The Unimaginable Mathematics of Borges' Library of Babel (41 page)

The end crowneth the work.

—Elizabeth
I, quoted in
The Sayings of Queen Elizabeth

The end crowns all;

And that old common
arbitrator, Time,

Will one day end it.

—William
Shakespeare,
Troilus and Cressida

1.
     
Did you look?

1.
             
For example, Lasswitz, who wrote "The
Universal Library," which profoundly influenced Borges, calculated the
number of books in his Library. Other mathematicians and critics who find the
number of books include Amaral, Bell-Villada, Rucker, Nicolas, Faucher, Salpeter,
and the anonymous encyclopediasts who wrote the page found at Wikipedia.org!
Amaral deserves special plaudits for finding influences of Lasswitz's "The
Universal Library" in the work of Lasswitz's mathematical contemporaries
Kummer, Fraenkel, pp. 7ff, and Hausdorff, pp. 61ff.

2.
             
The quote below appears in Borges'
expansive short story "Tlön, Uqbar, Orbis Tertius."

There are no
nouns in the conjectural
Ursprache
of Tlön, from which its "present-day" languages and
dialects derive: there are impersonal verbs, modified by monosyllabic suffixes
(or prefixes) functioning as adverbs. For example, there is no noun that
corresponds to our word "moon," but there is a verb which in English
would be "to moonate" or "to enmoon." "The moon rose
above the river" is
"hlor u fang
axaxaxas mlö','
or, as Xul Solar succinctly
translates:
Upward, behind the outstreaming it
mooned.

His use of
the phrase "
Axaxaxas
Mlö"
in "The Library of Babel" is presumably a reminder that
even books written in the
Ursprache
of Tlön, including all volumes of the first and second editions
of the
Encyclopedia of Tlön,
are in the Library. A careful reader may object that volume 11 of
the
First Encyclopedia of Tlön
consists of 1,001 pages, while Library books number only 410. Our
rejoinder is that three books of the Library, the last of which will contain
229 blank pages—blank spaces filling each slot—yield the necessary 1,001 pages.
Of course, they may not be shelved anywhere near each other, but this in no way
negates the fact that the 11th volume
is
in the Library. A variation of this observation refutes Rucker's
casual statement that "the minute history of the future" can't be
contained in the Library, in
Infinity and the
Mind,
pp. 121-22. The minute history of the
future is contained in the Library; it is found in volumes perhaps scattered
throughout the Library. There is no implicit promise that the information in
the Library is accessible or verifiable—it just must be there, somewhere.

3.
             
From the 1875 Grindon citation under the
definition of "septillion" in the
Oxford
English Dictionary.
"Thousands of plants
consist of nothing more than a few such cells as in septillions make up an
oak-tree..."

4.
             
In the words that comprise "The Library
of Babel," Borges adroitly finesses the fact that by being restricted to a
mere 25 symbols, the books of the Library could not contain both upper- and
lower-case symbols, let alone diacritical and punctuation marks beyond the comma
and the period. The story, as written, could not appear in the Library. In this
book, when we are referring to imagined entries in the Library, we will hew to
the standard set by Borges and not restrict ourselves to using only the
orthographic symbols of all uppercase letters (or lower-case letters), spaces,
periods, and commas.

1.
              
For a mathematically sophisticated reader: in
fact, one may imagine that at some distant point in the future, we might have a
superfast supercomputer running an algorithm that

1. Was able
to test for primality every number expressible in 100 digits.

2. Kept
a tally of the number of primes without listing them.

3. Output
simply the number of primes
N
expressible in 100 digits.

Then,

4. Determine
if
N
is odd or even.

5. Determine,
if
N
is odd, which
numbered prime is the median of the set.

6. Determine,
if
N
is even, which
two numbered primes average to the median of the set.

Then,

7. Run
the algorithm again, keeping count until the number(s) from step 5 (or 6) is
(are) achieved.

8. Output
the median!

At no point
was a list of all primes necessary.

1.
     
See Salpeter, for example, on "The Book
of Sand."

2.
     
Benardete, quoted in Merrell on page 58,
independently thinks along similar lines.

3.
     
Another fine point for the interested:
obviously we're capitalizing on the fact that the number of pages is countably
infinite.

4.
     
If a mathematically sophisticated reader is worried
about the need to invoke the axiom of choice, the issue is easily sidestepped
by assigning the same infinitesimal, 6, for the thickness of each page.

1.
             
For the mathematically adventurous reader: in
fact, the famous Hopf fibration of the 3-sphere decomposes the 3-sphere into
great circles over a base space equivalent to the 2-sphere.

2.
             
A point for a mathematically sophisticated
reader: earlier in this chapter, we observed that the gravitational field of
the Library needed to be imposed by the builders of the Library. Since the
Library presumably does not possess any regions of zero gravity, it is vital
that the 3-manifolds under consideration may be equipped with everywhere
nonzero vector fields. But of course they can, since all 3-manifolds have Euler
characteristic equal to 0, entailing the existence of everywhere nonzero vector
fields.

3.
             
A way to see that the surface of a coffee mug
is the same as the surface of a donut is to simply shrink the "cup"
part of the mug to the strip that lies between the joining spots of the handle
to the mug!

4.
             
Another way around this problem would be to
require that the orthographic symbols be symmetric under 90° rotations, too.
For example, the symbols O, X, +,
,
, and
 satisfy this criterion.

5.
             
Technical note: as the Library stands, we
couldn't actually use an exact cube to construct the Library, for the number of
hexagons is not a perfect cube. That is, the number of hexagons is not of the
form
x
³
for some
integer
x.
However,
by tinkering with the number of hexagons on each floor, it would be possible to
have a shape very close to a cube. If done carefully, it would be equally easy
to make the identifications between the sides of the near-cube, and the end
result would be just slightly less symmetric.

1.
              
An Argentine colleague, Martin Hadin, brought
to my attention a 1971 dialogue between Borges and Herbert Simon that suggests
Borges was intrigued by these kinds of ideas, but previously unaware of them.
The dialogue appears in
Primera Plana,
January 5, 1971.

1.
             
The interesting readings in Wheeler, Alazraki,
Bell-Villada, Barrenechea, Rodriguez Monegal, Slusser, Ammon, Eco, Keiser, Nicolas,
and Faucher, for example, fall outside this domain. Although I might differ
with the conclusions they draw, it seems to me that Nicolas and Faucher get the
math correct.

2.
             
An infinite set is
countable
(also called
denumerable
or more precisely, countably
infinite) if it can be placed into one-to-one correspondence with the positive
integers. In effect, this means that one may write down all the elements of the
set in an orderly (infinitely long) list.

1. 
↔
"first" element of
the set

2. 
↔
"second" element of
the set

3. 
↔
"third" element of
the set

etc.

It's easy
and not inaccurate, therefore, to think of "countable" as synonymous
with "listable." Cantor, the creator of set theory and the theory of
transfinite numbers, was presumably shocked to discover that the
rational numbers
are countable.
Regardless, in one of the most beautiful arguments in mathematics, he
demonstrated that the
irrational numbers
are
not
countable or listable. Any set that is not listable is called
uncountable
or uncountably
infinite.

3.
             
If, for example, I begin by assuming,
"The moon is made of green cheese," I can derive a whole host of
interesting implications from that premise, such as that dairy farmers and
cheese manufacturers may be responsible for the lack of missions to the moon in
recent years. That if we built cheese-harvesting factories on the moon and sent
back enormous loads of cheese, the world hunger situation might be abated. And
so on. However, the original premise is false, so it doesn't matter how
interesting or plausible the ensuing speculations are to us.