The Unimaginable Mathematics of Borges' Library of Babel (3 page)

Borges was a master of
understating ideas, allowing them the possibility of gathering heft and power,
of generating their own gravity. I'm under no delusion that he traced out all
the consequences of the dormant mathematics I uncover. I allow myselfthe
ambition, though, to paraphrase what Borges wrote in a forward and hope that
this book would have taught him many things about himself (see Barrenchea, p.
vii).

I request a
last indulgence from the reader. The introductory material, thus far, has been
written in the friendly and confiding first person singular voice. Starting in
the next paragraph, I will inhabit the first person plural for the duration of
the mathematical expositions. This should not be construed as a "royal
we." It has been a construct of the community of mathematicians for
centuries and it traditionally signifies two ideas: that "we" are all
in consultation with each other through space and time, making use of each
other's insights and ideas to advance the ongoing human project of mathematics,
and that "we"—the author and reader—are together following the
sequences of logical ideas that lead to inexorable, and sometimes poetic,
conclusions.

A word, too,
about the language in the book. We started our college years intending to be
some sort of creative writer. Beyond the insight mathematics offered into the
natural world and epiphenomena of life, and beyond the aesthetic joy at
understanding how the iron rules of logic crystallize a good proof into a work
of art, one of the reasons we turned to math was the lilt and rhythm of the
"if-then" syntax coupled with the musicality of words often repeated,
such as "thus," "hence," "suppose," and
"let." We hope our readers might develop an ear for this music, too.

We close the
introduction by offering several related disclaimers. Mathematics, like any
discipline, is not a monolith; it's a sprawling agglutination of overlapping
and intersecting fields and specialties: one's talents, tastes, and beliefs
determine individual focus. We carefully checked and rechecked our ideas,
mathematics, and figures. To the best of our knowledge, there are no mistakes.
However, a different mathematician might well expose divergent mathematical
themes from the story and utilize different sets of ideas to explain them.

Furthermore,
there's a natural tendency for an individual reaching across traditional
boundaries to be perceived as a universal embodiment of the foreign, the other.
Although our inductions and deductions are correct, some mathematicians might
issue philosophic challenges to underlying assumptions, especially in the
chapters "Real Analysis" and "More Combinatorics."
Consequently, no one, including the author, should be seen as a Representative
or Ambassador, speaking in one voice for an ideologically unified Entity of
Mathematicians: such an Entity of Mathematicians simply doesn't exist. (Lest
this be subject to misinterpretation, allow us to note that
all
mathematicians would agree on the centrality of logically consistent deductions
and derivations from agreed-upon axioms.)

It's
important to bear in mind that the mathematical expositions contained herein
are not rigorously developed, nor are they intended as comprehensive introductions
to the various theories. Just as a stirring musical performance will not
transform a concertgoer into a musician, composer, lyricist, musicologist, or
music critic, so this book won't transform a reader into any kind of a
mathematician. However, just as a concert may move, inspire, or transfigure a
listener, so we hope that this book will stimulate, dazzle, and expand its
readers.

Finally,
about the title of the book: why the word "unimaginable"? By way of
an answer, we note that in his sixth Meditation, Descartes makes clear the
distinction between simply naming a thing and visualizing it in a clear,
precise way that allows for mental manipulations.

I note first
the difference between imagination and pure intellection or conception. For
example, when I imagine a triangle, I not only conceive it as a figure composed
of three lines, but moreover consider these three lines as being present by the
power and internal application of my mind, and that is properly what I call
imagining. Now if I wish to think of a chiliagon, I indeed rightly conceive
that it is a figure composed of a thousand sides, as easily as I conceive that
a triangle is a figure composed of only three sides; but I cannot imagine the
thousand sides of a chiliagon, as I do the three of a triangle, neither, so to
speak, can I look upon them as present with the eyes of my mind.

Some of the ideas we'll talk
about, such as titanic numbers and higher dimensions, are unimaginable in this
sense. We can give names to the ideas, use metaphors to approach them, give
simple examples to substitute in as models, and try to find a consistent set of
rules and mathematical objects that encapsulate the essence of the ideas—but we
will never be able to visualize them any more than we could Descartes' thousand-sided
chiliagon. Indeed, our task as your guide is to trigger the processes by which
you build intuition and insight into the Unimaginable.

The Unimaginable Mathematics of Borges' Library of Babel

Jorge
Luis Borges

By this art you may
contemplate the variation of the 23 letters....

—Anatomy
of Melancholy,
Pt. 2, Sec. II, Mem. IV

THE UNIVERSE (WHICH OTHERS CALL THE
Library)
is composed of an indefinite, perhaps infinite number of hexagonal galleries.
In the center of each gallery is a ventilation shaft, bounded by a low railing.
From any hexagon one can see the floors above and below—one after another,
endlessly. The arrangement of the galleries is always the same: Twenty
bookshelves, five to each side, line four of the hexagon's six sides; the
height of the bookshelves, floor to ceiling, is hardly greater than the height
of a normal librarian. One of the hexagon's free sides opens onto a narrow sort
of vestibule, which in turn opens onto another gallery, identical to the
first—identical in fact to all. To the left and right of the vestibule are two
tiny compartments. One is for sleeping, upright; the other, for satisfying
one's physical necessities. Through this space, too, there passes a spiral
staircase, which winds upward and downward into the remotest distance. In the
vestibule there is a mirror, which faithfully duplicates appearances. Men often
infer from this mirror that the Library is not infinite—if it were, what need
would there be for that illusory replication? I prefer to dream that burnished
surfaces are a figuration and promise of the infinite. . . . Light is provided
by certain spherical fruits that bear the name "bulbs." There are two
of these bulbs in each hexagon, set crosswise. The light they give is insufficient,
and unceasing.

Like all the
men of the Library, in my younger days I traveled; I have journeyed in quest of
a book, perhaps the catalog of catalogs. Now that my eyes can hardly make out
what I myself have written, I am preparing to die, a few leagues from the
hexagon where I was born. When I am dead, compassionate hands will throw me
over the railing; my tomb will be the unfathomable air, my body will sink for
ages, and will decay and dissolve in the wind engendered by my fall, which
shall be infinite. I declare that the Library is endless. Idealists argue that
the hexagonal rooms are the necessary shape of absolute space, or at least of
our
perception
of space. They argue that a triangular or pentagonal
chamber is inconceivable. (Mystics claim that their ecstasies reveal to them a
circular chamber containing an enormous circular book with a continuous spine
that goes completely around the walls. But their testimony is suspect, their
words obscure. That cyclical book is God.) Let it suffice for the moment that I
repeat the classic dictum:
The Library is a sphere whose exact center is any
hexagon and whose circumference is unattainable.

Each wall of
each hexagon is furnished with five bookshelves; each bookshelf holds
thirty-two books identical in format; each book contains four hundred ten
pages; each page, forty lines; each line, approximately eighty black letters.
There are also letters on the front cover of each book; those letters neither
indicate nor prefigure what the pages inside will say. I am aware that that
lack of correspondence once struck men as mysterious. Before summarizing the
solution of the mystery (whose discovery, in spite of its tragic consequences,
is perhaps the most important event in all history), I wish to recall a few
axioms.

First:
The Library has existed
ab
æ
ternitate. That
truth, whose immediate corollary is the future eternity of the world, no
rational mind can doubt. Man, the imperfect librarian, may be the work of
chance or of malevolent demiurges; the universe, with its elegant appointments—
its bookshelves, its enigmatic books, its indefatigable staircases for the
traveler, and its water closets for the seated librarian—can only be the
handiwork of a god. In order to grasp the distance that separates the human and
the divine, one has only to compare these crude trembling symbols which my
fallible hand scrawls on the cover of a book with the organic letters
inside—neat, delicate, deep black, and inimitably symmetrical.

Second:
There are twenty-five orthographic symbols
.
¹
That discovery enabled mankind, three
hundred years ago, to formulate a general theory of the Library and thereby
satisfactorily solve the riddle that no conjecture had been able to divine—the
formless and chaotic nature of virtually all books. One book, which my father
once saw in a hexagon in circuit 15—94, consisted of the letters M C V
perversely repeated from the first line to the last. Another (much consulted in
this zone) is a mere labyrinth of letters whose penultimate page contains the
phrase
O Time thy pyramids.
This much is known: For every rational line
or forthright statement there are leagues of senseless cacophony, verbal
nonsense, and incoherency. (I know of one semibarbarous zone whose librarians
repudiate the "vain and superstitious habit" of trying to find sense
in books, equating such a quest with attempting to find meaning in dreams or in
the chaotic lines of the palm of one's hand. . .. They will acknowledge that
the inventors of writing imitated the twenty-five natural symbols, but contend
that that adoption was fortuitous, coincidental, and that books in themselves
have no meaning. That argument, as we shall see, is not entirely fallacious.)

For many
years it was believed that those impenetrable books were in ancient or
far-distant languages. It is true that the most ancient peoples, the first
librarians, employed a language quite different from the one we speak today; it
is true that a few miles to the right, our language devolves into dialect and
that ninety floors above, it becomes incomprehensible. All of that, I repeat,
is true—but four hundred ten pages of unvarying M C V's cannot belong to any
language, however dialectal or primitive it may be. Some have suggested that
each letter influences the next, and that the value of M C V on page 71, line
3, is not the value of the same series on another line of another page, but
that vague thesis has not met with any great acceptance. Others have mentioned
the possibility of codes; that conjecture has been universally accepted, though
not in the sense in which its originators formulated it.