GEBenGodel Escher Bach-永恒的金色辫子an Eternal Golden Braid.doc
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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
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