## Mathematics: A Concise History and Philosophy: A Concise History and PhilosophyThis is a concise introductory textbook for a one-semester (40-class) course in the history and philosophy of mathematics. It is written for mathemat ics majors, philosophy students, history of science students, and (future) secondary school mathematics teachers. The only prerequisite is a solid command of precalculus mathematics. On the one hand, this book is designed to help mathematics majors ac quire a philosophical and cultural understanding of their subject by means of doing actual mathematical problems from different eras. On the other hand, it is designed to help philosophy, history, and education students come to a deeper understanding of the mathematical side of culture by means of writing short essays. The way I myself teach the material, stu dents are given a choice between mathematical assignments, and more his torical or philosophical assignments. (Some sample assignments and tests are found in an appendix to this book. ) This book differs from standard textbooks in several ways. First, it is shorter, and thus more accessible to students who have trouble coping with vast amounts of reading. Second, there are many detailed explanations of the important mathematical procedures actually used by famous mathe maticians, giving more mathematically talented students a greater oppor tunity to learn the history and philosophy by way of problem solving. |

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### Contents

Mathematics for Civil Servants | 1 |

The Earliest Number Theory | 7 |

The Dawn of Deductive Mathematics | 13 |

The Pythagoreans | 17 |

The Pythagoreans and Perfection | 23 |

The Pythagoreans and Polyhedra | 29 |

The Pythagoreans and Irrationality | 35 |

The Need for the Infinite | 43 |

The Later Middle Ages | 131 |

Modern Mathematical Notation | 137 |

The Secret of the Cubic | 141 |

The Secret Revealed | 147 |

A New Calculating Device | 153 |

Mathematics and Astronomy | 157 |

The Seventeenth Century | 161 |

Pascal | 167 |

Mathematics in Athens Before Plato | 49 |

Plato | 57 |

Aristotle | 61 |

In the Time of Eudoxus | 69 |

Ruler and Compass Constructions | 75 |

The Oldest Surviving Math Book | 81 |

Euclids Geometry Continued | 87 |

Alexandria and Archimedes | 95 |

The End of Greek Mathematics | 105 |

Early Medieval Number Theory | 113 |

Algebra in the Early Middle Ages | 119 |

Geometry in the Early Middle Ages | 123 |

Khayyam and the Cubic | 127 |

The Seventeenth Century II | 175 |

Leibniz | 181 |

The Eighteenth Century | 185 |

Lagrange | 191 |

NineteenthCentury Algebra | 195 |

NineteenthCentury Analysis | 199 |

NineteenthCentury Geometry | 203 |

NineteenthCentury Number Theory | 209 |

Cantor | 213 |

Foundations | 217 |

TwentiethCentury Number Theory | 221 |

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Alexandria Algebra Anaximander Archimedes Aristotle Arithmetica Babylonian Book Brahmagupta calculus Cantor Cardano Cauchy centre Challenges for Experts circle circumference compass construction congruent contradiction cubic equation cyclic quadrilateral Descartes diameter Diophantus distinct unit fractions dodecahedron Egyptian equal Essay Question Euclid Euclid's algorithm Euclidean Eudoxus Euler example Exercises Fermat Fibonacci finite formula Gauss give Hence Hippias hyperbolic geometry icosahedron infinite number integer integer solution Kepler Lagrange Leibniz math mathematicians Mersenne primes Napier natural number number theory parabola parallel parallel postulate Pascal perfect numbers philosophy plane Plato Plimpton 322 polygon positive integer prime problem proof Proposition Prove pyramid Pythagoras quadrilateral quaternion radius rational real number regular right angles right triangle Second edition segment semicircle Show solution of x2 solve sphere square straight line straightedge and compass Suppose tangent Tartaglia Thales theorem of Pythagoras triangular numbers unit fractions vertex vertices volume Zeno