Models of Time
and Jorge Luis Borges
“Time,
then, cannot belong to the Real because it is a mere measure and its standard
is arbitrary. It is an ens rationis, a mode of thinking, or
rather of imagining (or misthinking) duration.
It is a mere ‘aid to the Imagination’.”[1]
Christopher
Shaw
Professor
Diane Isaacs
Thursday,
3 May 2001
General
Index
I.
General Index
II.
“Models of Time and Jorge Luis Borges”
III. Appendix I: Table of Endnotes
IV. Appendix
II: Supplementary Mathematical Equations
V.
Appendix III: List of Works Cited
Models
of Time and Jorge Luis Borges
I want to write something meaningful, and I so have
written a sentence: “Borges is dead.” I
wrote this sentence in the year 2001, and you have just read it. Supposing that Borges is an Argentine writer
who lived from 1899 to 1986, the statement itself has little or no meaning for
you: it is merely a declaration of fact, a fact that has been true for 15 years. There is very little I can do with this
sentence in my present situation to make it aesthetically meaningful. The only thing under my power to change is
the ordered set of words I have written.
Thus, most others who would want to write a significant remark would
choose a different statement than I have to satisfy their thirst for
profundity.
There is one way to change the meaning of this statement
without changing the words: I may write it again. “Borges is dead.” It is
still the year 2001 for me, and you are still you and you have still just read
my sentence. Now, however, it tells you
something else, perhaps that I am stubborn and refuse to make the effort to
change something I have written. Maybe
it tells you something about yourself, that you are inquisitive enough to
continue reading until you find the meaningful sentence I hope to write. But what if the reader of the sentence
changes? What if, when I wrote, “Borges
is dead,” it was read by a literary critic who believed that all contemporary
fiction in the Western hemisphere is merely a restatement of something already
written by Borges? Then, my sentence
would have the added metaphorical meaning that I believe Borges’s influence to
have ended. The metaphor, in this case,
did not come from me at all; it came from changing the audience. Together, you and I could easily achieve
this goal. All you have to do is find
this literary critic and hand him the article you are reading. Effecting this change in meaning relies on
your volition as the reader, not mine as the author.
But how else might I change the meaning of this
sentence? If Borges were alive, it
would have a completely different significance. A man who reads the words, “Borges is dead,” will interpret that
I believe Borges’s writing to be obsolete, even during the very moments in
which Borges is defining his writing style and adding to his body of work. Now, I would have accomplished my initial
goal of writing something meaningful: I would have synecdochially magnified
Borges’s writing to the scale of his entire life and then said that that life
is over. But it seems that while we
were quite easily able to change the reader of this text, neither of us was
successful in reviving Borges so that his existence may provide added meaning
to my sentence. To do that, we would
have to go back to a time in which Borges was still living, between 1899 and
1986. Although the object of both
changes is the same, one appears plausible while the other does not.
The above example is an illustration of the main
difference between spatial relativity and time: it is simple enough, in this
world, to alter the spatial resting place of the article in your hand in an
infinite number of ways, but an attempt to alter its slot in time without
destroying it is much more difficult.
For this reason, although time is considered the fourth dimension in our
space-time configuration, time has always seemed to be different in some way,
unchangeable. However, there is one
place in which the rules of time do not apply in this way: the space of literature. (Without much technical effort, a writer can
invoke scenes and situations from any historical time period, using words to
move objects around from time to time.)
Although Borges no longer writes, the world that exists within his work
manages to do what is impossible in this world: he changes time, bends it to
meet the needs of his esoteric story-lines.
Much of literature, while it does hold onto the basic principles of
physics, combines time with space in a way that is more fluid than science
allows us to do in this world.
All of the literature we will look at in this article
was originally written in Spanish, most of it by the same Jorge Luis Borges
mentioned above. His writing is
considered to be the precursor to what is known as modern Magic Realism, a
style of writing whose more outstanding contributors are Latin American. Since most of the literature will be taken
from translations that were not directly supervised by the original author, we
might be inclined to look at how the word time is translated to English, in
order to make sure that no meaning is lost.
But Borges answers this question for us in his story, “El jardín de
senderos que se bifurcan” (“The Garden of Forking Paths”) from the collection
of the same name:
—Sé que de todos los problemas,
ninguno lo inquietó y lo trabajé como el abismal problema del tiempo. Ahora bien, ése es el único problema que no figura en las páginas del Jardín.
Ni siquiera usa la palabra que quiere decir tiempo. ¿Cómo se explica usted
esa voluntaria omisión?…
—En una adivinanze cuyo tema es
el ajedrez, ¿cuál es la única palabra prohibida?
—La palabra ajedrez.
—Precisamente…. Omitir siempre una palabra, recurrir a
metáforas ineptas y a perífrasis evidentes, es quizá el modo más enfático de
indicarla.
(—I know that of all problems, none disturbed him more
greatly nor worked upon him so much as the abysmal problem of time. Now then, the latter is the only problem that does not figure in the
pages of the Garden. He does not even use the word that signifies
time. How do you explain this voluntary omission?….
—In a riddle whose answer is chess, what is the only
prohibited word?….
—The word chess.
—Precisely…. To omit
a word always, to resort to inept metaphors and obvious periphrasis, is perhaps
the most emphatic way of stressing it.)[2]
While his
metaphors are not necessarily inept, nor his descriptions periphrases, this
statement certainly applies to the literature we will analyze below. It is well-known that Borges himself had a
deep scholarly interest in the philosophy of time, and he has even written
essays on time and the way the world progresses. For the reason that he makes clear above, in his own fiction, it
is much more interesting to read and analyze the works that do not outwardly
mention time. With their furtive lack
of pretension, they silently bear the weight of Borges’s ideas and deliver them
carefully to us, wrapped in decorative labels of detective stories, historical
theses, or philosophical broodings. Of
more import than the words themselves are the ideas they express, which do not
require using the word time explicitly.
Let us begin with a short explanation of the concept of
time. In mathematical models, the
representation of time is usually a straight horizontal line, the ordinate
axis. Time (t) increases regularly from left to right, usually with some units
assigned to measure values of t as it
progresses. It always begins on the
left at t = 0, and increases at least
until the point where you are no longer interested in measuring anything,
ending with an arrow to indicate that time continues infinitely to the right
(see below).
Fig. 1 t
Time is
almost always the standard by which other quantities are measured: its rate of
change is constant, and the rate of change of other quantities is measured with
respect to time. Simple examples of
this use of time include measuring displacement, velocity, and acceleration of
a particle in space, all obtainable using basic calculus. Time, and the measurement of time, are taken
for granted in a similar way in almost any other situation: a certain length of
time measured today would cover exactly the same amount of distance on the t-axis as it would tomorrow or any other
day. In other words, a minute today
would be still a minute if counted tomorrow.
For this reason, it would not make sense to measure the change in time
with respect to some other factor. But
the problem with proving this to be true is that we cannot displace pieces of
time the way we could measures of spatial distance. Whereas one could certainly cut an infinitely long string into
four pieces and rearrange the two finite pieces, effectively maintaining the
length of the string, we could not do the same with a unit of time (see below).
Fig. 2 t
The concept of time is a problem that philosophers,
physicists, and mathematicians have long explored. The obstacle in defining it is that time is not a natural
phenomenon, not something unexplainable, but is treated like something mystical
and uncontrollable. There is a vague
perception that time would continue to pass, even if all life and activity were
to cease to exist. The positivist
necessity of science demands that time be treated like a fact[3]. But time is
merely a system of measuring duration, of measuring the distance between two
events. It depends on the existence of
those activities which it measures. The
words we use to describe the passage of time are simply constructs, defined in
circular relativity to each other. Thus, our system of time is merely a
mathematical model used to describe the distance between two events.
Standardizing the measurement of time requires the use
of units: countable and uniform lengths of time that can be shared between
multiple observers. The rudimentary way
we have of defining units of time is based on the rotation of our planet. A day is the distance between consecutive
instances in which the planet measured is at a certain position in its
rotation, the points between which correspond to the progress of the planet’s
motion (counter-clockwise when viewed from the positive z-axis, or from the north).
On Earth, a day is divided into smaller sub-units: 24 hours, 1440
minutes, or 86,400 seconds.
We also have larger units: a year is roughly 365.25
days, a century is 100 years, a millennium is 1,000 years, and so on. The straight line model of time is just a
visual translation of the rotational model (see below).
Fig. 3 t
Each
break in the line could represent an instance in which the Earth is at a
certain axial position p, and the
arrows to the left and right indicate that this is (and was) an infinitely
repeating pattern. Any other line
representing time is just a proportionally altered version of this one,
elongated or collapsed to represent the units of time we want to use. To make the diagram more practical, we set
zero at an arbitrary point on the graph, which we assume to be the starting
point of whatever event we are measuring, and exclude everything to the left of
it, which would represent any events that occurred before our measurement
period began.
Modern science has found that a more accurate way to
measure time is by recording the microwave light emission of a cesium-133
atom. According to the 13th
General Conference on Weights and Measures in 1967, exactly 9,192,631,770 wave
cycles constitute a second, also called the International System unit of time[4]. Of course,
cesium clocks are still known to have an error of one second per 1.5 million
years, a factor which depends on the measuring equipment, and the stability of
the atom being measured[5]. This
introduces an essential paradox: it seems that, in this case, the passing of time
causes time itself to be impossible to measure perfectly. Does time itself affect time? If the standard which we use to judge time
itself flawed, then can it, in fact, be called a standard? This measurement system, while more
sophisticated, is essentially the same idea as that used in the beginning of
civilization: we measure time based on something we can observe.
Note that the consideration of this model as “infinite”
is merely done for convenience. Since
the Earth, at some point, likely did not exist as we know it, and at some point
this planet will likely cease to turn, then there had to have been an initial
value some time ago, and there will likely be a terminal moment. If this is in fact the case, then our model
is not a globally perfect representation of time; rather it is only locally
authentic, extremely close to measuring time accurately for a period of some
millions of years, but not for an infinite length. In the same way that Einstein proposed that space exists only
insofar as it contains objects—air (an atmosphere), people,
structures—Aristotle said time exists merely as a consequence of the events
that take place in it.[6] Thus there can
be no absolute time, as time is only the name for a system of measurement of
events. Now we can begin to measure the
duration of time between two events, which is performed in much the same way as
distance is measured between two objects: creating a simple mathematical model,
we place both events on the same straight line, and count the units between
them.
So in one way, time is much like the three visible
dimensions of space. But in another
way, it is completely different. For
instance, if we establish the orientation of our three dimensions in a certain
way, the distance between two different objects must be measured using all
three dimensions (see below).
z
q
Fig. 4 p y
x
The
spatial distance between these two objects p
and q is measured by using a mathematical
formula (bulky enough to be omitted here) that utilizes each object’s distance
from the three axes, arbitrarily named x,
y, and z. Yet according to the
straight-line model, the distance in time between two events can only be
measured using a single dimension. So
if event a occurs at time t1, and event b occurs at time t2, then
the time between events a and b is the absolute value of the
difference t1 - t2, since a length of
time is never negative (see Equation 1).
This result is satisfactory for such a simple
mathematical model. But if we take
“distance” to have a more profound meaning, then this result is less
significant if event a occurs at time
t1 in Washington, DC and event b occurs at time t2 in Madrid. If t2 and t2 were only seconds apart, then the larger measure of distance
would be the spatial difference. The
mathematics above forces us to express the distance between any two events in
terms of space and time separately: event a
and event b are separated by some
number of minutes and some number of miles.
This formula doesn’t allow for the measurement to be taken in terms of a
single unit. A problem being tackled in
physics, mathematics, and philosophy today is how to take a meaningful
measurement of distance between two objects in space-time, which refers to two
objects at different points in space that are attached to two different events
in time. The formula used to solve for
the distance between p and q could be extended to a simple formula
for the distance between the two space-time objects a and b. But the end result would be an
unintelligible mess of useless units, which would be no help in recreating a
model if given the position of one of the space-time objects and its distance
from another space-time object. We
would not know where to place this object in order to satisfy the given
conditions. But science has attempted
to come up with an answer, using general relativity (largely based on research
done by Albert Einstein) and the more recent work of physicists like Stephen
Hawking, to determine a much more complicated model for how the universe
works.
For the purpose of this article, it is not necessary to
learn all of the technical mathematics and physics knowledge behind this
theory; rather a brief and broad description suffices to explain the purpose of
the theory and should be accessible to the non-mathematician. The explanation begins by noting the
apparent independence between space and time, perhaps in a more mathematical
and perhaps a more precise way than I have done above. From here, the theory says that the
measurement of time is firmly dependent on the speed of light (which is a
constant value at about 186,000 miles per second) in the following way:
measuring time is equivalent to measuring the time-distance between events,
which is equivalent to measuring the time between the observation of two
different events. The standard method
of observation is visual. But depending
on the location of the observer, any measurement of time will be at least
minusculely different for two different observers. No matter what the method of observation, the problem in
measurement will be the same, since auditory, sensory, and electronic methods
of observation are all limited by the velocity at which the information
indicating an event has passed approaches the observer: it is impossible to
record an instantaneous observation of an event. This all stems from the standards of human perception. We don’t perceive events directly, even when
they are related to our own actions. Our senses record and measure the different effects that an event
has on its surroundings. In the above
example, the visual perception of an event is in fact our sensory reaction to a
reflection of light, which takes a non-zero amount of time to travel. In this way, space and time are permanently
and inextricably linked.
Then, as physicist Jennifer Trusted states in her own
summary of the concept of Einstein’s theory and space-time, it is still natural
and unavoidable for the human mind to imagine spatial relativity and time
relativity as separate entities, a three-dimensional world concatenated, but
not thoroughly integrated, with the single dimension of time[7]. This means
that no matter what the relationship we eventually decide on between time and
space, the human mind will not instinctively be able to think in terms of
space-time relativity. Everyday life
still looks and feels exactly the same.
In order to make the integrated treatment of the universe plausible, we
say that events and distances can be measured accurately and repeatedly, but
only within a given frame of reference.
Empirically speaking, this is quite a plausible assumption to make. If I stand in one place and measure the time
between two events, and another observer standing about the same place performs
the same task, we will most likely get a similar result, provided our tools of
measurement are similar in accuracy. In
this situation, it is difficult to see why there should be a problem with the
consideration of time and space as independent and unrelated dimensions.
This leads us to the study of literature. It is a natural union of the concepts of
space and time as dimensions: distance and time affect each other
directly. With respect to the book as a
physical object, timing is strictly maintained by the turning of pages, a
semi-constant progression from some starting point to some ending point. This consideration is from the exterior
point of view of the world in which the book is written, not the interior world
which the book describes. For example,
in Germanic and Romance Languages (as well as some others), as in the standard
time-line, the left-most page is the first page, the beginning, and the reader
continues, in most cases, toward the right, until the reaching the right-most
(“last”) page on which action occurs, which ends the reading of the book.
What makes literature distinct from the world in which
we live is the default assumption that the action occurring in the story also
comes to an end at some point. The
world of a book is only supposed to exist for as long as someone reads it,
while life is assumed to go on after the book is closed. Miguel de Cervantes was perhaps the first to
circumvent this convention, creating a piece of literature that is conscious of
its own existence as a piece of literature.
In the second part of Don Quixote,
the hero discovers a copy of the book detailing his adventures in the first
part, published just two months after he completed them. The same concept appears some 350 years
later, in Gabriel García Márquez’s One
Hundred Years of Solitude. In this
case, rather than a published novel, the characters have in their possession an
untranslated manuscript. When Aureliano
Buendía discovers the key to understanding the language, he reads the text of
the actual book being written about him, which describe his death upon reading
the last words of the novel. The
abandonment of time as a linear constant, however, belongs to the writings of
Borges.
In “The Garden of Forking Paths,” a short story that
appears in a collection of the same name, Borges takes an initial step toward
loosening the reader’s assumed grip on the way time works. Housed within the story of a German war
veteran who must get a secret out to his countrymen, a legendary man named
Ts’ui Pên has written an infinite novel.
His work, called The Garden of
Forking Paths, begins with a set of initial circumstances, and continues on
to describe all of the infinite possibilities that lie ahead:
En todas las ficciones, cada vez
que un hombre se enfrenta con diversas alternativas, opta por una y elimina las
otras; en la del casi inextricable Ts’ui Pên, opta—simultáneamente—por
todas. Crea, así diversos porvenires, diversos tiempos, que también
proliferan y se bifurcan…. En la obra de
Ts’ui Pên, todos los desenlaces ocurren; cada uno es el punto de partida de
otras bifurcaciones. Alguna vez, los
senderos de ese laberinto convergen; por ejemplo, usted llega a esta casa, pero
en uno de los pasados posibles usted es mi enemigo, en otro mi amigo.
(In all fictional works, each time a man is confronted
with several alternatives, he chooses one and eliminates the others; in the
fiction of Ts’ui Pên, he chooses—simultaneously—all of them. He
creates, in this way, diverse futures, diverse times which themselves also
proliferate and fork…. In the work of Ts’ui Pên, all possible outcomes occur;
each one is the point of departure for other forkings. Sometimes, the paths of this labyrinth
converge: for example, you arrive at this house, but in one of the possible
pasts you are my enemy, in another, my friend.)[8]
For
readers of The Garden of Forking Paths,
reading the text a second time yields different words, a new story, which takes
place at the same time and uses the same characters as the old story. No two readings of the text are the
same.
Thus, “The Garden of Forking Paths”, through its
infinite novel, elegantly illustrates that every book is infinite. Consider again the statement, “Borges is
dead.” Just like at the beginning, each
time you re-read the words, their meaning can change. The words themselves have not been substituted or re-arranged in
any way, but they elicit a different response from you each time you read them. This defines the first concept of time for
which literature is responsible. If we
isolate from the infinite time-line the span of time in which things have been
written, we would get a line that looks exactly like our first model: the left
side represents the first time any writing was ever done, and the right side
stretches out into infinity to represent the perpetuation of all writing.
Let us call this the writing line. Then, the book itself that I am reading can
be looked at as a finite line segment, taken from this infinite writing
line. Each time I read a book, I visit
a particular finite line segment of the writing line. Each time I re-read a book, I am reliving the same period of time
again, effectively reversing time in this model. It is important to note that the process of revisiting a point on
the writing time-line is not a concatenation of this section of the line. In the larger frame, the time-line in which
I exist, it would be a continuation, but since the writing itself hasn’t
changed at all, I am still visiting the same point on the writing line. Of course, such a model is not applicable to
our existence, but it is a useful tool to make us begin to look at time in
different ways. Now, no two readings of
a particular book will be exactly the same, suggesting that the reversal of
time in the case of our Frame 1 does not necessarily guarantee that the events
I experienced in my first visit to a particular line segment will be replicated
on my return trip. This is the most
pertinent aspect of this time model: we are effectively changing the events
that occur in a point of time that has already happened.
While written as a fantasy, the infinite book within
“The Garden of Forking Paths” is an example of a text changing in meaning upon
multiple readings. The author does not
state that this infinite novel contains words or text. Rather, the book is an idea, and to read it
is to interpret this idea. Each time
the reader goes back to a point in the book and rereads it, he finds himself
reintepreting the idea and imagining and speaking different words. It is impossible that the text itself
physically changes or that the ink on the pages rearranges itself when the
reader is not looking: this would extend beyond the boundaries of Borgesian
magic realism, in which the magical elements are human, not physical. Thus the book itself does not change with
each reading; it is only the human perception of the book that changes.
Another story by Borges, “Pierre Menard, Autor del
Quijote” (“Pierre Menard, Author of Quixote”), explores this possibility in a
different way when the story’s title character rewrites the text of Don Quixote in a modern age. As is customary in many of Borges’s works,
the story begins with the description and abridged bibliography of a fictional
writer, whose name is Pierre Menard. Most
of the imagined works by Menard are literary criticisms, histories of
philosophers, or collections of verses.
But his masterpiece is described as:
…la subterránea, la
interminablemente heroica, la impar.
También, ¡ay de las posibilidades del hombre!, la inconclusa. Esa obra, tal vez la más significativa de
nuestro tiempo, consta de los capítulos noveno y trigésimo octavo de la primera
parte del don Quijote y de un fragmento del capítulo veintidos…. Su admirable ambición era producir unas páginas que
coincidieran—palabra por palabra y línea por línea—con las de Miguel de
Cervantes.
…subterranean, interminably heroic, and unequaled, and
which is also—oh, the possibilities inherent in the man!--inconclusive. This work, possibly the most significant of
our time, consists of the ninth and thirty-eighth chapters of Part One of Don Quixote and a fragment of the twenty
second chapter…. His admirable ambition
was to produce pages which would coincide—word for word and line for line—with
those of Miguel de Cervantes.)[9]
Menard’s
goal, then, was to re-write Don Quixote:
not to write a new Don Quixote, but
to reproduce the original book verbatim.
The reaction of Menard’s contemporary critics, including the story’s
narrator, is not one of contempt or scorn, as one might imagine to be in this
case of plagiarism, but rather astonishment and awe:
El fragmentario Quijote de Menard es más sutil que el de
Cervantes…. El texto de Cervantes y el
de Menard son verbalmente idénticos, pero el segundo es casi infinitamente más
rico.
(The fragmentary Don
Quixote of Menard is more subtle than that of Cervantes…. The text of
Cervantes and that of Menard are verbally identical, but the second is almost
infinitely richer.)[10]
Although
these descriptions are written in a way that sounds ludicrous, the point of the
exercise is the same as the second iteration of, “Borges is dead.” According to Menard’s logic, Cervantes had
written a novel that was, on the whole, uninteresting and easily explainable. The narrator remarks (referring to one of
the book’s more famous speeches) that Cervantes has merely written, “un mero
elogio retórico de la historia” (“a mere rhetorical eulogy of history.”)[11] Much like my
first iteration of “Borges is dead,” the novel was insignificant at the time it
was first written, before Menard chose to work with it. At this point, I should point out that it is
nearly impossible to imagine that Borges actually felt this way about the work
of Cervantes. Of course, Don Quixote is subjectively touted as
the most influential book written in the Spanish language, and this analysis
extends in no small way to the writings of Borges. Thus, in our world without Pierre Menard, Don Quixote is an aesthetically meaningful piece of literature by
itself.
Pierre Menard chose to make the novel meaningful in the
same way our original statement started out.
He wrote it again. He chose
several sections and re-wrote them exactly as they had been written before, to
get a new meaning from them. The new Quixote is critically successful in
lending weighty meaning to words already spoken, and for several reasons. First of all, for Cervantes, the backdrop
for the adventures of don Quixote is chosen to be 16th century
Spain, Cervantes’s own epoch, geographic location, and culture, with little
regard for artistic impression.
For Pierre Menard to have chosen such a setting,
however, was an unforeseeable act of genius:
¡Qué españoladas no habría aconsejado esa elección a
Maurice Barrès o al doctor Rodríguez Larreta!
Menard, con toda naturalidad, las elude.
(What hispanophile would not have advised Maurice Barrès
or Dr. Rodriguez Lauretta to make such a choice! Menard, as if it were the most natural thing in the world, eludes
them.)[12]
The words
themselves also seem to gain meaning after three centuries of
insignificance. The narrator quotes two
identical passages, one drawn from the Cervantes novel, and one from the Menard
novel. The conclusion of his analysis
is that, “El estilo arcaizante de Menard—extranjero al fin—adolece de alguna
afectación. No así el del su precursor,
que maneja con desenfado el español de su época.” (“The archaic style of
Menard—in the last analysis, a foreigner—suffers from a certain affectation. No so that of his precursor, who handles
easily the ordinary Spanish of his time.”)[13] According to
the narrative voice, the styling of Menard’s language is markedly interesting,
since he manages successfully to write in a version of the Spanish language
that has long been evolving to the present vernacular, whereas Cervantes merely
manages to write in the same way he would speak. His lack of effort makes his writing basely unprofound to the
reader, again similar to stating for once that “Borges is dead.” But Menard’s rewritten version affects the
reader in a different way, much like the repetition of “Borges is dead.”
Menard’s writing of Don
Quixote has such an impact on the narrator of the story that when he goes
back to read the original Cervantes Don
Quixote, he finds himself imagining that Menard is the original author of
the novel. This means that each time
the narrator revisits the particular section of the writing line devoted to Don Quixote, he undergoes a different
experience from the first time he read the novel. In this way, an anterior portion of the writing time-line has
been permanently altered, a feat that we considered impossible in our initial
contemplation of time. One of the basic
assumptions about the passing of time is that once an event is recorded, its
history cannot be altered in any way.
Considering our writing time-line, this is proven to be untrue.
In “Pierre Menard”, Borges makes use of one of the
different ways in which we changed our sentence at the beginning. His character repeats a specific set of
words that, when first written, were not meaningful. When Menard discovered the plainness of Don Quixote, he realized the only way to add significance was to
repeat the novel, word for word. Anyone
who missed it the first time would catch it the second. More importantly, Borges changes the time
frame in which Cervantes’s book exists.
If Menard had merely made photocopies of the original Don Quixote and asked the narrator to
read it, he would have succeeded only in making the narrator re-affirm
Cervantes’s genius at having written it.
That Menard himself actually re-writes the book and puts his own name on
it forces the reader to consider it anew, as if it had appeared for the first
time. Don Quixote is no longer an early 17th century novel; it
is an early 20th century novel.
The words of a book do not alone make the book a work of art. The art does not speak for itself, and the
art is not timeless in the sense that it maintains the same meaning as it
endures time. The timelessness of Don Quixote is that, viewed as a modern
novel, it is even more affecting than 300 years before.
Conspicuously, in “Pierre Menard,” a story that deals
with writing words from the past in the present, one word that the author never
inserts (in neither the English nor the Spanish versions) is “time.” The only iteration of the word is when the
narrator quotes Cervantes, and again of course when he quotes Menard’s version
of the same quote. This hearkens back
to “The Garden of Forking Paths,” when the narrator said that to omit a word
was the strongest way to emphasize it.
The example in his case was “time,” and sure enough, this very word is
also the one he chooses to omit in “Pierre Menard.” While this is not enough to guarantee that Borges had conceived
of time as the central theme in his story (it is possible that a reader might
notice there a whole host of words which are not written explicitly in the
story: if he didn’t mention a hippopotamus in his story, does that mean the
story is tacitly about a hippopotamus?), we might arrive at that conclusion
anyway, for what is the major disparity between Cervantes and Menard? Each is an author, a scholar of his own
culture’s popular literature. Each has
written a novel called Don Quixote,
both of which are identical. The
largest difference between them is that they are 300 years apart, a duration of
time. It would have been convenient,
perhaps even beneficial, for Borges to have mentioned the word time, but he
didn’t. Given his reference in “The
Garden of Forking Paths,” it is difficult not to imagine that Borges wanted us
to see his implicit theme of time.
Apart from viewing time in the philosophical and
mathematical sense, we should ask the question of whether we can really speak
of eternity. Martin Heidegger, in his
written explanation of the definition of time, begins with a rumination on
eternity. “Wenn die Zeit ihren Sinn
findet in der Ewigkeit, dann muis sie von daher verstanden werden…. Diese Verlegenheit ist für die Philosophie
ni zu beheben.” (“If time finds its
meaning in eternity, then it must be understood starting from eternity….
Philosophy can never be relieved of this perplexity.”)[14] Here, the
philosophical difficulty is that, without faith (which is the one thing that
philosophy, especially scientific philosophy, denies itself), one cannot use
the theological definition of eternity.
The religious concept is that God is eternity; however, the only means
we have to knowing God is through faith.
To make this concept more complete, let us revisit the
standard mathematical model for time, the infinite straight line (see Fig. 3
above). In order to grasp the concept
of infinity the way it is seen in mathematics, let us draw a circle on top of
the straight line, positioned so that the line bisects the circle into two
equal semicircles. Now we have a
two-dimensional graph, with an x-axis
and a y-axis. According to topology, the branch of
mathematics which is largely concerned with creating and distorting shapes, the
line we are using to represent time is nearly homeomorphic to the circle it is drawn through. A homeomorphism
is a rule for interchanging points on two different shapes in a way that
preserves local distances. This means
that you may pick any point on the circle, and by using my rule, I can show you
a corresponding point on the line and vice versa. When you pick two points that are arbitrarily close together on
the line, the two points on the circle that my rule will associate to them will
also be close together. Two mathematical
structures that are homeomorphic are considered equivalent in everything but
the way they look, for any operation performed on one of the objects can be
done in exactly the same way with the same result to the other. The operation of finding a homeomorphism
will be integral in understanding the different models of time that I will
present throughout the rest of the paper.
A commonly used and tangible example is the
homeomorphism between a donut and a coffee mug. The most definitive characteristic that these two objects share
is a hole at some place on the structure.
Thus, if you were given a donut made of some very flexible putty, you
could easily form it into the shape of a coffee mug. What was once the middle of the donut will then become the space
inside the handle of the coffee mug.
Our example of time is a bit more elusive. The way we get our particular homeomorphism
is by picking a point, t, from the
line. Then we draw a line segment from
that point to the north pole of the circle (see below).
“infinity”
(x, y)
- t “now” +
Fig. 5 For the associated mathematical proof, see
Equation 2.
The place
where the line intersects the rest of the circle is the image of t, and we express
it in the form (x, y), where x is the horizontal component of the circle and y is the vertical component. The reverse can be done if given a point on
the circle: we draw a straight line connecting the point on the circle to the
north pole. The place where this line
intersects the time-line is the image of the inverse homeomorphism. With this model, we can force the
intersection to occur at almost any point in the circle (See Equation 3). Using the center of the circle as our point t, which we can think of as “now” (while
everything to the left is “before” and everything to the right “after”), we end
up with a line that points straight downward to the bottom of the circle. I said that the line is only nearly
homeomorphic to the circle because there is one single point that we will never
be able to reach with this method. The
further we take our points to the left or right of “now,” the closer we come to
the north pole of the circle. However,
no matter how far we go out, we will never actually touch it with the
intersecting piece of the line. The
only line that would do the job would have to hit the circle at precisely one
point, which is the definition of a tangent
line to the circle. Such a line would
be parallel to the time-line, and thus they would never cross. The point at the north pole of the circle,
then, is infinity. Once we remove it
from the picture, our two structures are homeomorphic.
What makes this mathematical representation interesting
is that there is no difference between negative and positive infinity. The furthest points to the left are
infinitesimally close to the furthest points on the right. What this means is that the mathematical
straight-line model for time seems to correspond with the theological
perception that there is one eternity, one constant. Without calling this eternity “God,” as Heidegger does, we can
say that our model is strikingly similar to the way theology would write
eternity. The end is the same as the
beginning, insofar as there exists neither beginning nor end.
At first glance, an apparent flaw in this homeomorphism
is that it if we take two points that are distant from each other on opposite
sides of the “now”, they are very close together on the circle. But the method of measuring distance on the
circle is not the standard Euclidean distance of 2-dimensional space. The distance must be measured by following
the path of the circle, which means that, since there is no point at the north
pole, any distance must be measured by traveling around the bottom of the
circle. Thus, two points that may
appear close together toward the top are actually quite distant. But this is not relevant, since the main
point of this exercise is to show that, mathematically, positive and negative
infinity are equivalent.[15]
This mathematical evidence is not enough to suggest that
existence, if it is infinite and has no beginning or ending, is at least
heading in the direction from whence it came.
But it is enough to make us question the validity of a straight-line
model of time, for what system of measuring time would allow for the past to
take place at the same time as the future?
One distinct theory of time solves this problem by eliminating the
concepts of past and future. In the
preface to his treatise on time, Quentin Smith briefly defines what
philosophers agree to be the two main theories of time: the tenseless theory and the tensed theory: “The tenseless theory
holds that temporal determinations consist only of the relations of earlier than, later than, and simultaneous with. The tensed theory of time (at least on one
version) holds that temporal determinations also include the properties of
pastness, presentness, and futurity.”[16]
The tenseless theory corresponds with the regional
theory of time, which holds that time is divided up into different regions,
like space, but that no particular region of time is the present, past, or
future. So the set of all of these
regions is simply called time.[17] Now we can
compare the regions in terms of the relation “earlier than” (<), but doing
so does not yield an ordered set. An
ordered set is a set such that for any two distinct elements x and y, either it is true that x<y or that y<x. The set we call time allows for another
relation, that of simultaneity, or equivalency. This means that we can select some distinct pair of regions, r1 and r2, so that neither r1<r2, nor r2<r1 .
This set, then, is called a partially ordered set. Each of these regions is also a subset (a set
whose elements are also all contained in some larger set) of all of time. The general definition of a partially
ordered set is a set P that may be divided up into nontrivial subsets whose
intersection is empty, and at least one of which contains more than one
distinct element of P, such that for any two subsets (called classes) R and S
of P, it is true that given a binary relation (order) “<”, either for every r in R and every s in S, we have r<s, or that for every r in R and every s in S, s<r.
At this point, we can establish the other two relations, “later than”
(>), and simultaneous with (»), where r>s if and only if s<r, and r»s if and only if neither r<s nor s<r.
As a concrete example of a partially ordered set, let us
look at the x-y plane. As our partial order, we will say that given
two points x and y, x<y if the distance from x to the origin is less than the
distance from y to the origin. This is a partial order, because if we view
the plane as the collection of all circles centered at the origin, it is true
that all the circles centered at the origin will eventually fill the entire
plane, that no two circles contain any of the same point, and that given any
two different circles C and D, the circle C which has the minimal radius
contains points which are all closer to the origin than all of the points on
circle D. Each circle is a class of
points that are equidistant from the origin (see Equation 3). Similarly, another familiar partially
ordered set is three-dimensional space (or, as we have been calling it,
space). The order relations are the
same, referring to an object’s distance from the center, but this time, the
classes are all spheres centered at the origin.
This leads us to the work of Rafael Alvira, a
philosopher-psychologist from Spain, who builds on Heidegger’s tenseless theory
of time. Alvira says, similarly to
Heidegger, that time itself is “a ‘wholeness’, something ‘complete’, a
permanent synthesis of past, present, and future.”[18] This leads to
the same concept as before: if time is actually a synthesis of past, present,
and future, then there can be no differentiating between the three ideas. Thus, as before, we would have a set with a
partial order. Alvira further expands
his definition of a partially ordered time with the introduction of
three-dimensional time. The basis
vectors for the three dimensions are “a) the dynamism of being (time as origin), b) the measure of this dynamism
(time as mediation), and c) the
duration of being (time as end).[19] He says that
there are three different ways to understand time, depending on whether it is
linked to matter, space, or movement.
With respect to matter, time is a unit of change, symbolically representing the process of change. With respect to space, it is a unit of measure, and with respect to movement,
time is a unit of aim, which
represents the proximity of all processes to their ends. This view lays the essential groundwork for
time to be considered as a three dimensional, partially ordered set.
The significance of viewing time as a partially ordered
set, rather than an ordered set, is that there can be no arranging the whole
set into a straight line. This would
make it impossible for any of the classical representations of time to exist in
the manner they are represented in this article. I live right now; one hundred years ago, I did not, and in
another one hundred years’ time, I will not.
However, that does not guarantee that those three periods—one hundred
years ago, now, and one hundred years from now, are not all in the same region
of time and thus impossible to put in order of earliest to latest. To use the concentric circles from the
plane, each of us could be traveling on our own single circular orbit of the
center point, wherein everything that occurs to us happens in the same subset
of time. Without using the same
mathematical models, Heidegger arrives at this same point and asks:
Was ist die Zeit? Wurde zur Frage:
Wer ist die Zeit? Näher: sind wir
selbst die Zeit? Oder noch näher: bin
ich meni Zeit?
(What is time? became the question: Who is time? More closely: are we ourselves time? Or closer still: am I my time?)[20]
If time
is actually such a set of circular orbits, then do I only travel in one
direction on my orbit? Are there any
orbits that contain more than one person?
If there is, indeed, someone else in my orbit, then is that person also
me? And what is the relationship
between me and the other orbits that are adjacent to mine?
Echoing these thoughts is the story, “Las ruinas
circulares” (“The Circular Ruins”), by Borges.
When the hero of this story washes up on the shore of an ancient
mountainous village, surrounded by the ruins of a once-great society, he has no
intention of interacting with a single human being ever again. His plan, his calling, is to imagine a being
in his mind and infuse him in the real world, to literally make a dream come
true. His only interactions, outside
the necessary negotiations for food and water, take place in his sleep, when he
communicates with the pupils of an imaginary school. His plan fails at first, and then succeeds: the man he creates
steps out into the real world and performs the feats his master commands to
demonstrate his existence. The end of
his dreaming is accompanied by the absence of his pupil, in whom he has deleted
any memory of being created by the old man’s dream. This leads us as readers to question whether the old man had
actually been awake when his dreamed man supposedly entered the waking world.
In the midst of the man’s recovery from his failure, he
ruminates on the history of the mountain village he inhabits. The ruins were caused by the gods who
destroyed them before his lifetime. The
fire god, with whom the old man communicates several times, is primarily
responsible for the damage, which destroyed everything that had been built
before. After many years of resting
from his project, suddenly the old man awakens to find himself worrying about
the future of his progeny. He wonders
at the “unparalleled humiliation” he would encounter if he were to discover
that he exists only as a projection of another man’s dreams.
The solution to his worry will come in the form of the
same destruction of the temple that had come earlier: the fire was coming back
to destroy the temple again.
“Comprendió [luego] que la muerte venía a coronar su vejez y absolverlo
de sus trabajos.” [21] (“He realized [then] that death was coming to crown his
years and to release him from his labors.”)[22] This death
absolves him of all worry, because once he dies in the flames, his “son” will
carry on without the possibility of accidentally meeting his “father”. The final revelation of the story is that,
in his death, the old man discovers that he himself is the imagined progeny of
another dreamer: “Con alivio, con humillación, con terror, comprendió que él
también era una apariencia, que otro estaba soñandolo.”[23] (“In relief, in
humiliation, in terror, he understood that he, too, was an appearance, that
someone else was dreaming him.”)[24]
With that conclusion, in one sentence, Borges changes
the whole setting of the story. Until
now, it would have been possible to interpret that the essential plot is that a
man climbs ashore in a foreign land, only to suffer hallucinations until his
death in a forest fire caused by drought.
The last sentence validates all of the man’s ruminations, and explains
that he is a product of his own process, a victim of the very humiliation that
he feared for his son.
This story is a literary model of the theory that time
is really arranged as a set of circular orbits: first of all, it is unavoidable
that all three—dreamed, dreamer, and dreamer of the dreamer—exist in the same
spatial region, for they walk the same ground.
But their time frames are all mutually exclusive, even though they exist simultaneously. Even though they exist in the same moment and on the same pages
in the story, they do not exist at the same time, for the man that dreamt the
old man did not exist until the old man himself ceased to exist. So, as Heidegger says, the old man is his
own time, and he shares space with others simultaneously.
Now, though, the intersection of the lives of these
three men is enough to force the asking of this question: is it possible that
some of the circular orbits intersect at least one point? If this is the case, our partial ordering of
the plane no longer functions, and we get a chaotically organized set of
regions. However, with the addition of
Alvira’s three-dimensional time, we can try to make sense of the partial
ordering of three-dimensional space. It
turns out that using spheres as our class subsets, we can pick orbits that are
not completely circular, but which do revolve at a constant distance around the
origin. Two of these orbits may
intersect an infinite number of times, or they may not intersect at all (see
Equation 4).
With the elements we have gathered from philosophy, mathematics, and literature, we can construct an elementary homeomorphism that will demonstrate the concept of orbits of time. Let us take as our first set (X) the entirety of time, perhaps an infinite set. Now given a specific person (A) as a reference point, define the distance between two events as the relevance of the event to A. Strung along throughout this set are the events that A has personally encountered, each two events as close as their relevance to A. Now we let the set Y be the 3-dimensional sphere. The homeomorphism from X to Y will place all events from X in a specific place on the sphere, depending on their relevance to A. The image of A’s experience will be a curve that sits on the face of the sphere in some way, starting and ending at the same place (thus making a full orbit). If we take a similar image for all other people, we will get a series of orbits, some that intersect and some that do not. The points of intersection will be events of great relevance to more than one person. While esoteric, this model allows us to ponder a re-worded version of Heidegger’s question: “Am I mathematically equivalent to my time?” the answer to which is an emphatic yes.
At this point, we have achieved relative freedom from
the usage of terms like “before” and “after”, which have precious little
meaning in the partially ordered three-dimensional time model. Even in the case of the previous story from
Borges, there is no guarantee that the old man existed “before” the man he
dreamed. The only relationship we can
establish between the two men is that one caused
the other. But who caused whom? Physicist and philosopher Graham Nerlich is
quick to point out that one event does not have to happen before another in
order to have caused it. The idea of a cause is that A causes B if and only if
A is the reason for B. There is no
requirement that a causal event also be a creative one.[25] Thus, just as
it could be said that the old man is the cause for the existence of the young
man in that he dreamed him, it could also be said that the old man only existed
in order that his son be created. In
this way, the progeny caused the necessity for the father.
It seems that, in terms of philosophy, any continued
analysis of time is destructive rather than constructive. All of the rules and systems that we take
for granted become moot and useless under subjective scrutiny. After a continued analysis of our
understanding of the concept of time, it appears that we are left with an
utterly abstract notion that still, because of the direction in which our
memories proceed, feels unnatural to the human mind. Perhaps this is inevitable, for the human mind, inextricably
linked with the human body, senses the progress of its own growth and decay,
and knows when its death is near. Time
is the only way we know how to measure change.
And time, viewed as a system for measuring change, is an easy rule to
master, but assumes total understanding of the way time works in this
world.
This does not mean that the linear view of time should
be rejected; indeed, it is natural and convenient to think of time as an
unstoppable force that meters out change at every passing. Certainly, trying to understand time in the
way that philosophers imagine involves wading through counter-intuitive,
conflicting philosophical discourses.
But an analytical look at Borges, a writer who is both widely read and
critically acclaimed, reveals that the philosophy is implicit in his fiction
already, and the careful reader will find that he has already begun to
understand time in a non-traditional way.
Perhaps this knowledge will prove useless in its practical application to
daily life, but it is certainly true that achieving an understanding of the way
the world works provides some level of comfort in our continued orbits of time,
which is certainly relevant to daily life.
While it is true that Borges focuses on time more than
most, one can make the case that there are alternate views of time present in
practically any work of literature.
Once we have learned to interpret Borges in more than the usual
straightforward way, it is possible to enact a similar analysis with any other
writing. For why does literature
persist in the world if not for the purpose of intellectual challenge and
philosophical understanding of the world?
Searching for different models of time is just one of the ways in which
we can enhance our reading and perception of the great works of literature that
abound throughout the world, in any language.