The Unimaginable Mathematics of Borges' Library of Babel Read online

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  "The Library of Babel," from COLLECTED FICTIONS by Jorge Luis Borges, translated by Andrew Hurley, copyright © 1998 by Maria Kodama; translation copyright © 1998 by Penguin Putnam Inc. Used by permission of Viking Penguin, a division of Penguin Group (USA) Inc.

  Library of Congress Cataloging-in-Publication Data Bloch, William Goldbloom.

  The Unimaginable Mathematics of Borges' Library of Babel / William Goldbloom Bloch. p. cm.

  Includes bibliographical references and index. ISBN 978-0-19-533457-9

  1. Borges, Jorge Luis, 1899—1986—Knowledge—Mathematics.

  2. Mathematics and literature. 3. Mathematics—Philosophy. I. Title.

  PQ7797.B635Z63438 2008 868—dc22 2008017271

  987654321

  Printed in the United States of America on acid-free paper

  Contents

  We do not content ourselves with the life we have in ourselves and in our own being; we desire to live an imaginary life in the mind of others, and for this purpose we endeavor to shine. We labor unceasingly to adorn and preserve this imaginary existence and neglect the real.

  —Blaise Pascal, Pensées, no. 147

  Acknowledgments

  Preface

  Introduction

  The Library of Babel

  CHAPTER 1 Combinatorics: Contemplating Variations of the 23 Letters

  CHAPTER 2 Information Theory: Cataloging the Collection

  CHAPTER 3 Real Analysis: The Book of Sand

  CHAPTER 4 Topology and Cosmology: The Universe (Which Others Call the Library)

  CHAPTER 5 Geometry and Graph Theory: Ambiguity and Access

  CHAPTER 6 More Combinatorics: Disorderings into Order

  CHAPTER 7 A Homomorphism: Structure into Meaning

  CHAPTER 8 Critical Points

  CHAPTER 9 Openings

  Appendix—Dissecting the 3-Sphere

  Notations

  Notes

  Glossary

  Annotated Suggested Readings

  Bibliography

  Acknowledgments

  Pigmæos gigãtum humeris impositos plusquam ipsos gigantes videre.

  —Didacus Stella (Diego de Estella), In sacrosanctum Jesu Christi Domini nostri Evangelium secundum Lucam Enarrationum

  I say with Didacus Stella, a dwarf standing on the shoulders of a giant may see farther than a giant himself.

  —Robert Burton, Anatomy of Melancholy, "Democritus to the Reader"

  IT IS A PLEASURE TO ACKNOWLEDGE THE MANY DEBTS OF gratitude I owe; indeed, so much so that it's difficult to affix a starting point. Rather arbitrarily, I'll begin with Joe Roberts, the professor who introduced me to the concept of elegance in mathematics via the study of combinatorics. Around the same time, I read Rudy Rucker's Geometry, Relativity and the Fourth Dimension, which contains a lovely exposition of Riemann's century-old idea that a universe could be both finite and limitless. Another well-deserved "thank you" to the unremembered friend who, many years ago, put a copy of Labyrinths into my hand.

  Leaping to the present day, I thank Tricia Arnold for endowing the fellowship that enabled me to travel to Buenos Aires. In a similar vein, I thank Susanne Woods and Wheaton College for supporting this project with time, resources, and encouragement. Not surprisingly, two librarians, Martha Mitchell and TJ Sondermann, were extremely helpful in identifying and obtaining old books and journal articles linking mathematics and Borges. Another marvelous staff member at Wheaton, Kathy Rogers, consistently provided vital textual support.

  I am grateful to everyone in Buenos Aires who assisted me, most especially Fernando Palacio, cultural mediator and translator por excelencia. The Director of the National Library of Argentina, Silvio Maresca, and the Associate Director, Roberto Magliano, were kind enough to meet with me and do whatever was within their power to facilitate this project. Clara Baya, webmaster and semiofficial translator for the National Library, provided invaluable aid in guiding me first around the building and then around various rules that turned out to be surprisingly pliant. An anonymous guard at the old National Library and an anonymous librarian at the Miguel Cane Municipal Library were both also willing to bend rules and show me parts of their respective buildings that are generally off-limits to the public. The librarian, who was delighted that someone from the United States cared enough about Borges to visit the Miguel Cane Municipal Library, informed me that Argentine civil servants can't bear to read "The Library of Babel." Apparently, they take the Kafkaesque qualities of the tale quite personally, viewing the story as an extended slap against their daily work-life and their organizational systems.

  My colleagues from the Humanities, Michael Drout and Hector Medina, acted as a pushmi-pullyu—see Lofting, 77—85—in jump-starting the project, in devoting the time to read and comment on my manuscript, and to talk over many of its points with me. Drout also provided etymologies for me when necessary, encouraged me to create the word "slimber" out of "slim" and "limber," and reassured me whenever I feared that I was using too many infinitives.

  Eric Denton coordinated the first group reading and offered salient suggestions and collegial encouragement duly leavened with cynicism. ("Bill, your book is neither fish nor fowl.")

  Anni Baker, Bernard Bloch, Tom Brooks, Michael Chesla, Bev Clark, Betsey Dyer, Lisa Lebduska, Shelly Leibowitz, Shannon Miller, Laura Muller, Rolf Nelson, John Partridge, Joel Relihan, Dorothea Rockburne, Pamela Stafford, David Wulff, and Paul Zeitz read this book in manuscript form and provided worthy and meaningful feedback. Any errors or infelicities remaining are, of course, solely my own.

  Domingo Ledezma helped me out by translating some thorny passages in the story and Doug Jungreis confirmed my intuitions about Hopf fibrations. Julio Ortega encouraged me and introduced me to Borges' widow, the remarkable Maria Kodama.

  John Wronoski of Lame Duck Books in Cambridge, Massachusetts first let me hold Borges' autograph manuscript of "La biblioteca de Babel" in my shaking hands, and then kindly let me use images of it in this volume. (By the way, the manuscript is for sale for approximately $650,000. Prior to learning this, I never actually ached to be a multimillionaire, but now I hereby publicly promise that if this book sells over three million copies, I will cheerfully call Mr. Wronoski to negotiate a price.)

  Throughout the process, my editor, Michael Penn, combined abiding wisdom, keen grammatical insight, calming patience, and sly humor. Working with him was a continuous pleasure. Stefano Imbert, the illustrator, did a marvelous job capturing the ambience of the Library. Other people associated wi
th Oxford University Press who helped shape the final result are Ned Sears, Stephen Dodson, and Keith Faivre.

  On a number of occasions, my mother-in-law and my parents gave generously of their time and energy by watching my young children, allowing me to devote myself to this work. Speaking of my children, Dylan always loved the "pokey things" in the illustrations and Levi was always willing to cheer me up with a cartwheel performance. Finally, my wife Ingrid tolerated my obsessions, disjunctions, and corporeal absences as I wrangled with various parts of this book. Her multiform support was, and continues to be, vital and cherished.

  Thank you, one and all.

  Preface

  One feels right away that this is the kingdom of books. People working at the library commune with books, with the life reflected in them, and so become almost reflections of real-life human beings.

  —Isaac Babel, "The Public Library"

  "WHO IS THE INTENDED AUDIENCE FOR THIS WORK IN progress?" This question, asked almost apologetically by a friend, stumped me for only a fraction of a second. With the clarity and explosiveness usually reserved for a rare mathematical insight, the answer burst from me: Umberto Eco! Polymath, brilliant semiotician, editor of the journal Variaciones Borges, interpreter of "The Library of Babel," and a favorite author for many years—Eco struck me as the ideal reader of this writing. (And Umberto, I hope you do read and enjoy this, someday.)

  Of the more than six billion people who are not Umberto Eco, I imagine that those who'd find this work appealing would share, to varying degrees, the following traits: a familiarity with and affinity for Borges' works, especially "The Library of Babel"; a nodding, perhaps cautious, acquaintance with the thought that mathematics might not be the root of all evil; and the habit of rereading sentences, paragraphs, and stories for sheer delight, as well for playing with the superpositions of layers of available meanings.

  While it's possible to set up a straw man and use it to wonder which way of presenting information is "better," I take the view that the approaches are complementary; they aren't two opponents locked into a zero-sum game for which one side must prevail. So, since part of my not-so-hidden agenda is to persuade those of a literary temperament that mathematics can be more than the "problem/solution" model of much rudimentary education, I present a Venn diagram that visually encapsulates the speculations of the previous paragraph (figure 1).

  The intended audience is the intersection of the three different sets of character traits. Judging mainly from the steady sales of Borges' fiction, I have managed to convince myself that besides you (presumably), there are at least several hundred thousand people who fit this description.

  If, however, an unimaginative education or a particularly unpleasant teacher left a lingering distaste for all things mathematical, I hope this book acts as a corrective. Mathematics can be creative, whimsical, and revelatory all at once. More to the point, as embodied in the different meanings of the word "analysis," it is simultaneously a process and an intellectual structure. Borges, a great imbiber of mathematics, seems to have understood this idea and instantiated it in many of his stories—most especially "The Library of Babel." His imagination works in, through, out, about, and all around logical strictures.

  Conversely, for those of a mathematical bent who've not read Borges, I hope this volume inspires two things: a desire to explore more of Borges' work—there are many riches to be found—and, equally, a desire to learn more about the math tools I employ. We, as a society, are gifted these days; many books introducing math to the casual reader are readily available.

  The chapters that are mathematical in nature will generally begin with the introduction of a mathematical idea. Some exposition, and perhaps a few examples, are given to help concretize the concept. Finally, the ideas will be applied to some aspects of "The Library of Babel" towards the desired end of producing an unimaginable (or unimagined) result.

  Andrew Wiles, who proved Fermat's last theorem, memorably analogized the process of doing mathematics as follows:

  You enter the first room of the mansion and it's completely dark. You stumble around bumping into the furniture but gradually you learn where each piece of furniture is. Finally, after six months or so, you find the light switch, you turn it on, and suddenly it's all illuminated. You can see exactly where you were. Then you move into the next room and spend another six months in the dark. (Singh, pp. 236—37)

  Reading the math chapters of this work might be likened to stumbling around in a dark room, bumping into furniture, and finally, after finding the light switch, learning that you're not in a mansion after all, but rather facing away from the screen in a movie theater, and that the switch is really a fire alarm.

  After the suite of introductory material comes the touchstone for this work: Andrew Hurley's superb translation of "The Library of Babel." After the story, and unlike most math books, the chapters are logically independent and can be dipped and skimmed as fancy dictates. (Of course, some intratextual references are unavoidable.) Although I've endeavored to structure the book so that it may be enjoyed from start to finish, based on predilections, nonlinear routes may be better suited for different kinds of readers.

  In fact, it's safe to say that there are three main themes woven into this book. The first one digs into the Library, peels back layers uncovering nifty ideas, and then runs with them for a while. The second thread is found mostly in the "Math Aftermath" sections appended to the chapters: in them, I develop the mathematics behind the ideas to a greater degree and, in some cases, give step-by-step derivations for formulas used in the main body of the chapter. (Allow me to emphasize that the Math Aftermaths are—I hope—clear and engaging, but they certainly aren't required in order to understand and enjoy any other parts of the book.) The third focus is on literary aspects of the story and Borges; the chapters playing with these motifs come after those concerned with the math.

  In the first chapter, "Combinatorics: Contemplating Variations of the 23 Letters," I use millennia-old ideas, alluded to in the story itself, to calculate the number of books in the Library. Once the basic concept of exponential notation is absorbed, the number is unexpectedly easy to find; it is understanding the magnitude ofthat number that occupies the bulk of the chapter. A number of previous critics also calculate this number, and several have provided similar means of understanding its size. By contrast, I fully explain the underlying mathematics and, moreover, add a new twist to the calculation. Expanding on some of the ideas raised, the Math Aftermath shows how to use a property ofthe logarithm function to recast the number of distinct books of the Library in terms more familiar, more amenable to our understanding. The chapter ends with the derivation of an ancient counting formula.

  After that, in "Information Theory: Cataloging the Collection," I consider the meaning of a catalogue for the Library and the forms that it might take. The Math Aftermath takes some basic results in number theory and applies them to aspects of the Library and the unknowability of certain pieces of compressed information. Then, in "Real Analysis: The Book of Sand," I apply elegant ideas from the seventeenth century and counterintuitive ideas of the twentieth century to the "Book of Sand" described in the final footnote of the story. Three variations of the Book, springing from three different interpretations of the phrase "infinitely thin," are outlined.

  Next, in "Topology and Cosmology: The Universe (Which Others Call the Library)," I employ late nineteenth- and early twentieth-century mathematics to explore possible shapes ofthe Library. Ultimately, I propose a rapprochement between the apparently conflicting views outlined by the narrator of the story. In the Math Aftermath section of the chapter, the discussion moves into somewhat more sophisticated domains by introducing two possible variations of the Library, each of which possesses noteworthy traits, one example being nonorientability.

  Following this, in "Geometry and Graph Theory: Ambiguity and Access," I use Borges' descriptions of the Library to abstract the architecture of each floor of the Library and use
it to unfold a surprising consequence. Interested readers can continue the tale of the chapter by following along in the Math Aftermath as I unpack an even stronger mathematical result stemming from the story.

  The next chapter, "More Combinatorics: Disorderings into Order," is a kind of a fantasia on the possibilities inherent in ordering and disordering the distribution of books in the Library, and it concludes the mathematical section of the book.

  After this, despite a desire to resist interpretation of the story, by drawing on metaphors from Alan Turing and information theory, I propose a new reading in "A Homomorphism: Structure into Meaning." Following that, in "Critical Points," prior work on "The Library of Babel" serves as a springboard to some compelling ruminations about life in the Library and other topics. Finally, in "Openings," a "What did he know and when did he know it? How did he know it?" attitude is adopted vis-à-vis Borges and mathematics. Was he a mathematician? A philosopher? A visionary writer blithely unaware of the depth of his insights?

  The literary chapters are followed by a cortege of back matter, beginning with an appendix, "Dissecting the 3-Sphere," for those who want a refresher on how equations capture the characteristics and properties of multidimensional spheres. The appendix may sound scarier than it really is; I don't use much beyond the Pythagorean theorem, and I even provide a review of that.

  In general, I avoid mathematical notation beyond that encountered in middle school or perhaps the early years of high school. However, in case it is unfamiliar, following the appendix is a short list of notations with definitions. Speaking of definitions, there's a lot to say on the matter. Mathematics is an intellectual discipline built on definitions; indeed, the axioms of mathematics are exactly definitions that have been accepted as plausible and true by the concerted critical faculty of millions of thinkers around the world aggregated over the past several millennia. Moreover, these days great theoretical breakthroughs occur when brilliant mathematicians see new interrelations and make definitions that enable a cascade of untold consequences to be discovered by other workers in the field. For us, definitions will be considerably more prosaic; I italicize words that strike me as being of a technical nature, outside the usual range of quotidian use, and provide definitions in a glossary following the notations and the endnotes.