《GEBenGodel Escher Bach-永恒的金色辫子an Eternal Golden Braid.doc》由会员分享，可在线阅读，更多相关《GEBenGodel Escher Bach-永恒的金色辫子an Eternal Golden Braid.doc（799页珍藏版）》请在得力文库 - 分享文档赚钱的网站上搜索。
1、ContentsOverviewviiiList of IllustrationsxivWords of ThanksxixPart I: GEBIntroduction: A Musico-Logical Offering3Three-Part Invention29Chapter I: The MU-puzzle33Two-Part Invention43Chapter II: Meaning and Form in Mathematics46Sonata for Unaccompanied Achilles61Chapter III: Figure and Ground64Contrac
2、rostipunctus75Chapter IV: Consistency, Completeness, and Geometry82Little Harmonic Labyrinth103Chapter V: Recursive Structures and Processes127Canon by Intervallic Augmentation153Chapter VI: The Location of Meaning158Chromatic Fantasy, And Feud177Chapter VII: The Propositional Calculus181Crab Canon1
3、99Chapter VIII: Typographical Number Theory204A Mu Offering231Chapter IX: Mumon and Gdel246ContentsIIPart II EGBPrelude .275Chapter X: Levels of Description, and Computer Systems285Ant Fugue311Chapter XI: Brains and Thoughts337English French German Suit366Chapter XII: Minds and Thoughts369Aria with
4、Diverse Variations391Chapter XIII: BlooP and FlooP and GlooP406Air on Gs String431Chapter XIV: On Formally Undecidable Propositions of TNTand Related Systems438Birthday Cantatatata .461Chapter XV: Jumping out of the System465Edifying Thoughts of a Tobacco Smoker480Chapter XVI: Self-Ref and Self-Rep4
5、95The Magn fierab, Indeed549Chapter XVII: Church, Turing, Tarski, and Others559SHRDLU, Toy of Mans Designing586Chapter XVIII: Artificial Intelligence: Retrospects594Contrafactus633Chapter XIX: Artificial Intelligence: Prospects641Sloth Canon681Chapter XX: Strange Loops, Or Tangled Hierarchies684Six-
6、Part Ricercar720Notes743Bibliography746Credits757Index759ContentsIIIOverviewPart I: GEBIntroduction: A Musico-Logical Offering. The book opens with the story of Bachs Musical Offering. Bach made an impromptu visit to King Frederick the Great of Prussia, and was requested to improvise upon a theme pr
7、esented by the King. His improvisations formed the basis of that great work. The Musical Offering and its story form a theme upon which I improvise throughout the book, thus making a sort of Metamusical Offering. Self-reference and the interplay between different levels in Bach are discussed: this l
8、eads to a discussion of parallel ideas in Eschers drawings and then Gdels Theorem. A brief presentation of the history of logic and paradoxes is given as background for Gdels Theorem. This leads to mechanical reasoning and computers, and the debate about whether Artificial Intelligence is possible.
9、I close with an explanation of the origins of the book-particularly the why and wherefore of the Dialogues.Three-Part Invention. Bach wrote fifteen three-part inventions. In this three-part Dialogue, the Tortoise and Achilles-the main fictional protagonists in the Dialogues-are invented by Zeno (as
10、in fact they were, to illustrate Zenos paradoxes of motion). Very short, it simply gives the flavor of the Dialogues to come.Chapter I: The MU-puzzle. A simple formal system (the MIL-system) is presented, and the reader is urged to work out a puzzle to gain familiarity with formal systems in general
11、. A number of fundamental notions are introduced: string, theorem, axiom, rule of inference, derivation, formal system, decision procedure, working inside/outside the system.Two-Part Invention. Bach also wrote fifteen two-part inventions. This two-part Dialogue was written not by me, but by Lewis Ca
12、rroll in 1895. Carroll borrowed Achilles and the Tortoise from Zeno, and I in turn borrowed them from Carroll. The topic is the relation between reasoning, reasoning about reasoning, reasoning about reasoning about reasoning, and so on. It parallels, in a way, Zenos paradoxes about the impossibility
13、 of motion, seeming to show, by using infinite regress, that reasoning is impossible. It is a beautiful paradox, and is referred to several times later in the book.Chapter II: Meaning and Form in Mathematics. A new formal system (the pq-system) is presented, even simpler than the MIU-system of Chapt
14、er I. Apparently meaningless at first, its symbols are suddenly revealed to possess meaning by virtue of the form of the theorems they appear in. This revelation is the first important insight into meaning: its deep connection to isomorphism. Various issues related to meaning are then discussed, suc
15、h as truth, proof, symbol manipulation, and the elusive concept, form.Sonata for Unaccompanied Achilles. A Dialogue which imitates the Bach Sonatas for unaccompanied violin. In particular, Achilles is the only speaker, since it is a transcript of one end of a telephone call, at the far end of which
16、is the Tortoise. Their conversation concerns the concepts of figure and ground in variousOverviewIVcontexts- e.g., Eschers art. The Dialogue itself forms an example of the distinction, since Achilles lines form a figure, and the Tortoises lines-implicit in Achilles lines-form a ground.Chapter III: F
17、igure and Ground. The distinction between figure and ground in art is compared to the distinction between theorems and nontheorems in formal systems. The question Does a figure necessarily contain the same information as its ground% leads to the distinction between recursively enumerable sets and re
18、cursive sets.Contracrostipunctus. This Dialogue is central to the book, for it contains a set of paraphrases of Gdels self-referential construction and of his Incompleteness Theorem. One of the paraphrases of the Theorem says, For each record player there is a record which it cannot play. The Dialog
19、ues title is a cross between the word acrostic and the word contrapunctus, a Latin word which Bach used to denote the many fugues and canons making up his Art of the Fugue. Some explicit references to the Art of the Fugue are made. The Dialogue itself conceals some acrostic tricks.Chapter IV: Consis
20、tency, Completeness, and Geometry. The preceding Dialogue is explicated to the extent it is possible at this stage. This leads back to the question of how and when symbols in a formal system acquire meaning. The history of Euclidean and non-Euclidean geometry is given, as an illustration of the elus
21、ive notion of undefined terms. This leads to ideas about the consistency of different and possibly rival geometries. Through this discussion the notion of undefined terms is clarified, and the relation of undefined terms to perception and thought processes is considered.Little Harmonic Labyrinth. Th
22、is is based on the Bach organ piece by the same name. It is a playful introduction to the notion of recursive-i.e., nested structures. It contains stories within stories. The frame story, instead of finishing as expected, is left open, so the reader is left dangling without resolution. One nested st
23、ory concerns modulation in music-particularly an organ piece which ends in the wrong key, leaving the listener dangling without resolution.Chapter V: Recursive Structures and Processes . The idea of recursion is presented in many different contexts: musical patterns, linguistic patterns, geometric s
24、tructures, 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 describe
25、s 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 code
26、d 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
27、resemblance, except in title, to Bachs Chromatic Fantasy and Fugue. It concerns the proper way to manipulate sentences so as to preserve truth-and in particular the questionOverviewVof whether there exist rules for the usage of the word arid. This Dialogue has much in common with the Dialogue by Lew
28、is 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 Zente
29、nces-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
30、 his first appearance in this Dialogue. It is perhaps the densest Dialogue in the book in terms of formal trickery and level-play. Gdel, Escher, and Bach are deeply intertwined in this very short Dialogue.Chapter VIII: Typographical Number Theory. An extension of the Propositional Calculus called TN
31、T 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 actual
32、ly 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-refer
33、ential games.Chapter IX: Mumon and Gdel. 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 ma
34、thematics. After this Zennery, Gdels fundamental idea of Gdel-numbering is introduced, and a first pass through Gdels Theorem is made.Part II: EGBPrelude . This Dialogue attaches to the next one. They are based on preludes and fugues from Bachs Well-Tempered Clavier. Achilles and the Tortoise bring
35、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? Th
36、is 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
37、 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 an
38、y exist.OverviewVIAnt 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 H
39、OLISM, REDLCTIONIsM, and ML. The discussion veers off to a friend of the Anteaters 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
40、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 o
41、f 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 Carrolls nonsense poem Jabberwocky together with two translations: o
42、ne 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 al
43、l 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 Bachs Goldberg Variations, and whose content is related to numbe
44、r-theoretical problems such as the Goldbach conjecture. This hybrid has as its main purpose to show how number theorys 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
45、 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 thi
46、s Chapter is to give an intuition for the notions of primitive recursive and general recursive functions in number theory, for they are essential in Gdels proof.Air on Gs String. A Dialogue in which Gdels self-referential construction is mirrored in words.The idea is due to W. V. O. Quine. This Dial
47、ogue serves as a prototype for the next Chapter.Chapter XIV: On Formally Undecidable Propositions of TNT and Related Systems. This Chapters title is an adaptation of the title of Gdels 1931 article, in which his Incompleteness Theorem was first published. The two major parts of Gdels 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 w
黑公网安备:91230400333293403D