Part II: EGB
Prelude & Chapters X - XV
Prelude…
This Dialogue attaches to the next one. They are based on preludes and fugues from Bach's Well-Tempered Clavier. Achilles and the Tortoise bring a present to the Crab, who has a guest: the Anteater. The present turns out to be a recording of the W.T.C.; it is immediately put on. As they listen to a prelude, they discuss the structure of preludes and fugues, which leads Achilles to ask how to hear a fugue: as a whole, or as a sum of parts? This is the debate between holism and reductionism, which is soon taken up in the Ant Fugue.
Chapter X: Levels of Description, and Computer Systems.
Various levels of seeing pictures, chessboards, and computer systems are discussed. The last of these is then examined in detail. This involves describing machine languages, assembly languages, compiler languages, operating systems, and so forth. Then the discussion turns to composite systems of other types, such as sports teams, nuclei, atoms, the weather, and so forth. The question arises as to how man intermediate levels exist-or indeed whether any exist.
...Ant Fugue.
An imitation of a musical fugue: each voice enters with the same statement. The theme-holism versus reductionism-is introduced in a recursive picture composed of words composed of smaller words. etc. The words which appear on the four levels of this strange picture are "HOLISM", "REDLCTIONIsM", and "ML". The discussion veers off to a friend of the Anteater's Aunt Hillary, a conscious ant colony. The various levels of her thought processes are the topic of discussion. Many fugal tricks are ensconced in the Dialogue. As a hint to the reader, references are made to parallel tricks occurring in the fugue on the record to which the foursome is listening. At the end of the Ant Fugue, themes from the Prelude return. transformed considerably.
Chapter XI: Brains and Thoughts.
"How can thoughts he supported by the hardware of the brain is the topic of the Chapter. An overview of the large scale and small-scale structure of the brain is first given. Then the relation between concepts and neural activity is speculatively discussed in some detail.
English French German Suite.
An interlude consisting of Lewis Carroll's nonsense poem "Jabberwocky`' together with two translations: one into French and one into German, both done last century.
Chapter XII: Minds and Thoughts.
The preceding poems bring up in a forceful way the question of whether languages, or indeed minds, can be "mapped" onto each other. How is communication possible between two separate physical brains: What do all human brains have in common? A geographical analogy is used to suggest an answer. The question arises, "Can a brain be understood, in some objective sense, by an outsider?"
Aria with Diverse Variations.
A Dialogue whose form is based on Bach's Goldberg Variations, and whose content is related to number-theoretical problems such as the Goldbach conjecture. This hybrid has as its main purpose to show how number theory's subtlety stems from the fact that there are many diverse variations on the theme of searching through an infinite space. Some of them lead to infinite searches, some of them lead to finite searches, while some others hover in between.
Chapter XIII: BlooP and FlooP and GlooP.
These are the names of three computer languages. BlooP programs can carry out only predictably finite searches, while FlooP programs can carry out unpredictable or even infinite searches. The purpose of this Chapter is to give an intuition for the notions of primitive recursive and general recursive functions in number theory, for they are essential in Gödel’s proof.
Air on G's String.
A Dialogue in which Gödel’s self-referential construction is mirrored in words. The idea is due to W. V. O. Quine. This Dialogue serves as a prototype for the next Chapter.
Chapter XIV: On Formally Undecidable Propositions of TNT and Related Systems.
This Chapter's title is an adaptation of the title of Gödel’s 1931 article, in which his Incompleteness Theorem was first published. The two major parts of Gödel’s proof are gone through carefully. It is shown how the assumption of consistency of TNT forces one to conclude that TNT (or any similar system) is incomplete. Relations to Euclidean and non-Euclidean geometry are discussed. Implications for the philosophy of mathematics are gone into with some care.
Birthday Cantatatata...
In which Achilles cannot convince the wily and skeptical Tortoise that today is his (Achilles') birthday. His repeated but unsuccessful tries to do so foreshadow the repeatability of the Gödel argument.
Chapter XV: Jumping out of the System.
The repeatability of Gödel’s argument is shown, with the implication that TNT is not only incomplete, but "essentially incomplete The fairly notorious argument by J. R. Lucas, to the effect that Gödel’s Theorem demonstrates that human thought cannot in any sense be "mechanical", is analyzed and found to be wanting.
Edifying Thoughts of a Tobacco Smoker.
A Dialogue treating of many topics, with the thrust being problems connected with self-replication and self-reference. Television cameras filming television screens, and viruses and other subcellular entities which assemble themselves, are among the examples used. The title comes from a poem by J. S. Bach himself, which enters in a peculiar way.