The Unimaginable Mathematics of Borges' Library of Babel

by William Goldbloom Bloch

  The story, including spaces, is comprised of approximately 18,000 lexical symbols. These contiguous 18,000 symbols could occupy 1,294,001 different starting positions in a 410-page book. To see this, observe that the first such position would entail that the first symbol of the story occupied the first slot of the book's 1,312,000 slots. The last such position has the concluding period of the story occupying the last slot in the book. This entails that the first symbol of the story occupies the 1,294,000th slot. It may be helpful to visualize this process as a block of 18,000 red squares moving along a tape of 1,312,000 slots. If the first red square is at the first slot, then the last red square is at the 18,000th slot. If the last red square is at the last slot, then the first red square is at the [(1,312,000 — 18,000) + 1]th slot.

  If we remind ourselves of the work we did in the chapter "Combinatorics," we see that the number of books containing the story at a specific starting position is the number of different ways the slots not occupied by the story can be filled. The number of unfilled slots is

  From the first paragraph of this note, we see there are 1,294,001 such positions; thus there are approximately

  distinct books in the Library containing the story. So the probability of finding a Library volume containing the story is

  which, in turn, is approximately

  This is roughly equivalent to the likelihood of winning a major lottery 3,600 times in succession!

  There is, however, a profound sense in which Hayles is correct, a sense that Borges explicitly intended. You are in the Library. A multivolume set is scattered throughout the Library that details every single day of your life, including your death. In fact, a multi-quintillion volume set that details the lives, deaths, and protein transfers of each and every one of your cells is also scattered throughout the Library.

  By invoking this theme in conjunction with the idea of potential inaccuracies of a particular volume, Hayles opens the door to a stimulating line of thought. Suppose, miraculously, you were to find a grouped set of volumes, each of which had one page dedicated to one day of your life. Every single page, as far as your memory can recall and corroborate, is an accurate portrayal of that day. You read the page that corresponds to tomorrow. At the end of the next day, you reread the page: it, too, turns out to be an accurate description of the day. You continue this process for years; unimaginably, and despite your perverse and whimsical attempts to subvert their accuracy, the books continue to meticulously depict your days.

  Here's the question: can you now say, with certainty, that the page that corresponds to today's tomorrow will also be accurate? No! Based on the number of books in the Library and the number of ways in which the description may be inaccurate, despite the long streak of accurate descriptions, it is almost a certainty that the book will deviate. This is a disturbing and counterintuitive conjunction of probability with the comprehensiveness of the Library, yet it is unavoidable. Perhaps this example might help clarify the point. Suppose you flipped a fair coin 15,000 times, which is about once a day for 40 years, and it always came up heads. If you flipped the coin tomorrow, would you expect it to be heads or tails? Of course you'd expect it to be heads again—but if it's a fair coin, there's an equally likely chance it will be tails! A closer correspondence to the probabilities associated with such a book might be: suppose that every day for the past 20 years, you've won the big jackpot of the daily lottery Do you believe you'll win tomorrow, too? The odds are tremendously against it, but then again, the odds were even more incredible against your winning every day for 20 years. How can you rationally assess tomorrow?

  Merrell's book, Unthinking Thinking: Jorge Luis Borges, Mathematics, and the New Physics, is the most comprehensive attempt to link ideas of modern mathematics, physics, and philosophy with Borges, via the critical tools of literary analysis. As such, Unthinking Thinking contains a number of interesting insights and juxtapositions. For example, Merrell offers unique perspectives on the structure of the Library as seen through the lenses of the theory of special relativity and the expanding universe theory

  I think Merrell makes solid contributions in two areas. I particularly enjoyed his thoughts regarding enantiomorphic (mirror-reversed) forms. He provides a nice discussion of mirror-reversal in the Möbius band and applies his notion imaginatively to the "problem" of mirrors in the Library, especially in reference to his relativistic "world-lines" of librarians.

  Second, Merrell gives four arguments for the impossibility of deriving a global order of the Library from the local information that a librarian would have available. Merrell's arguments run the gamut from intertextual references to Borges' story "Averroes' Search" to an appeal to authorities on probability; in particular, Spencer-Brown and the astronomer Layzer, as quoted in Campbell.

  Here is a concrete way of thinking about this problem. Suppose I provide you with a rule to generate a sequence, something such as "Start with the two numbers 0, 1. Forever after, employ Rule Fib."

  Rule Fib: The next number in the sequence is defined to be the sum of the preceding two numbers.

  Rule Fib entails that to find the third number, you must add the first two numbers:

  0, 1, 0 + 1 = 0, 1, 1.

  To get the fourth number, you add the second and third terms, 1 + 1, and get:

  0, 1, 1, 2.

  To get the fifth number, you add the third and fourth terms, 1 + 2, and get:

  0, 1, 1, 2, 3.

  The sequence—actually a famous sequence, known as the Fibonacci sequence—begins to grow rapidly:

  0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89,...

  Given time and inclination, you or a computer could generate many numbers of this sequence. Conversely, if I provided you with the sequence above, you might well guess the rule that produces "the next term." However, the rule might be much more complicated; for example, it might be: "Let the first 12 terms correspond to the Fibonacci sequence; let the next 12 be the first 12 digits in the decimal expansion of ; the 47 digits after that should all be 7s; etc. etc. etc." Given a more complicated rule such as this last one, although your guess is "good," it relies on the false assumption that the rule generating the sequence must be as simple as possible. This has the unfortunate effect that your good guess produces the wrong answer. I conclude that without complete information, there is no way to ensure the successful induction of a unique generating-rule for a sequence.

  The problem is much worse in higher dimensions. The example of the Fibonacci sequence is one-dimensional; now let's look at one example of the "guessing-of-terms" problem in a two-dimensional setting. The numbers in the hexagonal array pictured in figure 69 are the first seven terms of the Fibonacci sequence, and let's imagine that all the digits to fill out the plane are exactly those appearing in the Fibonacci sequence. But how are the next twelve terms to be ordered? Your guess is as good as mine: There are a vast number of distribution rules utilizing the Fibonacci digits which would produce the pictured hexagonal array. Now imagine a librarian's dilemma, confronting 410-page collections of seemingly random lexical symbols—not even numbers—distributed in some sort of three-dimensional lattice. The mind balks at conceiving of any rule to order the books.

  A final pair of observations that fit this chapter best. First, Borges introduces the belief of the Book-Man.

  We also have knowledge of another superstition from that period: belief in what was termed the Book-Man. On some shelf in some hexagon, it was argued, there must exist a book that is the cipher and perfect compendium of all other books, and some librarian must have examined that book; this librarian is analogous to a god.

  A cipher is either a key or a code, and a compendium is, according to various dictionaries, a brief, a condensation, an epitome, or an abstract. My guess is that Borges meant that since it is possible to conceive of a book 410 pages in length that is a key to and an abridgement of the Library, that book must therefore exist in the Library. Such a book might, in today's parlance, be a computer algorithm for generating all
possible symbol sequences of length 1,312,000 from an alphabet of 25 orthographic symbols, for such an algorithm could actually be written in just a few lines of code. More of the Book might be devoted to the generating principles and topology of the Library; possibly a rule for ordering the books (although probably no such rule could fit in one volume); the motivations of the constructors of the Library; how the Library was built and where the materials for it came from; how librarians entered the system; etc. etc. Since such a book can be conceived, the import of Borges' footnote is precisely that it must appear in the Library. Of course, its refutations also exist in the Library, a fact that highlights and compounds the problem of interpretation of truth.

  Second, many critics, including some of those mentioned here, have speculated about the meaning and significance of one of Borges' parenthetical asides in the story:

  (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.)

  I won't presume to provide an exegesis of the cyclical book, but I offer the following insight for a future critic who might wish to interpret it: I believe that again Borges is winking at the reader.

  It would be impossible to remove such a book from the shelf!

  The only way to read the book would be to physically cut out sections; in other words, the only way for the mystics to attain the Book that is God would be to destroy It. The Book is closed (figure 70).

  NINE

  Openings

  To open a book brings profit.

  —Chinese proverb

  IN THIS CHAPTER, I ASSEMBLE SOME FACTS FOR THE purpose of sketching a picture of the mathematics Borges may have known and how it may have affected the story In his prologue to the first part of Ficciones, Borges winks yet again at the reader when he writes "I am not the first author of the narrative titled 'The Library of Babel'; those curious to know its history and its prehistory may interrogate a certain page of Number 59 of the journal Sur, which records the heterogenous names of Leucippus and Lasswitz, of Lewis Carroll and Aristotle." This is precisely the issue of Sur in which his essay "The Total Library" appears.1

  Perhaps few others have had the patience to ferret out the particulars of a hint of Borges' knowledge of combinatorics. Borges opens the story with the following fragment from Burton's The Anatomy of Melancholy: "By this art you may contemplate the variation of the 23 letters. . ." The entire section of Burton is concerned with ways of diverting and amusing oneself, ostensibly towards the end of avoiding or curing melancholy.2 For several pages before the excerpt, Burton waxes erudite on the pleasures of reading, especially scripture, and of libraries. Without even a paragraph break to ease the transition, Burton moves to pleasures mathematical (emphasis added):

  art of memory, Cosmus Rosselius, Pet. Ravennas, Scenkelius's Detectus, or practise Brachygraphy, &c., that will ask a great deal of attention; or let him demonstrate a proposition in Euclid, in his last five books, extract a square root, or study Algebra; than which, as Clavius holds, "in all human disciplines nothing can be more excellent and pleasant, so abstruse and recondite, so bewitching, so miraculous, so ravishing, so easy withal and full of delight," omnem humanum captum superare videtur. By this means you may define ex ungue leonem, as the diverb is, by his thumb alone the bigness of Hercules, or the true dimensions of the great Colossus, Solomon's temple, and Domitian's amphitheatre out of a little part. By this art you may contemplate the variation of the twenty-three letters, which may be so infinitely varied, that the words complicated and deduced thence will not be contained within the compass of the firmament; ten words may be varied 40,320 several ways; by this art you may examine how many men may stand one by another in the whole superficies of the earth .. .

  It's worth mentioning that the number of distinct ways to order eight words is

  8! = 40,320.

  Perhaps Burton had neither the skill nor the stomach to continue multiplying 40,320 by 9 and then again by 10, which would yield 3,628,800, the number of different ways to order 10 words. Whether or not Borges would have recognized this number is moot, yet in his 1936 essay "The Doctrine of Cycles," he correctly calculates the number of ways that the order of 10 atoms can be permuted.

  Regardless, he was aware that the passage alluded to combinations and permutations, and that "the words complicated and deduced thence will not be contained within the compass of the firmament." Later in the story, Borges' use of the phrase "the rudiments of combinatory analysis, illustrated with examples of endlessly repeating variations" shows that Borges understood the ideas well, even if a modern mathematician would more likely employ the phrase "variations with unlimited repetition."

  Beyond gleaning the story and Selected Non-Fictions for clues about his knowledge and predilections, I was fortunate to find another source of information. The chapter title, "Openings," stems from an intersection of optimism and pseudo-randomness. While visiting the National Library of Argentina, I had the great pleasure of perusing the math and science books Borges donated to the collection. I applied the principle that a book beloved by its owner, when held gently underneath the spine and allowed to fall open, will naturally reveal an oft-consulted page. My excitement at achieving interesting results was matched by my chagrin when, after multiple applications of this "opening" principle, I discovered that Borges marked the back end leaves of his volumes with his name, the year of acquisition, and the page numbers—coupled with a succinct phrase—of passages that especially interested him. My chagrin was tempered by the fact that his annotated page numbers unmistakably corresponded with my optimistic openings.

  I'll begin with a book that postdates "The Library of Babel," one that evinces that Borges hadn't lost interest in the idea of the Library. In 1949, Borges acquired Russell's Human Knowledge: Its Scope and Limits. One of his three annotations on the end leaf is "Eddington's monkeys." Here is the passage from page 484 (emphasis added):

  Eddington used to suggest as a logical possibility that perhaps all the books in the British Museum had been produced accidentally by monkeys playing with typewriters.3 There are here two kinds of improbability: in the first place some of the books in the British Museum make sense, whereas the monkeys might have been expected to produce only nonsense [.. . ] Suppose you have in your hands two copies of the same book, and suppose you are considering the hypothesis that the identity between them is due to chance: the chance that the first letter in the two books will be the same is one in twenty-six, so is the chance that the second letter will be the same, and so on. Consequently the chance that all the letters will be the same in two copies of a book of 700,000 letters is the 700,000th power of .

  Russell derives a viewpoint complementary to that of the Library. If there are 700,000 letters per book and an alphabet of 26 letters, then the total number of books is 26700,000. Therefore, the probability of picking a book that exactly matches another is one in 26700,000; that is

  Many commentators have pointed towards Borges' amiable review of Kasner and Newman's Mathematics and the Imagination as an indication of his interest in mathematics and also as a source of his knowledge. Unfortunately, it was not among the books from his personal library that were donated to the National Library. However, I was able to obtain a copy elsewhere and give it a professional reading. (An evocative aspect of the book is that the cover, as opposed to the dust jacket, is embossed with an aleph-nought, 0, which was Cantor's symbol for a countably infinite set.4) Borges' review, reprinted in Selected Non-Fictions, notes that the book includes

  . .. the endless map ofBrouwer,5 the fourth dimension glimpsed by More and which Charles Howard Hinton claims to have intuited, the mildly obscene Möbius strip, the rudiments of the theory of transfinite numbers, the eight paradoxes of Zeno, the parallel lines of Desargues that intersect in infinity, the binary notation Leibniz discovered in the diagrams of the I Ching, the beau
tiful Euclidean demonstration of the stellar infinity of the prime numbers, the problem of the tower of Hanoi, the equivocal or two-pronged syllogism.

  Most surprising to me, given that many today attribute an interest in fractals to Borges, is that Kasner and Newman's book examines the famous Koch snowflake curve in some depth on pages 344—55. The snowflake curve is a standard introductory example of a fractal—and for historical context, I mention that Kasner and Newman's discussion precedes the term "fractal" by almost 40 years. Apparently, though, Borges was sufficiently unimpressed by the snowflake curve that he neglected to mention it in his review.

  Perhaps Borges found the anti-Nazi gibes another appealing facet of the book, given his own strong—and unpopular—anti-Nazi stance during World War II. Despite these many commendable contents and qualities, given that the book was published in 1940, it seems unlikely that it was available for his consultation and degustation prior to the writing of "The Library of Babel."