The Unimaginable Mathematics of Borges' Library of Babel (27 page)

My
crestfallen look, lit only by my eyes eagerly seeking to read your face, is
enough to tell you: I failed, also. We sit uncomfortably by the air shaft and
each recount the particulars of our fruitless runs; there's not much to say,
"I ran in, I ran out; I ran in, I ran out; I ran in, I ran out; I ran in,
I ran out;..."

It is
irrelevant which one of us first exclaimed, "The stairs! The spiral
stairs! Maybe we could have gone up, over, and down to the hexagon. Or down,
over, and up? Or down, down, over, up, over, over, up, up, over, and back
down?" It is equally irrelevant which combination of whose thoughts
destroyed this diaphanous, infinitely permutable, insight:

Every
hexagon has an airshaft through the center—it is easy enough to look up and
down our airshaft and see what we always see: stacked hexagons. Moreover, each
hexagon has two sides singled out by the presence of a spiral staircase (figure
54). Looking up and down the spiral staircases of our hexagon confirms what we
already know, that the hexagons above and below must have the staircases in the
same sides as the one we are in. In other words, the hexagons above and below
are, in this regard, exact clones of our hexagon. Extending this reasoning, for
each and every hexagon on our floor, the hexagons above and below it are exact
clones. That means that the labyrinthine paths we ran are precisely the same
above and below—the stairs and shafts dictate this.

Each
floor plan is inevitably, invariably, precisely the same as every other floor
plan. There is no advantage gained by taking a set of stairs up or down.

We may, therefore, restrict
our investigation to the floor we are on. We might have been lucky; it could
have been the case that our starting-point hexagon comprised a part of the
floor plan that looked like figure 55. In such a case, one of us would have
reached the hexagon in seconds. On the other hand, we were unlucky, so unlucky
that we couldn't say how unlucky we were (figure 56). We didn't hazard a guess
as to how many hexagons we would have to pass through to reach the other
librarians; such a presumptive act would be tantamount to heresy. For that
matter, after thinking about it, we daren't even say if the other librarians
were really fighting; perhaps they belonged to an entirely different
civilization—perhaps an entirely different
species.

Our Stark
and Depressing Realization:
Our question was
decisively answered. We would not necessarily be able to reach the adjacent
hexagon in time to prevent a murder.

Our conception of the
Library's structure was so perturbed by these cascades of devastating insights
that it didn't even occur to us until later that the floor plan could plausibly
contain eternally inaccessible closed loops, such as the three in figure 57.

The
unexpected, against our desires, found us and found us wanting. Without the
barrier of even a single door, the adjacent hexagon—the source of noise,
confusion, and probable violence—was locked away from us forever.

(We were very
startled to realize this; indeed, it only became clear while flying to Buenos
Aires when we sketched out the library floor plan. Our readings of "The
Library of Babel" always left us with the impression that regardless of
which hexagon "we" were "in," we could reach any nearby
hexagon in a short period of time. The Realization of this section, in fact, a
minor
lemma
in the field of graph theory, is a prime example of the
unimagined math of the story. The Math Aftermath, "A Labyrinth, Not a
Maze," at the end of this chapter contains an extension of this story
providing a sense of why a stronger result about the inaccessibility of
adjacent hexagons must be true.)

Now let's examine the Library
by interpreting the first paragraph of the story to mean that only one
passageway in each hexagon is perforated by a spiral staircase (as in figure
51). Once again, the hexagons are forced to be stacked by the existence of the
ventilation shafts, but in this case, only one wall, one doorway, is specified
by the spiral staircase. This entails that although the hexagons' sides are
aligned, they are no longer cloned. Hexagons above and below each other must
share one entranceway, but the second may branch out in a different direction
(as in figure 58).

Thus, a pair
of floors may have labyrinth patterns such as these depicted in figure 59.
These illustrations include the spiral staircases, because the presence of a
staircase induces a connecting passage between hexagons on all other floors of
the Library. Combining the floor plans produces the pleasingly symmetric
picture of figure 60. Most importantly, this means that a librarian may reach
any adjacent hexagon by traveling through only two additional hexagons and two
flights of stairs, one up, one down. The sense of a bewildering array of
choices is omnipresent and factual.

It is our
considered opinion that Borges simultaneously intended for the Library to have
a spiral staircase in every doorway and also to present the librarians with a
bewildering array of options. The "stark and depressing realization"
of the librarians in the preceding section indicates the impossibility of such
a conjunction, whereas this minor modification allows for enormous mutability
in the floor plans and potentially a quick access to any nearby hexagon.

Math Aftermath: A Labyrinth, Not a Maze

The subject does not belong
to the world; rather, it is a limit of the world.

—Ludwig
Wittgenstein,
Tractatus Logico-Philosophicus

The goal of this Math
Aftermath is to provide grounds for believing an even stronger consequence than
the "stark and depressing conclusion" that followed from the first
case we deconstructed, that adjacent hexagons need not be accessible. The
stronger consequence has a relatively easy proof, but is too messy to offer up
in these pages due to the necessity of breaking down a number of related cases.
(Despite not offering much of a framework for the proof, it is tempting to
employ the standard trope appearing in works of mathematics at moments such as
this: "The reader may supply the details.") The librarians in this
pastiche will name it "our conjecture of extreme disconsolation," and
it is:

In any
Library constrained as this one,

on any given
floor,

for any
positive integer
n
less than, say, 1,000,000,000,000 = 10
¹²
,

there must
necessarily be pairs of abutting hexagons, Hi and H2,

such that a
librarian would need to walk through more than
n
distinct hexagons to
travel from H
₁
to H2.

That is, there are many, many
hexagons that possess effectively inaccessible adjacent hexagons.

(Now, if
these observations were deep mathematical insights worthy of publication in an
eminent journal—or even a second-rate journal—we'd probably label the
"Stark and Depressing Conclusion"
weak inaccessibility
and
"Our Conjecture ofExtreme Disconsolation"
strong inaccessibility.
Also, an aspect of the statement of the stronger result is worthy of comment:
notice that the amount of detail accrued in the service of excluding unwanted
interpretations renders it difficult to read and to understand. This seems to
yield the counterintuitive notion that the more precise the transmission of an
idea, the more opaque the language.)