The Unimaginable Mathematics of Borges' Library of Babel (2 page)

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.

As a reader,
when I encounter an endnote, I'm compelled almost against my will to flip to
the back of the book to learn what the endnote says.
¹
As I writer, I find that despite my best efforts to incorporate
them into the body of the book, my work includes diverting digressions, fine
points of mathematics that might interest only specialists, and citations to
other works. All of these are consigned to the endnotes.

After the
glossary, an annotated list of suggested readings is provided for those with
curiosity primed to learn more of the mathematics used in the book. A
bibliography of references cited or consulted rounds out the end matter.

We adore chaos because we
love to produce order.

—M.
C. Escher

 IT’S AN IRONIC JOKE THAT
BORGES WOULD HAVE
appreciated: I am a
mathematician who, lacking Spanish, perforce reads "The Library of
Babel" in translation. Furthermore, although I bring several thousand
years of theory to bear on the story, none of it is literary theory.

Having
issued these caveats, it is my purpose to make explicit a number of
mathematical ideas inherent in the story. My goal in this task is not to reduce
the story in any capacity; rather it is to enrich and edify the reader by
glossing the intellectual margins and substructures. Borges was a consummate
synthesist; his lapidary prose sparkles and reveals unexpected depths when
examined from any angle or perspective. I submit that because of his well-known
affection for mathematics, exploring the story through the eyes of a
mathematician is a dynamic, useful, and necessary addition to the body of
Borgesian criticism.

In what
follows, I assume no special mathematical knowledge. I only ask that the reader
trust that I am a tour guide through a labyrinth, like that marble pathway on
the floor of the cathedral at Chartres, not the gatekeeper of a Stygian maze
without center or exit (figure 2). Beyond enhancing the story, the reader's reward
will be an exposure to some intriguing and entrancing mathematical ideas.