Review: “Gödel, Escher, Bach: An Eternal Golden Braid”
“Gödel, Escher, Bach: An Eternal Golden Braid” by Douglas R. Hofstadter is a magnificent, utterly absorbing book. Somehow, Mr. Hofstadter blends Zen koans, Bach canons, Achilles and the Tortoise, Escher mosaics, and Gödel into a highly enjoyable read.
The book is not a light summer read; you must be ready to think. Not the kind of thinking necessary to prove Pythagoras’ theorem, but the kind to appreciate puns and double- and hidden meanings; word puzzles and mosaics; and lexical canons. When you are done with this book, I guarantee that you’ll have enough fascinatingly troubling thoughts running through your brain to spark innumerable lively discussions with friends.
I first saw the 777-page book in 1991 at the University of Massachusetts Amherst. I was acquainted with Bach and Escher but not Gödel, thus intrigued. The book’s structure drew me in further as it wove dialogues and chapters, each mirroring the other.
Kurt Gödel is most famous for his Incompleteness Theorem. This was a shocking blow to mathematicians in 1931. By looping number theory into talking about itself, he proved that any system that contains natural number arithmetic cannot be both consistent and complete. A consistent formal system has no contradictions. A complete formal system can distinguish all true and false statements allowed in the system. Mr. Gödel didn’t just show that our version of number theory is flawed but that any sophisticated logic system must be.
The book’s cover is what caught my attention. It depicts a block of wood carved so that if lights are shined on it along the three main axes, the shadows show G, E, and B. This merging presages the author’s style. Within the first 19 pages, we travel a Strange Loop through Bach’s canons, to Escher’s woodcuts, to Gödel’s self-referencing mathematics, and back to Bach.
Maurits Cornelis Escher’s name brings to mind depictions of loops and transformations, such as “Waterfall” (lithograph, 1961), “Hand with Reflecting Sphere” (lithograph, 1935), and “Swans” (wood engraving, 1956). A graphic artist by trade, his ability to display geometric principles through seeming paradoxes and intricate designs brought him worldwide fame.
The chapter-beginning dialogues are usually witty, tiny plays. Mr. Hofstadter’s main characters are Achilles and the Tortoise, originally introduced in Zeno’s paradox, and much later parodied by Lewis Carroll. The dialogues reflect the upcoming chapter material. A dialogue might be a canon. The characters might discuss logic. The connection might be subtle. They’re always entertaining.
Johann Sebastian Bach was an 18th-century musical genius. According to Mr. Hofstadter, Bach had an uncanny ability to improvise organ music, creating complex fugues and canons at command. A canon is music with two or more harmonizing parts. These parts are transformations of a theme. The simplest type of canon is a round as in “Row, Row, Row Your Boat.” The theme can also be transformed in pitch, speed, direction, or any combination of these and more. A canon can loop endlessly.
One of Mr. Hofstadter’s main theses is reductionism of consciousness. As he writes, “every aspect of thinking [is] a high-level description…governed by simple, even formal, rules.” (p. 559) There is a charged debate among those studying consciousness. Can it be best explained by studying its parts, i.e., neurons, or by understanding how real-world objects and concepts are mapped within the brain? Mr. Hofstadter comes down on the neuron side. But, he tinges this with an acceptance of mappings and meanings as required to understand a purely reductionist explanation. High level clarifies the low and the low explains the high.
Strange Loops permeate the book. Escher makes ants wind their way through a Möbius strip in “Möbius Strip II” (woodcut, 1963). Gödel describes number systems that can talk about themselves. Bach weaves melodies through canons. A Strange Loop occurs when you move through a hierarchy and end up where you began. Mr. Hofstadter uses Bach’s “Canon per Tonos” as a Strange Loop example. This canon begins in C minor, modulates through the entire chromatic scale, and ends in C minor, without most listeners noticing.
Mr. Hofstadter has created a book of endless fascination. It requires and you’ll gladly give it multiple readings. Without the readers noticing, Mr. Hofstadter educates us about how to question consciousness.