Chile has Neruda. Argentina has Borges. Borges was a poet who is most widely known for his short stories. He scoffed at the notion that one needed to write a full-length novel to make a point when the same could be done in a handful of pages. So his stories are short and to the point. Some say they are as word perfect and succinct as a mathematical proof. Math was something which obviously interested Borges as it does Chile’s Parra who was at one time a professor of math and physics. The mathematics of Borges short story “The Library of Babel” is the theme of William Goldbloom Boch’s book “The Unimaginable Mathematics of Borge’s Library of Babel“.

Borges imagines a library composed of each and every possible combination of the the letters in the alphabet a-z plus the comma, space, and period. Not only would this library contain the very boring book containing nothing but the letter p as in “pppppppppppppppppppp……” and so forth. But it would also contain the book whose only difference is the very first letter is different as in “qpppppppppppppppp….” and so forth. But since this library contains every possible variation of not only “p” and “q” but “a” and “b” and space and “z” it would contain not only these two books of non sensible letters it would contain a book on your life, a book on my life, and different versions of both, plus book reviews of each.

Borges puts it like this:*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.*

If you think about this, while this is not exactly plausible this is in fact possible. If you take every combination of letters and spaces you would not only produce a library of books containing “War and Peace”, you would find all of Tolstoy’s early drafts, you would find all the known literary criticism and analysis of said work. You would find Franz Kafka’s “The Trial”, “The Metamorphosis”, and “The Castle” plus all the works of Vladimir Nabokov. You would find “Don Quixote” in Spanish as well as any other book that could be expressed using the 22 letters of the Latin Alphabet. This includes all languages since like all can be reduced to Latin spelling through phonetics or like the Chinese pinyin or Japanese katakana technique of rendering these into something which will fit into the computer’s keyboard.

Borges says his library is a sphere comprised of hexagons each separated by a ventilation shaft and stairs. (Boch says he might have got his idea when he worked as a librarian at the Miguel Cané Municipal Library in Buenas Aires.) He imagines souls doomed to wander the corridors of this Library of Babel forever fighting with one another in the endless corridors looking for the following nirvana:*…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.*

Boch asks in his book how large would this library be. He walks the reader through some elementary notion of mathematics to explain that Borges library would be so large that someone wandering its stacks could spend a lifetime looking a particular book would not find it nor come anywhere close to finding it. He explains how the numbers quickly add up.

Suppose you consider just the first two letters of each book in the library. You can readily see that there are 25 * 25 = 625 combinations of these. Why it this? For each first letter of the book there are 25 possible letters that you can pick for the second letter of the book. So we have “aa”, “ab”, “ac”,….,”ba”, “bb”, “bc”,…,all the way to “za”, “zb”,…”zz” along with “.a”, “,a”, and “(space)a” equals 25 * 25. If there are three letters to be considered then for each 25 * 25 possible combinations of letters there are 25 more. This number quickly zooms to unimaginable proportions.

Consider two roll of two dice. Each dice has 7 sides so each dice can turn up as 1,2,3,4,5,6,7. Roll two of them and there are 7 * 7 = 49 = 7 ^ 2 possible combinatons. With three dice there are 7 * 7 * 7 = 7 ^3 and so forth.

Given that simple explanation how many books are in the library? Well as Borges explains: *each book contains four hundred ten pages; each page, forty lines; each line, approximately eighty black letters*

So there are 410 * 40 * 80 = 1,312,000 spots in each book which can contain the letters of the alphabet a through z plus the space, the period, and the comma. So the number of books in the library is:

25 ^ 1,312,000

This number is vast indeed and Boch spends much of the remainder of his book showing that not only would this booknot fit inside the size of the universe–neither would any index of the library which Borges calls the “catalogue of catalogues”. (A mathematician called Henri Poncairé showed that the universe however vast is not infinite in size. If you travel in any direction therein you would come back to where you start. Mathematicians call this a “closed manifold”. Borges acknowledges this in his story thus demonstrating again his interest in mathematics.)

Borge’s library is so vast that not only would all the books fit in the library–a number which we just showed to be finite–neither would they fit if each was shrunk the size of a grain of sand. If we shrink each to the size of an atom they would still not fit. So you can begin to see that while this library is indeed comphrensive–it contains every possible work of literature–it is somewhat incomprehensible.

Boch’s book is about the sublime beauty of mathematics and the beauty of the works of literature created by Borges. Other stories by Borges vary on this theme. The math here is not too difficult for the average reader and requires no math training. Yet the author does delve into some esoteric topics like the aforementioned manifolds. What exactly is a manifold? Not even those who teach the subject can explain it exactly and neither can I but it still interesting to contemplate.