Whenever Juzhi was asked a question, he simply raised one finger.
– Blue Cliff Record, Case 19
Numbers on the Mind
As a physicist, I find numbers companionable, even compelling. One aspect of numbers that reels me in is how sure-footedly they lead to a strange landscape. Put one logical foot in front of another, step by step, and pretty soon the mathematical roadside looks like Oz instead of Kansas. The mathematician Richard Dedekind said, “Numbers are the free creation of the human mind.” If that’s the case, it’s not surprising that logic can swiftly leave common sense in the dust. When mind freely creates, it reaches beyond what we imagine is possible. Juzhi’s single raised finger is pointing beyond the imaginable.
But it’s not necessary to engage in integral calculus or non-Euclidian geometry to encounter mathematical vistas that drop one’s intellectual jaw. Simple counting can do the trick—a trick that rests in the ability of mathematics to accurately represent a logical reality that exceeds our comprehensive ability. Here’s a beautiful example from the Argentine metafictionalist Jorge Luis Borges.
In his story “The Library of Babel,” Borges conjures a strange and terrifying collection of books. The Library contains a complete collection of possible texts—none are missing. Any text you can imagine is there, and more importantly, a far vaster number than anyone can imagine is there.
All books in the Library of Babel (LoB) are identical in size: 410 pages, with 40 lines per page and 80 characters per line. This means a book consists of 1,312,000 (or ~1.3 x 106) locations at which one can place a letter, space or punctuation mark. The LoB consists of all possible permutations of the letters in the alphabet, plus punctuation, one book for each permutation. Let’s say the books are in English and that we employ the period, comma, space, and exclamation point as punctuation. This totals 30 distinct characters to place at the 1,312,000 locations.
How many books is this? The number of books equals the number of characters raised to the power of the number of locations. That is, multiply 30 by itself 1,312,000 times: 30 x 30 x 30 x 30 (repeat 1,312,000 times). Folks that do mathematics have a way to write this number in a highly compressed form:
number of books = (number of characters)(number of locations).
So, the LoB contains 301,312,000 books, or (put another way) 101,937,983 books. This means that the number of books is a number containing almost two million decimal places.
While no two books in the LoB are identical, virtually all of them are meaningless. For example, there is only one book that contains the letter p in all 1,312,000 locations, but there are 1,312,000 books that contain all p’s except for one q—one book, that is, for each possible location of that q. In Borges’s story, the librarians spend their entire lives searching for a single book that possesses a scrap of meaning. Their searches are hopeless, since the probability of such a discovery is nill. Virtually all the possible permutations are random strings of letters. Nonetheless, all possible books are present in the LoB. Here is Borges’s description:
The library is total and its shelves register all the possible combinations of the twenty-odd orthographical symbols (a number which, though extremely vast, is not infinite): in other words, all that is given to express, in all languages. Everything: the minutely detailed history of the future, the archangels’ autobiographies, the faithful catalogue of the Library, thousands and thousands of false catalogues, the demonstration of the fallacy of those catalogues, the demonstration of the fallacy of the true catalogue, the Gnostic gospel of Basilides, the commentary on that gospel, the commentary on the commentary on that gospel, the true story of your death, the translation of every book in all languages, the interpolations of every book in all books.
This very essay is in the library, along with all previous versions of it, and all possible improvements of it.
One might think the number of books contained in the LoB—101,937,983—is simply a big number, and we’ve all seen big numbers before (such as the national debt of the U.S., at ~1013 dollars).
However, the number of books in the LoB is not just a big number, but one whose size lies beyond our comprehension. It’s easy to miss crossing the border from the merely large into the unimaginably large, and that’s part of the slipperiness present in number contemplation. First slip: the number is easily calculated, requiring only a few lines of reasoning. Second slip: it’s a cinch to write down. Here it is again: 101,937,983, voilà. As easy as pie and infinitely easier than pi (but we’ll get to that in Part II). Moral: just because something can be represented doesn’t mean it can be comprehended. Juzhi’s finger comes to mind.1
There are several ways to illustrate the bewildering vastness of the book count in the LoB. Probably the most concrete is to (try to) imagine the sheer physical size of the library itself. Since we’re just imagining, let’s make this easy and say each book is the size of a single hydrogen atom, and that they are close-packed: atoms cheek-to-cheek. If the entire universe were filled with nothing but these atom-sized books, we’d have accommodated 10101 , leaving ~101,937,882 still needing shelf space. In which case, the universe can hold almost none of the books present. 2
The number of books in the LoB is leviathan, it swallows all attempts to get it into perspective. How can this number so surpass our ability to visualize it? It’s due to the fact that the number of books is not an actual count of books. It’s a count of possible arrangements of letters. Counting arrangements, or permutations, is one step up the abstraction ladder from the counting of actual objects. When we try to map the counting of permutations onto the counting of objects we plunge into a conceptual abyss.
Mind can naturally move toward abstraction, and then just as naturally turn back, reflect on the move, attempt to re-cognize what has occurred. In the case of the LoB, this leads to a face-to-face encounter with a paradox: a logical construction exceeds rational comprehension. The rational mind steps from a precipice that one took, originally, for a curb.
Often our abstraction moves are not so simple and singular; they’re animate and shape-shifting, ubiquitous and resilient. From atoms to psychological typology, from moral principles to fatherhood, we can vacuum seal our abstractions onto life till there’s no room to breathe. Unlike the single elegant premise underlying the LoB, our daily abstractions are dense enough to form a lens, one through which the world does not appear unusual, but appears to make perfect sense.
In fact, if someone points out the simple fact-of-the-matter, and we follow that point, we could find ourselves in free fall over a cliff. This fall however is not within the realm of abstraction. It’s more dangerous—a fall out of abstraction.
- A more accurate moral to draw from Juzhi and his finger: what can be embodied cannot necessarily be comprehended. I don’t think Juzhi is attempting to represent something by raising his finger. In particular he’s not counting to one, nor attempting to communicate that ‘all is one.’
- What books could we fit into the universe? Obviously, we’ll need to greatly restrict the subset of LoB books in order to shoe-horn them into our itty-bitty universe. Consider only those books with the letter ‘A’ in the first 1,311,664 locations available in each book, leaving the last 336 locations to be filled with, and only with, either ‘A’ or ‘B’. These books will nicely fill our universe, assuming each book is the size of a hydrogen atom.