The Unimaginable Mathematics of Borges' Library of Babel (11 page)

A form a
catalogue might take in principle is: Book (identifiers), Hexagon (location),
Shelf (only 20 per hexagon), Position on Shelf (only 32 books per shelf).
Perhaps surprisingly,
self-referentiality
is not a problem. A volume of
the catalogue, say the tenth, residing in Hexagon 39, Shelf 20, Position 14,
could well be marked on the spine "Catalogue Volume Ten," and
correctly describe itself as the tenth volume of the catalogue and specify its
location in Position 14, on Shelf 20, in Hexagon 39: there is no paradox.
However, beginning with the obvious, here are some of the difficulties that
arise.

Clearly, the
Library holds far too many books to be listed in one volume; any catalogue
would necessarily consist of a vast number of volumes, which, perversely, are
apt to be scattered throughout the Library. Indeed, reminiscent of the approach
of another ofBorges' stories, "The Approach to Al-Mu'tasim," and of
the lines in "The Library of Babel,"

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. . . .

an immortal librarian trying
to track down a specific book likely has a better chance by making an orderly
search of the entire library, rather than finding a true catalogue entry for
the book. Every plausible entry from any plausible candidate catalogue volume
would have to be tracked down, including regressive scavenger hunts. An
immortal librarian would spend a lot of time traversing the Library,
ping-ponging back and forth between different books purporting to be volumes of
a true catalogue.

After
revealing the nature of the Library, the librarian notes that contained in the
Library are "the faithful catalog of the Library, thousands and thousands
of false catalogues, the proof of the falsity of those false catalogs, a proof
of the falsity of the true catalogue..." This, then, is the second problem
of any catalogue: the only way to verify its faithfulness would be to look up
each book. Furthermore, the likelihood of any book being located within a
distance walkable within the life span of a mortal librarian is, to all intents
and purposes,
zero.
Sadly, even if we were fortunate enough to possess a
true catalogue entry for our Vindication, presumably our Vindication would
merely give details of the death we encountered while spending our life walking
in a fruitless attempt to obtain the Vindication. (Recall in "The Library
of Babel," Borges describes Vindications as "books of apology and
prophecy which vindicated for all time the acts of every man in the universe
and retained prodigious arcana for his future.")

Let's
consider the first category of information found on library cards, that which
uniquely specifies the book.
Authorship is moot.
One might argue that
the God(s), or the Builder(s) of the Library, is (are) the author(s) of any
book. One might also make a claim that the author is an algorithm embodied in a
very short computer program which would, given time and resources, generate all
possible variations of 25 orthographic symbols in strings of length 1,312,000.
One could make the Borgesian argument that One Man is the author of
all
books.

For that
matter, the writer Pierre Menard, a quixotic character in Borges' story
"The Don Quixote of Pierre Menard," may as well be credited with
authorship of all the books in the Library.

Certainly there are many, many
books whose first page resembles the one in figure 4. How many such books?
Specifying one page means that 80 symbols for each of 40 lines are
"frozen." This means that out of the 1,312,000 symbols of a book, the
first 3,200 are taken, leaving 1,308,800 spaces to fill. By the work of the
"Combinatorics" chapter, there are thus precisely 25
^1,308,800
books with a first page exactly the same as the depicted title page. (Using
logarithms as in the first Math Aftermath, this number is seen to be
approximately 10
^1,829,623
books.) Viewed from a complementary angle,
there are 25
^3,200
possible first pages, and although significantly
smaller than the numbers we've been contemplating, it is yet another enormous
number. The chance of randomly selecting a book with this particular first page
is "only" 1 in 25
^3,200
, approximately 10
^4,474
,
which means, essentially, that it will never happen. For comparison's sake, the
chance of a single ticket winning a lottery is better than 1 in 100,000,000 =
10
⁸
. So finding such a book is equivalent to winning the lottery
more than 559 times in a row. (In the equation below, each factor of 10
⁸
signifies winning the lottery once.)

As a source of useful
information for a catalogue entry, a title on the spine of a book, such as
The Plaster Cramp,
is similarly moot, for there must still be something
like

distinct books with the exact
same orthographic symbols on the spine.

Edition,
publisher, city of publication, year of publication—all are meaningless in this
Library. The one sort of information we mentioned that may possibly prove
useful is that of a short description of the contents of the book. We'll take
"short" to mean "half-page or less." It's much more
difficult to say what we mean by "description." We'll take it to mean
"something that significantly narrows the possible contents of the
book." For example, "The book is utter gibberish, completely random
nonsense," doesn't significantly narrow the possible contents of the book.
(We are aware that this definition is problematic.)

Any book
published in the last 500 years likely has a short, reasonably limiting
description. A book whose contents consist of the letters MCV repeated over and
over evidently has a short description. A book whose entire contents are
similar to the 80-symbol line

unmenneo
.ernreiuht.naper,utuytgn or fgioe,no,e,dn .roih senoi.,erg n cprih npp

almost certainly doesn't have
a short description. Or does it? A fascinating area of study in the field of
information theory concerns the difficulty of deciding whether or not a line
such as the one above has some sort of algorithmic description that is shorter
than the line itself. Borges seems to have an intimation of this when he writes
"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." Perhaps
"hr,ns llrteee"
is a more concise description of the line, or
perhaps a succinct translation into English is "Call me Ishmael."

It does no
good to excerpt a passage as a short description; titanic numbers of books in
the Library will contain the same passage. In an important sense, then, for all
languages currently known by human beings, for the cataclysmic majority of
books in the Library,
the only possible description of the book is the book
itself.
This, in turn, leads to a lovely, inescapable, unimagined
conclusion:

The
Library is its own catalogue.

Let's restrict the
investigation to a slightly more agreeable collection of books: all those whose
entire contents cohere and are recognizably in English, and whose first page
contains precisely a short title and a half-page description, both of which
accurately reflect the contents. Any rule of selection will have problems. Some
associated with this one are: What does it mean to "cohere"? Would a
collection of essays on different topics constitute a coherent work? Would
sections of James Joyce's infamous novel
Finnegans Wake
register as
"recognizably English"? What if the book contains a non-English word,
such as "ficciones"? What if the title, as in the case of
Ulysses,
is more allusive than descriptive? Can any description "accurately
reflect" the contents of a book? Regretfully, we'll ignore these and other
legitimate, interesting concerns.

For example,
suppose the first page of a volume of the Library began with the following
description, modified slightly from the back cover of the 2002 Routledge Press
edition of Wittgenstein's
Tractatus Logico-Philosophicus.

Tractatus
Logico-Philosophicus by Ludwig Wittgenstein

Perhaps the
most important work of philosophy written in the twentieth century, Tractatus
Logico-Philosophicus was the only philosophical work that Ludwig Wittgenstein
published during his lifetime. Written in short, carefully numbered paragraphs
of extreme brilliance, it captured the imagination of a generation of
philosophers. For Wittgenstein, logic was something we use to conquer a reality
which is in itself both elusive and unobtainable. He famously summarized the
book in the following words, "What can be said at all can be said clearly,
and what we cannot talk about we must pass over in silence."

If next came the precise
contents of the book, including Bertrand Russell's introduction, followed by
the appropriate number of pages consisting of nothing but blanks, then that
Library volume would be included in the collection. We are also willing to
include books longer than 410 pages, so long as the title page includes
reference to an appropriate volume number. This allows, among other things, for
the inclusion of this Catalogue of Books in English into the putative catalogue
we are trying to define, which we may as well call
Books in English.