## Truth in MathematicsThe nature of truth in mathematics is a problem which has exercised the minds of thinkers from at least the time of the ancient Greeks. The great advances in mathematics and philosophy in the twentieth centuryand in particular the proof of Godel's theorem and the development of the notion ofindependence in mathematicshave led to new viewpoints on this question in our era. This book is the result of the interaction of a number of outstanding mathematicians and philosophersincluding Yurii Manin, Vaughan Jones, and Per Martin-Lofand their discussions of this problem. It provides anoverview of the forefront of current thinking, and is a valuable introduction and reference for researchers in the area. |

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### Contents

Truth and the foundations of mathematics | 1 |

Tarski | 11 |

The realismantirealism debate and the question about math | 20 |

Bibliography | 34 |

Truth and objectivity from a verificationist point | 41 |

Objectivity | 48 |

Varieties of constructive mathematics | 55 |

Complex analysis | 63 |

Bibliography | 214 |

Mathematical definability | 233 |

4 | 235 |

Applications | 242 |

Bibliography | 249 |

Seeing or interpreting? | 255 |

Truth | 261 |

Bibliography | 268 |

On founding the theory of algorithms | 71 |

Abstract machines and implementations | 79 |

Implementations | 89 |

Infinitary algorithms | 95 |

Bibliography | 102 |

Proofs | 120 |

Computers and mathematics | 136 |

Conclusion | 144 |

Truth rigour and common sense | 147 |

Materials for three case studies | 155 |

How to be a naturalist about mathematics | 161 |

The naturalistic philosopher | 172 |

Bibliography | 179 |

The style of formalists | 186 |

Arguments against realism | 194 |

11 | 203 |

First and secondorder reflection | 277 |

Higherorder reflection | 283 |

quotient and direct limit types | 289 |

Putnams Models and Reality and the concepts of finiteness | 296 |

Notes | 306 |

Two conceptions of natural number | 311 |

Notes | 318 |

Bibliography | 326 |

Firstorder logic | 332 |

The formula | 338 |

353 | |

363 | |

371 | |

372 | |

### Common terms and phrases

algebra algorithm argument arithmetic assertion Axiom of Choice Axiom of Infinity Borel classical closure computation consistent constructive contains Continuum Hypothesis countable defined determinacy determinate truth value discussion domain Dummett elements equations example exists fact Feferman finite ordinal finite sets first-order formal formula free variables fullest mathematical theory given Godel sentence hierarchy Hilbert implies impredicativity inaccessible cardinals induction infinite sets integers internal relation intuition intuitionistic iterative conception judgements Kant Kechris Kleene Kurt Godel language large cardinal Lemma logic Maddy Math mathematical statements mathematicians means mergesort Moschovakis natural numbers naturalist number theory objects philosophical Philosophy of mathematics physical platonism platonist plenitudinous predicate principle priori problem proof proposition provable prove question real number reason recursors satisfy second-order arithmetic sequence set theory set-theoretic Solovay stationary structure subset theorem transfinite transfinite recursion true Turing Turing degrees undecidable Wittgenstein