Chapters V - IX
Chapter V: Recursive Structures and Processes.
The idea of recursion is presented in many different contexts: musical patterns, linguistic patterns, geometric structures, mathematical functions, physical theories, computer programs, and others.
Canon by Intervallic Augmentation
Achilles and the Tortoise try to resolve the question, "Which contains more information-a record, or the phonograph which plays it This odd question arises when the Tortoise describes a single record which, when played on a set of different phonographs, produces two quite different melodies: B-A-C-H and C-A-G-E. It turns out, however, that these melodies are "the same", in a peculiar sense.
Chapter VI: The Location of Meaning.
A broad discussion of how meaning is split among coded message, decoder, and receiver. Examples presented include strands of DNA, undeciphered inscriptions on ancient tablets, and phonograph records sailing out in space. The relationship of intelligence to "absolute" meaning is postulated.
Chromatic Fantasy, And Feud.
A short Dialogue bearing hardly any resemblance, except in title, to Bach's Chromatic Fantasy and Fugue. It concerns the proper way to manipulate sentences so as to preserve truth-and in particular the question of whether there exist rules for the usage of the word "arid". This Dialogue has much in common with the Dialogue by Lewis Carroll.
Chapter VII: The Propositional Calculus.
It is suggested how words such as .,and" can be governed by formal rules. Once again, the ideas of isomorphism and automatic acquisition of meaning by symbols in such a system are brought up. All the examples in this Chapter, incidentally, are "Zentences"-sentences taken from Zen koans. This is purposefully done, somewhat tongue-in-cheek, since Zen koans are deliberately illogical stories.
Crab Canon.
A Dialogue based on a piece by the same name from the Musical Offering. Both are so named because crabs (supposedly) walk backwards. The Crab makes his first appearance in this Dialogue. It is perhaps the densest Dialogue in the book in terms of formal trickery and level- play. Gödel, Escher, and Bach are deeply intertwined in this very short Dialogue.
Chapter VIII: Typographical Number Theory.
An extension of the Propositional Calculus called "TNT" is presented. In TNT, number-theoretical reasoning can be done by rigid symbol manipulation. Differences between formal reasoning and human thought are considered.
A Mu Offering.
This Dialogue foreshadows several new topics in the book. Ostensibly concerned with Zen Buddhism and koans, it is actually a thinly veiled discussion of theoremhood and nontheoremhood, truth and falsity, of strings in number theory. There are fleeting references to molecular biology-particular) the Genetic Code. There is no close affinity to the Musical Offering, other than in the title and the playing of self-referential games.
Chapter IX: Mumon and Gödel.
An attempt is made to talk about the strange ideas of Zen Buddhism. The Zen monk Mumon, who gave well known commentaries on many koans, is a central figure. In a way, Zen ideas bear a metaphorical resemblance to some contemporary ideas in the philosophy of mathematics. After this "Zennery", Gödel’s fundamental idea of Gödel- numbering is introduced, and a first pass through Gödel’s Theorem is made.