The Unimaginable Mathematics of Borges' Library of Babel by William Goldbloom Bloch

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 n th 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 10 12 and 2 n , 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