


To Infinity and Beyond: 
PrefaceThe 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 (18621943) There is a story attributed to David Hilbert, the preeminent mathematician whose quotation appears above. A man walked into a hotel late one night and asked for a room. "Sorry, we don't have any more vacancies," replied the owner, "but let's see, perhaps 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 astonishment 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: This famous anecdote tells, in a way, the entire story of infinity. It is a story of intriguing paradoxes and seemingly impossible situations which have puzzled mankind for more than two millennia. The roots of these paradoxes lie in mathematics, and it is this discipline which has offered the most fruitful path towards their eventual resolution. The clarification and demystification of the infinite was fully accomplished only in our own century, and even this feat cannot be regarded as the final word. Like every science, 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 segmentor any quantity, for that matteris 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 scalethe 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 compendium recently compiled by the Mathematical Society of Japan,^{1} 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 mathematicshow to countis 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 assumptionthat we can, at least in principle, extend a line indefinitely in both directions. Even in such seemingly "finite" 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 assume that an infinite number of tosses would produce an equal outcome. My first encounter with infinity was as a young boy. I was given a bookit was the Haggadah, the story of the Exodus from Egypton whose cover was a picture of a young boy holding the very same book in his hand. When I looked carefully I could see the same picture on the cover of the small Haggadah the boy was holding. It may even be that the picture showed up again in the picture's pictureI can't remember for sure. But I do remember that my mind was boggled by the thought that if it were possible to continue this process, it would go on forever! An 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 limas 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 propellershaped object was hinged. It looked as though the elongated object was meant to turn freely on ins hinges, so I gently touched it, anticipating it to commence its motion. Instead, a hidden alarm went off, and with such a shrill sound that I was quite scared. After my initial shock was gone, I could hear an inner voice in 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 chronological order. My story is, to some extent, a subjective oneit 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 nonexpert, without at the same time compromising the standards of rigor his professional peers expect of him. This dilemma is all the more acute in mathematics, which relies almost entirely on a nonverbal language of symbols and equations. I hope I have succeeded in properly addressing this problem. 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 delegated 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 literary lines on infinity. Many friends helped me with this work, and I owe them many thanks. Particularly, I am indebted to my colleagues Wilbur Hoppe and Robert Langer, who have read large parts of the manuscript 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 dedicated; 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 WisconsinEau 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 Birkhäuser 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 nose. At the outset of every discussion, a mathematician must define his symbols and notation. Let it therefore be known that by "he" I mean "he or she," by "him," "him or her," etc. If I am using the more traditional language in this book, it is solely for the sake of brevity. Rochester, Michigan, ^{1}English translation published by The MIT Press, 1980. ^{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 place but a vital part of the sculpture, brings the solid and finite earth into the equation. That block of stone is our base from which to contemplate infinity."
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