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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...

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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.

Gödel, Escher, Bach

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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 Hofstadter, a “Strange Loop,” a phenomenon which occurs “whenever, by moving upwards (or downwards) through the levels of some hierarchial system, we unexpectedly find ourselves right back where we started.” This paradox can be applied to the comment above about intentional “errors” in a computer program. Is a “mistake” a mistake when it is planned? Can a true/false statement be applied in this case? Are there statements of truth which cannot be proved?

Such a strange loop led Kurt Gödel to develop his “Incompleteness Theorem” in response to Principia Mathematica, proposed by Bertrand Russell and Alfred North Whitehead as a way to “prove” all true statements of number theory. Gödel’s statement, as paraphrased by Hofstadter, is “All consistent axiomatic formulations of number theory include undecidable propositions.” That is, some “truths” cannot be proved. If one proves one’s system in terms of that system, it is like pulling one’s self up by one’s own bootstraps—another strange loop.

Gödel’s statement in 1931 came just before the development of the electronic digital computer. The Londoner Charles Babbage (1792-1871) was the first person to propose an “Analytical Engine,” a complex system of interlocking geared cylinders which would store data and use it to compute and, perhaps, to make rational decisions. He died before such a machine was ever built, but his friend, Lady Ada Lovelace (Lord Byron’s daughter) was aware he was proposing a machine which approached mechanized intelligence, especially if the engine could react on the basis of something other than numbers. Babbage himself recognized such a possibility, and said such a machine would be “eating its own tail,” or, in contemporary computer terminology, such a machine acting in such a way would alter its own program. Lady Lovelace did say, however, the “Analytical Engine has no pretensions whatever to originate anything. It can do whatever we know how to order it to perform.” Today, one would not make quite such a strong statement, and it is here the author moves into the most important issue of this book, important for all persons, not just mathematicians and computer programmers: Artificial Intelligence (AI), thinking and reasoning done by machines.

The “Computer-Which-Can-Think” is the contemporary equivalent of Mary Shelley’s Frankenstein’s monster, the stereotyped man-made, mechanical humanoid. Whereas the creature Dr. Frankenstein created in his laboratory was slow and awkward and had finite powers, the monster computers from IBM or Univac or any of a half-dozen other “laboratories” deftly perform their functions in terms of seconds or microseconds, and they may be infinite in their powers once they learn to “eat their own tails.” Such self-referencing computers are the ones which show up in science fiction novels and films, such as “HAL,” the evil computer in Stanley Kubrick’s 2001: A Space Odyssey.

Beyond self-referencing, which computers already can do—for example, by comparing an accumulation of data during a program function against a predetermined end point for computation—Hofstadter muses about computers being able to “jump out of the system.” Human beings cannot “transcend” themselves; that is, they cannot step out of their own skins to achieve true objectivity. They can, however, move out of ruts in thinking and perceive new approaches or solutions to problems. This stepping out is based upon new connections of the subsystems of the brain (hardware). It still operates on basic principles which the mind (software) has implanted. By the same token, Hofstadter theorizes that a computer, which can modify its program—“but such modifiability has to be inherent in the program to start with”—still is unable to violate its own instructions. Those who would create Artificial Intelligence today must face a basic problem: how to tell an inflexible machine to be flexible.

It is self-referencing, however, which allows us as humans to carry out routine functions, to solve problems, and to be creative. As the mind self-references, so, too, would AI in a computer, but just as it is difficult to determine the locus of a nonmechanical function in the biological structure, so, too, would it be difficult to know where or how to place such functions in a program for AI. That AI will be developed, the author does not doubt.

When it comes, what kind of intelligence will it be? Will it be a “super-intelligence” which will neither be able to nor desire to communicate with humans? If it desires such communication, will we understand? Even the experts can only theorize at this point.

On this basis, Hofstadter argues that there is no such thing as “computer music.” For a computer to compose as Bach or Chopin, “it would have to have known resignation and world weariness, grief and despair, determination and victory, piety and awe.” No machine does that today. Music produced by a computer ultimately is composed by the programmer; thus, the machine is a tool for a human idea, not a creative element in the composition. “It is a simple and single-minded piece of software with no flexibility, no perspective on what it is doing, and no sense of itself.” Only with AI, with self-referencing, with emotion can a machine—not its program—compose music.

While discussions of AI, strange loops, and other terms and catch-phrases—TNT (“Typographical Number Theory”), isomorphism (“an information-preserving transformation”), contracrostipunctus (“a study in levels of meaning”), and Hofstadter’s Law (“It always takes longer than you expect, even when you take into account Hofstadter’s Law”)—make for complex reasoning and interpretations relative to mathematics, there also are a great many profound, but humorous, comments in relationship to music and art.

Hofstadter uses thirty-five drawings by M. C. Escher (1902-1972), the Dutch graphic artist who created “some of the most intellectually stimulating drawings of all time.” It is not surprising that Hofstadter would be drawn to Escher’s works which are based upon paradox, illusion, and double meanings. Thus Escher’s “Mobius” strips become visual representations of strange loops, as do his “Waterfall,” where water seems to flow uphill to power its own waterwheel, and his “Drawing Hands,” where two hands draw themselves. These are all self-referencing images.

The works of J. S. Bach, especially his “Crab Canon” which goes back into itself, are also representations of the strange loop and self-referencing. In the last measure of the “Art of the Fugue,” written just before Bach died, there is a four-note melody which transcribes in the German notation system as B-A-C-H. If that melody is augmented in a certain way it comes out as C-A-G-E, that is, John Cage, composer of modern, aleatoric (“found”) music. In one of the Dialogues, Achilles comments on this augmentation phenomenon and pursues it further, pointing out that “when you augment CAGE over again, you get BACH back, except jumbled up inside, as if BACH had an upset stomach after passing through the intermediate stage of CAGE.” To this the Tortoise answers: “That sounds like an insightful commentary on the new art form of Cage.” Cage as well as abstract, that is, non-referencial, art express little if anything to Hofstadter, who feels they “exist as pure globs of paint, or pure sounds, but in either case drained of all symbolic value.”

Gödel, Escher, Bach is a work like a Bach canon, full of symbolic values. It gains new meaning with each new experiencing of the work. It is a work which requires a return or a replaying to achieve understanding. Significantly, it ends with the word “ricercar.”

Hofstadter’s comments relative to understanding the origin of life might apply also to the reader’s ability to understand totally his book. “For the moment, we will have to content ourselves with a sense of wonder and awe, rather than an answer. And perhaps experiencing that sense of wonder and awe is more satisfying than having an answer—at least for a while.”


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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.

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