Gödel, Escher, Bach Analysis
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...
(The entire section is 1,212 words.)