Let's discuss Jorge Luis Borges' short story "The Book of Sand" ("El Libro de Arena"). First, a synopsis. (You can also find an English translation here.)
The story opens with a few sentences about points, lines, and planes. More on this later. A man (the narrator — Borges?) receives a visit from a traveling Bible salesman. He expresses disinterest in buying a Bible, as he owns a Wycliff Bible, a copy of Cypriano de Valera's translation, Luther's translation, and the Latin Vulgate. The salesman — a man of indistinct features, dressed in gray, with a gray valise — then shows him the Book of Sand (so-named because sand has neither beginning nor end).
The narrator opens it. Each page is numbered with an Arabic numeral; the left-hand page might be numbered, say, 40514, or some eight-digit number, the left, 999, or a number raised to the ninth power. There are crude illustrations as well. "Study the page well," the salesman says, "for you'll never see it again." The narrator notes the page number, closes the book, then tries to find it again. He is unable to. He also discovers that he can't reach the beginning of the book. No matter how close he tries to open it to the cover, there are always pages between his thumb and the board.
In the end he purchases the book in exchange for his retirement fund and his Wycliff Bible. He hides it behind his volumes of the Arabian Nights. As he investigates the book over the coming months he concludes it to be monstrous. Rather than burning it — he's afraid that the smoke from an infinite book would suffocate the whole earth — he hides it on a shelf at the National Library.
A strange, enigmatic tale in its basic outline. Possibly I've left out some of the most significant details, but I've included those that strike me. Though a writer by night and a painter on Sundays, I'm a mathematician by day, and it's the mathematical aspects that I wish to speak about.
The opening seems to me to be the key to the whole thing, for in many ways the Book of Sand might be likened to a line segment of finite length, e.g., the segment from zero to one on the number line. I mean the segment between these two points, excluding zero and one themselves. For such a segment has neither beginning nor end. No matter how close to zero you choose a point to be, there will always be infinitely many numbers between zero and your point. Notable, too, is the fact that the narrator can never again find a page he's visited. This is true of any infinite set. Choose an element at random, then choose a second. The probability that the choices will be the same is precisely nil.
So, the book resembles the line segment to some extent. But there are many infinite sets within the line segment. We could, for instance, take all the points that correspond to fractions. For a number between zero and one to be a fraction we have to be able to write it as a/b, where a and b and whole numbers and a is smaller than b. Though infinite, it is possible to denumerate this set by taking them in the order
1/2 1/3 2/3 1/4 3/4 1/5 2/5 3/5 4/5 1/6 5/6 1/7
so that 1/2 is the first, 1/3 the second, 2/3 the third, and so on. We're skipping the fractions that can be reduced and thus have already appeared in the list. Because this set of rational numbers (so-called because they are ratios of whole numbers) is denumerable (can be numbered off without leaving any out), it seems to me to resemble the pages of the book. The page numbers would correspond to the labels affixed to the rational numbers. For instance, the page that occurs exactly two-fifths of the way through the book would be labeled as page 7, because 2/5 appears seventh in our list. Going against this interpretation is the fact that facing pages have different numbers, and that the narrator can turn the pages one at a time; for between each pair of rational numbers are infinitely many rational numbers. It isn't possible to find two that are right next to each other, so to speak. For the book to be exactly like the rationals, recto and verso would need to bear the same number (and thus be identified), and the pages would have to stick when the narrator turned them, much as the "first" pages cling to the front board, so that he can never quite turn a single one.
Another possibility is that the pages correspond to the points in the entire continuum from zero to one; but this seems to be ruled out by the pagination, because the continuum, as Georg Cantor showed, is not denumerable. It is "more infinite" than the infinite set of rational numbers. (Incidentally, Cantor's proof made use of the decimal system, which originated in India; the Bible salesman had bought the Book of Sand from an Indian untouchable.) What we noted about page-turning above also goes against this interpretation.
There are of course many other infinite sets on this line segment that could be considered.
It's interesting, by the bye, that the book is called the Book of Sand; for the number of grains of sand is in fact finite. Though counterintuitive to some, the fact was demonstrated by Archimedes in a little book called The Sand-Reckoner, to which I feel almost certain Borges must have referred in at least one of his stories.
In this book, Archimedes obtains an estimate for the size of the material universe, which was very, very large (it is a myth that the ancients believed in a small cosmos and a flat earth), and, using another estimate for the density of packed sand in terms of grains per unit volume, computes an upper bound for the number of grains that could possibly be contained in the cosmos, were it packed solid with sand. He arrives at the number 1063, which is very large, but still, of course, finite. The square of this number would be larger than a googol, if that puts it in perspective for you.
The fact that the narrator fears polluting the whole earth with the smoke of his book makes me think of a certain nineteenth-century mathematical controversy, of which Cantor, mentioned above, was at the center. Cantor showed that the points on the line segment could be paired in a one-to-one fashion with the points on a unit square and, thus, with the set of points in any such space of any dimension. This pairing, though not continuous, scandalized the mathematical community, for it seemed contrary to reason that a one-dimensional space could be equally as infinite as a two-dimensional space.
Later on, Guiseppe Peano and David Hilbert devised continuous mappings from the segment onto the square; these were no longer one-to-one. The first several steps in the construction of Hilbert's mapping are shown above; the "squiggle" eventually fills the square, with no points left out. There is a three-dimensional analogue of this mapping, which fills the cube, and could, in principle, fill all of space.
So, if we imagine the pages of Borges' book to correspond to (say) the rational numbers, then the particles of its burning might not fill all the atmosphere, but, since the rationals are dense on the segment (infinitely many lie between any two points, no matter how close), we could easily imagine that its smoke might be so dense as to entirely pervade the atmosphere, with infinitely many particles in every space, no matter how small.
Cantor, as I noted, was a figure of controversy; he was denounced as a "scientific charlatan" and "corrupter of youth" by other mathematicians, and the strange curves and mappings he and others of his "school" described were shunned by many as monstrous or pathological. His investigation into the infinite also had a bearing on metaphysics and theology, for medievals like Thomas Aquinas had held that there could be no such thing as an actual existent infinity, only a potential infinity. Cantor's research seemed to some to indicate the existence of actual infinities, and thus, for various reasons, to tend toward pantheism. Cantor believed that further distinctions had to be made, and that Thomas wouldn't have objected to his ideas, had he been able to explain them; he corresponded with philosophers, theologians, and cardinals of the church, and even addressed a pamphlet to Pope Leo XIII, whose encyclical Aeterni Patris had advocated a renewed interest in scholastic philosophy.
But to return to the "monstrous" Book of Sand, whose serial infinity inspires in the narrator the same horror Cantor's theories had struck in his contemporaries. The narrator traded for it a copy of scripture that was a historical and literary treasure in itself as well as his savings for retirement. Might not this be a statement about modern man, who has traded his patrimony and future for a kind of gorgon's head of infinity? How fitting, too, that the narrator hides the book in a library, making it one page in the monstrous Book of Sand that is the modern glut of information, our "Library of Babel."