argument arithmetic assertions axioms belief Benacerraf Borwein called category theory causal chapter characterization Chihara claim concept conjecture constitutive social constructs constructivism defined definition diagrammatic reasoning discussion empirical example exist fact fictionalism fictionalists formal Frege function functor geometry given Godel Hersh Hilbert human Hume's principle ideas integers interpretation intuition intuitionism isomorphism language logic Mac Lane Math mathematical domains mathematical knowledge mathematical objects mathematical platonism mathematical practice mathematical proof mathematical theories mathematicians mathematics education means mental act morphism natural numbers nature of mathematical nominalist notion ontological Oxford University Press particular philosophy of mathematics platonism platonists probability problem proof scheme properties propositions question real numbers realism refer relations Resnik Reuben Hersh rigorous role semantic sense sentence set theory Shapiro solution solve statements Stewart Shapiro structure symbols theorem thesis things thinking true understanding view of mathematics visual