Undergraduate Texts in Mathematics Readings in Mathematics Editors s. Axler F. W. Gehring K.A. Ribet Springer-Science+Business Media, LLC
Graduate Texts in Mathematics Readings in Mathematics EbbinghauslHenneslHirzebruchlKoecherlMainzerlNeukirchIPrestellRemmert: Numbers FultonlHarris: Representation Theory: A First Course Remmert: TheO/y of Complex Functions Walter: Ordinary Differential Equations Undergraduate Texts in Mathematics Readings in Mathematics Anglin: Mathematics: A Concise History and Philosophy AnglinlLambek: The Heritage of Thales Bressoud: Second Year Calculus HairerIWanner: Analysis by Its History HlimmerlinlHoffmann: Numerical Mathematics Isaac: The Pleasures of Probability LaubenbacherlPengelley: Mathematical Expeditions: Chronicles by the Explorers Samuel: Projective Geometry Stillwell: Numbers and Geometly Toth: Glimpus of Algebra and Geometry
W.S. Anglin MatheIllatics: A Concise History and Philosophy With 15 Illustrations Springer
W.S. Anglin Keane, Inc. 19 Old King's Highway South Darien, CT 06820 USA Editorial Board S.Axler Mathematics Department San Francisco State University San Francisco, CA 94132 USA EW. Gehring Mathematics Department East Hali University of Michigan Ann Arbor, MI 48109 USA K.A. Ribet Department of Mathematics University of California at Berkeley Berkeley, CA 94720-3840 USA Mathematics Subject Classification (1991): OI-OI, 01A05, OOA30 Library of Congress Cataloging-in-Publication Data Anglin, W. S. Mathematics, a concise history and philosophy / W.S. Anglin. p. cm.-(undergraduate texts in mathematics. Readings in mathematics) Includes bibliographical references and index. ISBN 978-1-4612-6930-4 ISBN 978-1-4612-0875-4 (ebook) DOI 10.1007/978-1-4612-0875-4 1. Mathematics-History. 2. Mathematics-Philosophy. 1. Title. II. Series. QA21.A54 1994 510'.9-dc20 94-8075 Printed on acid-free paper. 1994 Springer Science+Business Media New York Originally published by Springer-Verlag Berlin Heidelberg New York in 1994 Softcover reprint of the hardcover Ist edition 1994 AII rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher Springer-Science+Business Media, LLC, except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use of general descriptive names, trade names, trademarks, etc., in this publication, even if the former are not especially identified, is not to be taken as a sign that such names, as understood by the Trade Marks and Merchandise Marks Act, may accordingly be used freely by anyone. Production managed by Natalie Johnson; manufacturing supervised by Vincent Scelta. Photocomposed copy prepared from the author's UTEX files. 9 8 765 4 3 SPIN 10727955
To Jim Lambek
Preface This is a concise introductory textbook for a one-semester (40-class) course in the history and philosophy of mathematics. It is written for mathematics 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 acquire 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, students are given a choice between mathematical assignments, and more historical 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 mathematicians, giving more mathematically talented students a greater opportunity to learn the history and philosophy by way of problem solving. For example, there is a careful treatment of topics such as unit fractions, perfect numbers, linear Diophantine equations, Euclidean construction, Euclidean proofs, the circle area formula, the Pell equation, cubic equations, log table construction, the four square theorem, quaternions, and Cantor's set theory. Third, several important philosophical topics are pursued throughout
viii Preface the text, giving the student an opportunity to come to a full and consistent knowledge of their development. These topics include infinity and Platonism. In the essay questions, students are challenged to address a wide range of important topics. In short, this book offers, in fewer pages, a deep penetration into the key mathematical and philosophical aspects of the history of mathematics. The research for this book was carried out at McGill University from 1989 to 1992, and I should like to acknowledge the support of the Social Sciences and Humanities Research Council of Canada from 1989 to 1991. I am also greatly indebted to Jim Lambek, whose own course in this subject was the inspiration and basis of this work. The final version of this book was created at the University of Regina from 1992 to 1993, and I should like to thank the university for the opportunity of teaching the material in a liberal arts course entitled 'Mathematical Problems, Ideas and Personalities'. I should also like to thank Andonowati, J. Brown, E. Choueke, J. Denton, D. Hanson, 1. Rabinovitch, and D. Zhang for their help and encouragement. W. S. ANGLIN
Contents Preface vii 1 Mathematics for Civil Servants 1 2 The Earliest Number Theory 7 3 The Dawn of Deductive Mathematics 13 4 The Pythagoreans 17 5 The Pythagoreans and Perfection 23 6 The Pythagoreans and Polyhedra 29 7 The Pythagoreans and Irrationality 35 8 The Need for the Infinite 43 9 Mathematics in Athens Before Plato 49 10 Plato 57 11 Aristotle 61 12 In the Time of Eudoxus 69
x Contents 13 Ruler and Compass Constructions 14 The Oldest Surviving Math Book 15 Euclid's Geometry Continued 16 Alexandria and Archimedes 17 The End of Greek Mathematics 18 Early Medieval Number Theory 19 Algebra in the Early Middle Ages 20 Geometry in the Early Middle Ages 21 Khayyam and the Cubic 22 The Later Middle Ages 23 Modern Mathematical Notation 24 The Secret of the Cubic 25 The Secret Revealed 26 A New Calculating Device 27 Mathematics and Astronomy 28 The Seventeenth Century 29 Pascal 30 The Seventeenth Century II 31 Leibniz 32 The Eighteenth Century 33 Lagrange 34 Nineteenth-Century Algebra 35 Nineteenth-Century Analysis 36 Nineteenth-Century Geometry 37 Nineteenth-Century Number Theory 75 81 87 95 105 113 119 123 127 131 137 141 147 153 157 161 167 175 181 185 191 195 199 203 209
Contents xi 38 Cantor 39 Foundations 40 Twentieth-Century Number Theory References Appendix A Sample Assignments and Tests Appendix B Answers to Selected Exercises Index 213 217 221 227 229 245 257