The Unimaginable Mathematics of Borges' Library of Babel

by William Goldbloom Bloch

  3. From the 1875 Grindon citation under the definition of "septillion" in the Oxford English Dictionary. "Thousands of plants consist of nothing more than a few such cells as in septillions make up an oak-tree..."

  4. In the words that comprise "The Library of Babel," Borges adroitly finesses the fact that by being restricted to a mere 25 symbols, the books of the Library could not contain both upper- and lower-case symbols, let alone diacritical and punctuation marks beyond the comma and the period. The story, as written, could not appear in the Library. In this book, when we are referring to imagined entries in the Library, we will hew to the standard set by Borges and not restrict ourselves to using only the orthographic symbols of all uppercase letters (or lower-case letters), spaces, periods, and commas.

  Chapter 2

  1. For a mathematically sophisticated reader: in fact, one may imagine that at some distant point in the future, we might have a superfast supercomputer running an algorithm that

  1. Was able to test for primality every number expressible in 100 digits.

  2. Kept a tally of the number of primes without listing them.

  3. Output simply the number of primes N expressible in 100 digits.

  Then,

  4. Determine if N is odd or even.

  5. Determine, if N is odd, which numbered prime is the median of the set.

  6. Determine, if N is even, which two numbered primes average to the median of the set.

  Then,

  7. Run the algorithm again, keeping count until the number(s) from step 5 (or 6) is (are) achieved.

  8. Output the median!

  At no point was a list of all primes necessary.

  Chapter 3

  1. See Salpeter, for example, on "The Book of Sand."

  2. Benardete, quoted in Merrell on page 58, independently thinks along similar lines.

  3. Another fine point for the interested: obviously we're capitalizing on the fact that the number of pages is countably infinite.

  4. If a mathematically sophisticated reader is worried about the need to invoke the axiom of choice, the issue is easily sidestepped by assigning the same infinitesimal, 6, for the thickness of each page.

  Chapter 4

  1. For the mathematically adventurous reader: in fact, the famous Hopf fibration of the 3-sphere decomposes the 3-sphere into great circles over a base space equivalent to the 2-sphere.

  2. A point for a mathematically sophisticated reader: earlier in this chapter, we observed that the gravitational field of the Library needed to be imposed by the builders of the Library. Since the Library presumably does not possess any regions of zero gravity, it is vital that the 3-manifolds under consideration may be equipped with everywhere nonzero vector fields. But of course they can, since all 3-manifolds have Euler characteristic equal to 0, entailing the existence of everywhere nonzero vector fields.

  3. A way to see that the surface of a coffee mug is the same as the surface of a donut is to simply shrink the "cup" part of the mug to the strip that lies between the joining spots of the handle to the mug!

  4. Another way around this problem would be to require that the orthographic symbols be symmetric under 90° rotations, too. For example, the symbols O, X, +, , , and satisfy this criterion.

  5. Technical note: as the Library stands, we couldn't actually use an exact cube to construct the Library, for the number of hexagons is not a perfect cube. That is, the number of hexagons is not of the form x3 for some integer x. However, by tinkering with the number of hexagons on each floor, it would be possible to have a shape very close to a cube. If done carefully, it would be equally easy to make the identifications between the sides of the near-cube, and the end result would be just slightly less symmetric.

  Chapter 7

  1. An Argentine colleague, Martin Hadin, brought to my attention a 1971 dialogue between Borges and Herbert Simon that suggests Borges was intrigued by these kinds of ideas, but previously unaware of them. The dialogue appears in Primera Plana, January 5, 1971.

  Chapter 8

  1. The interesting readings in Wheeler, Alazraki, Bell-Villada, Barrenechea, Rodriguez Monegal, Slusser, Ammon, Eco, Keiser, Nicolas, and Faucher, for example, fall outside this domain. Although I might differ with the conclusions they draw, it seems to me that Nicolas and Faucher get the math correct.

  2. An infinite set is countable (also called denumerable or more precisely, countably infinite) if it can be placed into one-to-one correspondence with the positive integers. In effect, this means that one may write down all the elements of the set in an orderly (infinitely long) list.

  1. ↔ "first" element of the set

  2. ↔ "second" element of the set

  3. ↔ "third" element of the set

  etc.

  It's easy and not inaccurate, therefore, to think of "countable" as synonymous with "listable." Cantor, the creator of set theory and the theory of transfinite numbers, was presumably shocked to discover that the rational numbers are countable. Regardless, in one of the most beautiful arguments in mathematics, he demonstrated that the irrational numbers are not countable or listable. Any set that is not listable is called uncountable or uncountably infinite.

  3. If, for example, I begin by assuming, "The moon is made of green cheese," I can derive a whole host of interesting implications from that premise, such as that dairy farmers and cheese manufacturers may be responsible for the lack of missions to the moon in recent years. That if we built cheese-harvesting factories on the moon and sent back enormous loads of cheese, the world hunger situation might be abated. And so on. However, the original premise is false, so it doesn't matter how interesting or plausible the ensuing speculations are to us.

  Chapter 9

  1. For the reference to “The Total Library,” see Collected Fictions, p. 67. For the essay itself, see Selected Non-Fictions, p. 214.

  2. See Burton, p. 95. The quote is lifted from Part 2, Sec. 2, Memb. 4 of Burton’s colossus.

  3. A marvelous, witty experiment was performed at the Paignton Zoo in Devon in 2003. Six Sulawesi crested macaque monkeys were placed in a cage outfitted with a computer, ostensibly to see if, between them, the monkeys might produce some work of Shakespeare over the course of a week of random typing. As reported by David Adam, science correspondent for The Guardian, “The macaques—Elmo, Gum, Heather, Holly, Mistletoe, and Rowan—produced just five pages of text between them, primarily filled with the letter S. There were greater signs of creativity towards the end, with the letters A, J, L and M making fleeting appearances, but they wrote nothing even close to a word of human language.”

  4. Ferrero and Palacios, Hayles, and especially Hernández have written more about Borges and 0.

  5. This “endless map of Brouwer” goes by the mathematical name of Brouwer’s fixed point theorem. The main idea is that if a nice enough space is mapped into or onto itself, then there must be at least one point that the map does not move, a fixed point. An intuitive way of seeing this is fundamentally similar to Josiah Royce’s construction, which is briefly quoted and discussed on page 146. For Royce, an exact smaller image of England is on a map. But if the map is exact, then there is an unimaginably smaller version of the map on the map. And that version must also have a smaller version contained within. These images, each one contained in the previous, appear to shrink to a point. In fact, they do: the math capturing Royce’s idea was formally stated and proved by Banach and others in the early 1920s, and today the result goes by the name of the contraction mapping principle. In Variaciones Borges, John Durham Peters takes a deep look at philosophic connections between Borges, Royce, and William James. He also considers Royce-type maps from many perspectives, delineating some interesting mathematical and situational implications, and using these ideas to meditate on the real world in contrast to mathematics.

  Glossary

  Thus we may define the real as that whose characters are independent of what anybody may think them to be.

  —Charles Sanders Pierc
e, How to Make Our Ideas Clear

  You who read me—are you certain you understand my language?

  —Jorge Luis Borges, The Library of Babel

  In some cases, the phrases and clauses that follow should be considered more as gestures and less as definitions: they're not necessarily meant to be rigorous or precise, but rather to evoke a way of understanding the mathematical object. For those with internet access and inclination, Wikipedia at www.wikipedia.org stands up as a surprisingly good source for formal definitions. Wolfram's MathWorld at mathworld.wolfram.com is equally good, and perhaps less subject to malicious or mischievous hacks and prankings.

  Beyond standing the test of time and invoking chills of the mythologic, the stacks of libraries stocked with math books are invested with the pregnant allure of opening crisp new or musty old books, and then using indices to seek out appearances of the term or concept. By so doing, you may follow Borges' footsteps through dim-lit libraries, tracking the spoor left by the intellectual history of an idea and slowly netting it with your growing framework of context and insight. Libraries are cultural resources eroding byte by byte under the rising tide of digitization. I point this out partly as a lament, but mostly as a tedious reminder for those so inclined to seize the opportunity to use libraries before they change beyond recognition.

  1-space The Euclidean line. The real number line. One-dimensional space.

  1-sphere A circle contained in a plane. All the points in a plane that are the same fixed distance from a particular point.

  2-space The Euclidean plane. The Cartesian coordinate plane. Length by width. The x-y plane. Two-dimensional space.

  2-sphere A basketball. A soccer ball. The generalization of the 1-sphere to a higher dimension. All the points in 3-space that are the same fixed distance from a particular point.

  3-Klein bottle A three-dimensional analogue of the Klein bottle. A nonorientable object living in higher dimensions that is formed by identifying the faces of a solid cube or hexagonal prism. A somewhat improbable model for the universe that is the Library.

  3-space The space we appear to live in. Volume. Length by width by height. The x-y-z space. Three-dimensional space.

  3-sphere The generalization of the 2-sphere to a higher dimension. A geometric object that lives naturally in 4-space. All the points in 4-space that are the same fixed distance from a particular point. A model for the universe that is the Library which satisfies the particulars of the Librarian's classic dictum as well as those of the Librarian's solution.

  3-torus The generalization of the torus to higher dimensions. A geometrically flat object that lives most naturally in 6-space, although it may inhabit 4-space. A solid object living in higher dimensions that is formed by identifying the faces of a solid cube or hexagonal prism. The most sensible model (whatever that means) for the universe that is the Library.

  4-space In the context of our universe, 4-space is often called the space-time continuum, and can be thought of as (Volume) x (One time dimension). In this book, though, it's (Volume) x (Another Euclidean dimension). The w-x-y-z space. Four-dimensional space.

  annulus An annulus is the area between two concentric circles in the Euclidean plane. Topologically, it is the same as a cylinder, or a can that has had the top and bottom removed.

  Archimedean property Often stated in the form that there is no largest integer. This is then usually flipped, by taking reciprocals, to conclude that there is no smallest positive number. It's then easily generalized to point out that all real numbers are beset and besieged by other real numbers, none of which is "closest."

  axiom A statement so fundamentally in accord with our intuition and experience of the world that we are willing to accept it as a basis for all future developments. A logical "given."

  base of an exponent The number that is multiplying itself some fixed number of times. For example, in the expression 53 = 5 • 5 • 5 = 125, the number 5 is the base of the exponent.

  Brouwer's fixed point theorem In its simplest form, Brouwer's fixed point theorems says that if we take any closed disk in the plane and twist it, stretch it, contract it, rotate it, and do what we will with it, and then squish the transmogrified disk back down into the plane so that it lies within its original boundaries, then there must be at least one point that is unmoved. That is, despite all the distortions and contortions, there must be a fixed point.

  Cavalieri's principle Cavalieri's principle is a way to think of the volume of an object as the sum of infinitely many infinitesimally thin slices of the object. In calculus terms, for a sufficiently "nice" object, we can integrate the areas of the slices to find the volume of the object.

  chiliagon A thousand-sided polygon. A chiliagon on the page of this book would be virtually indistinguishable from a circle. Descartes used it as an example of a geometric object that's easy to define but impossible to visually imagine within the mind's eye.

  circular logic See "illegitimate deduction."

  circumference The circumference of an n-sphere, for any dimension n, is the distance around the equator of the sphere. The distance around any great circle of the sphere. If the radius of the n-sphere is r, then 2wr is the circumference.

  closed interval A closed interval of the real number line is the set of all points between two numbers, inclusive of the endpoints. For example, the closed interval between 1 and 7 is the set of all numbers x such that 1 < x < 7.

  codimension The codimension of a geometric object living in some n-space is the difference in dimensions between the object and the ambient space. For example, a line is a one-dimensional object. The codimension of the line in 2-space is equal to 1. The codimension of the line in 3-space is equal to 2. The codimension of the line in 4-space is equal to 3. The codimension of the line in n-space is equal to (n - 1).

  combinatorics Combinatorics is the art of counting something in two different ways, setting those equal to each other, and thereby finding a formula with general applicability. A lot of interesting combinatorics can be done by thinking carefully about the many ways different colored balls can be placed into barrels.

  countable A set is countable if it can be put into one-to-one correspondence with the positive integers. If the elements of a countable set are playing musical chairs and there is a chair for each positive integer, when the music stops every element will be able to find a seat, every time. Such a set is also called countably infinite.

  definition See "definition" or "self-referential."

  denominator The denominator of a fraction is the number dividing into the numerator. The bottom of the fraction. The basement of the fraction. In the expression 3/5, the denominator is 5.

  empty set, complete list of elements contained within See page xlii.

  Euclidean space A space satisfying Euclid's postulates. See also 1-space, 2-space, 3-space, 4-space, etc.

  exponential notation A remarkably condensed and useful notation that captures the idea of a number multiplying itself some specified number of times. In this book, we use it only for integer self-multiplications, but the ideas can be extended so that all real numbers are legitimate exponents. With somewhat more difficulty, the ideas may be further extended so that imaginary and complex numbers may also serve as exponents.

  factor (noun) A positive integer which, when multiplied by another positive integer, produces yet another positive integer of situational interest.

  factor (verb) Given a positive integer, it's the finding of the factors (noun) that multiply each other to produce the original integer.

  factorial A useful notation for positive integers that often crops up in combinatorial formulas. Generalized by the gamma function. Approximated by Stirling's formula. See the section "Notations" for a formal definition and an example.

  Fibonacci sequence Counts, for each successive generation, the number of immortal rabbits living in an infinite universe. Has terms whose ratios converge to the Golden Mean. Arises in surprising places in nature. Is related to logarithmic spirals.
Is the object of study of entire books.

  fixed point A fixed point of a function that maps from a space to itself doesn't move. For example, if we map the real numbers to themselves by the function f(x) = 3x, then f(0) = 3-0 = 0, entailing that 0 is a point fixed by the function.

  flat Describes a geometric object equipped with a notion of distance which may be precisely the same as in Euclidean space. For example, although curved, the surface of a cylinder is flat, because to find the distance between two points, the cylinder may be "cut open and unrolled" and "laid flat." Then the two points may be connected by a Euclidean straight line and then "rerolled."

  fractal An object with a dimension that is not an integer. An object that continues to present visual complexity under increased magnification. Clouds, bark, lungs, leaves, coastlines, strange attractors, the Koch snowflake curve, the Cantor set,. ..

  function A way of relating two spaces. A way of relating a space with itself. A way of corresponding elements of one set with elements of another set. A systematic process that inputs real numbers and outputs real numbers.