The Unimaginable Mathematics of Borges' Library of Babel

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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

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

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 with 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.

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.)