The Unimaginable Mathematics of Borges' Library of Babel

by William Goldbloom Bloch

  As a reader, when I encounter an endnote, I'm compelled almost against my will to flip to the back of the book to learn what the endnote says.1 As I writer, I find that despite my best efforts to incorporate them into the body of the book, my work includes diverting digressions, fine points of mathematics that might interest only specialists, and citations to other works. All of these are consigned to the endnotes.

  After the glossary, an annotated list of suggested readings is provided for those with curiosity primed to learn more of the mathematics used in the book. A bibliography of references cited or consulted rounds out the end matter.

  Introduction

  We adore chaos because we love to produce order.

  —M. C. Escher

  IT’S AN IRONIC JOKE THAT BORGES WOULD HAVE appreciated: I am a mathematician who, lacking Spanish, perforce reads "The Library of Babel" in translation. Furthermore, although I bring several thousand years of theory to bear on the story, none of it is literary theory.

  Having issued these caveats, it is my purpose to make explicit a number of mathematical ideas inherent in the story. My goal in this task is not to reduce the story in any capacity; rather it is to enrich and edify the reader by glossing the intellectual margins and substructures. Borges was a consummate synthesist; his lapidary prose sparkles and reveals unexpected depths when examined from any angle or perspective. I submit that because of his well-known affection for mathematics, exploring the story through the eyes of a mathematician is a dynamic, useful, and necessary addition to the body of Borgesian criticism.

  In what follows, I assume no special mathematical knowledge. I only ask that the reader trust that I am a tour guide through a labyrinth, like that marble pathway on the floor of the cathedral at Chartres, not the gatekeeper of a Stygian maze without center or exit (figure 2). Beyond enhancing the story, the reader's reward will be an exposure to some intriguing and entrancing mathematical ideas.

  Borges was a master of understating ideas, allowing them the possibility of gathering heft and power, of generating their own gravity. I'm under no delusion that he traced out all the consequences of the dormant mathematics I uncover. I allow myselfthe ambition, though, to paraphrase what Borges wrote in a forward and hope that this book would have taught him many things about himself (see Barrenchea, p. vii).

  I request a last indulgence from the reader. The introductory material, thus far, has been written in the friendly and confiding first person singular voice. Starting in the next paragraph, I will inhabit the first person plural for the duration of the mathematical expositions. This should not be construed as a "royal we." It has been a construct of the community of mathematicians for centuries and it traditionally signifies two ideas: that "we" are all in consultation with each other through space and time, making use of each other's insights and ideas to advance the ongoing human project of mathematics, and that "we"—the author and reader—are together following the sequences of logical ideas that lead to inexorable, and sometimes poetic, conclusions.

  A word, too, about the language in the book. We started our college years intending to be some sort of creative writer. Beyond the insight mathematics offered into the natural world and epiphenomena of life, and beyond the aesthetic joy at understanding how the iron rules of logic crystallize a good proof into a work of art, one of the reasons we turned to math was the lilt and rhythm of the "if-then" syntax coupled with the musicality of words often repeated, such as "thus," "hence," "suppose," and "let." We hope our readers might develop an ear for this music, too.

  We close the introduction by offering several related disclaimers. Mathematics, like any discipline, is not a monolith; it's a sprawling agglutination of overlapping and intersecting fields and specialties: one's talents, tastes, and beliefs determine individual focus. We carefully checked and rechecked our ideas, mathematics, and figures. To the best of our knowledge, there are no mistakes. However, a different mathematician might well expose divergent mathematical themes from the story and utilize different sets of ideas to explain them.

  Furthermore, there's a natural tendency for an individual reaching across traditional boundaries to be perceived as a universal embodiment of the foreign, the other. Although our inductions and deductions are correct, some mathematicians might issue philosophic challenges to underlying assumptions, especially in the chapters "Real Analysis" and "More Combinatorics." Consequently, no one, including the author, should be seen as a Representative or Ambassador, speaking in one voice for an ideologically unified Entity of Mathematicians: such an Entity of Mathematicians simply doesn't exist. (Lest this be subject to misinterpretation, allow us to note that all mathematicians would agree on the centrality of logically consistent deductions and derivations from agreed-upon axioms.)

  It's important to bear in mind that the mathematical expositions contained herein are not rigorously developed, nor are they intended as comprehensive introductions to the various theories. Just as a stirring musical performance will not transform a concertgoer into a musician, composer, lyricist, musicologist, or music critic, so this book won't transform a reader into any kind of a mathematician. However, just as a concert may move, inspire, or transfigure a listener, so we hope that this book will stimulate, dazzle, and expand its readers.

  Finally, about the title of the book: why the word "unimaginable"? By way of an answer, we note that in his sixth Meditation, Descartes makes clear the distinction between simply naming a thing and visualizing it in a clear, precise way that allows for mental manipulations.

  I note first the difference between imagination and pure intellection or conception. For example, when I imagine a triangle, I not only conceive it as a figure composed of three lines, but moreover consider these three lines as being present by the power and internal application of my mind, and that is properly what I call imagining. Now if I wish to think of a chiliagon, I indeed rightly conceive that it is a figure composed of a thousand sides, as easily as I conceive that a triangle is a figure composed of only three sides; but I cannot imagine the thousand sides of a chiliagon, as I do the three of a triangle, neither, so to speak, can I look upon them as present with the eyes of my mind.

  Some of the ideas we'll talk about, such as titanic numbers and higher dimensions, are unimaginable in this sense. We can give names to the ideas, use metaphors to approach them, give simple examples to substitute in as models, and try to find a consistent set of rules and mathematical objects that encapsulate the essence of the ideas—but we will never be able to visualize them any more than we could Descartes' thousand-sided chiliagon. Indeed, our task as your guide is to trigger the processes by which you build intuition and insight into the Unimaginable.

  The Unimaginable Mathematics of Borges' Library of Babel

  The Library of Babel

  Jorge Luis Borges

  By this art you may contemplate the variation of the 23 letters....

  —Anatomy of Melancholy, Pt. 2, Sec. II, Mem. IV

  THE UNIVERSE (WHICH OTHERS CALL THE Library) is composed of an indefinite, perhaps infinite number of hexagonal galleries. In the center of each gallery is a ventilation shaft, bounded by a low railing. From any hexagon one can see the floors above and below—one after another, endlessly. The arrangement of the galleries is always the same: Twenty bookshelves, five to each side, line four of the hexagon's six sides; the height of the bookshelves, floor to ceiling, is hardly greater than the height of a normal librarian. One of the hexagon's free sides opens onto a narrow sort of vestibule, which in turn opens onto another gallery, identical to the first—identical in fact to all. To the left and right of the vestibule are two tiny compartments. One is for sleeping, upright; the other, for satisfying one's physical necessities. Through this space, too, there passes a spiral staircase, which winds upward and downward into the remotest distance. In the vestibule there is a mirror, which faithfully duplicates appearances. Men often infer from this mirror that the Library is not infinite—if it were, what need would there be for
that illusory replication? I prefer to dream that burnished surfaces are a figuration and promise of the infinite. . . . Light is provided by certain spherical fruits that bear the name "bulbs." There are two of these bulbs in each hexagon, set crosswise. The light they give is insufficient, and unceasing.

  Like all the men of the Library, in my younger days I traveled; I have journeyed in quest of a book, perhaps the catalog of catalogs. Now that my eyes can hardly make out what I myself have written, I am preparing to die, a few leagues from the hexagon where I was born. When I am dead, compassionate hands will throw me over the railing; my tomb will be the unfathomable air, my body will sink for ages, and will decay and dissolve in the wind engendered by my fall, which shall be infinite. I declare that the Library is endless. Idealists argue that the hexagonal rooms are the necessary shape of absolute space, or at least of our perception of space. They argue that a triangular or pentagonal chamber is inconceivable. (Mystics claim that their ecstasies reveal to them a circular chamber containing an enormous circular book with a continuous spine that goes completely around the walls. But their testimony is suspect, their words obscure. That cyclical book is God.) Let it suffice for the moment that I repeat the classic dictum: The Library is a sphere whose exact center is any hexagon and whose circumference is unattainable.

  Each wall of each hexagon is furnished with five bookshelves; each bookshelf holds thirty-two books identical in format; each book contains four hundred ten pages; each page, forty lines; each line, approximately eighty black letters. There are also letters on the front cover of each book; those letters neither indicate nor prefigure what the pages inside will say. I am aware that that lack of correspondence once struck men as mysterious. Before summarizing the solution of the mystery (whose discovery, in spite of its tragic consequences, is perhaps the most important event in all history), I wish to recall a few axioms.

  First: The Library has existed ab æternitate. That truth, whose immediate corollary is the future eternity of the world, no rational mind can doubt. Man, the imperfect librarian, may be the work of chance or of malevolent demiurges; the universe, with its elegant appointments— its bookshelves, its enigmatic books, its indefatigable staircases for the traveler, and its water closets for the seated librarian—can only be the handiwork of a god. In order to grasp the distance that separates the human and the divine, one has only to compare these crude trembling symbols which my fallible hand scrawls on the cover of a book with the organic letters inside—neat, delicate, deep black, and inimitably symmetrical.

  Second: There are twenty-five orthographic symbols.1 That discovery enabled mankind, three hundred years ago, to formulate a general theory of the Library and thereby satisfactorily solve the riddle that no conjecture had been able to divine—the formless and chaotic nature of virtually all books. One book, which my father once saw in a hexagon in circuit 15—94, consisted of the letters M C V perversely repeated from the first line to the last. Another (much consulted in this zone) is a mere labyrinth of letters whose penultimate page contains the phrase O Time thy pyramids. This much is known: For every rational line or forthright statement there are leagues of senseless cacophony, verbal nonsense, and incoherency. (I know of one semibarbarous zone whose librarians repudiate the "vain and superstitious habit" of trying to find sense in books, equating such a quest with attempting to find meaning in dreams or in the chaotic lines of the palm of one's hand. . .. They will acknowledge that the inventors of writing imitated the twenty-five natural symbols, but contend that that adoption was fortuitous, coincidental, and that books in themselves have no meaning. That argument, as we shall see, is not entirely fallacious.)

  For many years it was believed that those impenetrable books were in ancient or far-distant languages. It is true that the most ancient peoples, the first librarians, employed a language quite different from the one we speak today; it is true that a few miles to the right, our language devolves into dialect and that ninety floors above, it becomes incomprehensible. All of that, I repeat, is true—but four hundred ten pages of unvarying M C V's cannot belong to any language, however dialectal or primitive it may be. Some have suggested that each letter influences the next, and that the value of M C V on page 71, line 3, is not the value of the same series on another line of another page, but that vague thesis has not met with any great acceptance. Others have mentioned the possibility of codes; that conjecture has been universally accepted, though not in the sense in which its originators formulated it.

  Some five hundred years ago, the chief of one of the upper hexagons2 came across a book as jumbled as all the others, but containing almost two pages of homogeneous lines. He showed his find to a traveling decipherer, who told him that the lines were written in Portuguese; others said it was Yiddish. Within the century experts had determined what the language actually was: a Samoyed-Lithuanian dialect of Guarani, with inflections from classical Arabic. The content was also determined: the rudiments of combinatory analysis, illustrated with examples of endlessly repeating variations. Those examples allowed a librarian of genius to discover the fundamental law of the Library. This philosopher observed that all books, however different from one another they might be, consist of identical elements: the space, the period, the comma, and the twenty-two letters of the alphabet. He also posited a fact which all travelers have since confirmed: In all the Library, there are no two identical books. From those incontrovertible premises, the librarian deduced that the Library is "total"—perfect, complete, and whole—and that its bookshelves contain all possible combinations of the twenty-two orthographic symbols (a number which, though unimaginably vast, is not infinite)—that is, all that is able to be expressed, in every language. All—the detailed history of the future, the autobiographies of the archangels, the faithful catalog of the Library, thousands and thousands of false catalogs, the proof of the falsity of those false catalogs, a proof of the falsity of the true catalog, the gnostic gospel of Basilides, the commentary upon that gospel, the commentary on the commentary on that gospel, the true story of your death, the translation of every book into every language, the interpolations of every book into all books, the treatise Bede could have written (but did not) on the mythology of the Saxon people, the lost books of Tacitus.

  When it was announced that the Library contained all books, the first reaction was unbounded joy. All men felt themselves the possessors of an intact and secret treasure. There was no personal problem, no world problem, whose eloquent solution did not exist—somewhere in some hexagon. The universe was justified; the universe suddenly became congruent with the unlimited width and breadth of humankind's hope. At that period there was much talk of The Vindications—books of apologia and prophecies that would vindicate for all time the actions of every person in the universe and that held wondrous arcana for men's futures. Thousands of greedy individuals abandoned their sweet native hexagons and rushed downstairs, upstairs, spurred by the vain desire to find their Vindication. These pilgrims squabbled in the narrow corridors, muttered dark imprecations, strangled one another on the divine staircases, threw deceiving volumes down ventilation shafts, were themselves hurled to their deaths by men of distant regions. Others went insane. . . . The Vindications do exist (I have seen two of them, which refer to persons in the future, persons perhaps not imaginary), but those who went in quest of them failed to recall that the chance of a man's finding his own Vindication, or some perfidious version of his own, can be calculated to be zero.

  At that same period there was also hope that the fundamental mysteries of mankind—the origin of the Library and of time—might be revealed. In all likelihood those profound mysteries can indeed be explained in words; if the language of the philosophers is not sufficient, then the multiform Library must surely have produced the extraordinary language that is required, together with the words and grammar of that language. For four centuries, men have been scouring the hexagons. . .. There are official searchers, the "inquisitors." I have seen them about their tasks: they
arrive exhausted at some hexagon, they talk about a staircase that nearly killed them—rungs were missing—they speak with the librarian about galleries and staircases, and, once in a while, they take up the nearest book and leaf through it, searching for disgraceful or dishonorable words. Clearly, no one expects to discover anything.

  That unbridled hopefulness was succeeded, naturally enough, by a similarly disproportionate depression. The certainty that some bookshelf in some hexagon contained precious books, yet that those precious books were forever out of reach, was almost unbearable. One blasphemous sect proposed that the searches be discontinued and that all men shuffle letters and symbols until those canonical books, through some improbable stroke of chance, had been constructed. The authorities were forced to issue strict orders. The sect disappeared, but in my childhood I have seen old men who for long periods would hide in the latrines with metal disks and a forbidden dice cup, feebly mimicking the divine disorder.

  Others, going about it in the opposite way, thought the first thing to do was eliminate all worthless books. They would invade the hexagons, show credentials that were not always false, leaf disgustedly through a volume, and condemn entire walls of books. It is to their hygienic, ascetic rage that we lay the senseless loss of millions of volumes. Their name is execrated today, but those who grieve over the "treasures" destroyed in that frenzy overlook two widely acknowledged facts: One, that the Library is so huge that any reduction by human hands must be infinitesimal. And two, that each book is unique and irreplaceable, but (since the Library is total) there are always several hundred thousand imperfect facsimiles—books that differ by no more than a single letter, or a comma. Despite general opinion, I daresay that the consequences of the depredations committed by the Purifiers have been exaggerated by the horror those same fanatics inspired. They were spurred on by the holy zeal to reach—someday, through unrelenting effort—the books of the Crimson Hexagon—books smaller than natural books, books omnipotent, illustrated, and magical.