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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 Vll PREFACE 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 ITWO-YEAR CCLLE6 ~, .,). 1 MATHEMATICS J()01tNAL =-!£II[lIIIII: ~u. --.r ___ _ THISlsstlE : ..._ "" ...... '................... .......... "'- ....... _--.- 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.

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