G?del, Escher, Bach: An Eternal Golden Braid by Douglas Hofstadter is an intricate and complex study of Strange Loops, Meta-Concepts in general and their connection to self-awareness and intelligence. Each chapter is separated from the next by a short fictional piece in the style of Lewis Carroll, which exemplifies the theme of the following chapter. The drawings of M.C. Escher and the Surrealist paintings of René Magritte are used as illustrations.
Mathematical, logical and computer-science concepts discussed in this book include:
- G?del's incompleteness theorem
- The Epimenides paradox
- Zeno's paradoxes
- Euclid's Theorem
- Non-Euclidean geometry
- Recursion
- Stacks
- Propositional Calculus
- Fermat's Last Theorem
- Quines
- The Church-Turing Thesis
- Cantor's diagonal method
- Goldbach's conjecture
The Dialogues contain examples of:
- All There in the Index: For example, to find where the true end of Aria with Diverse Variations is, search for 'typos'.
- Author Avatar: In the final dialogue, the characters figure out that they're fictional characters in a dialogue, and the author, Douglas Hofstadter, himself appears in the dialogue and talks about the book he's writing — the very same book the dialogue is included in.
- Breather Episode: The interlude "English French German Suite", consisting of Lewis Carroll's poem "Jabberwocky" in three languages, taking place between two chapters of theories on how the human mind works.
- A Dog Named "Dog": The book features characters named Tortoise, Crab, and Anteater who are talking versions of the animal of their name.
- Doorstopper: This book has well over 700 pages.
- Foreshadowing: Looking at the semantic network on page 370 reveals a lot about the latter half of the book.
- Fun with Acronyms:
- The chapter "Contracrostipunctus", which contains an acrostic which itself contains an acrostic.
- RICERCAR and CEGABBAB ("Babbage, C(harles)" backwards) show up in the last chapter.
- Gold–Silver–Copper Standard: ... Well, okay, not really. But the words "Copper", "silver", and "gold" appear (in that order) in several places.
- Harassing Phone Call: Achilles begins the dialogue "Air on G's String" by telling the Tortoise about an obscene phone call he received in which the caller merely shouted twice before hanging up, "Yields falsehood when preceded by its quotation!" When the Tortoise teaches him about quines, he realizes that the caller was stating a logical paradox.
- Hidden in Plain Sight: The answers to the puzzles in "Sonata for An Unaccompanied Achilles." Not only is the answer explicitly used in the text, the two puzzles make up the entire word.
- In Name Only: By the author's own admission (in the Overview), the dialogue "Chromatic Fantasy, And Feud" bears "hardly any resemblance, except in title, to Bach's Chromatic Fantasy and Fugue."
- Leaning on the Fourth Wall: In the final dialogue "Six-part Ricercar" the characters suspect their world could be fictional in a dialogue before they are absolutely sure of that.
- Metafiction: Some stories are so extremely meta that they have no content other than discussions of themselves discussing themselves discussing themselves ad infinitum, usually indirectly.
- Mind Screw: Many of the Dialogues and accompanying illustrations do this in one way or another. Indeed, the entire book is deliberately written to screw with the reader's perception of reality, using a blend of mathematics and Zen philosophy.
- Musical Theme Naming: Every dialogue is named after a piece by Johann Sebastian Bach, or a pun on one.
- Nested Story: In "Little Harmonic Labyrinth", Achilles and the Tortoise read a story that stars themselves, "Djinn and Tonic", in which they enter an Escher painting and begin reading a different story from the same storybook. The dialog ends after "Djinn and Tonic" and we never see what happens to the Achilles and Tortoise who were reading it.
- Nested Story Reveal: The Subjunc-TV dialogue is All Just A Hypothetical Situation.
- Our Genies Are Different:
- In the dialogue "Little Harmonic Labyrinth," Genies are allowed to grant wishes, but not wishes about wishes, which are known as meta-wishes. Meta-Genies (who come from Meta-Lamps) are allowed to grant meta-wishes, but not wishes about meta-wishes, which are within the authority of Meta-Meta-Genies. The word "Djinn" is generically used to designate Genies, Meta-Genies, Meta-Meta-Genies, and all others in GOD (which stands for "GOD Over Djinn").
- In the chapter "Typographical Number Theory," "djinn" is an undefined term used in place of "natural number" in setting out the five Peano postulates, with "genie" taking the place of zero.
- Portal Picture: In "Little Harmonic Labyrinth," the lower levels of stories are entered by entering pictures by drinking pushing potions.
- Public Domain Character: Achilles and the Tortoise, from the Greek philosopher Zeno (who also himself appears).
- Punny Name: The coppers Silva and Gould, as well as the ant collective Aunt Hillary.
- Recursive Acronym: "GOD" in "Little Harmonic Labyrinth" (short for "GOD Over Djinn").
- Running Gag:
- This dialogue, repeated, at least, some variation of, thrice; on pages 62, 200, and 724.
Achilles: Say, don't you play the guitar?
Tortoise: Fiddle. It makes a big difference, you know.
Achilles: Oh, well, it's all the same to me.- Also, in page 731 each of the characters (Babbage, Crab, Tortoise, Achilles, and the Author respectively) say "The grounds are excellent!" at the beginning of each line followed by Achilles saying "Yes, I know. One even might say the grounds were excellent."
- Schr?dinger's Butterfly: The book uses several of these, nesting several layers of drama. In the dialogue "Little Harmonic Labyrinth," Achilles and the Tortoise are on an airship and start reading a book about themselves. The bad news is that the story doesn't "pop back" all the way to the last level, and the initial story is still left hanging. The good news is that the Tortoise and Achilles can move up to a previous level using popcorn.
- She's a Man in Japan: The Tortoise is referred to as male in the original, but the French word for tortoise is the feminine tortue. Hofstadter, who is interested in the phenomenon of unconscious sexism in language, was delighted when this was pointed out, and gave the French translators the go-ahead to make the Tortoise a female character.
- Self-Deprecation: A book about "metal-logic", called Copper, Silver, Gold: an Indestructible Metallic Alloy is mentioned in the dialogues. The Crab says that it's "filled with strange Dialogues about many subjects, including molecular biology, fugues, Zen Buddhism, and heaven knows what else." Achilles responds that "probably some crackpot wrote it". The book is also listed in the bibliography, where it's called "a formidable hodge-podge, turgid and confused". Indeed very little praise is given to its author, Egbert B. Gebstadter.
- Self-Fulfilling Prophecy: Commented on if a psychic who could determine the minds of other people was allowed to choose his or her jury if accused.
- Shaped Like Itself: "Church-Turing Thesis, Tautological Version: Mathematics problems can be solved only by doing mathematics."
- Sick Episode: "Sonata for Unaccompanied Achilles," a dialogue for one in which Achilles is telephoned by the Tortoise, who is apparently suffering from a headache. (The readers only get Achilles' side of the conversation.)
- Significant Anagram: Terry Winograd, author of SHRDLU, becomes "Dr. Tony Earrwig" in the dialogue "SHRDLU, Toy of Man's Designing."
- Subverted Rhyme Every Occasion: In one of the dialogues, the Crab puts on a record of himself singing "A Song Without Time or Season." Here's how it goes:A turner of phrases quite pleasin',
Had a penchant for trick'ry and teasin'.
In his songs, the last line
Might seem sans design;
What I mean is, without why or wherefore.- This is lampshaded almost immediately:Achilles: Lovely! Only I'm puzzled by one thing. It seems to me that in your song, the last line is—
Crab: Sans design?
Achilles: No ... What I mean is, without rhyme or reason.
- This is lampshaded almost immediately:
- Textbook Humor: Well, the text wasn't all that serious to begin with, but you have to wonder when Hofstadter describes the DNA of a feline as CATCATCATCATCAT...
- To Be Continued: The two Dialogues Prelude... and ... Ant Fugue are, well, two sections of a separated story. The end of the former ends with TTortoise, while the latter begins with Achilles and CCrab, using "ATTACCA" note as a guide.
- The Treachery of Images: Subverted—one character takes the pipe out of the Magritte painting and smokes it.
- Turing Test: Parodied in the final chapter, where Alan Turing and Charles Babbage each try to prove that the other is a computer invented by them. Turing insists on calling it a "Babbage test."
- Viewers Are Geniuses: The dialogues, especially, conceal mathematical jokes that aren't explained in the text. For example, in a spoof of Fermat's Last Theorem, the Tortoise claims to have found a counterexample in which the exponent n is "the only positive integer which does not occur anywhere in the continued fraction for pi". Mathematically inclined readers will know that, while the famous constants e and sqrt(2) have highly patterned continued fractions, that for pi is quasi-random, so the Tortoise's claim is of no help at all in finding the value of n.
- Your Princess Is in Another Castle!: A unique version of a fake ending which takes place after the actual ending.
