The Unimaginable Mathematics of Borges' Library of Babel (44 page)

tiling of space
An object tiles a space if clones of the object completely fill
out the space with no interstices or overlaps. For example, it's not too hard
to see that squares tile the plane and that cubes tile 3-space. Bisecting the
squares along diagonals shows that triangles also tile the plane. Looking at a
beehive suggests how hexagons may tile the plane, which in turn suggests the
correct belief that hexagonal prisms tile 3-space.

topology
Very loosely, topology is the study of the possibilities and
immutable characteristics of spaces.

torus
The surface of a donut or bagel. A good example of a
two-dimensional locally Euclidean space.

transfinite numbers
These days, most mathematicians would call trans-finite numbers
either infinite cardinal numbers or infinite ordinal numbers. As the name
suggests, transfinite numbers are beyond the finite, and they are truly
unimaginable.

uncountable
Describes an infinity infinitely larger than countably infinite.
The cardinality of the set of irrational numbers.

unique factorization
The property enjoyed by the positive integers that they may be
decomposed into products of primes in essentially only one way.

upper bound
A maximal estimate. "There are at most thus-and-such."

Venn diagram
A way of viewing unions, intersections, and subsets of
collections of sets by representing the sets as circles.

well-ordering principle
A surprisingly contentious axiom or theorem (depending on the
system) that says every set of positive integers contains a least element. The
reason some mathematicians and logicians reject the well-ordering principle is
that it is used to facilitate kinds of deductions that may lead to disturbing
conclusions.

All books are divisible
into two classes, the books of the hour, and the books of all time.

—John
Ruskin,
Sesame and Lilies

I was impressed for the ten
thousandth time by the fact that literature illuminates life only for those to
whom books are a necessity. Books are unconvertible assets, to be passed on
only to those who possess them already.

—Anthony
Powell,
The Valley of Bones

In this section, I list a few
readings that in one way or another go deeper into ideas raised in this book.
I've loosely organized them, mostly by the chapter that they illuminate. Like
most of my book, the list is somewhat idiosyncratic; books and articles
appearing tend to have had a lasting impact on me, or, in a few cases, received
a strong recommendation from someone I respect. For more personalized
recommendations, feel free to write me at <
[email protected]
> describing your math background and the kinds of things you'd
like to learn. Publication details for each book may be found in the
Bibliography.

The Heart of Mathematics,
by Edward B. Burger and Michael Starbird.

Burger and Starbird produced a
funny, inspirational, eminently readable book pitched at the level of bright
high school students and college students who haven't (yet) had a lot of
training in mathematics. It's almost as if they thought, "What are the
niftiest ideas in math that don't need a deep theoretic background? How can we get
them all into one book?" and then went ahead and did it. Great problems
are found at the end of every chapter, and some answers are included. It's
worth mentioning that Starbird is a raconteur of the first order, and Burger
worked as a stand-up comedian before becoming a mathematician.

The Pleasures of Counting,
by T. W. Korner.

This remarkable book unites a
host of topics by the common theme of mathematics making a difference in
solving real-world problems. Korner opens the book with a discussion of how Dr.
John Snow essentially invented epidemiology when he analyzed data pertaining to
cholera outbreaks in the middle 1800s. Korner moves with ease from there
through contributions to thwarting submarine warfare, development of radar,
cracking the Enigma code, and a host of other fascinating applications.

"What Is Good
Mathematics?" by Terence Tao
(Bulletin of the American Mathematical
Society
44(2007): 623—34, available as a free .pdf download from the
American Mathematical Society at
http://www.ams.org/bull/2007-44-04/S0273-0979-07-01168-8/S0273-0979-07-01168-8.pdf
)

The Fields
Medal is often called the Nobel Prize for mathematics, although it differs from
the Nobel in several key ways. For one, the Fields is only awarded once every
four years—although in recent years there's been a tendency to award it to four
people each time. The second is that a recipient must be under the age of 40,
and the selection committee hews to this: Andrew Wiles' proof of Fermat's last
theorem was completed when he was slightly older than 40, and while he received
a special medal and recognition, he did not receive the Fields Medal. Terence
Tao is a 2006 Fields Medalist, and in this two-part article, he tackles an
elusive question, "What is
good
mathematics?" His thoughts in
the first part are quite interesting and accessible to all; in the second part,
he illustrates some of his categories of "good math" via a case study
of Szemeredi's theorem.

Symmetry,
by Herman Weyl.

Weyl was a mathematician who
did a lot of work in physics, notably quantum mechanics. This classic book
explores symmetry in nature and mathematics. Weyl once told Freeman Dyson,
"My work always tried to unite the true with the beautiful, but when I had
to choose one or the other, I usually chose the beautiful." I'm not sure
they're words to live by, but I find them profound.

The Mathematics of Ciphers,
by S. C. Coutinho.

Coutinho is a computer
scientist in Brazil. The book consists of engaging expositions of primality,
prime number testing, and the RSA cryptography scheme intended for a first-year
class in computer science. The translated work is relatively easy to read and
builds to some interesting ideas. Because it was slated for nonmathematicians,
Coutinho's perspective is that of a keen-eyed outsider.

"It Is Easy to Determine
Whether a Given Integer Is Prime," by Andrew Granville
(Bulletin of the
American Mathematical Society
42(2004): 3—38, available as a free .pdf
download from the American Mathematical Society at
http://www.ams.org/bull/2005-42-01/S0273-0979-04-01037-7/S0273-0979-04-01037-7.pdf
)

This article
summarizes and explains some of the huge breakthroughs that occurred in the
search for "large" prime numbers after Agrawal, Kayal, and Saxena's
paper "PRIMES is in P" appeared in 2004. By my highly subjective
rating, although very much worth the effort, this is the hardest reading
appearing on this list, and it probably requires the equivalent of an
undergraduate degree in mathematics. Because this field is exploding, and
because of the importance to e-commerce, I'd guess that all of these results
have since been extended and refined, but still it's worth a look.

The Pea and the Sun: a
Mathematical Paradox,
by Leonard Wapner.

Wapner's book is pitched at
the level of bright, mathematically inclined high school students who've
(perhaps) heard of the Banach-Tarski paradox. This counterintuitive
construction explains how to disassemble a small solid ball into a finite
number of nonmeasurable sets, and then reassemble the pieces into a very large
solid ball. Along the way, Wapner gets at some of the ideas of measure theory,
and gives nice proofs that lead up to the main result. I liked this book a lot.

Measure and Category,
by John C. Oxtoby.

This slender book is one of
the publisher Springer-Verlag's infamous yellow "Graduate Texts in
Mathematics." (Infamous among math graduate students, at any rate.)
Although it's probably necessary to have the equivalent of an undergraduate
math education to profit from reading it, the writing is so light, clean, and
lively, and the results are so enrapturing, I am pleased to recommend it.

The Problems of
Mathematics,
by Ian Stewart.

Although Stewart's book
encompasses many other nifty mathematical ideas, in particular it contains a
chapter outlining the nuances and the history of some of the issues surrounding
nonstandard analysis, including the subtle distinction between Leibniz's static
infinitesimals and Newton's variable fluxions.

Non-standard Analysis,
by Abraham Robinson.

Robinson's seminal work is for
an enterprising individual with the equivalent of, say, a master's-level
education in mathematics or logic.

The Shape of Space,
by Jeff Weeks.

Weeks has produced a luminous
work, comparable to Oxtoby's
Measure and Category,
that takes advanced
ideas and presents them so clearly and compellingly that it feels like everyone
could and should read it. At any rate, if you were gripped by the Math
Aftermath "Flat-Out Disoriented" and need more, Weeks is a good place
to start.

Contemporary Abstract
Algebra,
by Joe Gallian.

Gallian's book is the
friendliest introduction I've seen to algebraic groups and homomorphisms. Even
so, it's aimed at sophomore- and junior-level math majors, and, as such, may
require a large dollop of commitment.

Mathematical Foundations of
Information Theory,
by A. Ya. Khinchin.

Pretty technical, and redolent
with the spare language of the professional mathematician, but I learned a lot
about information theory from it when I was just beginning to be interested in
mathematics. In particular, I found the discussions of entropy and information
theory lucid and comprehensible.

An Introduction to
Information Theory,
by John R. Pierce.

I haven't read this, but I
stumbled across it when I was looking up the exact title of Khinchin's book. It
looks good and may be an easier read than Khinchin's book.

The Magical Maze: Seeing
the World Through Mathematical Eyes,
by Ian
Stewart.

Another book by Stewart, a
great explainer and popularizer of mathematics. This book, too, is filled with
all sorts of good things; among them is a solid introduction to Turing
machines.

Gödel, Escher, Bach: An
Eternal Golden Braid,
by Douglas Hofstader.

What praise can I apply to
this book that hasn't been already written? Winner of the Pulitzer Prize, it
leads nonspecialists to the ideas of Gödel, reframes self-referentiality and
paradox, and—I think—is the site of the first appearance of the evocative
phrase "strange loop." It's marvelously witty, profoundly deep, and
it heralded a new genre in letters.

Gödel's Theorem: An
Incomplete Guide to Its Use and Abuse,
by Torkel
Franzen.

Franzen's book concisely achieves
its goal of clearly demarcating the extent of applicability of Gödel's theorem.
Along the way of accomplishing that, he shows its power and majesty in the
fields of set theory and foundations, and brings into sharp focus many amusing
nuances.

Gödel's Proof,
by Ernst Nagel.

Nagel takes a dedicated reader
step by step through a proof of Gödel's theorem. A classic.

Any Book by Raymond
Smullyan,
by Raymond Smullyan.

Actually, there is no such
book by Smullyan (although I think he'd appreciate the self-referential title).
He's an influential mathematical logician who, in addition to publishing
serious works, has also written many highly readable books weaving together Gödel's
theorem, truth, lies, formal systems, knights and knaves, islands, detectives,
logic machines . . .

Contributions to the
Founding of the Theory of Transfinite Numbers,
by
Georg Cantor.