Form and Content

(Literary Essentials: Nonfiction Masterpieces)

Early in the twentieth century, Alfred North Whitehead and Bertrand Russell set about devising a mathematical system that would be consistent and complete, one that could generate every true statement about number theory without producing any false ones. The result was their monumental Principia Mathematica (1910-1913). In 1931, the twenty-five-year-old Czech mathematician Kurt Gödel undercut this massive work with a short paper demonstrating that while for practical purposes the Whitehead-Russell system achieved its goal, in fact certain true propositions in number theory remained undecidable in their scheme. Moreover, Gödel showed that no formal system could be consistent and complete; any formulation powerful enough to produce almost all truths about natural numbers (integers greater than zero) would necessarily be flawed. Douglas Hofstadter draws an analogy to illustrate this point. If a certain record player reproduced sound with sufficiently high fidelity, one could create a record capable of producing resonances within the machine that could destroy it. Thus, no phonograph will be able to play every record.

Gödel’s so-called Incompleteness Theorem deeply disturbed mathematicians seeking a perfectly logical, ordered universe. It was, in fact, the equivalent for mathematics of Werner Heisenberg’s uncertainty principle, Albert Einstein’s general theory of relativity, and Max Planck’s discovery of quantum mechanics. All revealed the mythical nature of the traditional view of science as fixed, orderly, and rational.

Hofstadter initially intended to write a short book about Gödel’s theorem, similar in content and brevity to Ernest Nagel and James R. Newman’s Gödel’s Proof (1958); in his bibliography, Hofstadter credits this work as the inspiration for...

(The entire section is 741 words.)

Gödel, Escher, Bach

(Literary Masterpieces, Volume 16)

Perhaps more frequently than he realizes (although that is difficult to accept), Douglas Hofstadter uses the verbs “evoke” and “provoke,” or their adjectival forms, in his complex weaving of strands of mathematics, music, art and philosophy into an interdisciplinary “golden braid.” Those two verbs summarize his attempt to evoke correspondences between and among formal systems (starting with Gödel’s Theorem), DNA, the brain (as hardware), the mind (as software), Bach canons and fugues, Escher prints, Artificial Intelligence, and computers. To provoke such connections, the author has developed a format which is almost as unique as the ideas he pursues.

In his Introduction, Hofstadter defines ricercar (an Italian word originally meaning “to seek”) as a designation, in Bach’s time, of “an erudite kind of fugue, perhaps too austerely intellectual for the common ear.” Gödel, Escher, Bach, too, in format is “a kind of fugue,” but Hofstadter’s goal is to present the information in such a way that it will not be “too austere.” Whether he succeeds or not, of course, will depend upon the individual reader and the amount of time and thought the reader wishes to give to this large volume, but the author deserves praise for his innovative manipulation of words and graphics, even if the words become almost too cute and too sprinkled with puns after a while. For example, a section which concerns a computer language, SHRDLU, is entitled “SHRDLU, Toy of Man’s Designing,” an almost painful pun upon Bach’s “Jesu, Joy of Man’s Desiring.”

Yet whatever cuteness is here is controlled, is intended. The author does not play with words merely because he does not know how to make his way out of a verbal or semantic blind alley, as is the case with many punsters. Rather, he plays with words because he is an assistant professor of computer science and a mathematician who, not incidentally, set the type for the book himself—on a computer, of course. Thus, as indicated above, it is difficult to accept that he would not know how many times certain words are used in his book; he may well have a printout of the frequency of occurrence of every word in the 777 pages. Thus all words are controlled by the author, controlled in a double sense of creation and of technology. On the other hand, he may not have such a printout. The methodical, boring counting of words, a purely mechanical process which a computer does so much better than humans with limited, wandering attention spans, may be too rudimentary a program for Hofstadter to bother writing. (The occasional old-fashioned, reactionary humanist who happens to read this book, then, may even take a kind of perverse pleasure in the occasional apparent typographical error in such a controlled manuscript—until he encounters Hofstadter’s offhand comment that computers can be programmed to make seemingly random errors.)

To make complex intellectual concepts more easily understood, the author precedes each chapter with a “Dialogue” involving Achilles and a Tortoise—an idea taken from Zeno via Lewis Carroll—along with occasional visits from a Crab, a Sloth, and, in the final Dialogue, the Author himself. These Dialogues present, metaphorically, the ideas to be discussed later. Whether the reader can follow the more abstract, formal presentations of number theory and formal systems of logic and the structure of DNA or not, he usually can grasp the author’s intentions from the Dialogues.

Another example, which may be nothing more than verbal cleverness or may be brilliance, comes in a discussion of “recursion,” in other words, “nesting and variations on nesting,” a variation of the process in computer terminology of the “pushdown stack,” here called “push, pop, and stack.” When a machine or the brain “pushes,” it suspends operations on a task, without forgetting the place in the operation where the functioning stopped, in order to take on a new task. The first task is “stacked,” that is, stored away temporarily. When the machine “pops,” it returns to the first task, using the “return address” established for it in the stack. An image the author does not use but which visualizes the process is of several airplanes “stacked” over an airport awaiting instructions to land, to complete the suspended operation. We all operate with this “push, pop, and stack” process in conversations as we interject asides and parenthetical comments. The brilliance—or cleverness—in the present book comes as Hofstadter discusses the process in several long, involved sentences full of “stacked” ideas.

Apart from the entertaining side excursions, there are, perhaps, two main ideas which permeate the book: the Epimenides paradox and the potential abilities of Artificial Intelligence.

The Cretan philosopher Epimenides once said “All Cretans are liars.” At a different level, this becomes “I am lying” which becomes “This statement is false.” If it is a false statement, then it is true; if it is a true statement, however, how can it be false? This Epimenides paradox “rudely violates the usually assumed dichotomy of statements into true and false.” Such a paradox becomes, for...

(The entire section is 2156 words.)


(Literary Essentials: Nonfiction Masterpieces)

Gardner, Howard. “Strange Loops of the Mind,” in Psychology Today. XIII (March, 1980), pp. 72-85.

Gardner, Martin. “Mathematical Games: Douglas R. Hofstadter’s Gödel, Escher, Bach,” in Scientific American. CCXLI (July, 1979), pp. 16-24.

Gleick, J. “Exploring the Labyrinth of the Mind,” in The New York Times Magazine. August 21, 1983, pp. 23-27.

Kendrick, Walker. “The Ulysses of Soft Science,” in The Village Voice. XXIV (November 19, 1979), pp. 48-52.

Levin, Michael. “Thinking About the Self,” in Commentary. LXXIV (September, 1982), pp. 55-57.

Mattingly, Ignatius G. “Epimenides at the Computer,” in The Yale Review. LXIX (Winter, 1980), pp. 270-276.