The Unimaginable Mathematics of Borges' Library of Babel

by William Goldbloom Bloch

  · Erase the symbol in the square; that is, for example, throw the books down the airshaft.

  · Write a new symbol—from the finite alphabet—in the square; that is, either reorder the books in the hexagon, move books from hexagon to hexagon, or actually, as the narrator of the story states, inscribe symbols on the flyleaf or in the margins of a book.

  · Change to a new internal state; that is, the librarian can go to the bathroom or go to sleep. Or, perhaps the librarian has a revelation and by acquiring new ideas, moves to a new cognitive view of the world. Or even simply that the librarian's mood changes.

  · Move to the left one square or move to the right one square; that is, move to a new hexagon.

  · Halt permanently; that is, expire.

  The librarian's life and the Library together embody a Turing machine, running an unimaginable program whose output can only be interpreted by a godlike external observer.

  A user.

  A reader.

  EIGHT

  Critical Points

  In the mountains of truth you will never climb in vain: either you will already get further up today or you will exercise your strength so that you can climb higher tomorrow.

  —Friedrich Nietzsche, Maxims, aphorism 358

  CRITICAL POINT IS A TERM THAT HAS ASSUMED A host of meanings in mathematics. Generally, it denotes a location where typical behavior breaks down and unusual and interesting phenomena occur. In the qualitative study of the solutions of differential equations, some critical points are singularities where flow lines of solutions may converge from many different regions and then, regardless of initial proximity, shoot off in a variety of divergent directions. This seems an apt metaphor for critical perspectives arising from the interpretation of a literary work.

  Taking advantage of Nietzsche's metaphor, at the base, mountains offer a multiplicity of approaches towards the summit. Frequently these trails converge to several natural routes, and oftentimes the descent must be taken via an unexpectedly different path. Thus it is no surprise that in my quest to read all commentaries by critics on "The Library of Babel," I learned that many predecessors have independently climbed and descended the mountain, some along paths with sections that closely parallel mine. This chapter is devoted towards outlining some of these trails. In particular, I am restricting myself to those that involve mathematics in one form or another. I begin by acknowledging those who independently found some of the same mathematics in the story.

  The eminent mathematician and pioneer German science fiction writer, Kurd Lasswitz, in his 1901 story "The Universal Library," not only calculates the number of books in his universal library, but also mentions that filling our known universe with books barely dents the total. As mentioned in the first endnote to chapter 1, Amaral, Rucker, Nicolas, Faucher, and Salpeter all calculate the number of books in the Library, while Bell-Villada excerpts a passage from Gamow's One, Two, Three. . . Infinity that shows he has a notion of how such a calculation should be performed.

  While looking for a reference, for the interested reader, to the topology and cosmology of the Library presented above, I discovered that in their 1999 article "Is Space Finite?," Luminet, Starkman, and Weeks listed "The Library of Babel" as a suggestion for further reading. At least one of them must have thought about the topology of the Library. Floyd Merrell also speculates about the topology of the Library and briefly discusses the possibility of a catalog, albeit mostly in the context of space-time physics, when he talks about light-cones for librarians and their world-lines, which are infinitesimal by contrast to the size of the Library. Furthermore, inspired by work of Bernadete, Merrell includes a brief discussion of a Book of Sand that is similar in spirit to my first interpretation.

  Whereas I have mainly sought to elucidate the mathematics in the story, most commentators endeavor to use mathematics to create a framework of analysis of the story and, more generally, Borges' oeuvre. It strikes me as an interesting philosophic pursuit to examine the project of importing mathematical and scientific terminology and systems into the field of literary analysis. I recuse myself from this study on the grounds that I am professionally neither a philosopher nor a critic. I am, however, qualified to comment on the correctness of the mathematics brought to bear on Borges, and, by dint of careful and extensive reading of Borges, to agree or disagree with various other interpretations of "The Library of Babel."1

  Given that literary critics mount defensible arguments for the primacy of interpretation over authorial intent, why should we worry whether or not a critic's use of mathematics is perfectly correct? By way of an answer, consider the following hypothetical misreading of Borges. The not-so-eminent critic William Goldbloom Blockhead, my not-so-bright alter ego, has confused "South America" with "South Africa." After all, they sound similar and are both, more or less, continents. Blockhead's stirring postcolonial analysis of Borges' work raises new and disturbing questions. For example, Blockhead wonders why Borges wrote in Spanish in lieu of English or Afrikaans, declines to mention apartheid, and exercises narrative space on Argentine border and independence wars rather than the Boer War and World War I. Does this mean that Borges' own political views must be coded, rather than overtly expressed? Next, Blockhead points out that the virtual invisibility of Africans in Borges' writing is a strong sign that, in fact, race is his most important agenda—what else could explain such an egregious omission? Blockhead's analysis triumphantly concludes with the unique insight that this interesting homme de lettres employs these strategies to avoid the literary trap of being read as merely a colonist writing in and about a colony.

  It's conceivable that some of Blockhead's remarks may be of independent interest, and some may even apply to Borges—after all, there are some real political and historic parallels between Argentina/Spain and South Africa/England. Nevertheless, one may well feel an irresistible urge to ignore wholesale the stream of mistaken inferences following from Blockhead's wrong assumption. Seen in this light, then, I offer bouquets of refractions arising from the light of other critic's works.

  Sarlo, in Jorge Luis Borges: A Writer on the Edge, tellingly sees the Library as a symbol of political oppression in the milieu of a totalitarian state. The librarians are, in today's vernacular, information serfs who will never be able to acquire the necessary data to transform their status. She writes that

  Structurally, the Library is also a panoptic, whose spatial distribution of masses and corridors allows one to see every place in it from any of its hexagons. The panoptic design of the Library brings to mind that of a prison where the guards should be able to see any cell from every possible perspective. Foucault has studied this layout as a spatialization of authoritarianism, as an image of a society where total control is possible and no private place (no private thought) is admitted. The universe described as the Library lacks any notion or possibility of privacy: all the activities are, by definition, public.

  The mental exercise implicitly asked now of the Reader is to imagine oneself in the Library and intuit the sensory and emotional experience: Is it dimly or brightly lit? Are the books musty or shockingly pristine? Would the airshafts induce a tremendous vertigo or could they be overlooked? Do the spiral staircases, placed either at every entrance or every other entrance, hem librarians as do the bars of a cage, or are they comforting mileposts along their life paths? Are there doors on the small rooms designed for sleeping and physical necessities, or have the librarians grown up accultured to a different kind of privacy than us?

  Once we establish our imagination in one of the hexagons, we see that the Library is not panoptic; the only available line of sight is in the air shaft central to the hexagon. Short of sticking a head into the airshaft while looking up or down, only a few hexagons would be visible before the convergence forced by the rules of perspective would hide all but a small part of the floors and ceilings.

  Furthermore, as discussed in the chapter "Geometry and Graph Theory," it isn't clear what the sight lines are like on
an individual floor; hexagons may be arranged in a straight line, or the entrances of the hexagons may well curve around nonlinearly. The most extensive vision afforded by the structure of the Library, then, would be a hexagon at the intersection of a cross formed by its airshaft and a straight-line corridor. However, even if a passage ran straight, the spiral staircases would block the views. By contrast, Bentham's Panopticon, as described in Foucault's famous work Discipline and Punish, enables full-time viewing of all inhabitants by obscured central scrutinizers: an altogether different geometry.

  Sarlo also writes that "the infinity of the Library cannot be empirically experienced, even if a traveller were granted infinite time." Even if the Library extends forever in the manner of Euclidean 3-space, there is a nifty way to see that the whole Library may, in fact, be visited. First, we make the reasonable assumption that the architecture of the Library allows all adjacent hexagons to be visited. Now, imagine starting in any hexagon, which we may now consider to be the Origin of the Library.

  1. Begin at the Origin and visit every hexagon adjacent to the Origin on the same floor as the Origin. Proceed up one floor and pass through all the hexagons on this floor that are above a previously visited hexagon. Now, go down two floors to the floor below that of the Origin and visit every hexagon on this floor that is below a previously visited hexagon. It is the case now that every hexagon a "distance" of one hexagon from the Origin has been visited.

  2. Next, beginning on the floor of the Origin, visit every hexagon adjacent to a previously visited hexagon. Hence, on the floor of the Origin, all hexagons within two hexagons of the Origin have been visited. Now use the spiral staircases to proceed up one and two floors and visit all the hexagons on them that are above a previously visited hexagon. Next, go down one floor below the Origin, then two floors below the Origin, and do the same. Note that every hexagon a "distance" of two hexagons from the Origin has been visited.

  3. Next, again starting on the floor of the Origin, visit every hexagon adjacent to a previously visited hexagon, then proceed up one and two and three floors, visiting all the hexagons on each of these floors that are above a previously visited hexagon. Do the same for the one, two, and three floors below the floor of the Origin, and it follows immediately that every hexagon a "distance" of three hexagons from the Origin has been visited.

  4. Etc.

  If this hexagon-visiting algorithm is carried out, it is not hard to see that at any stage, only a finite number of hexagons have been visited, and a traveler granted infinite time must eventually visit every single hexagon. This last assertion follows because even if the Library extends infinitely in all directions, it still must be the case that any hexagon in the Library is fixed at a finite number of hexagons from the Origin—the hexagon in which the traveler began.

  Although he never explicitly mentions "The Library of Babel," I include a discussion of Svend Østergaard because I believe the book proposed in the final footnote of "The Library of Babel" is of a similar structure to the book described in Borges' short story "The Book of Sand." In The Mathematics of Meaning, Østergaard discusses the Book of Sand, but proceeds under the unwarranted assumption that it possesses uncountably infinitely many pages.2

  I find it unlikely that there are uncountably many pages, for the narrator of "The Book of Sand" only mentions the finding of integer-numbered pages: if that was so, it would be extraordinary to find even a single page numbered by an integer, because the likelihood of randomly finding an integer in the real number line is zero—or if that sounds improbably absolute, "vanishingly small." This is because the set of integers is countable and therefore, when considered as a set contained inside of the real numbers, it is of measure 0. This entails that any integer is much harder to find than a single prespecified dust speck adrift in South America. (See my second interpretation of the Book of Sand in the chapter "Real Analysis" for a discussion on measure 0.)

  This is essentially the reason that while the narrator for "The Book of Sand" is looking at a particular page, the mysterious stranger adjures him to "Look at it well. You will never see it again." The probability of randomly picking the same integer twice is also vanishingly small. To see this, imagine opening the Book of Sand to page 17. If there were only 100 pages in the Book, each time it was opened again there would be a 1/100 chance of randomly opening it to page 17. If there were 1,000 pages, there would be a 1/1,000 chance. If there were a million pages, there would be a 1/1,000,000 chance. If there were infinitely many pages, it is tempting to write that there would be a 1/ chance, meaning "probability 0." But it wouldn't be correct to write that, and the story of the probability, while interesting and exciting, is beyond the scope of this book. (If the stranger and narrator were truly interested in seeing a page a second time, it's fair to wonder why they didn't simply insert a cardboard bookmark to reenter this bookish Heraclitean river twice.)

  Moreover, the narrator of "The Book of Sand" states that illustrations occur every 2,000 pages. If there were uncountably many pages—that is, the same number of pages as there are of points in the real number line— then there would be no way of counting the number of pages between two selected pages. In fact, there'd be no way to find a "next page," for in the real number line, numbers lap up against each other with no "closest" number. This is known, in various guises, as the Archimedean property.

  I have great sympathy for the last two critics I'm going to discuss, N. Katherine Hayles and Merrell; their project, as I understand it, is truly noble. They seek to create or expand upon a theory which accounts for all the complex interrelations between the perceivable universe, consciousness, all previous and current human works, the Zeitgeist, culture, language, author, text, interpretation, and reader. Such a theory would, by virtue of absorption, dwarf a Grand Unified Theory of Everything from physics. It is natural, therefore, that two literary critics, steeped in the disciplines of chemistry and physics, would appropriate the language and approaches of mathematics and science to employ them in this most ambitious theory.

  Hayles' work (Hayles, 138—67) primarily consists of associating ideas of self-referentiality and infinite sequences, infinite series, and infinite sets to Borges' work. Many of her insights are deep. Although some passages seek to persuade the reader of the meaninglessness and marginalization of mathematics, Hayles is content to use mathematics as a means for understanding Borges, perhaps in the same way a sponge, riddled with holes, is useful in sopping up fluid reality.

  After a précis of the story, on page 151 Hayles critiques the librarian's "elegant hope" by noting that "the narrator's 'solution' is of course an answer only in a very narrow sense. While it suggests a way to transform randomness into ordered sequence, it contains no hint of how that sequence may be rendered intelligible or meaningful." As I noted in the chapter "Topology and Cosmology," the patterning of a periodically repeating Library may be thought of as symmetric three-dimensional wallpaper. For example, the illustration in figure 67 is, in some sense, random and chaotic. However, when it repeats periodically, it takes on a pleasant enough symmetry; an order, if you will (figure 68). I contend again that this is the Order that in-formed the narrator's out-look.

  Hayles' main intent, in her reading of "The Library of Babel," is poetic: she wants a Borgesian "Strange Loop" to dissolve the boundary between the reader and the text by roping the reader into the story itself. However, to accomplish this lyrical agenda, Hayles writes on page 152 that "Logic demands that we conclude the present text in hand (which of course is printed) to be the Library's book. What we have is not the narrator's handwritten text but a mirror of it, or perhaps one of the 'several hundreds of thousands of imperfect facsimiles."' It is curious that a critic eager to limit logic should invoke it almost as a magic amulet, for "logic" doesn't "demand" anything. Rather, it seems to me that Hayles is attempting to have her theory of Strange Loops produce a variation of a result that Borges himself stresses in the story, "This useless and wordy epistle already exists in one of the t
hirty volumes of the five shelves in one of the uncountable hexagons—and so does its refutation." The story is in the Library, the book it originally appeared in, Ficciones, is in the Library, and the complete works of Borges are in the Library. Hayles' books, the words of this book, and anything that can be written using 25 orthographic symbols: all are necessarily in the Library. Nevertheless, none of these inclusions implies that we are at this moment reading a Library book or that we are librarians roaming a universe of hexagons.

  If we can legitimately assume as a premise that we are holding a Library book in our hands, then Hayles' next set of ideas, which are intriguing, do follow: "... even more important is the implication that we are reading the Library's book. This, in turn, implies that we, like the narrator, are within the Library examining one of its volumes, which means that we, no less than the narrator, are contained within one of the books we peruse." Since the premise is unfounded—my copy of Ficciones is not 410 pages, I'm not in a dimly lit hexagon—the chain of implications does not follow.3 If, on the other hand, Hayles was referring to a sort of narratological space created by the story, where we readers accept that by virtue of reading the story we are somehow in the story's confines, then it still doesn't follow that we are reading a printed text of the Library rather than a handwritten note of an avuncular librarian. In fact, given that we are human, inhabiting a miniscule section of the Library where humans reside, it is vastly more likely that we would stumble across a book which a human has inscribed than one which contains the story "The Library of Babel."