American Scientist

THE UNIMAGINABLE MATHEMATICS OF BORGES’ LIBRARY OF BABEL. William Goldbloom Bloch. xx + 192 pages. Oxford University Press, 2008. $19.95.

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.

Bloch works hard to convey some sense of the magnitude of this number—pointing out, for example, that if our own universe were packed solid with books, it would hold something like 10⁸⁴ of them. In the end, though, he argues that such quantities are truly unimaginable, justifying his title—The Unimaginable Mathematics of Borges’ Library of Babel. (In confirmation, Bloch himself has trouble coping with at least one such gargantuan number. In a later chapter he estimates the number of ways of arranging the books in the library. His answer is 10^10^33,013,740, which he then describes as a number with 33 million digits. In fact it has 10^33,013,740 digits, an “unimaginably” larger number.)

These combinatorial exercises are the most obvious instances where mathematics can illuminate the Borges story, but Bloch finds much else to comment on as well. Many of his notes concern the structure of the library itself rather than the books in it, focusing on details that I blithely passed over in my own readings of the story. Borges describes the library as a close-packed array of hexagonal rooms. Four walls of each hexagon are lined with bookshelves, holding a total of 640 books; the other two walls provide portals to adjacent hexagons. Vertically, the levels of the library are connected by ventilation shafts and spiral stairways. This design has some curious consequences. For example, Bloch points out that somewhere in the library there must be at least one hexagon whose shelves are not full. The reason is that 25^1,312,000 is not evenly divisible by 640.

For me the biggest surprise in Bloch’s commentary comes in a chapter that applies ideas from graph theory to the layout of the library. Another celebrated Borges story is titled “The Garden of Forking Paths,” but it turns out the library consists entirely of forkless passages; there can be no branch points in any route weaving through a floor of the library. If you view each hexagon as a node of a graph, then all the nodes are of degree 2, connected to just two other nodes. Branching requires nodes with at least three connections. Similarly, the vertical structure of the library is strongly constrained by the ventilation shafts and stairways that connect the levels. Depending on exactly how Borges’s description is interpreted, it may be that every floor is required to have exactly the same layout. It’s even possible that the graph of the library is divided into multiple disjoint components. If that’s the case, then although the library holds all possible 410-page books, there’s no guarantee you can get to them all.

Would Borges have cared about any of these mathematical curiosities? Perhaps not. For the reader of Borges, however, some of Bloch’s observations may offer a useful new way of engaging with the themes of the fiction. Many of Borges’ stories are nightmares, but they are written in such restrained, quiet prose that the horror can slip by unnoticed. In an essay that prefigured “The Library of Babel,” Borges referred to “the vast, contradictory library, whose vertical deserts of books run the incessant risk of metamorphosis, which affirm everything, deny everything, and confuse everything—like a raving god.” He called the library “a minor horror”; I find it interesting that it takes a bit of calculation to really grasp what the horror is all about.

I’ve just returned from a visit to my own local library, which is vast but far from universal. I discovered that the set of books by and about Borges would fill up roughly half a hexagon. Bloch’s little volume will add one more to those shelves. Sometimes the ceaseless proliferation of books makes me dizzy. With all those Borgesian commentaries on commentaries, there’s a literary Malthusian principle at work: Books are reproducing geometrically, readers only arithmetically. But Borges’ own view is even darker. When you sit down to write a book, you congratulate yourself on an act of creativity. In the library of Babel, however, you are not creating anything. No matter what you write—a commentary on the gospel of Basilides, perhaps?—your book already exists, somewhere among the 25^1,312,000 volumes on the shelves.

Brian Hayes is Senior Writer for American Scientist.