Gödel, Escher, Bach by Douglas R. Hofstadter

First published: 1979

Type of work: Philosophy/science

Form and Content

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 his own. Yet he instead produced a 777-page tome ranging in subject matter from artificial intelligence to Zen Buddhism, from the baroque fugues of Johann Sebastian Bach to the twentieth century prints of Maurits Cornelis Escher. The scope of the contents is suggested by the appearance of reviews in journals as diverse as the James Joyce Quarterly, Journal of Symbolic Logic, Ethics, Review of Metaphysics, American Journal of Psychiatry, Byte, and Perspectives on New Music. Hofstadter claimed that the book expressed his religion; it certainly seems to encapsulate his extensive education.

A work of half a million words treating such esoteric matters as the propositional calculus, typographical number theory, Zen koans, and operator languages might easily overwhelm the typical reader. Gödel, Escher, Bach nevertheless proved both a critical and popular success. Nominated for the National Book Critics Circle Award in 1979, it was awarded the Pulitzer Prize in general nonfiction and the American Book Award for 1980. After its release in paperback, it remained on the best-seller list for five months. In large part, such wide appeal is attributable to Hofstadter’s style and presentation. A teacher at Indiana University and the University of Michigan, he leads his audience gradually but systematically from simple but fundamental concepts to difficult theories. Along the way he offers pleasant but instructive digressions and challenging problems, allowing one to pause and reflect during his journey through the development of modern technology.

A key device in Hofstadter’s presentation is the use of preludes. The book itself is divided into two parts, with the first nine chapters serving as a prelude to the fugue of the longer, more intricate second section. Before each of the twenty chapters, Hofstadter also places a prelude that presents new concepts metaphorically, through imaginative dialogues and images, before he discusses them more rigorously and abstractly. These preludes derive from Bach’s music, as titles such as “Little Harmonic Labyrinth” and “SHRDLU, Toy of Man’s Designing” suggest. They are inventive, entertaining, and complex. “Air on G’s String” suggests not only Bach’s composition but also the strings of numbers that Gödel (“G”) created for his mathematical notation, and the succeeding chapter deals extensively with strings in typographical number theory. “The Crab Canon,” another prelude that takes its title from Bach, reads almost the same backward and forward, as do strands of crab DNA, in which adenine (A) and thymine (T) are paired. “A” and “T” also represent the two speakers in Zeno’s dialogue Achilles and the tortoise, so that the nucleotide sequence of the crab’s DNA provides an outline for the conversation in “The Crab Canon.”

Even if one is overwhelmed by the arguments, these linguistic tricks, as well as numerous biographical anecdotes, can be fascinating. One can also savor the tricks in perspective exhibited by the Escher prints distributed throughout.

Critical Context

Gödel, Escher, Bach provokes thought about thought. In this sense, as in its very structure, the book is recursive. When Hofstadter makes the following comment about Bach’s Musical Offering, he is speaking about his own work as well:

One cannot look deeply enough into the Musical Offering. There is always more after one thinks one knows everything. . . . Things are going on on many levels. . . . There are tricks with notes and letters. . . . There is beauty and extreme depth of emotion; even an exultation in the many-leveledness of the work comes through.

Clearly, the Pulitzer Prize committee and hundreds of thousands of readers who bought Gödel, Escher, Bach exulted in its many-leveledness, too, but the book has provoked controversy as well as admiration. Some reviewers complained about its length, its quirky style, and its scope. Other popularizers of scientific ideas— Stephen Jay Gould, Loren Eiseley, Lewis Thomas, Carl Sagan—may reveal less literary inventiveness but do not provoke a paraphrase of Gertrude’s plea to Polonius: “Less matter, with less art.” Hofstadter, certain critics claim, may bemuse rather than amuse his audience.

Some musical theorists have objected that Bach seems dragged into the book because Hofstadter wanted to discuss the composer, not because his music is truly recursive. The “Canon per Tonos” is clever, but it does not rely on paradox in the way that Gödel’s theorem or Escher’s prints do. Researchers in artificial intelligence are also divided on the merits of Hofstadter’s theories; some experts in this field refute his definition of machine intelligence as well as his proposed method of achieving that goal. Whereas Hofstadter would build his programs from the bottom up and let the computer combine elements freely, others believe that a program must contain all instructions, including the information about when to make a particular choice.

Underlying Hofstadter’s view of intelligence, both human and artificial, is the assumption that the brain is hardware, the mind software. Neurophysiology, biology in general, and past experiences of the individual and humankind are irrelevant to an understanding of creativity. This same philosophy informs The Mind’s I: Fantasies and Reflections of Self and Soul (1981), which Hofstadter edited with the philosopher Daniel C. Dennett.

The Mind’s I includes an article by John Searle, professor of philosophy at the University of California, Berkeley. Searle argues that even if one could program a computer to say, “I’m thirsty,” no one would think that the machine wants a drink. Similarly, when a computer solves a mathematical problem, it has no understanding of the numbers. Mathematician and computer pioneer Alan Turing established a test for intelligence: If a machine can convince a person that another human is responding to questions, then that machine is intelligent. Searle rejects that measure, arguing that sophisticated circuitry can now fool many people even though its creators would not claim intelligence for it. Searle concludes that Hofstadter and his followers mistake simulation for duplication, the ability to produce an answer with the capacity for understanding it.

In 1981, Hofstadter succeeded Martin Gardner as the mathematics columnist for Scientific American. In 1985, he collected the essays which had appeared in his column, together with seven new ones, in Metamagical Themas. The same wit, range of interest (from linguistic paradoxes to nuclear war), and concern about the mind apparent in Hofstadter’s earlier writings are evident here, but his optimism about reproducing human intelligence is tempered. Indeed, he seems to lean toward J. R. Lucas’ view that Gödel’s theorem limits the capacity of any program to reproduce human thought—though Hofstadter suggests that the theorem may also imply restrictions on the extent of human creativity. Hofstadter’s reservations about machine intelligence may derive in part from his own unsuccessful efforts to create programs that solve problems intuitively rather than mechanically. His writings show how much researchers have learned about the intricacies of the brain and mind, about Strange Loops and Tangled Hierarchies, but they also reveal that no one has yet unraveled what John Donne called “the subtle knot that makes us man.”

Bibliography

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.