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Last Updated on May 6, 2015, by eNotes Editorial. Word Count: 1212

These wide-ranging discussions of art, music, mathematics, philosophy, technology, and language at times suggest the stream-of-consciousness technique of James Joyce. Like Ulysses (1922) and Finnegans Wake (1939), Gödel, Escher, Bach is a maze, but a maze not without a plan. On the dust jacket and again before parts 1 and...

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These wide-ranging discussions of art, music, mathematics, philosophy, technology, and language at times suggest the stream-of-consciousness technique of James Joyce. Like Ulysses (1922) and Finnegans Wake (1939), Gödel, Escher, Bach is a maze, but a maze not without a plan. On the dust jacket and again before parts 1 and 2 Hofstadter presents a figure that he carved from a single block of wood. When light shines through this figure in three directions, it casts shadows that form the letters GEB (Gödel, Escher, Bach) or EGB (eternal golden braid). Physically, Gödel, Escher, and Bach are thus joined in a single carving; linguistically, their initials stand for the interlocking strands described in the book’s subtitle. More fundamentally, Hofstadter regards his wooden figure as the representation of a deep truth: “Gödel and Escher and Bach [are] only shadows cast in different directions by some central solid essence.”

For Hofstadter, art, music, mathematics, philosophy, linguistics—indeed, all disciplines—depend on a limited number of basic principles. He thus allies himself with structuralists rather than particularists such as Noam Chomsky, who argue that a task such as language acquisition is so complex that it can have nothing to do with any other field of knowledge. Structuralists such as Jean Piaget claim that only a few logical-mathematical operations underlie all thought. At Carnegie-Mellon University, researchers in artificial intelligence have had some success with a general problem-solver program that instructs a computer to apply basic techniques, such as breaking down problems into smaller parts, to any puzzle—ranging from chess to mathematics. Further support for this structuralist view comes from scientists’ knowledge of certain neurological syndromes. A Viennese neurologist has observed that patients with a particular lesion in the left parietal lobe of the brain have trouble writing, calculating, recognizing their fingers, and distinguishing left from right. All four activities require placing elements of a set (letters, numbers, fingers) in the proper relationship with other members. Thus, it seems that a single skill forms the basis of seemingly disparate tasks.

The unifying principle that links Bach’s music, Escher’s art, and Gödel’s theory is self-reference, which Hofstadter also calls recursion, or Strange Loops; a system that contains a Strange Loop is a Tangled Hierarchy. All these phenomena involve self-reference. Thus, Bach uses the letters of his name to create a theme in the final “Contrapunctus” of The Art of the Fugue. In his Musical Offering, he creates “Canon per Tonos,” an ever-rising canon, which begins in C minor and, after six modulations, concludes in the same key but an octave higher. Escher’s Print Gallery (1956) presents a young man in a gallery looking at a picture in which a woman looks down on a gallery where a young man is looking at a print. Drawing Hands (1948) depicts a left hand drawing a right hand drawing a left hand drawing a right hand ad infinitum, clearly a Strange Loop.

Just as Bach’s music and Escher’s prints refer to themselves, so Gödel’s theorem depends on self-reference: The Principia Mathematica could generate the statement, “I am not a theorem of this system.” If the statement is not true, the system is flawed because it produces falsehoods. If the statement is true, the system is incomplete because it cannot create all true statements about natural numbers. More than two thousand years earlier, the Cretan poet Epimenides had developed a similar paradox when he said, “All Cretans are liars.” Gödel’s achievement was to illustrate that formal mathematical systems are subject to the same paradoxes as the much less rigorous world of language.

The British philosopher J. R. Lucas argued from Gödel’s Incompleteness Theorem that because machines rely on formal systems of instructions and since, as Gödel proved, all formal systems are incomplete, artificial intelligence is impossible. Hofstadter draws a different lesson from the apparent paradoxes of Bach’s music, Escher’s art, and Gödel’s theorem. Drawing Hands ceases to be a paradox if one steps outside the world of the picture to realize that an artist, not the hands in the print, drew the image. Similarly, Zeno’s paradox, which supplied the two characters of Achilles and the tortoise for Hofstadter’s dialogues, is insoluble if one treats it in the terms given. If the tortoise has a head start, Achilles will never catch up because Achilles must first halve the distance between himself and the turtle, then halve that half, and so on. If one views the problem not in terms of constantly halving halves but as one of different speeds, Achilles will catch the tortoise, as he would in reality.

Intelligence involves the ability to step back from a problem and thereby either solve it or determine that no solution exists. As an example, Hofstadter cites a computer chess program that impressed observers not because of its high level of play—in fact, it played rather poorly—but because, unlike most programs, it could recognize a hopeless position and concede. In a sense, it exhibited intelligence because it was able to abstract itself from the mechanical process of making and analyzing individual moves to see the game as a whole.

Hofstadter, therefore, believes that computers will someday exhibit intelligence, but he is cautious about attributing intelligence to computers. In Hofstadter’s opinion, a program that plays chess at the level of world masters would not exhibit intelligence, because it could not adapt as well as a human to a board with more than sixty-four squares or a rule that allowed pawns to move backward. Nor could such a program solve mathematical problems, read, write, perform the myriad other tasks that true intelligence permits, or get bored. Hofstadter would also not grant intelligence to a program that combines two pieces of music into a third. For a computer to compose, Hofstadter believes that itwould have to wander around the world on its own, fighting its way through the maze of life and feeling every moment of it. It would have to understand the joy and loneliness of a chilly night wind, the longing for a cherished hand, the inaccessibility of a distant town, the heartbreak and regeneration after a human death. It would have to have known resignation and world-weariness, grief and despair, determination and victory, piety and awe.

Still, Hofstadter thinks that intelligent computers will be developed. Although creativity and mechanization seem antithetical, Hofstadter maintains that “every creative act is mechanical” at its deepest level, because the brain consists of some 10 billion neurons that perform in only one mechanical way: They either discharge or do not discharge, just as a circuit only turns on or off. Yet these individual neurons, firing in seemingly random order, generate patterned impulses that at a much higher level produce thought, even though neither the thinker nor any observer can explain how. As Hofstadter explains this mystery, in a paraphrase of Rene Descartes: “I think; therefore I have no access to the level where I sum.” Computer hardware and software will probably evolve to a point at which flexible groupings of circuits will create what Hofstadter calls signals. These, in turn, will produce symbols and subsystems, all of which will interact with one another through Strange Loops to parallel human thought.

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