How it is possible to create a line segment, a set of positive measure, out of points of measure 0? That is a long story for another day.
Third Interpretation
Perhaps the elusive nature of the preceding interpretation is unsatisfying; we never took a stand on how thick "infinitely thin" is; we merely observed that it is of measure 0. For the third interpretation, we will glimpse some of the basic elements of one of the most underutilized mathematical theories of the twentieth century: nonstandard analysis. The roots of the development of nonstandard analysis began with Leibniz, one of the inventors of the calculus. Both Leibniz and Newton used infinitely small quantities, infinitesimals (also known as fluxions), in their early calculations. In his foreword to the revised edition of Abraham Robinson's seminal work, Non-standard Analysis, the logician Wilhelmus Luxemburg notes that "Bishop Berkeley disdainfully referred to infinitesimals as the 'ghosts of departed quantities,"' and that in response to this and other attacks, "Leibniz proposed a program to conceive of a system of numbers that would include infinitesimally small as well as infinitely large numbers."
Because of the difficulties inherent in beginning Leibniz's bold program, and for other historical reasons, his ideas lay fallow for almost 300 years. In 1961, with the publication of Non-standard Analysis, Robinson rebutted Berkeley and fulfilled Leibniz's dream. Using various tools of logic and set theory developed in the late nineteenth and early twentieth centuries, Robinson was able to create a consistent, logical model of a number system that included infinitesimals.
It should be mentioned, with sincere respect, that adherents of non-standard analysis possess a striking combination of mystic fervor and matter-of-fact pragmatism about the topic. This may be because the mainstream of mathematics has, at least for now, marginalized nonstandard analysis due to its less intuitive constructions and technical complexities. Bearing this in mind, here are selections, originally excerpted by Mark McKinzie and Curtis Tuckey, from H. Jerome Keisler's college textbook, which approaches the calculus from the nonstandard viewpoint (emphases added by present author).
In grade school and high school mathematics, the real number system is constructed gradually in several stages. Beginning with the positive integers, the systems of integers, rational numbers and finally real numbers are built up . . .
What is needed [for an understanding of the calculus] is a sharp distinction between numbers which are small enough to be neglected and numbers which aren't. Actually, no real number except zero is small enough to be neglected. To get around this difficulty, we take the bold step of introducing a new kind of number, which is infinitely small and yet not equal to zero . . .
The real line is a subset of the hyperreal line; that is, each real number belongs to the set of hyperreal numbers. Surrounding each real number r , we introduce a collection of hyperreal numbers infinitely close to r. The hyperreal numbers infinitely close to zero are called infinitesimals. The reciprocals of nonzero infinitesimals are infinite hyperreal numbers. The collection of all hyperreal numbers satisfies the same algebraic laws as the real numbers . . .
We have no way of knowing what a line in physical space is really like. It might be like the hyperreal line, the real line, or neither. However, in applications of the calculus it is helpful to imagine a line in physical space as a hyperreal line. The hyperreal line is, like the real line, a useful mathematical model for a line in physical space.
In nonstandard analysis, there are infinitely many hyperreal infinitesimals clustered around 0, every one smaller than any positive real number. Each signifies an infinitely small distance. We may simply assign any infinitesimal we wish to each page of the Book.4 By the rules of nonstandard analysis, we compute the thickness of the Book by adding together all of the infinitesimals. For a summation such as this one, adding the infinite number of infinitesimals produces yet another infinitesimal, so the Book is, again, infinitely thin: never to be seen, never to be found, never to be opened. This time, though, we may elegantly console ourselves that the infinite thinness is a precisely calculable nonstandard thickness.
Regardless of which interpretation we assume, if the pages are 'infinitely thin,' then by necessity the Book of Sand itself is infinitely thin.
Math Aftermath: Logarithms Redux
Reason looks at necessity as the basis of the world; reason is able to turn chance in your favor and use it. Only by having reason remain strong and unshakable can we be called a god of the earth.
—Johann Wolfgang Von Goethe, Wilhelm Meister's Apprenticeship, bk. I, ch. 17
Recall that in the first Math Aftermath, we used logarithms to solve an equation involving exponentials. This is another example, only slightly more complicated, of using logarithms to solve an equation. Earlier in this chapter, we claimed that if the Book of Sand started with a normal page thickness, say one millimeter, 10-3 meters, and each successive page was half the thickness of the preceding page, then the 41st page would be thinner than a proton, which measures a little more than 10-15 meters across. How did we find the number 40?
Let's set it up as an equation. Each page is half the thickness of the preceding page, so if we measure the nth page after the first page, it will be the thickness of the first page cut in half n times. That is, it will be
meters across.
Since the size of the proton is approximately 10 15, we set these two terms equal to each other and then simplify the equation.
, which implies ;
therefore,
Solving this last equation without logarithms would be very difficult. (In fact, in 2004, powerful mathematical software running on my late-model computer crashed the computer in a failed, naïve, brute-force attempt to solve for such an n.) Since 1012 and 2n, although written differently, are the same number, it should again be the case that any function applied to both of them will output the same number. Thus,
,
which, by using the remarkable property of the logarithm, entails that
.
Dividing both sides by log(2) yields
which can quickly be solved with a computer, a calculator, or—for traditionalists—logarithm tables. When we do so, we find that n is about equal to 39.9, so to ensure we get the result we want, we round upwards. Thus, if we cut the initial page's thickness in half 40 times, it will be the case that the 41st page is thinner than a proton.
FOUR
Topology and Cosmology
The Universe (which Others Call the Library)
A fact is the end or last issue of spirit. The visible creation is the terminus or the circumference of the invisible world.
—Ralph Waldo Emerson, "Nature"
TOPOLOGY IS A BRANCH OF MATHEMATICS THAT explores properties and invariants of spaces, and for the purposes of this book we consider a space to be a set of points unified by a description. Cosmology is quite literally the study of our cosmos. If we consider the Library to constitute a universe and the universe to be the Library, it is not unreasonable to combine these notions and speculate as to a conceivable topology of the Library that best reflects the anonymous librarian's received wisdom and secret hopes.
Early in the story—and many commentators have noted the connection between the italicized phrase and Borges' essay "Pascal's Sphere"— Borges writes
Let it suffice for the moment that I repeat the classic dictum: The Library is a sphere whose exact center is any hexagon and whose circumference is unattainable.
The final sentences of the story invite us to reopen the question of the topology of the Library:
I am perhaps misled by old age and fear, but I suspect that the human species—the only human species—teeters at the verge of extinction, yet that the Library—enlightened, solitary, infinite, perfectly unmoving, armed with precious volumes, pointless, incorruptible, and secret—will endure.
I have just written the word "infinite." I have not included that adjective of out of mere rhetorical habit; I hereby state that it is not illogical to think that
the world is infinite. Those who believe it to have limits hypothesize that in some remote place or places the corridors and stairs and hexagons may, inconceivably, end—which is absurd. And yet those who picture the world as unlimited forget that the number of possible books is not. I will be bold enough to suggest this solution to the ancient problem: The Library is unlimited but periodic. If an eternal voyager should journey it in any direction, he would find after untold centuries that the same volumes are repeated in the same disorder—which, repeated, becomes order: the Order. My solitude is cheered by that elegant hope.
Collecting the properties of the classic dictum (CD) and the Librarian's solution (LS), we obtain the following list:
1. Spherical (CD)
2. Center can be anywhere—uniform symmetry (CD)
3. Circumference is unattainable. (CD)
4. No boundaries (LS)
5. Limitless (LS)
6. Periodic (LS)
Is there a space that embodies all six of these properties? If so, how can we best envision it and grasp it with our intellect? We claim there is an excellent candidate that encompasses these properties, if we are willing to refine our interpretations just a smidge. In the Math Aftermath to this chapter, we discuss two other compelling ways ofconfiguring the Library that each significantly expand our conceptions of the possible.
Let's begin with the space most familiar to our intuitive geometric sense: Euclidean three-dimensional space (henceforth, 3-space). It is a space we think of as possessing volume, as having three axes of orientation with ourselves as the central point; we may move forward or backwards, we may move left or right, and we may move up or down. And, ofcourse, we may also move in combinations of these directions. Notice that from this description, there is no fixed preferred center point: we are our own central points.
Indeed, one of Descartes' deepest ideas was to specify a point—some point, any point—in 3-space and call it the origin. Three axes intersecting at the origin, typically called the x, y, and z axes, are set with each axis at right angles to the other two. They abstract our innate, intuitive orientation and, with the introduction of a unit length, which naturally induces a numbering of the axes, give rise to a coordinatization of space. Algebra can now conjoin geometry, creating analytic geometry, and later spawn calculus.
But there are no distinguished points of any kind in Euclidean 3-space; in fact, the view from any point is the same as from any other point. There are no walls, no boundaries, and no limits. It seems at the end of the story the librarian envisioned this kind of space, partitioned into hexagons, filled with books, extending infinitely throughout the totality of 3-space. The books' shelving pattern repeats endlessly along each of the three axes, much as a symmetric wallpaper pattern does in two dimensions. While this conception of the Library satisfies points 2, 4, 5, and 6, it also induces a vertiginous disorientation born of trying to imagine a thing extending away forever. For example, if the Library goes down forever, what do the hexagons rest on? More hexagons? Rather remarkably, the architectural, model of the Library that we propose provides a satisfying answer to this question.
A note regarding the gravity of the situation. If the universe and the Library are synonymous, and if we make the reasonable assumption that the universe is neither expanding nor contracting, it follows that the natural gravitational field would be identically zero everywhere. Even though there are unimaginable amounts of matter in the universe/Library, its homogeneous distribution entails that the gravitational effect from any one direction would be canceled out by precisely the same effect from the opposite direction. Since the builders of a Library must be, at least from our perspective, omnipotent, their talents surely must include the ability of imposing a useful constant gravitational field on the Library.
Euclidean 3-space embodies some of the qualities of interest in our quest to understand the large-scale structure of the Library. We need to limn two more ideas, one mathematical, one mystical, before we can describe the form of a Library that reconciles the characteristics of the classic dictum and the Librarian's solution.
The mathematical idea is relatively recent—it comes from the early part of the twentieth century. For the purposes of this book, we'll say that a manifold is a space that is locally Euclidean but that on a global scale may be non-Euclidean. Perhaps the simplest possible example is that of a sphere, or globe, or surface of a cantaloupe, or of the earth, balloon, soccer ball; take your pick. Locally, assuming that we are so small we can't detect the curvature, each micropatch of a sphere is, in essence, a two-dimensional Euclidean plane (2-space). One need think only of the steppes of Central Asia, the corn belt of the United States, the Sahara desert, or any large, calm body of water to engage vivid testimony on this point. Globally, despite the essential flatness of each little patch, we find non-Euclidean behavior: if we begin at a point, pick a direction, and continue moving in that direction, we circumscribe the sphere and return to our starting point. This can't occur in 2-space, where we perforce travel forever in one direction and can't ever come close to a previously visited point.
Again, a manifold is locally Euclidean. If we start at any point in space, look around and take a few steps in any direction, do we think we are in Euclidean space? If the answer is yes, then we are in a manifold. If we continue walking, and some unusual phenomenon occurs, such as returning to our starting point, then we realize we are in a nontrivial manifold; that is, one with global non-Euclidean properties. Our universe, for example, seems to be a manifold, although interesting questions arise at black holes. Certainly one cannot imagine standing at a black hole and taking a step in any direction! Researchers are trying to devise methods of determining the global structure; a readable introduction to this area of research can be found in Luminet et al.
The mystical idea is relatively ancient—I leave it to a Borgesian intellect to trace its roots and agelong echoes. Let's begin in a familiar place, our own universe. If we talk about an object in our universe— for example, a desk or chair—we view it as embedded in a larger space. Consequently, we often use our relative coordinate system to refer to objects, as when we say "It's on my right," or "Over there! Directly behind you, to the left," or "Scratch my back.. . lower.. . lower... to the right. . . now up .. . that's it!" Over the millennia, primarily as navigation aids, we've settled upon somewhat less arbitrary reference points, such as the North Star, the magnetic North Pole, and the true North Pole. The point is, though, that these references, these origins, are all within our universe. "Outside the universe" is a phrase beyond normal comprehension. Some theories place our universe in a larger matrix, such as a superheated gas cloud containing an infinite number of inaccessible universes, or in a higher-dimensional space, or in a multiplicitous welter of bifurcating universes. However, these theories raise the question,
What is outside of the larger universe?
Really, now, though, "What is outside of the universe?" The answer is no thing; nothing; non-space; indescribability; un-thing-ed-ness; Void beyond vacuum: all these non-things are the "outside" of our universe.
These two ideas, the mathematical and the mystical, are woven together in this question and its answer.
Where is the center of a sphere?
If the sphere is considered as an everyday object embedded in our universe, the answer may take a form such as "at the intersection of two diameters," or, pointing at it dramatically, saying with particular emphasis, "There! In the middle, in the interior!" See figure 9. If, though, we consider the surface of the sphere as a manifold, as a space in itself and of itself, then the question and answer are subtler. As in the case of our universe, as if we were points residing in the sphere itself, there is no legitimate referral to a point outside the universe of the surface of the sphere. There is only the sphere; every thing else is no thing. Where is the center of a sphere? Considered as a manifold, then, the answer is
Everywhere and nowhere.
Every point has the property that locally, it l
ooks like Euclidean space, and regardless of the direction taken, consistently moving in any chosen direction returns us to the starting point. No point is distinguished in any way.
One more idea is necessary to provide a satisfying topology for the Library. The example of a manifold we used was a two-dimensional sphere (2-sphere). There are a number of ways to rigorously define a 2-sphere. Euclid might write something like, "Given a point p in 3-space, a sphere with center p is the collection of all points a specified uniform distance away from p." An analytic geometric equation for the standard unit sphere is x2 + y2 + z2 = 1. (If you're interested in seeing why this equation specifies a sphere, please turn to the appendix "Dissecting the 3-Sphere".) Here, using words and pictures, we provide a topological construction of a 2-sphere.
Start with a disk in the Euclidean plane and while preserving the interior of the disk except for bending and stretching, crimp the entire boundary circle up out of2-space, and then contract the boundary to one point. This point, the contraction of the boundary, becomes the north pole and vanishes into the surface of the sphere created as the process is completed (figure 10). An interesting point: the way we've described it, and the way the picture shows this process, it seems as though a disk is being modified over time. By contrast, though, one should simply say, "Identify the boundary of the disk to a point." Thus, in some sense, the creation of the sphere is a timeless step that happens "all at once."
The Unimaginable Mathematics of Borges' Library of Babel Page 7
