THE UNIMAGINABLE MATHEMATICS OF BORGES’ LIBRARY OF BABEL

Brian Hayes in American Scientist:

Among the South American writers well known to North American readers, Jorge Luis Borges is the cool, cerebral one. His austere little fable titled “The Library of Babel,” first published in 1944, describes a universal library—universal both in the sense that it fills the universe and also in the sense that it contains at least one copy of everything:

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 true story of your death, the translation of every book into every language . . .

The list goes on, but you get the idea. The secret behind this limitless collection of implausible texts lies in simple combinatorics. All those documents—and plenty more!—are guaranteed to be present somewhere on the shelves of the library because the books include every possible sequence of symbols that can be assembled from a fixed alphabet in a certain number of pages.

In William Goldbloom Bloch’s mathematical companion to “The Library of Babel,” the first task is to calculate the number of distinct books that can be created in this way. There’s not much to it. Borges tells us that the alphabet of the books is restricted to 25 symbols (22 letters, the comma, the period and the word space). He also mentions that each book has 410 pages, with 40 lines of 80 characters on each page. Thus a book consists of 410 × 40 × 80 = 1,312,000 symbols. There are 25 choices for each of these symbols, and so the library’s collection consists of 25^1,312,000 books.

More here.