Gödel, Escher, Bach: An Eternal Golden Braid (2 page)

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.

Chapter XVI: Self-Ref and Self-Rep
. This Chapter is about the connection between self-reference in its various guises, and self-reproducing entities e.g., computer programs or DNA molecules).

The relations between a self-reproducing entity and the mechanisms external to it which aid it in reproducing itself (e.g., a computer or proteins) are discussed-particularly the fuzziness of the distinction. How information travels between various levels of such systems is the central topic of this Chapter.

The Magnificrab, Indeed.
The title is a pun on Bach's Magnifacat in D. The tale is about the Crab, who gives the appearance of having a magical power of distinguishing between true and false statements of number theory by reading them as musical pieces, playing them on his flute, and determining whether they are "beautiful" or not.

Chapter XVII: Church, Turing, Tarski, and Others
. The fictional Crab of the preceding Dialogue is replaced by various real people with amazing mathematical abilities. The Church-Turing Thesis, which relates mental activity to computation, is presented in several versions of differing strengths. All are analyzed, particularly in terms of their implications for simulating human thought mechanically, or programming into a machine an ability to sense or create beauty. The connection between brain activity and computation brings up some other topics: the halting problem of Turing, and Tarski's Truth Theorem.

SHRDLU, Toy of Man's Designing
. This Dialogue is lifted out of an article by Terry Winograd on his program
SHRDLU
: only a few names have been changed. In it. a program communicates with a person about the so-called "blocks world" in rather impressive English. The computer program appears to exhibit some real understanding-in its limited world. The Dialogue's title is based on
Jesu, joy of Mans Desiring
, one movement of Bach's Cantata 147.

Chapter XVIII: Artificial Intelligence: Retrospects
, This Chapter opens with a discussion of the famous "Turing test"-a proposal by the computer pioneer Alan Turing for a way to detect the presence or absence of "thought" in a machine. From there, we go on to an abridged history of Artificial Intelligence. This covers programs that can-to some degree-play games, prove theorems, solve problems, compose music, do mathematics, and use "natural language" (e.g., English).

Contrafactus
. About how we unconsciously organize our thoughts so that we can imagine hypothetical variants on the real world all the time. Also about aberrant variants of this ability-such as possessed by the new character, the Sloth, an avid lover of French fries, and rabid hater of counterfactuals.

Chapter XIX: Artificial Intelligence: Prospects
. The preceding Dialogue triggers a discussion of how knowledge is represented in layers of contexts. This leads to the modern Al idea of "frames".

A frame-like way of handling a set of visual pattern puzzles is presented, for the purpose of concreteness. Then the deep issue of the interaction of concepts in general is discussed, which leads into some speculations on creativity. The Chapter concludes with a set of personal

"Questions and Speculations" on Al and minds in general.

Sloth Canon
. A canon which imitates a Bach canon in which one voice plays the same melody as another, only upside down and twice as slowly, while a third voice is free. Here, the Sloth utters the same lines as the Tortoise does, only negated (in a liberal sense of the term) and twice as slowly, while Achilles is free.

Chapter XX: Strange Loops, Or Tangled Hierarchies
. A grand windup of many of the ideas about hierarchical systems and self-reference. It is concerned with the snarls which arise when systems turn back on themselves-for example, science probing science, government investigating governmental wrongdoing, art violating the rules of art, and finally, humans thinking about their own brains and minds. Does Gödel’s Theorem have anything to say about this last "snarl"? Are free will and the sensation of consciousness connected to Gödel’s Theorem? The Chapter ends by tying Gödel, Escher, and Bach together once again.

Six-Part Ricercar.
This Dialogue is an exuberant game played with many of the ideas which have permeated the book. It is a reenactment of the story of the Musical Offering, which began the book; it is simultaneously a "translation" into words of the most complex piece in the Musical Offering: the Six-Part Ricercar. This duality imbues the Dialogue with more levels of meaning than any other in the book. Frederick the Great is replaced by the Crab, pianos by computers, and so on. Many surprises arise. The Dialogue's content concerns problems of mind, consciousness, free will, Artificial Intelligence, the Turing test, and so forth, which have been introduced earlier.

It concludes with an implicit reference to the beginning of the book, thus making the book into one big self-referential loop, symbolizing at once Bach's music, Escher's drawings, and Gödel’s Theorem.

FIGURE 1. Johann Sebastian Bach, in 1748. From a painting by Elias Gottlieb Hanssmann.

Introduction:

A Musico-Logical Offering

Author:

FREDERICK THE GREAT, King of Prussia, came to power in 1740. Although he is remembered in history books mostly for his military astuteness, he was also devoted to the life of the mind and the spirit. His court in Potsdam was one of the great centers of intellectual activity in Europe in the eighteenth century. The celebrated mathematician Leonhard Euler spent twenty-five years there. Many other mathematicians and scientists came, as well as philosophers-including Voltaire and La Mettrie, who wrote some of their most influential works while there.

But music was Frederick's real love. He was an avid flutist and composer. Some of his compositions are occasionally performed even to this day. Frederick was one of the first patrons of the arts to recognize the virtues of the newly developed "piano-forte" ("soft-loud"). The piano had been developed in the first half of the eighteenth century as a modification of the harpsichord. The problem with the harpsichord was that pieces could only be played at a rather uniform loudness-there was no way to strike one note more loudly than its neighbors. The "soft-loud", as its name implies, provided a remedy to this problem. From Italy, where Bartolommeo Cristofori had made the first one, the soft-loud idea had spread widely. Gottfried Silbermann, the foremost German organ builder of the day, was endeavoring to make a "perfect" piano-forte. Undoubtedly King Frederick was the greatest supporter of his efforts-it is said that the King owned as many as fifteen Silbermann pianos!

Bach

Frederick was an admirer not only of pianos, but also of an organist and composer by the name of J. S. Bach. This Bach's compositions were somewhat notorious. Some called them "turgid and confused", while others claimed they were incomparable masterpieces.

But no one disputed Bach's ability to improvise on the organ. In those days, being an organist not only meant being able to play, but also to extemporize, and Bach was known far and wide for his remarkable extemporizations. (For some delightful anecdotes about Bach's extemporization, see The Bach Reader, by H. T. David and A. Mendel.) In 1747, Bach was sixty-two, and his fame, as well as one of his sons, had reached Potsdam: in fact, Carl Philipp Emanuel Bach was the Capellmeister (choirmaster) at the court of King Frederick. For years the King had let it be known, through gentle hints to Philipp Emanuel, how

pleased he would be to have the elder Bach come and pay him a visit; but this wish had never been realized. Frederick was particularly eager for Bach to try out his new Silbermann pianos, which lie (Frederick) correctly foresaw as the great new wave in music.