The Unimaginable Mathematics of Borges' Library of Babel (32 page)

·
        
Erase the symbol in the square; that is, for
example, throw the books down the airshaft.

·
        
Write a new symbol—from the finite alphabet—in
the square; that is, either reorder the books in the hexagon, move books from
hexagon to hexagon, or actually, as the narrator of the story states, inscribe
symbols on the flyleaf or in the margins of a book.

·
        
Change to a new internal state; that is, the
librarian can go to the bathroom or go to sleep. Or, perhaps the librarian has
a revelation and by acquiring new ideas, moves to a new cognitive view of the
world. Or even simply that the librarian's mood changes.

·
        
Move to the left one square or move to the
right one square; that is, move to a new hexagon.

·
        
Halt permanently; that is, expire.

The librarian's life and
the Library together embody a Turing machine, running an unimaginable program
whose output can only be interpreted by a godlike external observer.

A
user.

A
reader.

Critical Points

In the mountains of truth
you will never climb in vain: either you will already get further up today or
you will exercise your strength so that you can climb higher tomorrow.

—Friedrich
Nietzsche,
Maxims,
aphorism 358

CRITICAL POINT
IS A TERM THAT HAS
ASSUMED A
host of meanings in mathematics. Generally,
it denotes a location where typical behavior breaks down and unusual and
interesting phenomena occur. In the qualitative study of the solutions of
differential equations, some critical points are singularities where flow lines
of solutions may converge from many different regions and then, regardless of
initial proximity, shoot off in a variety of divergent directions. This seems
an apt metaphor for critical perspectives arising from the interpretation of a
literary work.

Taking
advantage of Nietzsche's metaphor, at the base, mountains offer a multiplicity
of approaches towards the summit. Frequently these trails converge to several
natural routes, and oftentimes the descent must be taken via an unexpectedly
different path. Thus it is no surprise that in my quest to read all
commentaries by critics on "The Library of Babel," I learned that
many predecessors have independently climbed and descended the mountain, some
along paths with sections that closely parallel mine. This chapter is devoted
towards outlining some of these trails. In particular, I am restricting myself
to those that involve mathematics in one form or another. I begin by
acknowledging those who independently found some of the same mathematics in the
story.

The eminent
mathematician and pioneer German science fiction writer, Kurd Lasswitz, in his
1901 story "The Universal Library," not only calculates the number of
books in his universal library, but also mentions that filling our known
universe with books barely dents the total. As mentioned in the first endnote
to chapter 1, Amaral, Rucker, Nicolas, Faucher, and Salpeter all calculate the
number of books in the Library, while Bell-Villada excerpts a passage from
Gamow's
One, Two, Three. . . Infinity
that shows he has a notion of how
such a calculation should be performed.

While
looking for a reference, for the interested reader, to the topology and
cosmology of the Library presented above, I discovered that in their 1999
article "Is Space Finite?," Luminet, Starkman, and Weeks listed
"The Library of Babel" as a suggestion for further reading. At least
one of them must have thought about the topology of the Library. Floyd Merrell
also speculates about the topology of the Library and briefly discusses the
possibility of a catalog, albeit mostly in the context of space-time physics,
when he talks about light-cones for librarians and their world-lines, which are
infinitesimal by contrast to the size of the Library. Furthermore, inspired by
work of Bernadete, Merrell includes a brief discussion of a Book of Sand that
is similar in spirit to my first interpretation.

Whereas I
have mainly sought to elucidate the mathematics in the story, most commentators
endeavor to use mathematics to create a framework of analysis of the story and,
more generally, Borges' oeuvre. It strikes me as an interesting philosophic
pursuit to examine the project of importing mathematical and scientific
terminology and systems into the field of literary analysis. I recuse myself
from this study on the grounds that I am professionally neither a philosopher
nor a critic. I am, however, qualified to comment on the correctness of the
mathematics brought to bear on Borges, and, by dint of careful and extensive
reading of Borges, to agree or disagree with various other interpretations of
"The Library of Babel."
¹

Given that
literary critics mount defensible arguments for the primacy of interpretation
over authorial intent, why should we worry whether or not a critic's use of
mathematics is perfectly correct? By way of an answer, consider the following
hypothetical misreading of Borges. The not-so-eminent critic William Goldbloom
Blockhead, my not-so-bright alter ego, has confused "South America"
with "South Africa." After all, they sound similar and are both, more
or less, continents. Blockhead's stirring postcolonial analysis of Borges' work
raises new and disturbing questions. For example, Blockhead wonders why Borges
wrote in Spanish in lieu of English or Afrikaans, declines to mention
apartheid, and exercises narrative space on Argentine border and independence
wars rather than the Boer War and World War I. Does this mean that Borges' own
political views must be coded, rather than overtly expressed? Next, Blockhead
points out that the virtual invisibility of Africans in Borges' writing is a
strong sign that, in fact, race is his most important agenda—what else could
explain such an egregious omission? Blockhead's analysis triumphantly concludes
with the unique insight that this interesting
homme de lettres
employs
these strategies to avoid the literary trap of being read as merely a colonist
writing in and about a colony.

It's
conceivable that some of Blockhead's remarks may be of independent interest,
and some may even apply to Borges—after all, there are some real political and
historic parallels between Argentina/Spain and South Africa/England.
Nevertheless, one may well feel an irresistible urge to ignore wholesale the
stream of mistaken inferences following from Blockhead's wrong assumption. Seen
in this light, then, I offer bouquets of refractions arising from the light of
other critic's works.

Sarlo, in
Jorge Luis
Borges: A Writer on the Edge,
tellingly sees the Library as a symbol of
political oppression in the milieu of a totalitarian state. The librarians are,
in today's vernacular, information serfs who will never be able to acquire the
necessary data to transform their status. She writes that

Structurally,
the Library is also a panoptic, whose spatial distribution of masses and
corridors allows one to see every place in it from any of its hexagons. The
panoptic design of the Library brings to mind that of a prison where the guards
should be able to see any cell from every possible perspective. Foucault has
studied this layout as a spatialization of authoritarianism, as an image of a
society where total control is possible and no private place (no private
thought) is admitted. The universe described as the Library lacks any notion or
possibility of privacy: all the activities are, by definition, public.

The mental exercise implicitly
asked now of the Reader is to imagine oneself in the Library and intuit the
sensory and emotional experience: Is it dimly or brightly lit? Are the books
musty or shockingly pristine? Would the airshafts induce a tremendous vertigo
or could they be overlooked? Do the spiral staircases, placed either at every
entrance or every other entrance, hem librarians as do the bars of a cage, or
are they comforting mileposts along their life paths? Are there doors on the
small rooms designed for sleeping and physical necessities, or have the
librarians grown up accultured to a different kind of privacy than us?

Once we
establish our imagination in one of the hexagons, we see that the Library is
not panoptic; the only available line of sight is in the air shaft central to
the hexagon. Short of sticking a head into the airshaft while looking up or
down, only a few hexagons would be visible before the convergence forced by the
rules of perspective would hide all but a small part of the floors and
ceilings.

Furthermore,
as discussed in the chapter "Geometry and Graph Theory," it isn't
clear what the sight lines are like on an individual floor; hexagons may be
arranged in a straight line, or the entrances of the hexagons may well curve
around nonlinearly. The most extensive vision afforded by the structure of the
Library, then, would be a hexagon at the intersection of a cross formed by its airshaft
and a straight-line corridor. However, even if a passage ran straight, the
spiral staircases would block the views. By contrast, Bentham's Panopticon, as
described in Foucault's famous work
Discipline and Punish,
enables
full-time viewing of all inhabitants by obscured central scrutinizers: an
altogether different geometry.

Sarlo also
writes that "the infinity of the Library cannot be empirically
experienced, even if a traveller were granted infinite time." Even if the
Library extends forever in the manner of Euclidean 3-space, there is a nifty
way to see that the whole Library may, in fact, be visited. First, we make the
reasonable assumption that the architecture of the Library allows all adjacent
hexagons to be visited. Now, imagine starting in any hexagon, which we may now
consider to be the Origin of the Library.

1.
     
Begin at the Origin and visit every hexagon adjacent to the Origin
on the same floor as the Origin. Proceed up one floor and pass through all the
hexagons on this floor that are above a previously visited hexagon. Now, go
down two floors to the floor below that of the Origin and visit every hexagon
on this floor that is below a previously visited hexagon. It is the case now
that every hexagon a "distance" of one hexagon from the Origin has
been visited.

2.
     
Next, beginning on the floor of the Origin, visit every hexagon
adjacent to a previously visited hexagon. Hence, on the floor of the Origin,
all hexagons within two hexagons of the Origin have been visited. Now use the
spiral staircases to proceed up
one
and
two
floors and visit all
the hexagons on them that are above a previously visited hexagon. Next, go down
one floor below the Origin, then two floors below the Origin, and do the same.
Note that every hexagon a "distance" of two hexagons from the Origin
has been visited.

3.
     
Next, again starting on the floor of the Origin, visit every
hexagon adjacent to a previously visited hexagon, then proceed up
one
and
two
and
three
floors, visiting all the hexagons on each of
these floors that are above a previously visited hexagon. Do the same for the
one, two, and three floors below the floor of the Origin, and it follows
immediately that every hexagon a "distance" of three hexagons from
the Origin has been visited.