The Unimaginable Mathematics of Borges' Library of Babel
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100 = 10 10 = (2 5) (2 5) = (2 2) (5 5) = (22) (52)
and
100 = 4 25 = (2 2) (5 5) = (22) (52).
Because 100 is so familiar, it's probably not surprising to you that both of the initial factorizations lead to the unique one. And perhaps it is equally intuitive that no matter how large an integer we begin with, no matter how we might try, there will be only one way to factor it into powers of primes. Still, it's nice to know that Euclid showed that it must always be true.
By the work above, 52,624,000 is a unique factorization of 251,312,000 into primes, each raised to a power greater than or equal to one. In this case, plainly the number of distinct books uniquely decomposes to one prime (5) raised to a power greater than one (2,624,000). It follows that the only numbers that can divide 251,312,000 are powers of five. Now, as is easily inferred from the story, each hexagon in the Library contains 640 books. The number 640 uniquely factors into 275, and so the number 640 does not divide 251,312,000, for
and none of the seven 2s in the denominator may divide any of the millions of 5s in the numerator. This means that the books do not exactly fill out all the hexagons, which entails that either the Library is not complete (!!!), or that there is a special hexagon that is not full, or that at least one hexagon is differently configured, or that at least one hexagon contains exact copies of other books in the Library. We can't imagine that Borges considered this—or would have cared—when he assigned numbers to the quantity of shelves on a wall or the number of books per shelf in the Library.
Also, it may seem easy to juggle and tweak the numbers of shelves and books to make each hexagon hold, say, 625 = 54 books. After all, as written in the story, each hexagon holds 640 books, and 625 is very close to 640. But this is an opportunity to admire the power of Euclid's unique factorization theorem: if each of the four non-doorway walls has the same number of shelves, and if each shelf holds the same number of books, then each hexagon must hold
(4 walls) x (m shelves per wall) x (n books per shelf) = 4mn books.
The prime factors 22 = 4 will always be there; neither adjusting the number of shelves per wall, nor the tally of books per shelf will budge those 2s, which means that 4mn can never cleanly divide 251,312,000.
How, then, might we arrange matters so that the total number of distinct volumes may be evenly distributed throughout the hexagons? One possible solution is to expand the alphabet to 25 letters and, as Borges did, include the space, the comma, and the period to round the total up to 28 = (22) 7 orthographic symbols. Then, if the other (admittedly arbitrarily chosen) numbers for each book stay the same, there will be 281,312,000 distinct books.
Next, hire infinitely many cabinetmakers to rebuild the bookshelves in the hexagons, so that each of the four walls holds four shelves, and each shelf holds 49 books. Then a total of 4 4 49 = 784 = (24) (72) books furnish each hexagon, and since
after the renovation, the 281,312,000 books exactly fill (22,623,996) (71,311,998) hexagons.
For this last section, the aim is to explain concisely why we are currently, and for the foreseeable future, unequal to the task of determining the median of the prime numbers expressible in 100 digits. The median of the set of primes expressible in 100 digits is, in a sense, the "middle" of all of those primes. To compute the median, arrange the numbers sequentially from the smallest to the largest prime less than 10100 (which is called one googol).
Now, if there are an odd number of primes in the list, the median is the absolute middle of the list. If there are an even number of primes in the list, the median is the average of the two primes appearing in the middle of the list. (The average of these two numbers is guaranteed to be an integer, for the sum of two odd numbers is even, and we conclude the calculation of the average by dividing by two.)
The only way to find the median would be, in one way or another, to account for the complete list of prime numbers expressible in 100 digits. Including 0, there are exactly one googol numbers expressible in 100 digits. By the famous prime number theorem—which we'll outline in a moment—there are more than 1097 prime numbers smaller than 10100. This number may sound manageable, but 1097 is trillions of times larger than the number of subatomic particles in our universe. There simply isn't any imaginable way to list and keep track of 1097 numbers, which precludes the possibility of finding the median.1
The prime number theorem was first conjectured in various forms by Euler and others beginning in the late eighteenth century and was finally proved about a hundred years later in 1896 by Hadamard (and independently that same year by Poussin). Part of the beauty of the prime number theorem is that it provides an excellent estimate of how many primes there are that are smaller than 10100 without explicitly naming a single one!
The prime number theorem says that if n(n) is equal to "the number of primes less than or equal to n," then as n grows very large,
where ln(n) is the natural log function. (The natural log has the same remarkable properties as the log function, log(n), that we looked at earlier, and indeed, after multiplication by a constant, they are the same function.) We are interested in knowing approximately the number of primes expressible in 100 digits, so we compute π(10100) for a good estimate:
THREE
Real Analysis
The Book of Sand
To see a World in a Grain of Sand
And a Heaven in a Wild Flower
Hold Infinity in the palm of your hand
And Eternity in an hour.
—William Blake, "Auguries of Innocence"
REAL ANALYSIS IS THE BRANCH OF MATHEMATICS that explores, among other ideas, the nuances of the arbitrarily small. Paradoxically, in this chapter, thinking about the very small will prove decisive in understanding the very large: the Book that embodies the entire Library.
Borges' last insight regarding the Library is cloaked in a footnote adorning the conclusion of the last sentence. The footnote reads:
Letizia Álvarez de Toledo has observed that the vast Library is pointless; strictly speaking, all that is required is a single volume, of the common size, printed in nine- or ten-point type, that would consist of an infinite number of infinitely thin pages. (In the early seventeenth century, Cavalieri said that every solid body is the superposition of an infinite number of planes.) Using that silken vademecum would not be easy: each apparent page would open into other similar pages; the inconceivable middle page would have no "back."
Others have independently noticed that Borges continued to play with the idea of such a Book in his evocative short story "The Book of Sand."1
The mathematical analysis of a Book of Sand hinges on what is meant by the phrase "infinitely thin pages." Three different interpretations of "infinitely thin" lead to three Books similar in spirit, but disparate in the details. We'll examine them in ascending order of exoticness.
First Interpretation
If we take "infinitely thin" to mean merely "thinner than any subatomic particle," there are several refreshing possibilities. First, there are (410) (251,312,000) pages in the Library, a very large number, but still finite. Thus, if every page is the same thickness, say
th of an inch,
then the Book, sans cover, will be exactly one inch thick. Such a Book, though, would defraud the anonymous librarian of his "elegant hope" that the Library is repeated in its disorder, and also contravene the explicit statement in the footnote that the book would consist of an infinite number of pages. If, as above, the pages were all the same thickness, then an infinite periodic repetition of all the books of the Library would force the Book of Sand to be infinitely thick.
If we insist on each page having a definite thickness, and we equally insist upon infinite repetitions for the pages of the Book, we must therefore allow for ever-thinner pages. To make sense of such a Book, we need to understand an idea from the theory of infinite sums.2 We'll begin this short journey by treading parallel to the tiny footfalls, echoing loudly through the ages, of the Paradox of Zeno so beloved
by Borges.
Suppose, starting at one end of a room, we were to walk halfway across towards the opposite wall. After a brief pause, we walk half the distance from the midpoint towards the opposite wall. After another brief pause, we walk half the distance .. . (see figure 5).
In the coarse world we inhabit, we'll stub our toes on the wall in short order. In the idealized world of mathematics, we may always halve the distance between one point and an endpoint. (Zeno's and Parmenides' paradoxes exploit this chasm between the world of our perceptions and the mathematical vision of a line segment.)
For the purposes of this book, without offering a rigorous proof, note that by adding up the lengths symbolized by the arcs, the information encoded in figure 5 is equivalent to this equation:
This equation encapsulates a striking fact: by adding up infinitely many segments, each smaller than the previous by one-half, a form of unity is achieved. A gauge of the depth and profundity of this insight is that for several centuries, most thinkers conceded that this hammered home the final nail in the coffin for Zeno's Paradox. (Nowadays, thinkers have again complexified the picture, thereby casting doubt, raising questions, and essentially resurrecting the dead.)
Following the example set by the equation, choose the first page to be one-half of a standard page's thickness, then the next page half that thickness, the next half that thickness, and so on and so on. Then the entire Book, infinitely periodically repetitive, will be exactly one standard page thickness.
In the Math Aftermath following the chapter, we provide a bit more background on this next calculation, which is estimating the thickness of the 41st page. We conclude that the 41st page is
which is thinner than the diameter of a proton. Since each successive page is one-half the thickness of the preceding page, all the rest of the pages are also thinner than a proton. Of course, in this interpretation, though almost every page is invisible to the naked eye, or even an electron microscope, it is not the case that any page is actually "infinitely thin."
Second Interpretation
Here, we take "infinitely thin" in the sense indicated by the reference to Cavalieri's principle in the footnote: the thickness of a Euclidean plane. The thickness of a plane is the same as the length of a point, which is tricky to define. Consider a point in the line. It is clear that a Euclidean point is thinner than a line segment of any positive length. It is somewhat disquieting, though, to say that a point has length 0; if so, how does massing together sufficiently many 0-length entities create a line of positive length? Doesn't adding together 0s always produce another 0? How could an object be of length 0?
A subtle way of evading these traps was crafted at the beginning of the twentieth century, primarily through the work of Henri Lebesgue, whose theory is now a vast edifice with ramifications permeating much of modern mathematics. Fortunately, we need only a small cornerstone of the theory: the idea of a set of measure 0 contained in the real number line.
Recall that the real number line consists of all rational and irrational numbers, each representing a point on the line, each also signifying the distance from the origin to the point. It may be confusing that we are explicitly identifying the "length of an interval" with a "number," for again, a real-world idea, that of length, is interpenetrating a mathematical idealization. We inhabit this limbo for the rest of the chapter.
We need two definitions. A closed interval includes both endpoints of an interval; as an example, the notation [0, 1] means "all numbers between 0 and 1, inclusive." Now, let S be any set contained in the real number line. One says that S is a set of measure 0 if S can be contained in a union, possibly infinite, of closed intervals whose lengths add up to an arbitrarily small number. Several examples will help clarify this definition.
Example 1. A single point p in the real number line. Clearly p can be contained in a closed interval of arbitrarily small length (figure 6). Thus p is a set of measure 0. Note the fine distinction: we are not saying "the point p is of length 0"; rather we are saying that p is a set whose measure is 0. It turns out—and we'll see an example soon—that there are sets of measure 0 which are quite counterintuitive.
Example 2. Three points a, b, and c in the real number line. Let
a be contained in an interval of length 1/2,
b be contained in an interval of length 1/4, and
c be contained in an interval of length 1/8.
(It doesn't matter if the intervals overlap.) Then the three points are contained in a union of intervals whose sum-length is
Not arbitrarily small yet! But now, let
a be contained in an interval of length 1/4,
b be contained in an interval of length 1/8, and
c be contained in an interval of length 1/16.
Then, since each interval is half the length of its corresponding predecessor, the sum is also halved.
If we play this game again, starting with an interval of length 1/8, we find that
If we continue to put a, b, and c in intervals half of the lengths of the previous go-round, the triple of intervals will also sum to half the preceding length: first 7/64, then 7/128, and so on. By starting with a sufficiently small interval, we ensure the sum of the three intervals is arbitrarily small—that is, the set S = {a ,b ,c} is a set of measure 0 (figure 7).
Example 3. It is a curious fact that it is difficult to show that the interval of numbers between 1 and 4 is not of measure 0. Certainly our intuition informs us that the minimum length of intervals necessary to cover [1, 4] will sum to 3, but demonstrating it rigorously is a nontrivial exercise, well beyond the scope of this book. See figure 8.
Back to the infinitely thin pages of the Book. We interpret "infinitely thin" as meaning that each page has a thickness of measure 0. We also assume, as we did at the end of the first interpretation, that within this Tome, the books of the Library repeat over and over, enacting the anonymous librarian's "elegant hope" of a periodically repeating order. We are therefore confronted with an intriguing problem:
There are infinitely many pages, each of which has thickness of measure 0. How thick is the Book?
The answer might run counter to your intuition:
The thickness of the Book is of measure 0.
In other words, if we looked at the Book sideways, we would not be able to see it, let alone open it. How does this unexpected, unimagined, unimaginable state of affairs arise? Once we think to look for it, it turns out to be sitting there, almost as if it was waiting to be discovered.
The goal is to show that the thickness of the Book can be contained in a collection of closed intervals which can be chosen so that the sum of their lengths can be made arbitrarily small. If this can be done, then by definition the Book is of measure 0. We'll accomplish this by covering the thickness of each page in ever-smaller intervals in a sneaky way that exploits the infinite sum that embodied Zeno's Paradox.
First, though, another counterintuitive point, followed by a technical one. Although it's conceivable that the Bookbinder bound the infinitely many pages of the Book together in a straightforward order, it is also possible that the pages of the Book wash up against themselves similar to the rational numbers, meaning there is no more a "first" page of the Book than there is a "first" positive rational number. If so, we simply choose one of the 251,312,000 books to be the first, another to be the second, and so on, until we have a complete list of the books and their pages. Since the Book repeats, we are thus able to give numbers to its pages.3So, let
the first page be contained in an interval of length 1/2,
the second page be contained in an interval of length 1/4,
the third page be contained in an interval of length 1/8,
and so on,
and so on.
We saw in the first interpretation that
so the thickness of the Book can contained in an infinite union of intervals which sum to 1. Here's where the sneaky part comes in. Now, let
the first page be contained in an interval o
f length 1/4,
the second page be contained in an interval of length 1/8,
the third page be contained in an interval of length 1/16,
and so on,
and so on.
This time, the infinite union of intervals sums to
This is seen by simply subtracting 1/2 from both sides of the previous equation. Notice how we are exploiting an aspect of the idea of infinity: we are throwing away a term from the left side of the equation, but still have infinitely many terms to account for the infinite number of pages.
If we start by letting the thickness of the first page be contained in an interval of length 1/8, then the sum becomes:
Clearly, by continuing to play this game of lopping the intervals in half, we ensure that we may always find a union of intervals that contains the thickness of the Book and sums to an arbitrarily small number. This means that the thickness of the Book is of measure 0, an outcome surely unimagined by Borges.