The Unimaginable Mathematics of Borges' Library of Babel (4 page)

Some five
hundred years ago, the chief of one of the upper hexagons
²
came across a book as jumbled as all the
others, but containing almost two pages of homogeneous lines. He showed his
find to a traveling decipherer, who told him that the lines were written in
Portuguese; others said it was Yiddish. Within the century experts had
determined what the language actually was: a Samoyed-Lithuanian dialect of
Guarani, with inflections from classical Arabic. The content was also
determined: the rudiments of combinatory analysis, illustrated with examples of
endlessly repeating variations. Those examples allowed a librarian of genius to
discover the fundamental law of the Library. This philosopher observed that all
books, however different from one another they might be, consist of identical
elements: the space, the period, the comma, and the twenty-two letters of the
alphabet. He also posited a fact which all travelers have since confirmed:
In all the Library, there are no two identical books.
From those
incontrovertible premises, the librarian deduced that the Library is
"total"—perfect, complete, and whole—and that its bookshelves contain
all possible combinations of the twenty-two orthographic symbols (a number
which, though unimaginably vast, is not infinite)—that is, all that is able to
be expressed, in every language.
All
—the detailed history of the future,
the autobiographies of the archangels, the faithful catalog of the Library,
thousands and thousands of false catalogs, the proof of the falsity of those
false catalogs, a proof of the falsity of the
true
catalog, the gnostic
gospel of Basilides, the commentary upon that gospel, the commentary on the commentary
on that gospel, the true story of your death, the translation of every book
into every language, the interpolations of every book into all books, the
treatise Bede could have written (but did not) on the mythology of the Saxon
people, the lost books of Tacitus.

When it was
announced that the Library contained all books, the first reaction was
unbounded joy. All men felt themselves the possessors of an intact and secret
treasure. There was no personal problem, no world problem, whose eloquent
solution did not exist—somewhere in some hexagon. The universe was justified;
the universe suddenly became congruent with the unlimited width and breadth of
humankind's hope. At that period there was much talk of The Vindications—books
of
apologia
and prophecies that would vindicate for all time the actions
of every person in the universe and that held wondrous arcana for men's
futures. Thousands of greedy individuals abandoned their sweet native hexagons
and rushed downstairs, upstairs, spurred by the vain desire to find their
Vindication. These pilgrims squabbled in the narrow corridors, muttered dark
imprecations, strangled one another on the divine staircases, threw deceiving
volumes down ventilation shafts, were themselves hurled to their deaths by men
of distant regions. Others went insane. . . . The Vindications do exist (I have
seen two of them, which refer to persons in the future, persons perhaps not
imaginary), but those who went in quest of them failed to recall that the
chance of a man's finding his own Vindication, or some perfidious version of
his own, can be calculated to be zero.

At that same
period there was also hope that the fundamental mysteries of mankind—the origin
of the Library and of time—might be revealed. In all likelihood those profound
mysteries can indeed be explained in words; if the language of the philosophers
is not sufficient, then the multiform Library must surely have produced the
extraordinary language that is required, together with the words and grammar of
that language. For four centuries, men have been scouring the hexagons. . ..
There are official searchers, the "inquisitors." I have seen them
about their tasks: they arrive exhausted at some hexagon, they talk about a
staircase that nearly killed them—rungs were missing—they speak with the
librarian about galleries and staircases, and, once in a while, they take up
the nearest book and leaf through it, searching for disgraceful or dishonorable
words. Clearly, no one expects to discover anything.

That
unbridled hopefulness was succeeded, naturally enough, by a similarly
disproportionate depression. The certainty that some bookshelf in some hexagon
contained precious books, yet that those precious books were forever out of
reach, was almost unbearable. One blasphemous sect proposed that the searches
be discontinued and that all men shuffle letters and symbols until those
canonical books, through some improbable stroke of chance, had been
constructed. The authorities were forced to issue strict orders. The sect
disappeared, but in my childhood I have seen old men who for long periods would
hide in the latrines with metal disks and a forbidden dice cup, feebly
mimicking the divine disorder.

Others,
going about it in the opposite way, thought the first thing to do was eliminate
all worthless books. They would invade the hexagons, show credentials that were
not always false, leaf disgustedly through a volume, and condemn entire walls
of books. It is to their hygienic, ascetic rage that we lay the senseless loss
of millions of volumes. Their name is execrated today, but those who grieve
over the "treasures" destroyed in that frenzy overlook two widely
acknowledged facts: One, that the Library is so huge that any reduction by
human hands must be infinitesimal. And two, that each book is unique and
irreplaceable, but (since the Library is total) there are always several
hundred thousand imperfect facsimiles—books that differ by no more than a
single letter, or a comma. Despite general opinion, I daresay that the
consequences of the depredations committed by the Purifiers have been
exaggerated by the horror those same fanatics inspired. They were spurred on by
the holy zeal to reach—someday, through unrelenting effort—the books of the
Crimson Hexagon—books smaller than natural books, books omnipotent,
illustrated, and magical.

We also have
knowledge of another superstition from that period: belief in what was termed
the Book-Man. On some shelf in some hexagon, it was argued, there must exist a
book that is the cipher and perfect compendium
of all other books,
and
some librarian must have examined that book; this librarian is analogous to a
god. In the language of this zone there are still vestiges of the sect that
worshiped that distant librarian. Many have gone in search of Him. For a
hundred years, men beat every possible path—and every path in vain. How was one
to locate the idolized secret hexagon that sheltered Him? Someone proposed
searching by regression: To locate book A, first consult book B, which tells
where book A can be found; to locate book B, first consult book C, and so on,
to infinity. ... It is in ventures such as these that I have squandered and
spent my years. I cannot think it unlikely that there is such a total book
³
on some shelf in
the universe. I pray to the unknown gods that some man—even a single man, tens
of centuries ago—has perused and read that book. If the honor and wisdom and
joy of such a reading are not to be my own, then let them be for others. Let
heaven exist, though my own place be in hell. Let me be tortured and battered
and annihilated, but let there be one instant, one creature, wherein thy
enormous Library may find its justification.

Infidels
claim that the rule in the Library is not "sense," but
"nonsense," and that "rationality" (even humble, pure
coherence) is an almost miraculous exception. They speak, I know, of "the
feverish Library, whose random volumes constantly threaten to transmogrify into
others, so that they affirm all things, deny all things, and confound and confuse
all things, like some mad and hallucinating deity." Those words, which not
only proclaim disorder but exemplify it as well, prove, as all can see, the
infidels' deplorable taste and desperate ignorance. For while the Library
contains all verbal structures, all the variations allowed by the twenty-five
orthographic symbols, it includes not a single absolute piece of nonsense. It
would be pointless to observe that the finest volume of all the many hexagons
that I myself administer is titled
Combed Thunder,
while another is
titled
The Plaster Cramp,
and another,
Axaxaxas mlo.
Those
phrases, at first apparently incoherent, are undoubtedly susceptible to
cryptographic or allegorical "reading"; that reading, that
justification of the words' order and existence, is itself verbal and,
ex
hypothesi,
already contained somewhere in the Library. There is no
combination of characters one can make
—dhcmrlchtdj,
for example—that the
divine Library has not foreseen and that in one or more of its secret tongues
does not hide a terrible significance. There is no syllable one can speak that
is not filled with tenderness and terror, that is not, in one of those
languages, the mighty name of a god. To speak is to commit tautologies. This
pointless, verbose epistle already exists in one of the thirty volumes of the
five bookshelves in one of the countless hexagons—as does its refutation. (A
number
n
of the possible languages employ the same vocabulary; in some
of them, the
symbol
"library" possesses the correct definition
"everlasting, ubiquitous system of hexagonal galleries," while a
library—the thing—is a loaf of bread or a pyramid or something else, and the
six words that define it themselves have other definitions. You who read me—are
you certain you understand my language?)

Methodical
composition distracts me from the present condition of humanity. The certainty
that everything has already been written annuls us, or renders us phantasmal. I
know districts in which the young people prostrate themselves before books and
like savages kiss their pages, though they cannot read a letter. Epidemics,
heretical discords, pilgrimages that inevitably degenerate into brigandage have
decimated the population. I believe I mentioned the suicides, which are more
and more frequent every year. I am perhaps misled by old age and fear, but I
suspect that the human species—the
only
species—teeters at the verge of
extinction, yet that the Library—enlightened, solitary, infinite, perfectly
unmoving, armed with precious volumes, pointless, incorruptible, and
secret—will endure.

I have just
written the word "infinite." I have not included that adjective out
of mere rhetorical habit; I hereby state that it is not illogical to think that
the world is infinite. Those who believe it to have limits hypothesize that in
some remote place or places the corridors and staircases and hexagons may,
inconceivably, end—which is absurd. And yet those who picture the world as
unlimited forget that the number of possible books is
not.
I will be
bold enough to suggest this solution to the ancient problem:
The Library is
unlimited but periodic.
If an eternal traveler should journey in any
direction, he would find after untold centuries that the same volumes are
repeated in the same disorder—which, repeated, becomes order: the Order. My
solitude is cheered by that elegant hope.
⁴

Mar
del Plata, 1941

1)
     
The original manuscript has neither numbers
nor capital letters; punctuation is limited to the comma and the period. Those
two marks, the space, and the twenty-two letters of the alphabet are the
twenty-five sufficient symbols that our unknown author is referring to. [Ed.
note.]

2)
     
In earlier times, there was one man for every three
hexagons. Suicide and diseases of the lung have played havoc with that
proportion. An unspeakably melancholy memory: I have sometimes traveled for
nights on end, down corridors and polished staircases, without coming across a
single librarian.

3)
     
I repeat: In order for a book to exist, it is
sufficient that it be
possible.
Only the impossible is excluded. For example, no book is also a
staircase, though there are no doubt books that discuss and deny and prove that
possibility, and others whose structure corresponds to that of a staircase.

4)
     
Letizia Alvarez de Toledo has observed that the vast
Library is pointless; strictly speaking, all that is required is
a single volume,
of the common
size, printed in nine- or ten-point type, that would consist of an infinite
number of infinitely thin pages. (In the early seventeenth century, Cavalieri
stated that every solid body is the super-position of an infinite number of
planes.) Using that silken
vademecum
would not be easy: each apparent page would open into other
similar pages; the inconceivable middle page would have no "back."

Combinatorics

Contemplating Variations
of the 23 Letters

There are some, King Gelon,
who think that the number of the sand is infinite in multitude; and I mean by
the sand not only that which exists about Syracuse and the rest of Sicily but
also that which is found in every region whether inhabited or uninhabited.
Again there are some who, without regarding it as infinite, yet think that no
number has been named which is great enough to exceed its multitude.

—Archimedes,
The Sand Reckoner