The Unimaginable Mathematics of Borges' Library of Babel (42 page)

1.
             
For the reference to “The Total Library,” see
Collected Fictions
,
p. 67. For the essay itself, see
Selected Non-Fictions
, p. 214.

2.
             
See Burton, p. 95. The quote is lifted from Part 2, Sec. 2, Memb. 4 of
Burton’s colossus.

3.
             
A marvelous, witty experiment was performed at the Paignton Zoo in Devon
in 2003. Six Sulawesi crested macaque monkeys were placed in a cage outfitted with
a computer, ostensibly to see if, between them, the monkeys might produce some
work of Shakespeare over the course of a week of random typing. As reported by
David Adam, science correspondent for
The Guardian
, “The macaques—Elmo,
Gum, Heather, Holly, Mistletoe, and Rowan—produced just five pages of text
between them, primarily filled with the letter S. There were greater signs of
creativity towards the end, with the letters A, J, L and M making fleeting
appearances, but they wrote nothing even close to a word of human language.”

4.
             
Ferrero and Palacios, Hayles, and especially Hernández have written more
about Borges and
₀
.

5.
             
This “endless map of Brouwer” goes by the mathematical name of Brouwer’s
fixed point theorem. The main idea is that if a nice enough space is mapped into
or onto itself, then there must be at least one point that the map does not
move, a fixed point. An intuitive way of seeing this is fundamentally similar
to Josiah Royce’s construction, which is briefly quoted and discussed on page
146. For Royce, an exact smaller image of England is on a map. But if the map
is exact, then there is an unimaginably smaller version of the map on the map.
And that version must also have a smaller version contained within. These
images, each one contained in the previous, appear to shrink to a point. In
fact, they do: the math capturing Royce’s idea was formally stated and proved
by Banach and others in the early 1920s, and today the result goes by the name
of the contraction mapping principle. In
Variaciones Borges
, John Durham
Peters takes a deep look at philosophic connections between Borges, Royce, and
William James. He also considers Royce-type maps from many perspectives,
delineating some interesting mathematical and situational implications, and
using these ideas to meditate on the real world in contrast to mathematics.

Thus we may define the real
as that whose characters are independent of what anybody may think them to be.

—Charles
Sanders Pierce,
How to Make Our Ideas Clear

You who read me—are you
certain you understand my language?

—Jorge
Luis Borges,
The Library of Babel

In some cases, the phrases and
clauses that follow should be considered more as gestures and less as
definitions: they're not necessarily meant to be rigorous or precise, but
rather to evoke a way of understanding the mathematical object. For those with
internet access and inclination, Wikipedia at
www.wikipedia.org
stands
up as a surprisingly good source for formal definitions. Wolfram's MathWorld at
mathworld.wolfram.com
is equally
good, and perhaps less subject to malicious or mischievous hacks and prankings.

Beyond
standing the test of time and invoking chills of the mythologic, the stacks of
libraries stocked with math books are invested with the pregnant allure of
opening crisp new or musty old books, and then using indices to seek out
appearances of the term or concept. By so doing, you may follow Borges'
footsteps through dim-lit libraries, tracking the spoor left by the
intellectual history of an idea and slowly netting it with your growing
framework of context and insight. Libraries are cultural resources eroding byte
by byte under the rising tide of digitization. I point this out partly as a
lament, but mostly as a tedious reminder for those so inclined to seize the
opportunity to use libraries before they change beyond recognition.

1-space
The Euclidean line. The real number line. One-dimensional space.

1-sphere
 A circle contained in a plane. All the points in a plane
that are the same fixed distance from a particular point.

2-space
 The Euclidean plane. The Cartesian coordinate plane. Length
by width. The
x-y
plane. Two-dimensional space.

2-sphere
 A basketball. A soccer ball. The generalization of the 1-sphere
to a higher dimension. All the points in 3-space that are the same fixed
distance from a particular point.

3-Klein bottle
A three-dimensional analogue of the Klein bottle. A nonorientable
object living in higher dimensions that is formed by identifying the faces of a
solid cube or hexagonal prism. A somewhat improbable model for the universe
that is the Library.

3-space
The space we appear to live in. Volume. Length by width by
height. The
x-y-z
space. Three-dimensional space.

3-sphere
The generalization of the 2-sphere to a higher dimension. A
geometric object that lives naturally in 4-space. All the points in 4-space
that are the same fixed distance from a particular point. A model for the
universe that is the Library which satisfies the particulars of the Librarian's
classic dictum as well as those of the Librarian's solution.

3-torus
 The generalization of the torus to higher dimensions. A geometrically
flat object that lives most naturally in 6-space, although it may inhabit
4-space. A solid object living in higher dimensions that is formed by
identifying the faces of a solid cube or hexagonal prism. The most sensible
model (whatever that means) for the universe that is the Library.

4-space
 In the context of our universe, 4-space is often called the
space-time continuum, and can be thought of as (Volume)
x
(One time dimension). In this book,
though, it's (Volume)
x
(Another Euclidean dimension). The
w-x-y-z
space. Four-dimensional
space.

annulus
An annulus is the area between two concentric circles in the
Euclidean plane. Topologically, it is the same as a cylinder, or a can that has
had the top and bottom removed.

Archimedean property
Often stated in the form that there is no largest integer. This
is then usually flipped, by taking reciprocals, to conclude that there is no
smallest positive number. It's then easily generalized to point out that all
real numbers are beset and besieged by other real numbers, none of which is
"closest."

axiom
A statement so fundamentally in accord with our intuition and
experience of the world that we are willing to accept it as a basis for all
future developments. A logical "given."

base of an exponent
The number that is multiplying itself some fixed number of times.
For example, in the expression 5
³
= 5 • 5 • 5 = 125, the number 5 is
the base of the exponent.

Brouwer's fixed point
theorem
In its simplest form, Brouwer's fixed
point theorems says that if we take any closed disk in the plane and twist it,
stretch it, contract it, rotate it, and do what we will with it, and then
squish the transmogrified disk back down into the plane so that it lies within
its original boundaries, then there must be at least one point that is unmoved.
That is, despite all the distortions and contortions, there must be a fixed
point.

Cavalieri's principle
Cavalieri's principle is a way to think of the volume of an
object as the sum of infinitely many infinitesimally thin slices of the object.
In calculus terms, for a sufficiently "nice" object, we can integrate
the areas of the slices to find the volume of the object.

chiliagon
A thousand-sided polygon. A chiliagon on the page of this book
would be virtually indistinguishable from a circle. Descartes used it as an
example of a geometric object that's easy to define but impossible to visually
imagine within the mind's eye.

circular logic
See "illegitimate deduction."

circumference
The circumference of an
n
-sphere, for any dimension
n
,
is the distance around the equator of the sphere. The distance around any great
circle of the sphere. If the radius of the
n
-sphere is
r
, then 2
wr
is the circumference.

closed interval
A closed interval of the real number line is the set of all
points between two numbers, inclusive of the endpoints. For example, the closed
interval between 1 and 7 is the set of all numbers
x
such that 1
<
x
<
7.

codimension
The codimension of a geometric object living in some
n-
space
is the difference in dimensions between the object and the ambient space. For
example, a line is a one-dimensional object. The codimension of the line in
2-space is equal to 1. The codimension of the line in 3-space is equal to 2.
The codimension of the line in 4-space is equal to 3. The codimension of the
line in
n
-space is equal to (n - 1).

combinatorics
Combinatorics is the art of counting something in two different
ways, setting those equal to each other, and thereby finding a formula with
general applicability. A lot of interesting combinatorics can be done by
thinking carefully about the many ways different colored balls can be placed
into barrels.

countable
A set is countable if it can be put into one-to-one
correspondence with the positive integers. If the elements of a countable set
are playing musical chairs and there is a chair for each positive integer, when
the music stops every element will be able to find a seat, every time. Such a
set is also called
countably infinite.

definition
See "definition" or "self-referential."

denominator
The denominator of a fraction is the number dividing into the
numerator. The bottom of the fraction. The basement of the fraction. In the
expression 3/5, the denominator is 5.

empty set, complete list of
elements contained within
See page xlii.

Euclidean space
A space satisfying Euclid's postulates. See also 1-space,
2-space, 3-space, 4-space, etc.

exponential notation
A remarkably condensed and useful notation that captures the idea
of a number multiplying itself some specified number of times. In this book, we
use it only for integer self-multiplications, but the ideas can be extended so
that all real numbers are legitimate exponents. With somewhat more difficulty,
the ideas may be further extended so that imaginary and complex numbers may
also serve as exponents.

factor (noun)
A positive integer which, when multiplied by another positive
integer, produces yet another positive integer of situational interest.

factor (verb)
Given a positive integer, it's the finding of the factors (noun)
that multiply each other to produce the original integer.