## The Heritage of ThalesThis is intended as a textbook on the history, philosophy and foundations of mathematics, primarily for students specializing in mathematics, but we also wish to welcome interested students from the sciences, humanities and education. We have attempted to give approximately equal treatment to the three subjects: history, philosophy and mathematics. History We must emphasize that this is not a scholarly account of the history of mathematics, but rather an attempt to teach some good mathematics in a historical context. Since neither of the authors is a professional historian, we have made liberal use of secondary sources. We have tried to give ref cited facts and opinions. However, considering that this text erences for developed by repeated revisions from lecture notes of two courses given by one of us over a 25 year period, some attributions may have been lost. We could not resist retelling some amusing anecdotes, even when we suspect that they have no proven historical basis. As to the mathematicians listed in our account, we admit to being colour and gender blind; we have not attempted a balanced distribution of the mathematicians listed to meet today's standards of political correctness. Philosophy Both authors having wide philosophical interests, this text contains perhaps more philosophical asides than other books on the history of mathematics. For example, we discuss the relevance to mathematics of the pre-Socratic philosophers and of Plato, Aristotle, Leibniz and Russell. We also have vi Preface presented some original insights. |

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### Contents

Introduction | 1 |

History and Philosophy of Mathematics | 5 |

Egyptian Mathematics | 7 |

Scales of Notation | 11 |

Prime Numbers | 15 |

SumerianBabylonian Mathematics | 21 |

More about Mesopotamian Mathematics | 25 |

The Dawn of Greek Mathematics | 29 |

The Law of Quadratic Reciprocity | 169 |

Foundations of Mathematics | 173 |

The Number System | 175 |

Natural Numbers Peanos Approach | 179 |

The Integers | 183 |

The Rationals | 187 |

The Real Numbers | 191 |

Complex Numbers | 195 |

Pythagoras and His School | 33 |

Perfect Numbers | 37 |

Regular Polyhedra | 41 |

The Crisis of Incommensurables | 47 |

From Heraclitus to Democritus | 53 |

Mathematics in Athens | 59 |

Plato and Aristotle on Mathematics | 67 |

Constructions with Ruler and Compass | 71 |

The Impossibility of Solving the Classical Problems | 79 |

Euclid | 83 |

NonEuclidean Geometry and Hilberts Axioms | 89 |

Alexandria from 300 BC to 200 BC | 93 |

Archimedes | 97 |

Alexandria from 200 BC to 500 AD | 103 |

Mathematics in China and India | 111 |

Mathematics in Islamic Countries | 117 |

New Beginnings in Europe | 121 |

Mathematics in the Renaissance | 125 |

The Cubic and Quartic Equations | 133 |

Renaissance Mathematics Continued | 139 |

The Seventeenth Century in France | 145 |

The Seventeenth Century Continued | 153 |

Leibniz | 159 |

The Eighteenth Century | 163 |

The Fundamental Theorem of Algebra | 199 |

Quaternions | 203 |

Quaternions Applied to Number Theory | 207 |

Quaternions Applied to Physics | 211 |

Quaternions in Quantum Mechanics | 215 |

Cardinal Numbers | 219 |

Cardinal Arithmetic | 223 |

Continued Fractions | 227 |

The Fundamental Theorem of Arithmetic | 231 |

Linear Diophantine Equations | 233 |

Quadratic Surds | 237 |

Pythagorean Triangles and Fermats Last Theorem | 241 |

What Is a Calculation? | 245 |

Recursive and Recursively Enumerable Sets | 251 |

Hilberts Tenth Problem | 255 |

Lambda Calculus | 259 |

Logic from Aristotle to Russell | 265 |

Intuitionistic Propositional Calculus | 271 |

How to Interpret Intuitionistic Logic | 277 |

Intuitionistic Predicate Calculus | 281 |

Intuitionistic Type Theory | 285 |

Godels Theorems | 289 |

Natural Transformations | 303 |

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