To Infinity and Beyond Eli Maor To Infinity and Beyond

A Cultural History of the Infinite

With 162 Illustrations and 6 Color Plates

Birkhauser Boston· Basel· Stuttgart Eli Maor Associate Professor Department of Mathematical Sciences Oakland University Rochester, Michigan 48063, U.S.A.

On the front jacket: Fig. 13.4. M. C. Escher: Sphere Spirals (1958). © M. C. Escher Heirs c/o Cordon Art·Baaen-Holland

Library of Congress Cataloging in Publication Data Maor, Eli. To infinity and beyond. Bibliography; p. Includes index. 1. Infinite. I. Title. QA9.M316 1986 511.3 86-18791 ISBN-13:978-1-4612-5396-9

CIP-Kurztitelaufnahme der Deutschen Bibliothek Maar, Eli: To infinity and beyond: a cultural history of the infinite / Eli Maar. - Boston; Basel; Stuttgart: Birkhauser, 1986. ISBN-13:978-1-4612-5396-9 e-ISBN-13:978-1-4612-5394-5 DOl: 10.1007/978-1-4612-5394-5

Copyright © 1987 by Birkhauser Boston, Inc. 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 written permission of the publisher, Birkhauser Boston, Inc., 380 Green Street, Cambridge, Massachusetts 02139, U.S.A. Softcover reprint of the hardcover 1st edition 1987

ISBN-13:978-1-4612-5396-9 Manufactured in the United States of America. In memory of my teacher, Professor Franz Ollendorff (born in Germany, 1900; died in Israel, 1981) Preface

The infinite! No other question has ever moved so profoundly the spirit of man; no other idea has so fruitfully stimulated his intellect; yet no other concept stands in greater need of clarification than that of the infinite. . . - David Hilbert (1862-1943)

There is a story attributed to David Hilbert, the preeminent mathe• Infinity is a fathomless gulf, matician whose quotation appears above. A man walked into a into which all things vanish. hotel late one night and asked for a room. "Sorry, we don't have o Marcus Aurelius (121- any more vacancies," replied the owner, "but let's see, perhaps 180), Roman Emperor and philosopher I can find you a room after alL" Leaving his desk, the owner reluctantly awakened his guests and asked them to change their rooms: the occupant of room #1 would move to room #2, the occupant of room #2 would move to room #3, and so on until each occupant had moved one room over. To the utter astonish• ment of our latecomer, room #1 suddenly became vacated, and he happily moved in and settled down for the night. But a numbing thought kept him from sleep: How could it be that by merely moving the occupants from one room to another, the first room had become vacated? (Remember, all of the rooms were occupied when he arrived.) And then the answer dawned on our visitor: The hotel must be Hilbert's Hotel, the one hotel in town known to have an infinite number of rooms! By shifting each occupant one room over, room #1 became vacated:

What is that thing which does not give itself, and which if it were to give itself would not exist? It is the This famous anecdote tells, in a way, the entire story of infinity. infinite.' o Leonardo da Vinci It is a story of intriguing paradoxes and seemingly impossible (1452-1519), Italian situations which have puzzled mankind for more than two millennia. artist and engineer

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Wh~n we say anything is The roots of these paradoxes lie in mathematics, and it is this infinite, we signify only that discipline which has offered the most fruitful path towards their we are not able to conceive the ends and bounds of the eventual resolution. The clarification and demystification of the thing named. infinite was fully accomplished only in our own century, and even o Thomas Hobbes (1588- 1679), English this feat cannot be regarded as the final word. Like every science, philosopher mathematics has a refreshing air of incompleteness about it; no sooner has one mystery been solved, than a new one is already being introduced. The goal of achieving an absolute and final understanding of science is an elusive one. But it is this very elusiveness that makes the study of any scientific discipline so stimulating, and mathematics is no exception. Many thinkers have dealt with the infinite. The philosophers of ancient Greece argued endlessly about whether a line seg• ment-or any quantity, for that matter-is infinitely divisible, or whether an indivisible point, an "atom," would ultimately be reached. Their modern followers, the physicists, are still struggling with the same question today, using huge particle accelerators to search for the "elementary particles," those ultimate building blocks from which the entire universe is made. Astronomers have been pondering about infinity on the other extreme of the scale• the infinitely large. Is our universe infinite, as it seems to anyone watching the sky on a clear, dark night, or does it have a boundary beyond which nothing exists? The possibility of a finite universe seems to defy our very common sense, for isn't it clear that we can go forever in any direction without ever reaching the "edge"? But as we shall see, "common sense" is a very poor guide when dealing with infinity! Artists, too, have dealt with the infinite, depicting it on canvas and in lines that became literary treasures. "I am painting the infinite," exclaimed van Gogh as he gazed at the plains of France stretching before him as far as his eyes could see. "The eternal silence of these infinite spaces terrifies me" lamented Blaise Pascal in his characteristically gloomy outlook of the world, while another man of letters, Giordano Bruno, rejoiced in the thought of an infinite universe: "Open the door through which we may look out into the limitless firmament" was his motto, for which he was arrested by the Inquisition and sentenced to die. But however we look at the infinite, we are ultimately led back to mathematics, for it is here that the concept of infinity has its deepest roots. According to one view, mathematics is the science of infinity. In the Encyclopedic Dictionary of Mathematics, a compen• Infinity is a dark illimitable dium recently compiled by the Mathematical Society of Japan,l ocean, without bound. o John Milton (1608- 1674), English poet 1 English translation published by The MIT Press, 1980.

Vl11 PREFACE

Figure L Courtesy of the Mathematical Association of America.

FOUNDED BY PRINDLE. WEBER & __ JOURNAL SCHMIDT. INC. USSN 0049-49251

VOLUME 13, NUMBER 2 March 1982

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THIS ISSUE : Self· reference, TNT, Shunling, Truth Machines, and GCidel's Incompleteness Theorem 0 Tarr and Fether and Asylums 0 the words "infinity," "infinite," and "infinitesimal" appear no fewer than 50 times in the index, Indeed, it is hard to see how mathematics could exist without the notion of infinity, for the very first thing a child learns about mathematics-how to count• is based on the tacit assumption that every integer has a successor. The notion of a straight line, so fundamental in geometry, is based on a similar assumption-that we can, at least in principle, extend a line indefinitely in both directions, Even in such seemingly "fin• ite" branches of mathematics as probability, the notion of infinity plays a subtle role: when we toss a coin ten times, we may get five "heads" and five "tails," or we may get six "heads" and four "tails," or in fact any other outcome; but when we say that the probability of getting "heads" or "tails" is even, we tacitly Infinity is only a figure of assume that an infinite number of tosses would produce an equal speech, meaning a limit to outcome, which certain ratios may My first encounter with infinity was as a young boy. I was given approach as closely as desired, when others are a book-it was the Haggadah, the story of the Exodus from permitted to increase Egypt-on whose cover was a picture of a young boy holding indefinitely. o Carl Friedrich Gauss the very same book in his hand, When I looked carefully I could (1777-1855), German see the same picture on the cover of the small Haggadah the mathematician

IX PREFACE

A n infinite set is one that boy was holding. It may even be that the picture showed up again can be put into a one-to-one in the picture's picture-I can't remember for sure. But I do re• correspondence with a proper subset of itself member that my mind was boggled by the thought that if it were o Georg Cantor (1845- possible to continue this process, it would go on forever! An 1918), German mathematician intriguing possibility it was; little did I know that a relatively unknown Dutch artist, Maurits C. Escher, had been fascinated with the same idea and conveyed it in his graphic work, carrying the process to the very limits attainable with his drawing tools. I had another encounter with infinity much later in life, an encounter of an entirely different sort. Strolling one evening along Connecticut Avenue in Washington, D.C., I found myself standing in front of a large abstract sculpture, erected right on the sidewalk. A plaque identified it as Limits of Infinity III. It consisted of a large elliptical bronze ring, from whose extreme points a propeller• shaped object was hinged. It looked as though the elongated object was meant to turn freely on its hinges, so I gently touched it, Infinity converts the possible anticipating it to commence its motion. Instead, a hidden alarm into the inevitable. went off, and with such a shrill sound that I was quite scared. o Norman Cousins, After my initial shock was gone, I could hear an inner voice in Saturday Review, Apri115, 1978 me saying: "Thou shalt not touch infinity! "2 In the following chapters I have tried to share with the reader the excitement and awe that the infinite has inspired in men of all times. I took the title, To Infinity and Beyond, from a telescope manual that listed among the many virtues of the instrument the following: ''The range of focus of your telescope is from fifteen feet to infinity and beyond." As the subtitle "A Cultural History of the Infinite" indicates, my aim is to unfold the story of infinity throughout the ages, without necessarily following a strict chrono• logical order. My story is, to some extent, a subjective one-it is told from the point of view of a mathematician. This meant that I had to confront the same dilemma that every scientist faces when writing a book for the educated layman: How to express the author's ideas in a language understandable to the non-expert, without at the same time compromising the standards of rigor his professional peers expect of him. This dilemma is all the more

2 The artist, John Safer, was kind enough to send me a most beautiful book describing his work. Referring to Limits of Infinity III, he says: "That turned form of bronze in the center of the piece hangs within its bronze enclosure as if floating in space. It is, the shape reminds us, the symbol of infinity." Referring to the large base that supports the work, he says: "The granite block, which is not merely a suitable resting Infinity is where things place but a vital part of the sculpture, brings the solid and finite earth happen that don't. into the equation. That block of stone is our base from which to contem• o An anonymous schoolboy plate infinity."

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Figure 2. Limits of Infinity III, by John Safer (Washington, D.C.). Courtesy of John Safer.

Xl PREFACE

Figure 3. Infinity, by jOJe de Rivera (WaJhington, D. C.). CourteJY of the eJtate of jOJe de Rivera, Grace Borgenicht Gallery, New York, and the National MUJeum of American HiJtory, WaJhington, D.C.

acute in mathematics, which relies almost entirely on a non-verbal language of symbols and equations. I hope I have succeeded in properly addressing this problem.

xu PREFACE

As this book is intended for the general reader, I have refrained as much as possible from using "higher" mathematics in the text itself. (Of course, some familiarity with elementary algebra never does any harm.) Some specific mathematical topics have been dele• gated to the Appendix, thereby maintaining the continuity of the general discussion. The various chapters are, mostly, only loosely connected, so that skipping over a few of them would not impede the reading. Finally, the reader who prefers to browse through the book only casually will still enjoy the many illustrations and photographs, as well as the numerous quotations, poems, and liter• ary lines on infinity. Many friends helped me with this work, and lowe them many Infinity is that dimension thanks. Particularly, I am indebted to my colleagues Wilbur Hoppe without end which the human mind cannot grasp. and Rol;>ert Langer, who have read large parts of the manuscript o Anonymous and came up with numerous suggestions; to Blagoy Trenev, whom I incessantly bothered with questions of language and style; to Hilde Bacharach and Raffaella Borasi, for bringing to my attention two beautiful poems describing the infinite; to Ruth Ollendorff, who made available to me many unpublished writings of her late husband, Professor Franz Ollendorff, to whom this book is dedi• cated; to Mary Besser, who edited most of the manuscript and greatly helped in its final draft; to Lynn Metzker, who prepared most of the line drawings; to the University of Wisconsin-Eau Claire and to Oakland University in Rochester, Michigan, for two generous grants that greatly helped me in my work; and finally to the editorial and production staff of Birkhiiuser Boston for their special efforts to make this work a reality. But above all, I am indebted to my mother, Luise Metzger, for the intellectual enrichment she has given me over the years, and to my wife Dalia for her encouragement and patience during the many nights when I left her alone while working on the book in my office. If it were not for their support, this work would have never been completed. One final note. At the outset of every discussion, a mathemati• cian must define his symbols and notation. Let it therefore be known that by "he" I mean "he or she," by "him," "him or Infinity is a floorless room her," etc. If I am using the more traditional language i~ this without walls or ceiling. book, it is solely for the sake of brevity. o Anonymous

Rochester, Michigan, June 13, 1986

X111 Contents

Preface ...... Vll Color Plates...... 143

Part I. Mathematical Infinity

1. First Steps to Infinity ...... 2 Zero, One, Infinity ...... 6 2. Towards Legitimation ...... 10 Numbers Large and Small...... 14 3. Convergence and Limit...... 17 The Prime Numbers ...... 21 4. The Fascination of Infinite Series ...... 25 5. The Geometric Series ...... 29 6. More about Infinite Series...... 34 7. Interlude: An Excursion into the Number Concept .. 40 8. The Discovery of Irrational Numbers...... 44 1T, <1>, and e-Three Celebrated Irrationals...... 49 9. Cantor's New Look at the Infinite ...... 54 10. Beyond Infinity ...... 60

Part II. Geometric Infinity

11. Some Functions and Their Graphs ...... 68 Some Geometric Paradoxes Involving Infinity ...... 83 12. Inversion in a Circle ...... 88 13. Geographic Maps and Infinity...... 95 14. Tiling the Plane...... lO2

xv CONTENTS

15. A New Look at Geometry ...... 108 16. The Vain Search for Absolute Truth...... 118

Part III. Aesthetic Infinity

17. Rejoice the Infinite! ...... 136 18. The Mobius Strip...... 139 19. The Magic World of Mirrors ...... 149 20. Horror Vacui, Amor Infiniti ...... 155 21. Maurits C. Escher-Master of the Infinite...... 164 22. The Modern Kabbalists ...... 179

Part IV. Cosmological Infinity

23. The Ancient World...... 184 24. The New Cosmology ...... 190 25. The Horizons Are Receding...... 199 26. A Paradox and Its Aftermath ...... 204 27. The Expanding Universe...... 212 28. The Modern Atomists ...... 224 29. Which Way from Here? ...... 227

Epilogue...... 232 Appendix ...... 235 Bibliography...... 257 Index...... 265

XVi To Infinity and Beyond