Balckwell Rising!

In mathematics, the word “identity” refers to two expressions that produce the same value. The most obvious example I can think of is x + 0 = x. No matter what you put as x, the two expressions on either side of the equals sign will have the same value.

As with all English words that have found themselves tangled up in mathematics, “identity” here has taken on an unfamiliar nuance. When I first asked my friend who knows about mathematics to explain this concept, I mistakenly used the phrase “share an identity” when referring to the two above expressions. But that’s not what’s happening at all. They do not share an identity; they are an identity.

At first glance, this might seem opposed to normal usage. We generally think of our identity as something that is unique to us. Most commonly, we are identified via the data displayed on government licenses: our name, our date of birth, our address, our face. These personal identifiers signal that we are us, and not someone else. Another common way we are identified is through usernames and passwords, or, when it comes to more modern technology, thumbprints or facial recognition.

However, what is happening here is not so different from mathematical identification. When a security officer at the airport checks our ID, what they are doing is confirming the equivalence of our data and our corporeal form. The person that stands before them and the data on the card are an identity.

An even cruder form of this phenomenon is when we log into a website using a username and password. As far as the website is concerned, anyone who knows that username and that password is the same person as whoever controls the account. If two people share the same account, the website still considers them the same person. They are thus an identity in the eyes of the website.

In this final example, we see clearly that the form of identification is vitally important here. People often use the word identity in a broader sense to refer to ethnicity, gender, or race. When people do this, what they are postulating is that if we were to identify people solely by their ethnicity, for example, then this group of people would be equivalent. Obviously, by any other standard, they are not equivalent at all, but are instead each a unique person. But any form of categorization involves finding identities in this mathematical sense.

In mathematics, all that is required for identity is for two expressions to be functionally equivalent, i.e. that given the same input, they will produce the same output. When my friend says to me that sin(x) = 1/csc(x), all he is saying is that both of these functions, if given the same input x, would produce the same result. How this output is actually derived is irrelevant. Each function reaches this value through a completely different route, but they both end up in the same place. For geometers, that’s a handy thing to know, in the same way that knowing a shortcut to the local grocery store is handy, even though there are several different routes that would get you there eventually.¹

But this equivalence is extremely limited. The result of each human life is death; you start with a living baby (the input), and end up with a dead body (the output). Functionally, then, we are all the same person. This information is useful perhaps in a philosophical sense, as something to keep in mind as we ponder the transience of our existence, but it’s not a practically useful means of identifying people. You can’t recognize Bob across the street by distinguishing him as “the person who is going to die,” because if we define Bob as “the person who is going to die,” then we are all Bob.

Similarly, in this example we can’t identify Bob using his date of birth or his profession (he’s a geometrist, as it turns out), because we are identifying him using our visual perception. Instead, we should use identifiable physical traits: his pale skin, long hair, and short stature. These three traits don’t describe Bob in any holistic sense; they merely identify him. They allow us to pick him out from a limited group of people², and that’s about as much as they’re useful for.

I bring all this up to lead us to Jorge Luis Borges, for whom this concept of identity held a great fascination. Throughout his short stories, he explores a variety of ways in which two people can be the same person.

The most explicit example is “The Theologians,” which tells of the lifelong antagonism between two characters named Aurelian and John Pannonia. These two men spend their lives involved in the same theological disputes, primarily concerning emerging heresies. While they are both on the same side in every dispute, Aurelian competes with John to refute the heresies more effectively, using different sources and arguments. Aurelian’s arguments are more complex and chaotic than John’s. They are labyrinthine and deeply idiosyncratic; in a sense, extremely personal. John’s, on the other hand, are measured, precise, and universal — they “[seem] written not by a particular person, but by any man — or perhaps all men.” (Collected Fictions, 203) In Aurelian’s eyes, he and John could not be more different, and eventually, Aurelian underhandedly has John convicted of heresy, for which he is put to death.

Years later, when Aurelian himself dies and takes his place in the Kingdom of Heaven, he finds that in the eyes of God, he and John are the same person. The story points out that this is not a matter of confusion on God’s part, for of course, the deity is incapable of such folly. What’s actually happening is that God is using a vastly different method of identification than we do. This method might be as simple as distinguishing those with faith from those without. In this sense, all of the faithful would be the same person.

But the fact that John (the convicted heretic) and Aurelian (the orthodox theologian) are grouped together points to the idea that God might not see the difference between so-called heretics and true believers, and therefore might not even distinguish Christians from Muslims, Jews, or Buddhists. Perhaps, in the broadest sense, all the human beings that we consider individuals are, in the eyes of God, the same man.

In another story in the same collection, “The Immortal,” Borges posits a similar idea: that a single person with an infinite lifespan is equivalent to any number of mortal men. That this immortal, within their infinite lifespan, will write all books, compose all poems, experience all events, and live all lives.

This motif appears again and again: a 19th c. Frenchman attempts to write Don Quixote word-for-word (“Pierre Menard, Author of Don Quixote”); an Irish revolutionary tells the story of a coward, only to reveal the coward to be himself (“The Shape of the Sword”); a man manifests a person from his dreams only to find that he, himself, was born from another man’s dream (“The Circular Ruins”).

We even see its negative sense portrayed in “Funes, His Memory,” the story of a man who is so remarkably lucid that he is able to capture in his memory every second of every day — “not only every leaf of every tree of every patch of forest, but every time he had perceived or imagined that leaf.” (136) Funes has no need of categorization or generalizations, these processes being mere aids to memory. Instead, he sees each phenomena as entirely unique: “it irritates him that the ‘dog’ of three-fourteen in the afternoon, seen in profile, should be indicated by the same noun as the ‘dog’ of three-fifteen, seen frontally.” (136) In this way, he shatters the possibility of any identity at all, since no two things can ever possibly be equivalent to each other, and in fact no one thing can ever even be equivalent to itself.

Of course, Funes’ way of viewing the world is untenable. If you refuse to identify the dog of three-fourteen with the dog of three-fifteen, then you can never understand anything about that dog. You’d be like Drew Barrymore in “Fifty First Dates”, who has to spend each morning re-learning who her husband is. Knowledge is the discovery of patterns via the recognition of similarities. Knowledge is, in a very important way, about discovering identities.

In the metatextual epilogue to his story “Averroes’ Search,” Borges provides an insight into his philosophy regarding identity. In the story, Borges attempts to imagine a day in the life of the Arab polymath Abū Al-Walīd Muḥammad Ibn ʾAḥmad Ibn Rushd, also known as Ibn Rushd, or, to later Latin-speaking scholars, “Averroes.” On the day Borges chooses to imagine, Averroes is working on his commentaries of Aristotle, specifically tackling the Poetics. Working from a translation of a translation of the original Greek, Averroes is stumped by two unfamiliar words, tragedy and comedy, for the simple reason that he doesn’t know what a play is. In fact, he’s never even seen a theatre. At the end of the story, no closer to understanding, Averroes disappears in a puff of smoke.

In the epilogue, Borges explains that he chose to tell this story because he wanted to write about failure. He goes on to say:

“I felt that Averroes, trying to imagine what a play is without ever having suspected what a theater is, was no more absurd than I, trying to imagine Averroes yet with no more material than a few snatches from Renan, Lane, and Asin Palacios. I felt, on the last page, that my story was a symbol of the man I had been when writing it, and that in order to write that story I had to be that man, and that in order to be that man I had to write that story, and so on, ad infinitum.”

Borges finds that his inability to understand Averroes mirrors Averroes’ inability to understand Aristotle; in realizing this, Borges finds that he can now understand Averroes, if only in the sense of understanding what it feels like to try to understand someone and fail. His goal was to identify with Averroes, and the only way for him to truly do so was to fail to identify with Averroes.

Jorge Luis Borges and Abū Al-Walīd Muḥammad Ibn ʾAḥmad Ibn Rushd lived centuries apart, an ocean apart, within two entirely different cultures. They spoke different languages, and wrote in different genres. And yet could we not say, if we were to bend time and space to set these two events — Averroes’ misunderstanding of Aristotle and Borges’ misunderstanding of Averroes — side by side, or perhaps one on top of the other, that the two of them are, in some sense, the same man? If God only had the chance to check in on the world twice, at these two particular moments, might he not think that the world is made up of but one man, endlessly repeating the absurd process of failing to recognize his own self?

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For a more rigorous look at the intersection of Borges and Mathematics, I recommend William Goldbloom Bloch’s The Unimaginable Mathematics of Borges’ Library of Babel.

1. In the world of pure mathematics, time is not actually an issue: in the real world, removing 253 apples from a group of 255 in order to end up with 2 would be a lot less efficient than removing 1 from a group of 3, but in the world of mathematics, these two activities take the same amount of time. Thus, a route to the grocery store that covers every square inch of the globe before reaching the store is functionally equivalent to (and therefore forms an identity with) cutting through the alley behind the post office, in the sense that the input (aka departure point) is “home” and the output (aka destination) is “the grocery store.”
   However, mathematicians don’t actually live in the world of pure mathematics; they occupy the same fallible mortal bodies as the rest of us, and therefore do still need to take time into consideration.) ↩︎
2. It is statistically unlikely that Bob will be among a crowd that includes more than one person fitting this description, but of course if our domain were to be the entire Earth, then there would once again be millions of Bobs. ↩︎