Gödel, Escher, Bach: an eternal golden braid

804 Pages·1980·92 MB·English

by  Hofstadter, Douglas R.

WINNER OF THE PULITZER PRIZE
AN ETERNAL GOLDEN BRAID
[i)(I)|¥]@[yi\g B„
A METAPHORICAL FUGUE ON MINDS AND MACHINES
IN THE SPIRIT OF LEWIS CARROLL
Digitized by the Internet Archive
in 2009
http://www.archive.org/details/gdelescherbachOOhofs
GODEL
ESCHER,
BACH:
an EternalGolden Braid

• •
GODEL,
ESCHER.
BACH:
an Eternal Golden Braid
Douglas R.Hofstadter
VINTAGE BOOKS
ADivisionofRandomHouse • NewYork
VintageBooksEdition,September1980
Copyright©1979byBasicBooks,Inc.
AllrightsreservedunderInternationalandPan-AmericanCopyrightConventions.
PublishedintheUnitedStatesbyRandomHouse,Inc.,NewYork,andinCanadaby
RandomHouseofCanadaLimited,Toronto.OriginallypublishedbyBasicBooks,
Inc.,NewYork,inApril1979.
LibraryofCongressCataloginginPublicationData
Hofstadter,DouglasR.1945-
Godel,Escher,Bach.
Reprintoftheed.publishedbyBasicBooks,NewYork.
Bibliography:p.
Includesindex.
1.Metamathematics. 2.Symmetry. 3.Artificialintelligence.
4.BachjohannSebastian,1685-1750.
5.Escher,MauritsCornelis,1898-1971.
6.Godel,Kurt. I.Title.
(QA9.8.H631980] SlO'.l 80-11354
ISBN0-394-74502-7
ManufacturedintheUnitedStatesofAmerica
To M. and D.
Contents
Overview vtti
List of Illustrations xiv
Words of Thanks xix
GEB
Part I:
Introduction: A Musico-Logical Offering 3
Three-Part Invention 29
Chapter 1: The MU-puzzle 33
Two-Part Invention 43
Chapter 11: Meaning and Form in Mathematics 46
Sonatafor UnaccompaniedAchilles 61
Chapter III: Figure and Ground 64
Contracrostipunctus 75
Chapter IV: Consistency, Completeness, and Geometry 82
Little Harmonic Labyrinth 103
ChapterV: Recursive Structures and Processes 127
Canon by Intervallic Augmentation 153
ChapterVI: The Location ofMeaning 158
Chromatic Fantasy, AndFeud 177
Chapter VII: The Propositional Calculus 181
Crab Canon 199
Chapter VIII: Typographical Number Theory 204
A Mu Offering 231
Chapter IX: Mumon and Godel 246
vi Contents
1
Part II: EGB
Prelude ... 275
Chapter X: Levels of Description, and Computer Systems 285
. . . Ant Fugue 311
Chapter Xi: Brains and Thoughts 337
English French German Suite 366
Chapter XII: Minds and Thoughts 369
Aria with Diverse Variations 391
Chapter XIII: BlooP and FIooP and GlooP 406
Air on G's String 43
Chapter XIV: On Formally Undecidable Propositions ofTNT
and Related Systems 438
Birthday Cantatatata ... 461
ChapterXV: Jumping out of the System 465
Edifying Thoughts ofa Tobacco Smoker 480
Chapter XVI: Self-Refand Self-Rep 495
The Magnificrab, Indeed 549
Chapter XVII: Church, Turing, Tarski, and Others 559
SHRDLU, Toy ofMan's Designing 586
Chapter XVIII: Artificial Intelligence: Retrospects 594
Contrafactus 633
Chapter XIX: Artificial Intelligence: Prospects 641
Sloth Canon 681
Chapter XX: Strange Loops, Or Tangled Hierarchies 684
Six-Part Ricercar 720
Notes 743
Bibliography 746
Credits 757
Index 759
Contents
Overview
Part I: GEB
Introduction: A Musico-Logical Offering. The book opens with the story of
Bach'sMusicalOffering. Bach made an impromptu visit to King Frederick, the
GreatofPrussia,andwasrequestedtoimproviseuponathemepresentedbythe
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
interplaybetweendifferentlevelsinBacharediscussed;thisleadstoadiscussion
ofparallelideasinEscher'sdrawingsandthenGodel'sTheorem.Abriefpresen-
tation ofthe history oflogicand paradoxes is given as background forGodel's
Theorem. This leads to mechanical reasoning and computers, and the debate
about whether Artificial Intelligence is possible. I close with an explanation of
the originsofthe book—particularly the whyand wherefore ofthe Dialogues.
Three-PartInvention. Bachwrotefifte—enthree-partinventions. Inthisthree-part
Dialogue,—the Tortoise and Achilles the main fictional protagonists in the
Dialogues are "invented" by Zeno (as in fact they were, to illustrate Zeno's
paradoxes ofmotion). Veryshort, itsimplygives the flavorofthe Dialogues to
come.
Chapter I: The MU-puzzle. A simple formal system (the MlU-system) is pre-
sented, and the reader is urged to work out a puzzle to gain familiarity with
formal systems in general. A number offundamental notions are introduced:
string, theorem, axiom, rule ofinference, derivation, formal system, decision
procedure, working inside/outside the system.
Two-PartInvention. Bach also wrote fifteen two-part inventions. This two-part
Dialoguewaswrittennotbyme,butbyLewisCarrollin 1895.Carrollborrowed
AchillesandtheTortoisefromZeno,andI inturnborrowedthemfromCarroll.
Thetopicistherelationbetweenreasoning, reasoningaboutreasoning,reason-
ing about reasoning about reasoning, and so on. It parallels, in a way, Zeno's
paradoxesabouttheimpossibilityofmotion, seemingtoshow, byusinginfinite
regress,thatreasoningisimpossible. Itisabeautifulparadox,andisreferredto
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 MlU-system ofChapter I. Ap-
parentlymeaninglessatfirst,itssymbolsaresuddenlyrevealedtopossessmean-
ingbyvirtueofthe formofthetheorems theyappearin. This revelationisthe
firstimportantinsightinto meaning: itsdeepconnectiontoisomorphism. Vari-
ous issues related to meaning are then discussed, such as truth, proof, symbol
manipulation, and the elusive concept, "form".
Sonatafor UnaccompaniedAchilles. ADialoguewhichimitatestheBachSonatas
forunaccompaniedviolin. Inparticular,Achillesistheonlyspeaker,sinceitisa
transcriptofoneendofatelephonecall,atthefarendofwhichistheTortoise.
Their conversation concerns the concepts of"figure" and "ground" in various
Overview