algebra Revised Edition THE HISTORY OF algebra sets, symbols, and the language of thought Revised Edition John Tabak, Ph.D. ALGEBRA: Sets, Symbols, and the Language of Thought, Revised Edition Copyright © 2011, 2004 by John Tabak, Ph.D. All rights reserved. No part of this book may be reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying, recording, or by any information storage or retrieval systems, without permission in writing from the publisher. For information contact: Facts On File, Inc. An imprint of Infobase Learning 132 West 31st Street New York NY 10001 Library of Congress Cataloging-in-Publication Data Tabak, John. Algebra : sets, symbols, and the language of thought / John Tabak—Rev. ed. p. cm.—(The history of mathematics) Includes bibliographical references and index. ISBN 978-0-8160-7944-5 (alk. paper) ISBN 978-1-4381-3585-4 (e-book) 1. Algebra—History. I. Title. 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Date printed: May 2011 Printed in the United States of America 10 9 8 7 6 5 4 3 2 1 This book is printed on acid-free paper. To Diane Haber; teacher; mathematician, and inspirator CONTENTS Preface x Acknowledgments xvi Introduction xvii 1 The First Algebras 1 Mesopotamia: The Beginnings of Algebra 1 Mesopotamians and Second-Degree Equations 5 The Mesopotamians and Indeterminate Equations 7 Clay Tablets and Electronic Calculators 8 Egyptian Algebra 10 Chinese Algebra 12 Rhetorical Algebra 15 2 Greek Algebra 18 The Discovery of the Pythagoreans 20 The Incommensurability of √2 24 Geometric Algebra 25 Algebra Made Visible 27 Diophantus of Alexandria 31 3 Algebra from India to Northern Africa 34 Brahmagupta and the New Algebra 36 Mahavira 40 Bhaskara and the End of an Era 42 Poetry and Algebra 45 Islamic Mathematics 45 Al-Khwa¯rizmı¯ and a New Concept of Algebra 48 A Problem and a Solution 52 Omar Khayyám, Islamic Algebra at Its Best 52 Leonardo of Pisa 57 4 Algebra as a Theory of Equations 58 The New Algorithms 61 Algebra as a Tool in Science 67 François Viète, Algebra as a Symbolic Language 68 Thomas Harriot 73 Albert Girard 75 The Fundamental Theorem of Algebra 77 The Importance of Polynomials 81 5 Algebra in Geometry and Analysis 83 René Descartes 85 Descartes on Multiplication 88 Pierre de Fermat 92 Fermat’s Last Theorem 95 The New Approach 97 Polar Coordinates 98 6 The Search for New Structures 102 Niels Henrik Abel 104 Évariste Galois 106 Galois Theory and the Doubling of the Cube 109 Doubling the Cube with a Straightedge and Compass Is Impossible 112 Algebraic Structures 114 The Importance of Group Theory 120 7 The Laws of Thought 124 Aristotle 124 George Boole and the Laws of Thought 127 Boolean Algebra 131 Aristotle and Boole 135 Refining and Extending Boolean Algebra 136 Boolean Algebra and Computers 139 8 The Theory of Matrices and Determinants 142 Early Ideas 144 Spectral Theory 148 The Theory of Matrices 157 Matrix Multiplication 163 Emmy Noether and the Theory of Rings 165 Nicolas Bourbaki and Mathematical Structure 169 The Problem Solvers 172 Conclusion 179 Afterword: An Interview with Dr. Bonita Saunders on the Digital Library of Mathematical Functions 181 Chronology 193 Glossary 210 Further Resources 215 Index 225 PREFACE Of all human activities, mathematics is one of the oldest. Mathematics can be found on the cuneiform tablets of the Mesopotamians, on the papyri of the Egyptians, and in texts from ancient China, the Indian subcontinent, and the indigenous cul- tures of Central America. Sophisticated mathematical research was carried out in the Middle East for several centuries after the birth of Muhammad, and advanced mathematics has been a hallmark of European culture since the Renaissance. Today, mathematical research is carried out across the world, and it is a remarkable fact that there is no end in sight. The more we learn of mathematics, the faster the pace of discovery. Contemporary mathematics is often extremely abstract, and the important questions with which mathematicians concern them- selves can sometimes be difficult to describe to the interested nonspecialist. Perhaps this is one reason that so many histories of mathematics give so little attention to the last 100 years of dis- covery—this, despite the fact that the last 100 years have probably been the most productive period in the history of mathematics. One unique feature of this six-volume History of Mathematics is that it covers a significant portion of recent mathematical history as well as the origins. And with the help of in-depth interviews with prominent mathematicians—one for each volume—it is hoped that the reader will develop an appreciation for current work in mathematics as well as an interest in the future of this remarkable subject. Numbers details the evolution of the concept of number from the simplest counting schemes to the discovery of uncomputable numbers in the latter half of the 20th century. Divided into three parts, this volume first treats numbers from the point of view of computation. The second part details the evolution of the concept of number, a process that took thousands of years and culminated in what every student recognizes as “the real number line,” an x Preface xi extremely important and subtle mathematical idea. The third part of this volume concerns the evolution of the concept of the infi- nite. In particular, it covers Georg Cantor’s discovery (or creation, depending on one’s point of view) of transfinite numbers and his efforts to place set theory at the heart of modern mathematics. The most important ramifications of Cantor’s work, the attempt to axi- omatize mathematics carried out by David Hilbert and Bertrand Russell, and the discovery by Kurt Gödel and Alan Turing that there are limitations on what can be learned from the axiomatic method, are also described. The last chapter ends with the discov- ery of uncomputable numbers, a remarkable consequence of the work of Kurt Gödel and Alan Turing. The book concludes with an interview with Professor Karlis Podnieks, a mathematician of remarkable insights and a broad array of interests. Probability and Statistics describes subjects that have become cen- tral to modern thought. Statistics now lies at the heart of the way that most information is communicated and interpreted. Much of our understanding of economics, science, marketing, and a host of other subjects is expressed in the language of statistics. And for many of us statistical language has become part of everyday dis- course. Similarly, probability theory is used to predict everything from the weather to the success of space missions to the value of mortgage-backed securities. The first half of the volume treats probability beginning with the earliest ideas about chance and the foundational work of Blaise Pascal and Pierre Fermat. In addition to the development of the mathematics of probability, considerable attention is given to the application of probability theory to the study of smallpox and the misapplication of probability to modern finance. More than most branches of mathematics, probability is an applied dis- cipline, and its uses and misuses are important to us all. Statistics is the subject of the second half of the book. Beginning with the earliest examples of statistical thought, which are found in the writings of John Graunt and Edmund Halley, the volume gives special attention to two pioneers of statistical thinking, Karl Pearson and R. A. Fisher, and it describes some especially impor- tant uses and misuses of statistics, including the use of statistics xii ALGEBRA in the field of public health, an application of vital interest. The book concludes with an interview with Dr. Michael Stamatelatos, director of the Safety and Assurance Requirements Division in the Office of Safety and Mission Assurance at NASA, on the ways that probability theory, specifically the methodology of probabi- listic risk assessment, is used to assess risk and improve reliability. Geometry discusses one of the oldest of all branches of mathe- matics. Special attention is given to Greek geometry, which set the standard both for mathematical creativity and rigor for many cen- turies. So important was Euclidean geometry that it was not until the 19th century that mathematicians became willing to consider the existence of alternative and equally valid geometrical systems. This 19th-century revolution in mathematical, philosophical, and scientific thought is described in some detail, as are some alter- natives to Euclidean geometry, including projective geometry, the non-Euclidean geometry of Nikolay Ivanovich Lobachevsky and János Bolyai, the higher (but finite) dimensional geometry of Riemann, infinite-dimensional geometric ideas, and some of the geometrical implications of the theory of relativity. The volume concludes with an interview with Professor Krystyna Kuperberg of Auburn University about her work in geometry and dynamical systems, a branch of mathematics heavily dependent on ideas from geometry. A successful and highly insightful mathematician, she also discusses the role of intuition in her research.