The Unimaginable Mathematics of Borges' Library of Babel

by William Goldbloom Bloch

  There are at least two candidates from Borges' personal library to which it is tempting to assign influential status in the development of his mathematical thought. The first is Henri Poincaré's 1908 book Science et Méthode. Borges' end leaf notations, dated 1939, indicate an interest in Lesage's discredited theory of gravitation and, more tellingly, in geometry and Cantor. One paragraph, taken from pages 380—81, is a passage on geometry worth quoting (emphases added):

  A great advantage of geometry lies in the fact that in it the senses can come to the aid of thought, and help find the path to follow, and many minds prefer to put the problems of analysis into geometric form. Unhappily, our senses can not carry us very far, and they desert us when we wish to soar beyond the classical three dimensions. Does this mean, beyond the restricted domain wherein they seem to wish to imprison us, we should rely only on pure analysis and that all geometry of more than three dimensions is vain and objectless? [.. . ] We may also make an analysis situs of more than three dimensions. The importance of analysis situs is enormous and can not be too much emphasized; the advantage obtained from it by Riemann, one of its chief creators, would suffice to prove this. We must achieve its complete construction in the higher spaces; then we shall have an instrument which will enable us really to see in hyperspace and supplement our senses.

  Again, I don't imagine that Borges considered exotic cosmologies for the Library, but it interests me to think that he was aware of things living in higher-dimensional spaces.

  The sections pertaining to Cantor mainly restrict themselves to exuberant denunciations of set theory via what Poincaré terms "the Cantorian antinomies"—paradoxes arising from Cantor's theory of transfinite numbers. In many ways, Poincaré prefigures a movement towards constructivism in mathematics, which I briefly discuss in the Math Aftermath "Libits, Uniqueness, and Jumping from the Finite to the Infinite." Since Borges was evidently fascinated by transfinite numbers and the concept of infinity, it's striking that as an autodidact, he pursued the arguments and weighed the objections of Poincaré, a formidable opponent of all things infinite.

  The other book from Borges' library, philosophically opposed to Poincaré's, is Bertrand Russell's Principles of Mathematics. The book was originally published in 1903, and Borges' copy is a 1938 printing. Borges dated his copy "1939," and his annotations further indicate that it was a gift from "Adolfo" (presumably his life-long friend, colleague, and coauthor Bioy Casares). The easiest opening of this volume, and the first page singled out by Borges, concerns a resolution of Parmenides' paradox. The next page pleasantly segues into a discussion of Zeno's paradox of Achilles and the tortoise. The argument contained therein is similar to Russell's refutation in Mathematical Philosophy, which Borges outlined in his 1929 essay "The Perpetual Race of Achilles and the Tortoise." Indeed, Borges' annotation includes the phrase "(cf. Mathematical Philosophy, 138)."

  Borges' essay on Zeno's paradox not only betrays a fondness for and knowledge of Cantor's transfinite numbers; it also demonstrates that Borges understood at least the basics of summing infinite series. That Borges persisted in using Russell as a mathematical touchstone is further evidenced by the brilliant 1939 essay "When Fiction Lives in Fiction," which appears in Selected Non-Fictions, pages 160—62. Here, Borges writes

  . . . Fourteen or fifteen years later, around 1921, I discovered in one of Russell's works an analogous invention by Josiah Royce, who postulates a map of England drawn on a portion of the territory of England: this map—since it is exact—must contain a map of the map, which must contain a map of the map of the map, and so on to infinity. . .

  Principles of Mathematics is rife with Russell's perspectives on Cantor, transfinite numbers, infinitesimals, the meaning of zero, and a host of other Borgesian obsessions. Despite his cavil found on page 46 of Selected Non-Fictions that some of Russell's works are "unsatisfactory, intense books, inhumanly lucid," Borges returned to them again and again. Russell's book, although dry, discursive, and monolithic in conception and execution, contains poetic phrases, one of which Borges singled out with an end leaf notation:

  ... the infinite regress is harmless.

  A point needs to be stressed. Mathematics is a body of lore and an art that requires years of study and practice to understand and appreciate. Just as with twentieth-century atonal music, repeated exposure is required to acculturate the novice to the aesthetics of beauty and elegance particular to mathematics. Grappling with problems and attempting to produce one's own proofs using the licit logical structures are essential to internalize an understanding of the many subtleties inherent in mathematics. I contend that, in this sense, Russell's books are not mathematics; rather they are the philosophy of mathematics. Therein lay their appeal to Borges, and that is why Russell, not Kasner and Newman, remained Borges' inspiration and touchstone of mathematical thought.

  I'll close the book with a last opening: Borges' solitary annotation on the end leaf of Kesten's Copernicus and his World. There Borges inscribed a Latin phrase from Copernicus's De revolutionibus orbium coelestium and referenced the page containing the English translation.

  Mathemata mathematicis scribuntur. "On mathematics, you write for mathematicians only."

  It is my hope that this book belies that sentiment.

  Appendix

  Dissecting the 3-Sphere

  We sail within a vast sphere, ever drifting in uncertainty, driven from end to end. When we think to attach ourselves to any point and to fasten to it, it wavers and leaves us; and if we follow it, it eludes our grasp, slips past us, and vanishes for ever. Nothing stays for us.

  —Blaise Pascal, Pensées

  The aim here is to see that three-dimensional slices of a 3-sphere are, in fact, either points or 2-spheres. (We employed this notion in our discussion in the chapter "Topology and Cosmology" when we relied on lower-dimensional analogues to yield insight into the nature of the 3-sphere.) For those whose are interested in this kind of inquiry but whose memory of the equations of spheres and circles is confined to a misty past, we recommend first reading the second section of this appendix, which carefully uses the Pythagorean theorem and the notion of distance in Euclidean space to derive the analytic equations for a circle, 2-sphere, and 3-sphere.

  A way to understand three-dimensional slices is to use the analytic equation that defines the unit 3-sphere,

  which should be understood as "the set of all points (w, x, y, z) in coordinatized four-dimensional space that satisfy the above equation." For example the point (1, 0, 0, 0) satisfies the equation, as do the points (0, 1, 0, 0) and (1/2, 1/2, 1/2, 1/2). For the latter point, note that

  If we fix w, the coordinate for the fourth dimension, at 0, the equation becomes

  ;

  in other words, the equation of the standard unit 2-sphere. If we fix w = 1 (or -1) we are at the top or bottom of the unit 3-sphere, and the equation becomes

  which implies that

  The only way that three nonnegative numbers can add up to 0 is if they themselves are all 0. In other words, the three-dimensional slice at the coordinate w = 1 yields only the point (x, y, z) = (0, 0, 0).

  On the other hand, let w be any number strictly between —1 and 1. For a concrete example, let w be 1/2. Then the equation becomes

  , which implies that

  By taking the square root of both sides, we arrive at

  and this is equivalent to the statement "the set of all points in three-dimensional Euclidean space located at a distance from the origin (0, 0, 0)." In other words, the equation specifies a 2-sphere of radius .

  In the above argument, there was nothing special about letting w be 1/2. We could have chosen any number strictly between —1 and 1, and we would again end up with an equation specifying a sphere.

  In general, let w = R where -1 < R < 1. Then the examination of the three dimensional slice at w = R is facilitated by the equation

  , which implies that

  By taking the square root of both sides, we arrive at


  ,

  which is the equation for a 2-sphere of radius .

  Deriving the Equations for Circles and Spheres Via the Pythagorean Theorem

  The Pythagorean theorem states that for a right triangle with legs of lengths x and y and with hypotenuse of length h contained in a Euclidean plane, the equation always holds (figure 71). (There are dozens, maybe hundreds, of proofs of this theorem.) If the length of the hypotenuse is, say, 1/2, then the equation becomes

  ,

  and taking the square root of both sides of the equation yields

  .

  In fact, it follows instantly that if h is the length of the hypotenuse, then Pythagoras implies

  Thus the key point is the realization that the length of the hypotenuse is expressible in this form. Now, think of the bottom-left point of the hypotenuse as the origin (0, 0) of the plane and reimagine the lengths of the legs, x and y, as representing the horizontal and vertical coordinates for the right-top point of the hypotenuse. Then the length of the hypotenuse signifies the distance from the origin to the point p = (x, y), and applying the Pythagorean theorem reveals the distance to be . See figure 72.

  Now, we want to use these ideas in order to derive an equation equivalent to Euclid's intuitively satisfying definition of a circle. He defined a circle to be the set of points in a plane that are equidistant from a given point. If we set the given point to be the origin, and choose the distance to be equal to one, then a circle is the set of all points (x, y) that satisfy the equation.

  After squaring both sides, we see that it must be the case that a unit circle is precisely all points (x, y) that fit this equation:

  This is how the analytic equation for the circle arises, and figure 73 indicates a way of viewing a circle as a composition of distances from the origin, that is, as hypotenuses of right triangles.

  The equation for a 2-sphere is very similar in concept, and thus we need only adapt our notion of distance—and therefore, the Pythagorean theorem—to work in three-dimensional space. A standard way is a typically incisive mathematical maneuver which requires the clever use of the Pythagorean theorem twice.

  To see this, let's find the distance from the origin (0, 0, 0} to the point p = (x, y, z) in coordinatized 3-space. The point p naturally determines a right triangle, with the first leg of the triangle being the line segment contained in the x - y plane (for which z = 0} that connects the origin to the point (x, y, 0}. The second leg is the vertical line segment connecting the points p and (x, y, 0}. The hypotenuse of this right triangle is the distance we want—see figure 74.

  Observe that the length of the leg that connects p to the point (x, y, 0) is simply the height, z. Since the other leg is contained in the x-y plane for which z is constantly 0, at a critical juncture below we will ignore the z coordinate and blithely apply the Pythagorean theorem as we did above in the Euclidean plane. First, though, using the Pythagorean theorem on the dark gray triangle in figure 74 gives

  Again, because the second leg of the dark gray triangle is the hypotenuse of the light gray triangle in (essentially) the x-y plane, the Pythagorean theorem allows us to replace "distance from origin to (x, y, 0)" with "." We also may think of "height z" as just "z," and making these substitutions transforms the previous equation into

  Squaring the square root in the first term of the above equation and dropping the parentheses leaves us with

  Taking the square root of both sides of the equation yields

  One may now analytically define the unit 2-sphere in the same way the circle was defined; it is the set of points (x, y, z) contained in 3-space that all are of distance one from the origin. This translates into the fact that the 2-sphere is the set of all points (x, y, z) that satisfy the distance equation

  And thus, by squaring both sides, we arrive at the analytic equation for the 2-sphere:

  Generalizing these ideas to coordinatized four-dimensional Euclidean space is similar—we need only adapt our notion of distance to 4-space. We do this by again bootstrapping ourselves into a higher dimension by cleverly using Pythagoras twice.

  Let p = (w, x, y, z) be a point in 4-space—and notice this time that the "new" coordinate is added in front of, rather than behind, the previous coordinates. Once again, the point p naturally determines a right triangle in 4-space (which unfortunately we are unable to draw) with the first leg of the triangle being the line segment connecting the origin to the point (0, x, y, z), and this segment is completely contained in the x-y-z Euclidean 3-space for which the w coordinate is constantly equal to 0. The second leg of the triangle is the line segment "vertically" connecting (w, x, y, z) to (0, x, y, z); in other words, a leg of "height" equal to w. The hypotenuse of the triangle connects the origin of 4-space to the point p, and is the distance we want. So applying Pythagoras to the right triangle yields

  The distance formula in three dimensions derived earlier allows us to replace "distance from origin to (0, x, y, z)" with “," and "height w" is simply equal to "w." Making these substitutions transforms the equation into:

  Again, squaring the square root in the first term of the above equation and dropping the parentheses leaves us with

  Taking the square root of both sides yields

  The unit 3-sphere is the set of all points in 4-space uniformly a distance one from the origin. This is equivalent to the set of all points (w, x, y, z) satisfying the distance equation

  and, after squaring both sides, we end up with

  Notations

  The twentieth-century artist who uses symbols is alienated because the system of symbols is a private one. After you have dealt with the symbols you are still private, you are still lonely, because you are not sure anyone will understand it except yourself

  —Louise Bourgeois, quoted in Lives and Works

  : This says that for our intents and purposes, we can think of a as being roughly equivalent to b: "a is approximately b."

  : One way of signifying the product of a with b; that is, a way of notating the act of multiplying a by b.

  : Another way of signifying the product of a with b; that is, another way of notating the act of multiplying a by b.

  Read "n factorial," and .

  : The integer a raised to the bth power, for example, .

  : The closed interval between a and b. All numbers between a and b, inclusive.

  : A symbol for infinity. It's important to note that although connotes a kind of magnitude, and is sometimes shorthand for the idea of "arbitrarily large number," the symbol is not a number.

  Notes

  The end crowneth the work.

  —Elizabeth I, quoted in The Sayings of Queen Elizabeth

  The end crowns all;

  And that old common arbitrator, Time,

  Will one day end it.

  —William Shakespeare, Troilus and Cressida

  Preface

  1. Did you look?

  Chapter 1

  1. For example, Lasswitz, who wrote "The Universal Library," which profoundly influenced Borges, calculated the number of books in his Library. Other mathematicians and critics who find the number of books include Amaral, Bell-Villada, Rucker, Nicolas, Faucher, Salpeter, and the anonymous encyclopediasts who wrote the page found at Wikipedia.org! Amaral deserves special plaudits for finding influences of Lasswitz's "The Universal Library" in the work of Lasswitz's mathematical contemporaries Kummer, Fraenkel, pp. 7ff, and Hausdorff, pp. 61ff.

  2. The quote below appears in Borges' expansive short story "Tlön, Uqbar, Orbis Tertius."

  There are no nouns in the conjectural Ursprache of Tlön, from which its "present-day" languages and dialects derive: there are impersonal verbs, modified by monosyllabic suffixes (or prefixes) functioning as adverbs. For example, there is no noun that corresponds to our word "moon," but there is a verb which in English would be "to moonate" or "to enmoon." "The moon rose above the river" is "hlor u fang axaxaxas mlö',' or, as Xul Solar succinctly translates: Upward, behind the outstreaming it mooned.

>   His use of the phrase "Axaxaxas Mlö" in "The Library of Babel" is presumably a reminder that even books written in the Ursprache of Tlön, including all volumes of the first and second editions of the Encyclopedia of Tlön, are in the Library. A careful reader may object that volume 11 of the First Encyclopedia of Tlön consists of 1,001 pages, while Library books number only 410. Our rejoinder is that three books of the Library, the last of which will contain 229 blank pages—blank spaces filling each slot—yield the necessary 1,001 pages. Of course, they may not be shelved anywhere near each other, but this in no way negates the fact that the 11th volume is in the Library. A variation of this observation refutes Rucker's casual statement that "the minute history of the future" can't be contained in the Library, in Infinity and the Mind, pp. 121-22. The minute history of the future is contained in the Library; it is found in volumes perhaps scattered throughout the Library. There is no implicit promise that the information in the Library is accessible or verifiable—it just must be there, somewhere.