THE CULT OF PYTHAGORAS Math and Myths ALB E RTO A. MARTÍ N EZ University of Pittsburgh Press Martinez Cultltext4.indd 3
Published by the University of Pittsburgh Press, Pittsburgh, Pa., 15260 Copyright 2012, Alberto A. Martínez All rights reserved Manufactured in the United States of America Printed on acid-free paper 10 9 8 7 6 5 4 3 2 1 Library of Congress Cataloging-in-Publication Data Martinez, Alberto A. The cult of Pythagoras : math and myths / Alberto A. Martinez. p. cm. Includes bibliographical references and index. ISBN 978-0-8229-4418-8 (hardcover : alk. paper) 1. Mathematics History. 2. Geometry History. I. Title. QA21.M335 2012 910.938 dc23 2012022366> Martinez Cultltext4.indd 4
From of old it has been the custom, and not only in our time, for vice to make war on virtue. Thus Pythagoras, with three hundred others, was burned to death. Cohort ad Græcos, ca. 150 CE Martinez Cultltext4.indd 5
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CONTENTS List of Myths and Apparent Myths ix Acknowledgments xiii Introduction xvii 1. Triangle Sacrifice to the Gods 1 2. An Irrational Murder at Sea 16 3. Ugly Old Socrates on Eternal Truth 29 4. The Death of Archimedes 43 5. Gauss, Galois, and the Golden Ratio 59 6. From Nothing to Infinity 82 7. Euler s Imaginary Mistakes 101 8. The Four of Pythagoras 113 9. The War over the Infinitely Small 134 10. Impossible Triangles 156 11. Inventing Mathematics? 181 12. The Cult of Pythagoras 201 Notes 217 Illustration Sources and Credits 257 Index 259 Martinez Cultltext4.indd 7
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LIST OF MYTHS AND APPARENT MYTHS Did Pythagoras discover the hypotenuse theorem? 1 Was Pythagoras a mathematician? 2 Did Pythagoras prove the hypotenuse theorem? 8 Did Pythagoras believe that the world is made of numbers? 16 Did Pythagoras discover the theory of irrationals? 20 Was Hippasus murdered for revealing irrational numbers? 21 Did Hippasus discover irrationality in a pentagram? 21 Did Socrates value mathematics as an end in itself? 35 Did Archimedes solve the weight of a golden crown? 48 Did a soldier kill Archimedes for defending a diagram? 53 Did the bright boy Gauss add all numbers from 1 to 100? 59 Did Galois create group theory the night before dying by gunshot? 61 Did the Egyptians use the golden ratio in the Great Pyramid? 76 Did Pythagoras discover the golden ratio in a pentagram? 77 Is the golden ratio found in ancient architecture? 77 Do most people prefer the shape of golden rectangles? 78 Does the navel divide the body according to the golden section? 78 ix Martinez Cultltext4.indd 9
x L I S T O F M Y T H S A N D A P PA R E N T M Y T H S Is the golden ratio in the spiral of the nautilus shell? 79 Is the mathematics in The Da Vinci Code true? 79 Did Hippasus discover the golden ratio in a pentagram? 80 Is it impossible to divide by zero? 82 Are multiple solutions impossible in mathematics? 86 Is division the inverse of multiplication? 87 Did Pythagoras initiate the study of infinity? 90 Do computers always agree with mathematicians? 93 Was Euler confused about how to multiply imaginary numbers? 102 Was Euler confused about how to divide imaginary numbers? 108 Are squares and square roots inverse operations? 110 Do equal operations on both sides always preserve equalities? 111 Do numbers correspond to geometry? 113 Is the commutative rule of multiplication always valid? 118 Is the associative rule of multiplication always valid? 125 Did Pythagoras discover quaternions? 126 Can the rules of imaginary numbers be changed? 129 Is i the only and true value of the square root of 1? 131 Can a single operation have multiple solutions? 131 Did Hamilton tragically sacrifice his life on quaternions? 132 Is it true that 0.999... is equal to 1? 134 Can a number remain unchanged when more than zero is added? 143 Did the theory of limits banish infinitesimals from mathematics? 151 Can the sum of the angles of a triangle differ from 180 degrees? 156 Is Euclidean geometry perfect? 160 Do Euclid s triangles exist in nature? 168 Did Gauss test Euclid s geometry on mountaintops? 168 Did astronomers omit data to confirm Einstein s gravity? 175 Martinez Cultltext4.indd 10
L I S T O F M Y T H S A N D A P PA R E N T M Y T H S xi Did Galileo say that objects in space move in straight lines? 177 Is mathematics really the language of nature? 180 Do two positives always make a positive? 182 Does minus times minus have to be positive? 184 Can numbers exist that are larger than finite numbers? 191 Can philosophies affect mathematics? 194 Did Pythagoras initiate a golden age of reason? 201 Did Pythagoras plant the seeds of Western civilization? 204 Do innovative pioneers get the credit they deserve? 207 Did Pythagoras claim that the Earth circles the Sun? 209 Should we blame Pythagoras for Pythagorean myths? 214 Martinez Cultltext4.indd 11
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