Gabriel Solari
Borges’s Palimpsest, or how to hide the infinite in finite
short stories
“To see a world in a grain of sand
And a heaven in a wild flower,
Hold infinity in the palm of your hand
And eternity in an hour”
William Blake
Surely when watching “Circle limit IV” created by M. C. Escher , most
people would appreciate some mathematical concepts such as
reflections or rotational symmetry of order 3, but not many will realise
that the picture is also illustrating a model of non Euclidean geometry
developed by Henri Poincarè.
Circle limit IV
A deep mathematical concept is hidden from the profane.
In the same way someone can read some short stories written by the
argentine writer Jorge Luis Borges and not being aware of many ideas
related to mathematical topics dealing with the infinite.
Both artists found a way to develop metaphors of complex mathematical
concepts throughout their artistic creations.
We can say that they created some sort of palimpsest (old process used
to cover an original writing or drawing with new layers of parchment).
Jorge Luis Borges (1899 - 1986) was a famed writer, considered by
general critics as the eternal candidate for the Nobel Prize, which he
never received. Besides his mastery of the literary world, his knowledge
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of Mathematics is really astonishing, as I hope the three selected short
stories for this article will prove.
I invite the reader to join me on a journey through Borges’s literary
works, looking for hidden clues that will reveal how Zeno’s paradoxes,
the real number line and transfinite arithmetic are present in his work.
Let’s begin our journey with the short detective story “Death and the
compass”.
When trying to solve the case of a serial killer (Scharlach), a clever
detective (Lonrot) deduces that a concealed equilateral triangle disguised
within a town will lead him to the latest crime scene. Later on, he
realizes that he was framed by Scharlach, and actually the symmetrical
shape was a rhombus. So he moves quickly to Triste-Le-rois, the last
vertex of the rhombus, meeting Scharlach, who obviously… was waiting
for him:
“Lonnrot avoided Scharlach's eyes. He was looking at the trees and the sky divided into
rhombus of turbid yellow, green and red. He felt a little cold, and felt, too, an
impersonal, almost anonymous sadness. It was already night; from the dusty garden
arose the useless cry of a bird. For the last time, Lonnrot considered the problem of
symmetrical and periodic death. "In your labyrinth there are three lines too many," he
said at last. "I know of a Greek labyrinth which is a single straight line. Along this line so
many philosophers have lost themselves that a mere detective might well do so too.
Scharlach, when, in some other incarnation you hunt me, feign to commit (or do commit)
a crime at A, then a second crime at B, eight kilometres from A, then a third crime at C,
four kilometres from A and B, halfway enroute between the two. Wait for me later at D,
two kilometres from A and C, halfway, once again, between both. Kill me at D, as you are
now going to kill me at Triste-le-Roy." "The next time I kill you," said Scharlach, "I
promise you the labyrinth made of the single straight line which is invisible and
everlasting." He stepped back a few paces. Then, very carefully, he fired.” (( 1))
For those of us familiar with Zeno’s paradox known as “The Dichotomy”,
the tribute paid by Borges in the last paragraphs of “Death and the
Compass” becomes quite obvious.
Zeno of Elea, disciple of Parmenides, in the 4th century B.C. intended to
make their contemporaries realize that their concept of motion was
inconsistent. He maintained that “There is no motion, because what
moves must arrive at the middle of its course before it reaches the end”.
This argument can be translated into mathematical terms as a man,
who while trying to move from one point to another, must first cover half
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the length between the two points, then half of what remains left, and so
on time and time again.
1
2
1
4
1
8
So, a finite segment was divided into infinite segments. ("I know of a Greek
labyrinth which is a single straight line”.)
The wrong belief that the addition of infinite terms(1/2 + 1/ 4 + 1/8 +
1/16 + … ) would lead to an infinite result, unleashed a contradiction
that baffled mathematicians and philosophers for more than twenty
centuries (“Along this line so many philosophers have lost themselves that a mere
detective might well do so too”). Only after Calculus was born in the 17th
century was it possible to reveal the solution to this paradox.
We can continue the journey through Borgesland with “The book of
sand”.
In this story a man tries to sell different books to the main character
and after being rejected once and again, he makes one final attempt to
sell him a strange Bible…
“He told me his book was called the Book of Sand, because neither the book nor the sand
has any beginning or end."
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“Examining it, I was surprised by its unusual weight. On the spine were the words "Holy
Writ" and, below them, "Bombay."
"Nineteenth century, probably," I remarked.” ((2))
Why the reference to the “unusual” weight? May be a hint as to its
content?
Why “Bombay”? It may be a tribute to Indian civilization, to which we
can thank for our positional number system?
Why “nineteenth century”? We will have a clue later on.
“The stranger asked me to find the first page.
I laid my left hand on the cover and, trying to put my thumb on the flyleaf, I opened the
book.
It was useless. Every time I tried, a number of pages came between the cover and my
thumb. It was as if they kept growing from the book.
"Now find the last page."
Again I failed. In a voice that was not mine, I barely managed to stammer, "This can't
be."
Still speaking in a low voice, the stranger said, "It can't be, but it is. The number of
pages in this book is no more or less than infinite. None is the first page, none the last.
”((3))
I believe that after reading this excerpt it is possible to visualize the real
number’s line, and on it, any particular segment, for example the one
with extremes at 0 and 1.
0.99
1
0
0.9
What would happen if somebody asked us to find the point, or the
number nearest to 0? If you were thinking at 0.1, I can think in 0.01, or
0.001, or 0.00001, and so on. I always will be able to find one nearer to
0 than the one you thought of.
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Exactly the same would happen regarding the number nearest to 1, we
can find the numbers 0.9, 0.99, 0.999… (“None is the first page, none the
last”)
The book of sand is the metaphor Borges created to represent the real
continuum. He found a way to materialize the abstract concept of the
infinity hidden in a segment.
Returning to the story…finally the main character is persuaded to buy
the book, but becomes bewitched and tormented by its monstrous
beauty, to the point that some time later he ponders different ways to
get rid of the book. Probably quite similar feelings , to the early Greek
mathematicians who experimented with the “Horror Infiniti” from Zeno’s
paradox up to the discovery of the “alogos”, or irrational numbers.
“I thought of fire, but I feared that the burning of an infinite book might likewise
prove infinite and suffocate the planet with smoke” ((4))
In this paragraph, the suspicion of the biunivocal relationship between
two sets (the book of sand on the one hand, and the smoke produced by
it, on the other) is a clear reference to the method developed by Cantor
to study infinite sets.
And with that concept in mind we arrive to the final destination of our
trip, the short story “The Aleph”, where scratching the palimpsest
would allow us to marvel at Borges’s capacity of exposing Cantor’s
revolutionary ideas regarding infinite sets.
In the late 19th century the German mathematician George Cantor
sparked a great controversy when dealing with sets consisting of infinite
elements. Succinctly, it can be said that a way to compare two sets is by
trying to match them through a one-to-one correspondence between the
elements of one of them with the elements of the other.
By doing this he came to surprising conclusions such as that the set of
even numbers have the same quantity of elements as the set of the
natural numbers.
Why?
Just because you can find an even number for each natural number
thanks to the formula P = 2xN, and also because every even number can
be assigned a natural number through the formula N = P / 2.
A biunivocal correspondence was established.
Cantor believed that the cardinal of the Natural numbers should have a
name, and he chose the first letter of the Hebrew alphabet: Aleph.
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So, any set that can be put in a one-to-one correspondence with the set
of the Natural numbers will have as cardinal the Aleph.
Following the same principle, and in spite of what our intuition may be
tells us, it can be shown that the integer numbers and the rational
numbers share the same cardinal, the amazing Aleph.
However, the real numbers have a different cardinal. Yes!! There are
different kinds of transfinite numbers.
In the field of Geometry astonishing conclusions can also be
drawn…Let’s see the following examples:
A) Two segments of different length have exactly the same quantity of
points. How can we demonstrate this? Basically, following the
matching one-to-one principle. If we draw the two segments, and join
their extremes in pairs as suggested by the image below, we can see
that those lines intersect each other at a point, let’s say P.
Given any point Q in the shorter segment, we can always find a point
Q’ in the longer segment, providing that P, Q and Q’ belong to the
same line. Likewise, given any point of the longer segment we can
always find its “image” in the shorter segment.
P
Q
Q’
b) A segment and a line have the same quantity of points
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c) A square has as many as points as one of it sides.
Believe it or not…it is true.
Let’s consider a square with a side whose length is equal to one unit
within the Cartesian plane, and with its vertices placed at the points of
coordinates (0, 0), (0, 1), (1, 0), (1, 1).
It is easy to verify that any point belonging to the interior of the Square
will have as coordinates expressions of the type:
(0.abcd…, 0.mnpq…). Only by mixing the decimal expression of those
coordinates it is possible to obtain a single number such as:
0.ambncp….
That number can be interpreted as the coordinate of one of the points
resting on one side, so it is clear that for any point inside of the square
we can find another point belonging to one of its sides.
Y
1
P
0.mnpq
1
X
0
0.abcd…
New Number: 0.ambncpdq
d) A cube has as many as points as one of it edges (Same reasoning as
the detailed explanation above, but in three dimensions)
e) A tesseract (hypercube) has as many as points as one of it edges.
According to the Argentine professor Leopoldo Varela, it seems that
Borges considered the possibility of matching any instant in the life of
our universe with a segment of length equal to one inch.
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How? Let’s see…any point of the three-dimensional space can be
represented as (x, y, z) where the three letters represent coordinates
related to three axes.
What would happen if we want to represent a particular instant?
As the main character of H.G. Wells’ “The time machine” experienced
sitting on the device that allowed him to travel through time, the spatial
coordinates did not change (the machine did not move at all), but the
time coordinates did!!
So as we need three coordinates to identify a point in space, we would
need four coordinates to fix an instant, three for the position and one for
the time: (x, y, z, t). If all of us agree on a system of reference (three axes
and one particular moment), any moment in human history would be
determined by four numbers.
Therefore, if for any point of a tesseract or hypercube (“cube” of four
dimensions) it is possible to find a point in a segment of length equal
one inch, then every single moment of our lives would be represented by
some point in that segment.
In “The Aleph” Borges himself is portrayed as one of the characters in
the story, who is invited to watch a strange phenomenon in the cellar of
a friend’s house.
The following are some excerpts from the story that illustrate some of
the topics discussed above:
“…because down in the cellar there was an Aleph. He explained that an Aleph is one of
the points in space that contains all other points.”
“Let me warn you, you'll have to lie flat on your back. Total darkness, total immobility,
and a certain ocular adjustment will also be necessary. From the floor, you must focus
your eyes on the nineteenth step”
“. I felt a shock of panic, which I tried to pin to my uncomfortable position and not to
the effect of a drug. I shut my eyes -- I opened them. Then I saw the Aleph.
I arrive now at the ineffable core of my story. And here begins my despair as a writer.
All language is a set of symbols whose use among its speakers assumes a shared past.
How, then, can I translate into words the limitless Aleph, which my floundering mind can
scarcely encompass?”
“In that single gigantic instant I saw millions of acts both delightful and awful; not one
of them occupied the same point in space, without overlapping or transparency. What my
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eyes beheld was simultaneous, but what I shall now write down will be successive,
because language is successive.”
“On the back part of the step, toward the right, I saw a small iridescent sphere of
almost unbearable brilliance. At first I thought it was revolving; then I realized that
this movement was an illusion created by the dizzying world it bounded. The Aleph's
diameter was probably little more than an inch, but all space was there, actual and
undiminished”
“I saw the Aleph from every point and angle, and in the Aleph I saw the earth and in the
earth the Aleph and in the Aleph the earth; I saw my own face and my own bowels; I saw
your face; and I felt dizzy and wept, for my eyes had seen that secret and conjectured
object whose name is common to all men but which no man has looked upon -- the
unimaginable universe.
I felt infinite wonder, infinite pity.”((5))
Borges was able through this short story to dress some of the
fundamental concepts of infinity in Mathematics with the fine clothes of
an excellent literary production.
After reading the three short stories commented on this article, we can
not avoid asking ourselves: Did Borges manage to speak about
Mathematics using literary techniques, or was he developing literary
pieces using mathematical concepts?
Which ever answer we find more applicable, it is essential to
comprehend the extent to which his work builds a bridge between
Mathematics and Literature.
Bibliography
((1)) Jorge Luis Borges, “Death and the compass” in “Ficciones” (English
translation), Grove Press
((2)) ((3)) Jorge Luis Borges, “The book of sand and Shakespeare’s
Memory”, Penguin Classics
((4)) ((5)) Jorge Luis Borges, “The Aleph and other stories”, Penguin
Classics
Eli Maor, “To infinity and beyond”, Princeton University Press
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E. Maziarz, T. Greenwood, “Greek Mathematical Philosophy”, Barnes &
Noble
Alfredo Palacios, P. Barcia, J. Bosch, N. Otero, “Los Matematicuentos”,
Magisterio del Río de la Plata
Alfredo Palacios, P. Barcia, J. Clemente, “La Matemagia del laberinto”,
Magisterio del Río de la Plata
Alfredo Palacios, José María Ferrero, “Borges algunas veces
matematiza”, Ediciones del 80