Borges and Infinity

I left Argentina in 1986, the year of Borges’ death. Although I had lived in Argentina for more than three decades, I did not discover Borges until a quarter of a century later. Of course I knew who he was. While in high school in the late 1960’s, we read Borges, Cortázar and a few other (at the time) contemporary writers, prodded by an avant-garde literature teacher. What I thought of Borges back then I cannot recall, but I am certain that I did not understand his writing. For, you see, I was – I am – a hopeless nerd. A geek. A dweeb. Interested in mathematics and in an excruciatingly logical and scientific understanding of the physical world, my view of literature was that it was no more than entertainment. My reading list, as that of any good geek, consisted chiefly of science-fiction and spy thrillers. This sorry state of affairs lasted until the early 2000’s when, seemingly overnight, I realized that I could not take one more sentence of that atrocious stuff. It also dawned on me that I would never again be able to read a novel, any novel. I discovered, as many have before, that good writing comes in small packages. And I discovered Borges.

My initial exploration of Borges grew out of trying to reclaim the sights, sounds and smells of the time and place that I had grown up in, and to which I would never return. These are present in Borges’ work, but they are always entangled with his genius to such an extent that somebody who did not live in mid- to late-century Argentina may find it difficult to separate reality from imagination. Borges was much more than a chronicler of a time and a place, however. His mind was preoccupied with mathematical concepts. For instance, of the seven tales that make up El Jardín de Senderos que se Bifurcan, which is the first part of the collection Ficciones, four deal with ideas of infinity: Tlön, Uqbar, Orbis Tertius, The Circular Ruins, The Library of Babel and the title story, translated to English as The Garden of Forking Paths. One can use these stories as jump-off points for personal explorations of infinity.

Tlön, Uqbar, Orbis Tertius is a seemingly infinite collection of infinities and circular references. It begins with a passing reference to an unknown country, Uqbar, that appears as an entry in only one copy of a (fictitious) encyclopedia, which is itself described as a reprint, perhaps rendered through frosted glass, of the Encyclopaedia Britannica. All that we know about Uqbar, including a nebulous description of its geographic location, comes from this single copy of the encyclopedia. Otherwise identical copies of it that Borges and his friends consult do not contain an entry for Uqbar. But in this one and only attestation of Uqbar its literature is said never to dwell on reality, but rather on imaginary places, one of which is Tlön.

A single volume is subsequently found of another, mysterious encyclopedia of unknown origin, though bearing an imprint from Orbis Tertius – which, one might assume, means Earth. This (purportedly real) First Encyclopedia of Tlön describes the imaginary land of Tlön, itself a product of the imagination of the inhabitants of the shadowy country of Uqbar. Since only the eleventh volume of the First Encyclopedia of Tlön is found, Borges discusses, and dismisses, the possibility of reconstructing the missing volumes by means of a collaboration of an infinite number of real authors, each of them making an infinitesimal contribution.

The eleventh volume of The First Encyclopedia of Tlön appears to contain enough information to allow Borges to discuss the languages of the imaginary planet. For instance, nouns in the languages of Tlön’s northern hemisphere (fictitious languages of fictitious countries set on a fictitious planet described in a supposedly real encyclopedia that, however, exists only in Borges’ mind) are constructed by a potentially infinite layering of adjectives. We also glean insights into the philosophies of Tlön. There is the school of thought that asserts, not without merit (my emphasis), that the only time that exists is the present, and that the present is infinite. There is the idea, perhaps drawn from quantum mechanics, that a quantity is indefinite (infinite in Borges’ words) until it is counted by an observer. Also the proposition that all literature has been written by a single, anonymous and temporally infinite author, and that fictive authors have been invented by critics a posteriori, to fit the style and content of different books written by the same anonymous author. There is the indefinite proliferation of hrönir, where each hrön is a separate and slightly different embodiment of a single object, lost by one person and found by a different person. But if a hrön is lost, it in turn gives rise to the propagation of its own sequence of hrönir. Borges (for one must remember that it is Borges, not the fictitious authors of a non-existent encyclopedia) also discusses the impossibility of rigorously demonstrating that all space exists simultaneously at the same time. I can’t avoid thinking that he was tinkering with special relativity and Minkowski’s light cones. For he also writes about the complementary notion: that it is impossible to know what exists in a given place at any time other than when the observer is present.

This story, written quite early in Borges’ career, may be one of his most complex pieces. As if it was the preface or summary of topics that he would turn to in individual stories, each of them dealing with a finite subset of Tlön’s mind boggling universe of ideas.

Thus, in The Circular Ruins we are introduced to a mysterious hermit who arrives, in a canoe, at a ruined circular temple. This man comes from a country located upstream, in which, we are told, there is an infinite number of villages. The temple has burned in the past and is now being swallowed by the jungle. The hermit has a single purpose: to dream another man, completely and in every detail, and then to have this man become real.

We are given hints that the wanderer who arrives in the canoe is somehow different. His wounds heal overnight and he is unable to recall who he is. Inhabitants of a nearby village have not seen him arrive, yet they know that he is now in the ruined temple and they bring food and drink offerings, that the man takes as his only sustenance. He eventually succeeds in constructing, in his dreams, a perfect being that is perceived as real by everybody except by himself and by fire. He knows that his dream is not real, and because the dream is not real it cannot burn. After erasing from this dream, his son, the memory of who he is and of how he came into being, he sends him downstream, to another identical ruined temple that he knows exists there, and tasks him to dream another man.

Word eventually gets to him of the existence of a man downstream, who is immune to fire. His own ruined temple soon burns again, but he himself cannot burn. He understands then that he is as unreal as his creation. He is a dream that was dreamt by another one like him, upstream. But the sequence repeats itself ad infinitum, because an infinite number of villages exist upstream. And the mysterious hermit who is the focus of the story must be a single randomly chosen element of an infinite set that extends to infinity in both directions, upstream and downstream from the circular ruins in which he dreams. Each of these ruins burned an infinite number of times, after an infinite number of visits by an infinite number of dreamers.

The impenetrability of infinity is what, to me, makes The Library of Babel one of Borges’ most remarkable stories. It describes the universe as a sphere of infinite size, filled by an infinite number of hexagonal galleries stacked upon one another and joined laterally by narrow corridors, and such that any hexagon that one chooses is at the center of the sphere – necessarily so in an infinite universe. There are books stacked in shelves in all galleries, exactly six hundred and forty books in each gallery – but of course the total number of books in the library is infinite. The library has always existed and, thus, will always exist. This is Borges’ version of the Perfect Cosmological Principle: the observer (in this case the unnamed librarian who has composed the narration) does not occupy any special place nor time. Hence, if the universe (i.e., the library, for they are one and the same) extends infinitely into the past, it must also extend infinitely into the future.

The books are written in a script that consists of a finite number of characters (twenty five) but the vast majority of books are unintelligible. The reason soon becomes clear. In an infinite library every single possible combination of characters exists, every single possible book exists. In an infinite library of infinite age there is an infinite number of librarians (i.e. inhabitants of the Universe), and thus an infinite number of languages. An obscure book, for instance, is said to be written in a samoyed-lithuanian-arabic dialect of the guaraní language. One can imagine that, by amalgamating four unrelated languages (of the Uralic, Indo-European, Semitic and Amerindian families), Borges is also implying that in an infinite Universe every possible historical and cultural evolutionary path has taken, or will take, place.

But human intelligence is finite, so only a finite set of character combinations will correspond to actual words in the language of one librarian, or of a finite set of librarians who speak the same language. Even if there is an infinite number of languages, the total number of meaningful character combinations (words) in one language will be finite. Therefore, an infinitesimally small fraction of the total number of words in the library will be intelligible to a given human. Some books are fully readable to our narrator, but most are fully unintelligible, and yet others contain only a few meaningful words immersed in a sea of gibberish.

The library contains books that justify the existence of each librarian, and that tell of their future heroic deeds. Many librarians have searched for their own book, yet no one has ever found it. The reason is mathematically obvious: since the total number of books is infinite, the probability of finding one specific book is zero. Librarians have also searched in vain for the books that explain the origin of the library itself – they must exist, but the probability of finding them is also zero.

An infinite library must also contain a book that is the perfect summary of all other books – the one book that contains all of the library. A librarian may have found this book, and this librarian would thus be a kind of all-knowing god. But the probability of finding this librarian and his book (i.e., of finding god) is also zero. This is as elegant a justification for agnosticism as I have ever seen. The complementary notion, that human existence is as meaningless as assuming the existence of a god, also arises from the infinite extent of the library. For with an infinite number of words written in an infinite number of books comes the certainty that everything that can possibly be written has already been written. There is no purpose to the existence of man.

In a characteristic example of Borgesian logic, the narrator realizes that the account that he is writing must already exist elsewhere in the library – in an infinite universe, an identical text must have also been written by another librarian – in fact by an infinite number of other librarians. Yet the narrator also wonders whether his words have the same meaning in the reader’s language, or in the language of another writer of an identical text. For it is possible that what he writes is readable and understandable in another language, but the meaning of those words is different in different languages. How are we to know that this is the case?

Borges follows a different route to infinity in The Garden of Forking Paths. On its surface, this story purports to be about a minor detail of the 1916 Somme offensive. A Chinese scholar by the name of Yu Tsun is living in England and is spying for Germany. He has discovered the location of a British artillery emplacement and must find a way of conveying this information to his handler in Berlin. Only in the very last paragraph of the story we learn that the way he accomplishes this is by killing an Englishman whose last name, Albert, is also the name of the French town where the gun emplacement is located. When his handler reads in the British newspapers that a Chinese man by the name of Yu Tsun has committed the murder, he understands the secret message.

A writer of spy thrillers would have been content with embellishing this story line with some cloak and dagger antics and narrow escapes, and little else. But this is Borges, so the apparently straightforward plot is only a thin veneer. Yu Tsun has looked up in the phone book the address of somebody, anybody, with a name that will convey the secret message. His murder plot thus appears to have a random target – no more than a man who is unlucky enough to carry the surname Albert. Yet when he gets off the train on his way to his victim’s house, some children near the station know that he is looking for Dr. Stephen Albert’s house, and offer directions to it. When Yu Tsun arrives at Albert’s house he discovers that he is a sinologist, and an expert on one of his own ancestors, Ts’ui Pên, who is famous for having attempted to build a labyrinth in which “all men would be lost”, and for having written a novel composed of seemingly random chapters. Stephen Albert has discovered that the labyrinth and the novel are one and the same, and that the labyrinth is one of time, not of space. In an uncanny, and certainly coincidental, foretelling of the “many worlds” interpretation of quantum mechanics, what Albert has discovered is that Ts’ui Pên’s thesis is that the universe consists of an infinite number of different sequences of events (“temporal lines” if one wishes) that exist side by side, or simultaneously. This is the labyrinth from which no escape is possible, for we are forced to be and do all that there is to be and do, in one world or another among an infinity of worlds.

The story that we are told is one among an infinite number of different stories involving Stephen Albert and Yu Tsun. The fact that a random search for a name in a telephone book leads Yu Tsun to kill an unknown man who has unraveled the mystery of his own ancestor’s work, and that some children at the train station know that he is looking for precisely this man, are thus not fanciful coincidences, nor do they have some banal explanation, as would have been concocted by a lesser writer. They are events that must take place in an infinitely complex universe in which everything happens somewhere or sometime.

I mentioned earlier that I thought of these four tales as starting points for my own explorations of infinity, and I may have been carried away beyond what Borges meant to say. I can only hope that, if he was still around, he would take the time to either agree with my extrapolations or, more likely and more interestingly, explain why he disagreed.