The Unimaginable Mathematics of Borges' Library of Babel
Page 17
Funes-like Ireneo Funes is a character in a remarkable Borges short story who is gifted and afflicted with essentially perfect memory. Funes spends more than a day reliving every detail of a day.
gamma function An elegant way of generalizing the concept of the factorial of a positive integer to that of all real numbers.
glossary See "definition" or "self-referential."
great circle An equator of an n -sphere. A circle of maximal size that can be contained in an n-sphere.
hexagonal prism A hexagon is a symmetric six-sided object contained in a plane. A hexagonal prism is a three-dimensional object whose horizontal slices are filled-in hexagons.
homomorphism A function maps elements of one set to elements of another set. A homomorphism is a function that also preserves algebraic relations during the mapping; for example, we can think of integers as points in the set of real numbers, but we can also think ofnumbers as things that do algebraic stuff, such as addition and subtraction. A homomorphism maps integers both as elements and as algebraic objects. If this intrigues, see Gallian's Contemporary Abstract Algebra.
hyperreal number An infinitesimal affiliated with any nonzero real number.
hyperreal number line The real number line, combined with all the hyperreals affiliated with each real number.
illegitimate deduction See "circular logic."
infinitesimal An idea used by Euler, Newton, and Leibniz in thinking about calculus. Infinitesimals may be thought of as actual numbers in a logically consistent way, and may loosely be thought of as entities that have "magnitude" greater than 0 but are smaller than every positive real number. Every real number x can logically be thought of as being surrounded by infinitesimal hyperreal numbers that are closer to x than any other real number.
initial position The starting point for a Turing machine.
integers The set of whole numbers { . .. -2, -1, 0, 1, 2,... }.
internal state A particular set of instructions for a Turing machine, telling the Turing machine what should be done in response to each possible input.
irrational numbers The set of real numbers that can't be written in the form of a fraction p/q, where p and q are both integers. When an irrational number is written out in decimal form, the digits in the expansion neither terminate nor turn into a repeating pattern.
Klein bottle A torus that has lost it's way in 4-space. A boundaryless nonorientable two-dimensional object.
Koch snowflake curve A pleasantly symmetric example of a fractal that appears in a math book that Borges reviewed. One unusual property that it possesses is that the distance between any two points is infinite. The more closely we look at any portion of the snowflake curve, the more detail emerges.
lemma A lemma is an assertion not quite important enough to be called a theorem.
libit Short for "library unit." A library unit is a collection of contiguous hexagons sufficiently large to hold all 251,312,000 distinct volumes and sufficiently symmetric that copies of it are able to tile the infinite 3-space model of the Library.
locally Euclidean A space is locally Euclidean if at every point of the space, a severely myopic individual is convinced that they are, in fact, inhabiting a Euclidean space. As an example, consider the circle. It is clearly not a Euclidean line, but if you have access to a math program or drawing program that allows you to zoom in on an object, and you zoom in on any point, you'll find that what began as looking like a curve looks a lot like a straight line. Thus, a circle is locally Euclidean.
logarithm The logarithm is a function characterized by several remarkably useful properties. All of these stem from the fact that it is the inverse function to the exponential function in base 10. (An inverse function cancels the effect of its corresponding function.)
lower bound A minimal estimate. "At least thus-and-such."
manifold A shorter name for a locally Euclidean space. map Another name for a function, for we can think of a function not only as taking inputs and returning outputs but also as taking a point and mapping it, or moving it, to another point. median The median of a finite set of numeric data is a kind of a middle number: half of the data will be larger than the median, and half will be smaller.
Mobius band A Mobius band is a nonorientable, one-sided surface with one boundary circle. Taking two Mobius bands and gluing them together along their boundary circles produces a Klein bottle! (This is not an obvious construction.)
non-Euclidean Any space that is not a Euclidean space. For example, all manifolds, including spheres, tori, and Klein bottles, are non-Euclidean. A cylinder, a figure eight, and a spiral are all non-Euclidean. Typically, though, we'd only refer to a space as non-Euclidean if it's everywhere locally Euclidean.
nonorientable space Best defined in opposition to an orientable space: in an orientable two-dimensional manifold, at any point we may choose a definition of "up" and "right," and then, regardless of the path we navigate through the space, when we return to our beginning point our notions of "up" and "right" will agree with those that we originally chose. By contrast, in a nonorientable two-dimensional space, after making choices of "up" and "right," there are circuitous paths we may follow such that when we return to the starting point, either "up" will look like "down" or "right" will appear "left." In a three-dimensional manifold, we'd also have to choose a "front" to make a legitimate definition.
nonstandard analysis Logically sound mathematics done with infinitesimals and hyperreal numbers.
numerator The numerator of a fraction is the number being divided by the denominator. The top of the fraction. The attic of the fraction. In the expression 3/5, the numerator is 3.
one-to-one correspondence This is a map between sets A and B such that every element in A is sent to a distinct element in B and every element in B has exactly one element of A mapped to it. If A and B are finite sets, it means that they each have the same number of elements. If A and B are infinite sets, the implication is that they have the same cardinality.
origin The point in coordinatized n-space such that all coordinates are 0. The point where the axes all intersect. An arbitrarily chosen point that serves as the center of the space.
periodic A pattern is periodic if it repeats over and over. For example, the pattern of letters MCVMCVMCVMCV is periodic of period 3. The earth orbiting the sun is an example of periodic motion. A wallpaper pattern may be periodic.
power of 10 An exponential expression with a base of 10. Examples include 103, 10100, and, more abstractly, 10n, which signifies "some power often."
prime number A number p whose factors are limited to 1 and p. No other positive integer may divide a prime number.
product Another name for the act of multiplication.
raised to a power Another phrase for raising a base by an exponent. Another way of saying that a number is being multiplied by itself a specified number of times.
rational numbers All numbers of the form p/q, where p and q = 0 are both integers.
real numbers The set of all rational and irrational numbers.
real number line Euclidean 1-space. The set of real numbers identified with points on the Euclidean line.
self-referential See "self-referential."
set A collection of objects, usually called elements. If memory serves correctly, "set" is the word in the English language with the most definitions—at any rate, the 2nd edition of the Oxford English Dictionary runs to 23 pages of definitions and citations for the word "set."
set of measure 0 An inconsequential set. A set that essentially occupies none of the ambient space that it lives in. Pick an arbitrarily small number c: a set of measure 0 can be covered by—contained in— countably many sets whose diameters sum to a number smaller than c. (The diameter of a set may be thought of as the maximal distance across it.)
set theory One of the foundations of modern mathematics. One of the underlying languages of modern mathematics. A collection of seemingly unassailable intuitions about objects in our world.
space A collection of points, often equipped with some notion of distance between points.
Stirling's approximation to the factorial A way of approximating n! using Euler's constant e, exponentials, square roots, and 2w. See, for example, page 616 of Apostol's Calculus, Volume II for a derivation of the formula.
subset A subcollection of a set. A subset of a set can be the whole set, some of the set, or none of the set. The subset consisting of no elements is called the empty set. See "empty set."
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.
Annotated Suggested Readings
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
Generally Delightful
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.
Generally Thoughtful
"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.
Patterns
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.
Number Theory
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.
Real Analysis and Measure Theory
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 ma
th 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.
Nonstandard Analysis
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.
Topology, Manifolds, and Cosmology
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.