The Unimaginable Mathematics of Borges' Library of Babel

by William Goldbloom Bloch

  A Homomorphism

  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 . . .

  Critical Points

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

  Today's notation looks different, and the formulation of the ideas is simpler and cleaner, but it's wonderful to seize the opportunity to read, albeit in translation, the seminal work on infinity from the creator of set theory and the first imaginer of different sizes of infinity

  To Infinity and Beyond, by Eli Maor.

  A user-friendly introduction to many forms of infinity, including Cantor's notions.

  The Divine Proportion, by H. E. Huntley.

  Huntley's work is an accessible reference which rewards a patient reader with a multiplicity of extrapolations from Fibonacci's sequence to the real world.

  Openings

  Mathematics and the Imagination, by Edward Kasner and James Newman.

  A gift from Bioy Casares to Borges, this book is still surprisingly readable. Today, perhaps most notable for being the introduction of the word "googol" into the English language. It was coined by Kasner's nine-year-old nephew.

  Fixed Points, by Yu. A. Shashkin.

  Aimed at bright high-schoolers, this is the most elementary exposition of fixed points and Brouwer's theorem of which I am aware.

  Bibliography

  In sum, about all we can really say in paper's favor is that it's lighter than clay, less awkward than papyrus, and cheaper than parchment.

  —Gregory Rawlins, Moths to the Flame

  Abbott, Edward A. Flatland: A Romance of Many Dimensions. 1884. Reprint, New York: Dover, 1992.

  Adam, David. "Give six monkeys a computer, and what do you get? Certainly not the Bard." The Guardian, 9 May 2003. http://www.guardian.co.uk/arts/news/story/0,11711,952259,00.html (accessed 26 February 2008).

  Alazraki, Jaime. Borges and the Kaballah: and Other Essays on his Fiction and Poetry. New York: Cambridge University Press, 1988.

  Amaral, Pedro. "Borges, Babel y las matematicas." Revista Iberoamericana 37(1971): 421-28.

  Ammon, Ted. "A Note on a Note in 'The Library of Babel.' " Romance Notes 33(1993): 265-69.

  Apostol, Tom. Calculus, Volume II. 2nd ed. New York: John Wiley & Sons, 1969.

  Archimedes. The Works of Archimedes. Translated by T. L. Heath. New York: Dover Publications, 1950.

  Babel, Isaac. The Complete Works of Isaac Babel. Edited by Nathalie Babel, translated by Peter Constantine. New York: Norton, 2002.

  Barrenechea, Ana Maria. Borges: the Labyrinth Maker. Translated by Robert Lima. New York: New York University Press, 1965.

  Barth, John. Further Fridays. Boston: Little, Brown and Co., 1995.

  Bell-Villada, Gene. Borges and His Fiction. Rev. ed. Austin: University of Texas Press, 1999.

  Benardete, Jose. Infinity: An Essay in Metaphysics. Oxford: Clarendon Press, 1964.

  Bioy Casares, Adolfo, and Jorge Luis Borges. Chronicles of Bustos Domecq. Translated by Norman Thomas di Giovanni. New York: Dutton, 1976. Bioy Casares, Adolfo, and Jorge Luis Borges. Six Problems for Don Isidro Parodi. Translated by Norman Thomas di Giovanni. New York: Dutton, 1981.

  Borges, Jorge Luis, and Norman Thomas di Giovanni. The Aleph and Other Stories.

  New York: Dutton, 1978. Borges, Jorge Luis. Collected Fictions. Translated by Andrew Hurley. New York: Viking, 1998.

  __Ficciones. 1944. Rev. ed. 1956. Reprint: Buenos Aires: Emece, 2001.

  __Ficciones. Edited by Anthony Kerrigan, translated by Anthony Bonner.

  New York: Grove, 1999.

  __Labyrinths: Selected Stories and Other Writings. Edited by Donald A. Yates

  and James E. Irby. London: The Folio Society, 2007.

  __Labyrinths: Selected Stories and Other Writings. Edited by Donald A. Yates

  and James E. Irby. New York: New Directions, 1964.

  __The Library of Babel. Translated by Andrew Hurley, illustrated by Erik

  Desmazieres. Boston: David R. Godine, 2000.

  __Selected Non-Fictions. Edited by Eliot Weinberger. New York: Penguin

  Books, 1999.

  Burger, Edward, and Michael Starbird. The Heart of Mathematics. Emeryville: Key

  College Publishing, 2005. Burton, Robert. The Anatomy ofMelancholy. Introduction by Holbrook Jackson.

  New York: New York Review Books Classics, 2001. Campbell, Jeremy. Grammatical Man: Information, Entropy, Language, and Life. New

  York: Simon and Schuster, 1982. Cantor, Georg. Contributions to the Founding of the Theory of Transfinite Numbers.

  New York: Dover Publications, 1952. Capobianco, Michael. "Mathematics in the Ficciones of Jorge Luis Borges." The

  International Fiction Review 9(1982): 51—54. Christensen, James J. The Structure of an Atom. London: Wiley, 1990. Coutinho, S. C. The Mathematics of Ciphers. Natick: A. K. Peters, 1999. Delsemme, Armand. Our Cosmic Origins: From the Big Bang to the Emergence of Life

  and Intelligence. Cambridge: Cambridge University Press, 1998. Descartes, Rene. Discourse on Method and the Meditations. Translated by F. E.

  Sutcliffe. New York: Penguin Books, 1983. Eco, Umberto. "Between La Mancha and Babel." Variaciones Borges 4(1997): 51— 62.

  Escher, M. C. The Graphic Work. Translated by John Brigham. Berlin: B. Taschen, 1990.

  Ferrero, Jose Maria, and Alfredo Raul Palacios. Borges algunas veces matematiza.

  Buenos Aires: Ediciones del 80, 1986. Faucher, Kane X. "A Few Ruminations on Borges' Notions of Library and Metaphor." Variaciones Borges 12(2001): 126—37.

  Foucault, Michel. Discipline and Punish. Tr
anslated by Alan Sheridan. New York: Pantheon, 1977.

  Fraenkel, Abraham A. Abstract Set Theory. Amsterdam: North-Holland, 1953.

  Franzen, Torkel. Gödel's theorem: an incomplete guide to its uses and abuses. Wellesley: A. K. Peters, 2005.

  Gallian, Joseph. Contemporary Abstract Algebra. 5th ed. Boston: Houghton Mifflin, 2002.

  Gamow, George. One, Two, Three. . . Infinity. 1947. Reprint, New York: Viking, 1954.

  Grau, Christina. Borges y la arquitectura. Madrid: Catedra, 1989.

  Hausdorff, Felix. Grundzuge der Mengenlehre. 1914. Reprint, New York: Chelsea, 1949.

  Hayles, N. Katherine. The Cosmic Web: Scientific Field Models and Literary Strategies in the Twentieth Century. Ithaca: Cornell University Press, 1984.

  Hernandez, Juan Antonio. Biografia del infinito: la nocion de transfinitud en George Cantor y su presencia en la prosa de Jorge Luis Borges. Caracas: Comala.com, 2001.

  Hofstader, Douglas. Gödel, Escher, Bach: an Eternal Golden Braid. New York: Basic Books, 1979.

  Huntley, H. E. The Divine Proportion: A Study in Mathematical Beauty. New York: Dover, 1970.

  Kasner, Edward, and James Newman. Mathematics and the Imagination. New York: Simon and Schuster, 1940.

  Keiser, Graciela. "Modernism/Postmodernism in 'The Library of Babel': Jorge Luis Borges's Fiction as Borderland." Hispanofila 115(1995): 39—48.

  Keisler, H. Jerome. Elementary Calculus: An Infinitesimal Approach. Boston: Prindle, Weber & Schmidt, 1986.

  Kesten, Hermann. Copernicus and his World. Translated by E. B. Ashton and Norbert Guterman. New York: Roy Publishers, 1945.

  Khinchin, A. I. Mathematical Foundations of Information Theory. New York: Dover Publications, 1957.

  Kirk, G. S., J. E. Raven, and M. Schofield. The Presocratic Philosophers, A Critical History with a selection of texts. 2nd ed. Cambridge: Cambridge University Press, 1995.

  Korner, T. W. The Pleasures of Counting. Cambridge: Cambridge University Press, 2000.

  Lasswitz, Kurd. "The Universal Library." Translated by Willy Ley. In Fantasia Mathematica, edited by Clifton Fadiman. New York: Copernicus, 1997.

  Lofting, Hugh. The Story of Doctor Dolittle. 1920. Reprint, Philadelphia: Lippin-cott, 1948.

  Luminet, Jean-Pierre, Glenn D. Starkman, and Jeffrey Weeks. "Is Space Finite?" Scientific American, April 1999: 90-97.

  Maor, Eli. To Infinity and Beyond. Boston: Birkhauser, 1987.

  McKinzie, Mark, and Curtis D. Tuckey. "Higher Trigonometry, Hyperreal Numbers, and Euler's Analysis of Infinities." Mathematics Magazine 74(2001): 339— 68.

  Merrell, Floyd. Unthinking Thinking: Borges, Mathematics and the New Physics. West Lafayette: Purdue University Press, 1991.

  Miller, Lynn F., and Sally S. Swenson, Lives and Works: Talks with Women Artists. Metuchen: Scarecrow Press, 1981.

  Nagel, Ernest. Gödel's Proof. New York: New York University Press, 2001.

  Nicolas, Laurent. "Borges et l'infini." Variaciones Borges 7(1999): 88—146.

  0stergaard, Svend. The Mathematics of Meaning. Aarhus: Aarhus University Press, 1997.

  Oxford English Dictionary. 2nd ed. Oxford: Clarendon Press, 1989.

  Oxtoby, John. Measure and Category. New York: Springer-Verlag, 1971.

  Peters, John Durham. "Resemblance Made Absolutely Extact: Borges and Royce on Maps and Media." Variaciones Borges 25(2008): 1—24.

  Pierce, John R. An Introduction to Information Theory. New York: Dover Publications, 1980.

  Poincaré, Henri. The Foundations of Science. Translated by George Halstead. Lancaster: Science Press, 1946.

  Rawlins, Gregory. Moths to the Flame: The Seductions of Computer Technology. Cambridge: MIT Press, 1996.

  Robinson, Abraham. Non-standard Analysis. Rev. ed. Princeton: Princeton University Press, 1996.

  Rodriguez Monegal, Emir. Jorge Luis Borges: a Literary Biography. Dutton: New York, 1978.

  Rucker, Rudolph v. B. Geometry, Relativity, and the Fourth Dimension. New York: Dover, 1977.

  __Infinity and the Mind. Princeton: Princeton University Press, 1995.

  Russell, Bertrand. Human Knowledge: Its Scope and Limits. London: Routledge, 1948.

  __Principles of Mathematics. Cambridge: Cambridge University Press, 1903.

  Salpeter, Claudio. "La matematica biblioteca de Babel." Temakel. http://www.temakel.com/artborgesbabel.htm (accessed 26 February 2008).

  Sarlo, Beatriz. Jorge Luis Borges: A Writer on the Edge. New York: Verso, 1993.

  Shannon, Claude. "A mathematical theory of communication." Bell System Technical Journal 27(1948): 379-423, 623-56.

  Shashkin, Yu. A. Fixed Points. Providence: American Mathematical Society, 1991.

  Singh, Simon. Fermat's Enigma: the Quest to Solve the World's Greatest Mathematical Problem. New York: Walker, 1997.

  Slusser, George. "Bookscapes: Science Fiction in the Library of Babel." In: Mindscapes: The Geographies of Imagined Worlds, edited by George Slusser and Eric Rabkin. Carbondale: Southern Illinois University Press, 1989.

  Spencer-Brown, G. Probability and Scientific Inference. London: Longmans, Green and Co., 1957.

  Stewart, Ian. The Magical Maze: Seeing the World Through Mathematical Eyes. New York: John Wiley & Sons, 1998.

  __The Problems of Mathematics. New York: Oxford University Press, 1987.

  Sturrock, John. Paper Tigers: The Ideal Fictions of Jorge Luis Borges. Oxford: Clarendon Press, 1977.

  Stoicheff, Peter. "Chaos of Metafiction." In Chaos and Order: Complex Dynamics in Literature and Science, edited by N. Katherine Hayles. Chicago: University of Chicago Press, 1991.

  Tao, Terence. "What is good mathematics?" Bulletin of the American Mathematical Society 44(2007): 623-34.

  Toca, Antonio F. "The Architecture of J. L. Borges." Architecture and Urbanism 123(1984): 104-07.

  Wapner, Leonard. The Pea and the Sun: A Mathematical Paradox. Wellesley: A. K. Peters, 2005.

  Weeks, Jeffrey. The Shape of Space: How to Visualize Surfaces and Three-Dimensional Manifolds. New York: M. Dekker, 1985.

  Weyl, Herman. Symmetry. Princeton: Princeton University Press, 1952.

  Wheelock, Carter. The Mythmaker: A Study of Motif and Symbol in the Short Stories of Jorge Luis Borges. Austin: University of Texas Press, 1969.

  Williamson, Edwin. Borges, A Life. New York: Viking, 2004.

  Woodall, James. Borges: A Life. New York: Basic Books, 1996.

  Table of Contents

  Chapter 1

  Chapter 2

  Chapter 3

  Chapter 4

  Chapter 7

  Chapter 8

  Chapter 9