And All That PHILIP J. DAVIS WILLIAM G. CHINN. Birkhauser. Boston. Basel. Stuttgart

Transcription

1 And All That PHILIP J. DAVIS WILLIAM G. CHINN Birkhauser Boston. Basel. Stuttgart

2 TO OUR FAMILIES All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without prior permission of the copyright owner. Second edition. First edition published in 1%9 by Simon & Schuster, New York. Birkhauser Boston, 1985 ABCDEFGHIJ ISBN-13: DOl: / e-isbn-13: Library of Congress Cataloging in Publication Data Davis, Philip J., and all that. Bibliography: p. I. Mathematics-Popular works. I. Chinn, William G. II. Title. QA93.D CIP-Kurztitelaufnahme der Deutschen Bibliothek [Three onejour one six] and all that / Philip J. Davis and William G. Chinn.-2. ed. Boston; Basel; Stuttgart: Birkhiiuser, AnI!. im VerI. Simon and Schuster, New York HE: Davis, PhilipJ. [Mitverf.]; Chinn, William G. [Mitverf.]

3 CONTENTS Foreword vii Introduction ix 1. The Problem That Saved a Man's Life 1 2. The Code of the Primes 7 3. Pompeiu's Magic Seven What Is an Abstraction? Postulates-The Bylaws of Mathematics The Logical Lie Detector Number The Philadelphia Story Poinsot's Points and Lines Chaos and Polygons Numbers, Point and Counterpoint The Mathematical Beauty Contest The House That Geometry Built Explorers of the Nth Dimension The Band-Aid Principle The Spider and the Fly A Walk in the Neighborhood Division in the Cellar The Art of Squeezing The Business of Inequalities The Abacus and the Slipstick Of Maps and Mathematics "Mr. Milton, Mr. Bradley-Meet Andrey Andreyevich Markov" and All That 172 Bibliography 177 Ancient and Honorable Society of Pi Watchers: 1984 Report 177 Bibliography 181

4 ACKNOWLEDGMENTS Most of the articles in this book originally appeared in Science World. Grateful acknowledgment is made to Science World for permission to use them here. We thank the Scientific American for permission to reprint the article entitled "Number." The authorship of the individual articles is as follows: Philip J. Davis: 1,3,4,7,8,9, 12, 14, 15, 18,22,23, 24. William G. Chinn: 2, 5, 6, 10, 11, 13, 16, 17, 19,20,21. The first author would like to acknowledge his indebtedness to Professor Alexander Ostrowski and to Drs. Barry Bernstein and John Wrench for information incorporated in some of his articles. The second author would like to acknowledge his indebtedness to Professor G. Baley Price for permission to use materials from one of his articles.

5 FOREWORD LYTTON STRACHEY tells the following story. In intervals of relaxation from his art, the painter Degas used to try his hand at writing sonnets. One day, while so engaged, he found that his inspiration had run dry. In desperation he ran to his friend Mallarme, who was a poet. "My poem won't come out," he said, "and yet I'm full of excellent ideas." "My dear Degas," Mallarme retorted, "poetry is not written with ideas, it is written with words." If we seek an application of Mallarme's words to mathematics we find that we shall want to turn his paradox around. We are led to say that mathematics does not consist of formulas, it consists of ideas. What is platitudinous about this statement is that mathematics, of course, consists of ideas. Who but the most unregenerate formalist, asserting that mathematics is a meaningless game played with symbols, would deny it? What is paradoxical about the statement is that symbols and formulas dominate the mathematical page, and so one is naturally led to equate mathematics with its formulas. Is not Pythagoras' Theorem a 2 + b 2 = c 2? Does not the Binomial Theorem say that (a + b) 2 = a 2 + 2ab + b 2? What more need be said-indeed, what more can be said? And yet, as every devotee who has tried to be creative in mathematics knows, formulas are not enough; for new formulas can be produced by the yard, while original thought remains as remote as hummingbirds in January. To appreciate mathematics at its deeper level we must pass from naked formulas to the ideas that lie behind them. In the present volume, we have selected for reprinting a number of pieces that appeared principally in Science World, a periodical with a wide circulation among students and teachers. In writing these articles it was our aim to deal with a number of diverse areas of current mathematical interest and, by concentrating on a limited aspect of each topic, to expose in a modest way the mathematical ideas that underlie it. It has not been possible, in the few pages allotted to each essay, to present the topics in the conventional textvii

6 viii Foreword book sense; our goal has been rather to provide a series of appetizers or previews of coming attractions which might catch the reader's imagination and attract him to the thoroughgoing treatments suggested in the bibliographies. Each article is essentially self-contained. What reason can we put forward for the study of mathematics by the educated man? Every generation has felt obliged to say a word about this. Some of the reasons given for this study are that mathematics makes one think logically, that mathematics is the Queen of the Sciences, that God is a geometer who runs His universe mathematically, that mathematics is useful in surveying fields, building pyramids, launching satellites. Other reasons are that life has become increasingly concerned with the manipulation of symbols, and mathematics is the natural language of symbols; that "Euclid alone has looked upon beauty bare"; that mathematics can be fun. Each of these has its nugget of truth and must not be denied. Each undoubtedly can be made the basis of a course of instruction. But we should like to suggest a different reason. Mathematics has been cultivated for more than four thousand years. It was studied long before there were Democrats and Republicans or any of our present concerns. It has flourished in many lands, and the genius of many peoples has added to its stock of ideas. Mathematics dreams of an order which does not exist. This is the source of its power; and in this dream it has exhibited a lasting quality that resists the crash of empire and the pettiness of small minds. Mathematical thought is one of the great human achievements. The study of its ideas, its past and its present, can enable the individual to free himself from the tyranny of time and place and circumstance. Is not this what liberal education is about? Summer, 1968

7 INTRODUCTION AND ALL THAT is a beautiful book written by two masters of popularization. Philip J. Davis and William G. Chinn are mathematicians of many dimensions, who are interested in much more than mathematics. In the following twenty-four essays, you obtain pleasant introductions to powerful ideas of mathematics. In fact, you will have fun reading this book of mathematics. Essayists Davis and Chinn write with style, vigor, and a sense of the relationship of mathematics to other fields, including biology, history, music, philosophy, and even golf and woodworking. Professors Chinn and Davis are much aware of mathematics as a human activity. In their essays, they provide precious insights into how mathematicians think. They also show us that mathematics is alive and growing. If you like mathematics, this book will cause you to like it more. If you are wary of mathematics, then reading this book is likely to make you a mathematical convert. Donald J. Albers Second Vice-President The Mathematical Association of America ix