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. In the language of this zone there are still vestiges of the sect that worshiped that distant librarian. Many have gone in search of Him. For a hundred years, men beat every possible path—and every path in vain. How was one to locate the idolized secret hexagon that sheltered Him? Someone proposed searching by regression: To locate book A, first consult book B, which tells where book A can be found; to locate book B, first consult book C, and so on, to infinity. ... It is in ventures such as these that I have squandered and spent my years. I cannot think it unlikely that there is such a total book3 on some shelf in the universe. I pray to the unknown gods that some man—even a single man, tens of centuries ago—has perused and read that book. If the honor and wisdom and joy of such a reading are not to be my own, then let them be for others. Let heaven exist, though my own place be in hell. Let me be tortured and battered and annihilated, but let there be one instant, one creature, wherein thy enormous Library may find its justification.
Infidels claim that the rule in the Library is not "sense," but "nonsense," and that "rationality" (even humble, pure coherence) is an almost miraculous exception. They speak, I know, of "the feverish Library, whose random volumes constantly threaten to transmogrify into others, so that they affirm all things, deny all things, and confound and confuse all things, like some mad and hallucinating deity." Those words, which not only proclaim disorder but exemplify it as well, prove, as all can see, the infidels' deplorable taste and desperate ignorance. For while the Library contains all verbal structures, all the variations allowed by the twenty-five orthographic symbols, it includes not a single absolute piece of nonsense. It would be pointless to observe that the finest volume of all the many hexagons that I myself administer is titled Combed Thunder, while another is titled The Plaster Cramp, and another, Axaxaxas mlo. Those phrases, at first apparently incoherent, are undoubtedly susceptible to cryptographic or allegorical "reading"; that reading, that justification of the words' order and existence, is itself verbal and, ex hypothesi, already contained somewhere in the Library. There is no combination of characters one can make—dhcmrlchtdj, for example—that the divine Library has not foreseen and that in one or more of its secret tongues does not hide a terrible significance. There is no syllable one can speak that is not filled with tenderness and terror, that is not, in one of those languages, the mighty name of a god. To speak is to commit tautologies. This pointless, verbose epistle already exists in one of the thirty volumes of the five bookshelves in one of the countless hexagons—as does its refutation. (A number n of the possible languages employ the same vocabulary; in some of them, the symbol "library" possesses the correct definition "everlasting, ubiquitous system of hexagonal galleries," while a library—the thing—is a loaf of bread or a pyramid or something else, and the six words that define it themselves have other definitions. You who read me—are you certain you understand my language?)
Methodical composition distracts me from the present condition of humanity. The certainty that everything has already been written annuls us, or renders us phantasmal. I know districts in which the young people prostrate themselves before books and like savages kiss their pages, though they cannot read a letter. Epidemics, heretical discords, pilgrimages that inevitably degenerate into brigandage have decimated the population. I believe I mentioned the suicides, which are more and more frequent every year. I am perhaps misled by old age and fear, but I suspect that the human species—the only species—teeters at the verge of extinction, yet that the Library—enlightened, solitary, infinite, perfectly unmoving, armed with precious volumes, pointless, incorruptible, and secret—will endure.
I have just written the word "infinite." I have not included that adjective out of mere rhetorical habit; I hereby state that it is not illogical to think that the world is infinite. Those who believe it to have limits hypothesize that in some remote place or places the corridors and staircases and hexagons may, inconceivably, end—which is absurd. And yet those who picture the world as unlimited forget that the number of possible books is not. I will be bold enough to suggest this solution to the ancient problem: The Library is unlimited but periodic. If an eternal traveler should journey in any direction, he would find after untold centuries that the same volumes are repeated in the same disorder—which, repeated, becomes order: the Order. My solitude is cheered by that elegant hope.4
Mar del Plata, 1941
1) The original manuscript has neither numbers nor capital letters; punctuation is limited to the comma and the period. Those two marks, the space, and the twenty-two letters of the alphabet are the twenty-five sufficient symbols that our unknown author is referring to. [Ed. note.]
2) In earlier times, there was one man for every three hexagons. Suicide and diseases of the lung have played havoc with that proportion. An unspeakably melancholy memory: I have sometimes traveled for nights on end, down corridors and polished staircases, without coming across a single librarian.
3) I repeat: In order for a book to exist, it is sufficient that it be possible. Only the impossible is excluded. For example, no book is also a staircase, though there are no doubt books that discuss and deny and prove that possibility, and others whose structure corresponds to that of a staircase.
4) Letizia Alvarez de Toledo has observed that the vast Library is pointless; strictly speaking, all that is required is a single volume, of the common size, printed in nine- or ten-point type, that would consist of an infinite number of infinitely thin pages. (In the early seventeenth century, Cavalieri stated that every solid body is the super-position of an infinite number of planes.) Using that silken vademecum would not be easy: each apparent page would open into other similar pages; the inconceivable middle page would have no "back."
ONE
Combinatorics
Contemplating Variations of the 23 Letters
There are some, King Gelon, who think that the number of the sand is infinite in multitude; and I mean by the sand not only that which exists about Syracuse and the rest of Sicily but also that which is found in every region whether inhabited or uninhabited. Again there are some who, without regarding it as infinite, yet think that no number has been named which is great enough to exceed its multitude.
—Archimedes, The Sand Reckoner
WE BEGIN WITH A PAEAN TO THE MODERN method of denoting numbers, especially the convention of exponential notation, employed first by Descartes in 1637, then extended over the next few decades, primarily by Napier and Newton. (These days, it's commonly also called scientific notation.) In one of his most famous works, Archimedes, a singularly brilliant intellect of the classical world, needed approximately 12 pages (in English translation) to create names of numbers and methods of multiplication to produce an upper bound—a maximal estimate, a cap—on the number of grains of sand in the world. By using modern notation, particularly the idea of the exponential, it will take us less than one paragraph to produce an upper bound on the number of grains of sand in the universe. Furthermore, in short order these exponential conventions confer the power to accomplish a task that might well have stymied Archimedes: calculating the precise number of distinct books in the Library
A positive integer exponent signifies, "the amount of times some number is multiplied by itself." For example,
and .
are concise ways to express a "small" number
.
and a very large number.* There are only two rules regarding the manipulation of exponentials that concern us. The first:
Rule 1: Multiplying numbers written in exponential notation is equivalent to adding the exponents.
For example:
.
The second rule nicely complements the first.
Rul
e 2: Dividing numbers written in exponential notation is equivalent to subtracting the denominator's exponent from the numerator's.
For example:
The second rule leads to the useful convention of using a negative exponent to represent a power in the denominator, for instance,
.
Thus the previous example may concisely be written
It is remarkable that such relatively simple notation can transform relatively complicated tasks, multiplication and division, into the relatively easy and intuitive computations of addition and subtraction.
While pondering previous critical responses to "The Library of Babel," we discovered that a number of people either calculated the number of books or gave some indication of how one might go about it.1 Our intent in providing the lightning review of exponential notation is to demystify the calculation, and then, more importantly, to give a sense of the enormity of the Library. Then, after the calculation, we tease out a previously overlooked detail from the story and use it to set a new lower bound on the number of books in the Library. (For us, a lower bound will be number that says, "We guarantee that there are at least this many books in the Library.")
For the purposes of this book, combinatorics is the branch of mathematics that counts the number of ways objects can be combined or ordered. Before using combinatorics to calculate the number of the books, let's consider 10 familiar orthographic objects, the symbols we use as representations for digits: 3, 8, 9, 1, 6, 2, 0, 5, 7, 4. We deliberately disordered them to help you see them not as you usually do, as numbers, but rather as symbolic representatives of the numbers 0 through 9.
Using these symbols, we'd like to occupy exactly one slot with one symbol, and so we ask: how many distinct ways can we fill one slot? Hopefully, the answer is clear—there are 10 ways to fill one slot with one of the symbols.
1. 0
2. 1
3. 2
4. 3
5. 4
6. 5
7. 6
8. 7
9. 8
10. 9
Now, how many distinct ways are there to fill two slots, such that each slot contains one symbol? One complete list of answers, ordered in a familiar way, reads: 00, 01, 02, 03, ... , 97, 98, 99. So we see that there are 100 ways to fill the two slots, given that each slot contains one symbol and that repetition is allowed (enabling such combinations as 00, 11, 22, 33, etc.). Deliberately blurring the distinction between the orthographic symbols and the numbers they represent, we note that there are
ways to fill the two slots. If we ask how many distinct ways there are to fill three slots, such that repetition is allowed and each slot contains one symbol, we generalize our work from above and produce a complete list that reads: 000, 001, 002, 003,..., 99Z, 998, 999. This time, we see that there are 1,000 ways to fill the three slots. Continuing to blur the distinction between the orthographic symbols and the numbers they represent, it follows that there are
distinct ways to fill the three slots. By seizing on these ideas, by sensing that a simple pattern has been established and can be used to predict what we couldn't possibly list, we may ask how many distinct ways there are to fill, for example, 36 slots, where each slot contains one of our 10 allowed orthographic symbols and repetition of symbols is allowed. By applying the reasoning we established above, we see that there must be 1036 ways; that is, a 1 followed by thirty-six 0s—a thousand billion, billion, billion, billion, billion, billion ways:
Just for a lark, here are the first few and last few slot-fillings of the usual way one would list the fillings.
1. 000000000000000000000000000000000000
2. 000000000000000000000000000000000001
3. 000000000000000000000000000000000002
(Quite a few more!)
(1036 – 2). 99999999999999999999999999999999997
(1036 – 1). 99999999999999999999999999999999998
1036. 99999999999999999999999999999999999
And that's the end of the list.
In an article in the academic journal Variaciones Borges, our ideal reader, Umberto Eco, argues that the exact number of distinct volumes in the Library is irrelevant to both the story and to the reader. To the extent that the numbers of pages, lines, and letters in each book were chosen arbitrarily by Borges, we agree with him. (See the beginning of the chapter "Geometry and Graph Theory" for a quote from Borges regarding this matter.) However, we assert that understanding the combinatorial process that produces the exact number of distinct volumes is both important and relevant to an understanding of the story So let's apply these ideas to the story and, given the numbers and constraints Borges provides, use them to calculate the number of distinct volumes in the Library.
In "The Library of Babel," Borges writes:
...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; these letters neither indicate nor prefigure what the pages inside will say.
From these lines, we conclude each book consists of 410 40 80 = 1,312,000 orthographic symbols; that is, we may consider a book as consisting of 1,312,000 slots to be filled with orthographic symbols. Here a few more excerpts from the next few paragraphs:
There are twenty-five orthographic symbols. That discovery enabled mankind, three hundred years ago, to formulate a general theory ofthe Library and thereby satisfactorily resolve the riddle that no conjecture had been able to divine—the formless and chaotic nature of virtually all books.. .
Some five hundred years ago, the chief of one of the upper hexagons 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 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. These 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.
How many distinct books constitute the Library? Each book has 1,312,000 slots, each of which may be filled with 25 orthographic symbols—this is the "variations with unlimited repetition" mentioned above. Again, by employing the ideas outlined above, there are
25 ways to fill one slot,
25 25 = 252 ways to fill two slots,
25 25 25 = 253 ways to fill three slots,
and so on,
and so on for 1,312,000 slots.
It follows immediately that there are
251,312,000
distinct books in the Library That's it.
Somehow, it feels all too easy, even anticlimactic, as though instead we should have had to write pages and pages of dense, technical, high-level mathematics, overcoming one complex puzzle after another, before arriving at the answer. But most of the beauty—the elegance—of mathematics is this: applying potent ideas and clean notation to a problem much as the precise taps of a diamond-cutter cleave and husk the dispensable parts of the crystal, ultimately revealing the fire within. (Perhaps we should have ended the calculation by writing "That's it!" instead of "That's it.")
Our new twist on these calculations involves what Hurley translates as the "letters on the front cover of each
book." For the sake of precision, we note that the Spanish reads "el dorso de cada libro," which translates literally as "the back of the book." Idiomatically and bibliographically, however, the sense of this phrase is that the letters are on the spine of the Library's books. As such, the interpretation we use for the rest of this book is that the letters are on the spine.
Now, the number 251,312,000 we calculated above doesn't account for these spinal letters. It strikes us as likely that, within the imaginary universe of the Library, a book with the letters The Plaster Cramp written on the spine, whose 1,312,000 slots are filled by the repeated sequence of orthographic symbols MCV, should be considered as a book distinct from one with the exact same pages which is instead imprinted with the letters Axaxaxas Mlö on the spine.2 Scanning through the original Spanish version, "La biblioteca de Babel," we find a book described with the 19 orthographic symbols El calambre de yeso on its spine. This means that there are a minimum of 19 slots to fill on each spine, and accounting for these variations with repetition expands the Library by a factor of at least
The Unimaginable Mathematics of Borges' Library of Babel Page 3
