ContentsIIContents Overviewviii List oI Illustrationsxiv Words oI Thanksxix Part I: GEB Introduction: A Musico-Logical OIIering3 Three-Part Invention29 Chapter I: The MU-puzzle33 Two-Part Invention43 Chapter II: Meaning and Form in Mathematics46 Sonata Ior Unaccompanied Achilles61 Chapter III: Figure and Ground64 Contracrostipunctus75 Chapter IV: Consistency, Completeness, and Geometry82 Little Harmonic Labyrinth103 Chapter V: Recursive Structures and Processes127 Canon by Intervallic Augmentation153 Chapter VI: The Location oI Meaning158 Chromatic Fantasy, And Feud177 Chapter VII: The Propositional Calculus181 Crab Canon199 Chapter VIII: Typographical Number Theory204 A Mu OIIering231 Chapter IX: Mumon and Gdel246 ContentsIII Part II EGB Prelude ...275 Chapter X: Levels oI Description, and Computer Systems285 Ant Fugue311 Chapter XI: Brains and Thoughts337 English French German Suit366 Chapter XII: Minds and Thoughts369 Aria with Diverse Variations391 Chapter XIII: BlooP and FlooP and GlooP406 Air on G's String431 Chapter XIV: On Formally Undecidable Propositions oI TNT and Related Systems438 Birthday Cantatatata ...461 Chapter XV: Jumping out oI the System465 EdiIying Thoughts oI a Tobacco Smoker480 Chapter XVI: SelI-ReI and SelI-Rep495 The Magn Iierab, Indeed549 Chapter XVII: Church, Turing, Tarski, and Others559 SHRDLU, Toy oI Man's Designing586 Chapter XVIII: ArtiIicial Intelligence: Retrospects594 ContraIactus633 Chapter XIX: ArtiIicial Intelligence: Prospects641 Sloth Canon681 Chapter XX: Strange Loops, Or Tangled Hierarchies684 Six-Part Ricercar720 Notes743 Bibliography746 Credits 757 Index759 OverviewIVOverview Part I: GEB Introduction:AMusico-LogicalOffering.ThebookopenswiththestoryoIBach'sMusical OIIering.BachmadeanimpromptuvisittoKingFredericktheGreatoIPrussia,andwas requested to improvise upon a theme presented by the King. His improvisations Iormed the basis oIthatgreatwork.TheMusicalOIIeringanditsstoryIormathemeuponwhichI"improvise" throughoutthebook,thusmakingasortoI"MetamusicalOIIering".SelI-reIerenceandthe interplaybetweendiIIerentlevelsinBacharediscussed:thisleadstoadiscussionoIparallel ideas in Escher's drawings and then Gdel`s Theorem. A brieI presentation oI the history oI logic and paradoxes is given as background Ior Gdel`s Theorem. This leads to mechanical reasoning andcomputers,andthedebateaboutwhetherArtiIicialIntelligenceispossible.Iclosewithan explanation oI the origins oI the book-particularly the why and whereIore oI the Dialogues. Three-PartInvention.BachwroteIiIteenthree-partinventions.Inthisthree-partDialogue,the Tortoise and Achilles-the main Iictional protagonists in the Dialogues-are "invented" by Zeno (as in Iact they were, to illustrate Zeno's paradoxes oI motion). Very short, it simply gives the Ilavor oI the Dialogues to come. Chapter I: The MU-puzzle. A simple Iormal system (the MIL'-system) is presented, and the reader isurgedtoworkoutapuzzletogainIamiliaritywithIormalsystemsingeneral.AnumberoI Iundamental notions are introduced: string, theorem, axiom, rule oI inIerence, derivation, Iormal system, decision procedure, working inside/outside the system. Two-Part Invention. Bach also wrote IiIteen two-part inventions. This two-part Dialogue was written not by me, but by Lewis Carroll in 1895. Carroll borrowed Achilles and the Tortoise Irom Zeno, and I in turn borrowed them Irom Carroll. The topic is the relation between reasoning, reasoning aboutreasoning,reasoningaboutreasoningaboutreasoning,andsoon.Itparallels,inaway, Zeno'sparadoxesabouttheimpossibilityoImotion,seemingtoshow,byusinginIiniteregress, that reasoning is impossible. It is a beautiIul paradox, and is reIerred to several times later in the book. ChapterII:MeaningandForminMathematics.AnewIormalsystem(thepq-system)is presented,evensimplerthantheMIU-systemoIChapterI.ApparentlymeaninglessatIirst,its symbolsaresuddenlyrevealedtopossessmeaningbyvirtueoItheIormoIthetheoremsthey appearin.ThisrevelationistheIirstimportantinsightintomeaning:itsdeepconnectionto isomorphism. Various issues related to meaning are then discussed, such as truth, prooI, symbol manipulation, and the elusive concept, "Iorm". SonataforUnaccompaniedAchilles.ADialoguewhichimitatestheBachSonatasIor unaccompaniedviolin.Inparticular,Achillesistheonlyspeaker,sinceitisatranscriptoIone endoIatelephonecall,attheIarendoIwhichistheTortoise.Theirconversationconcernsthe concepts oI "Iigure" and "ground" in various OverviewV contexts-e.g.,Escher'sart.TheDialogueitselIIormsanexampleoIthedistinction,since Achilles' lines Iorm a "Iigure", and the Tortoise's lines-implicit in Achilles' lines-Iorm a "ground". Chapter III: Figure and Ground. The distinction between Iigure and ground in art is compared to thedistinctionbetweentheoremsandnontheoremsinIormalsystems.Thequestion"Doesa Iigure necessarily contain the same inIormation as its ground" leads to the distinction between recursively enumerable sets and recursive sets. Contracrostipunctus.ThisDialogueiscentraltothebook,IoritcontainsasetoIparaphrasesoI Gdel`s selI-reIerential construction and oI his Incompleteness Theorem. One oI the paraphrases oItheTheoremsays,"Foreachrecordplayerthereisarecordwhichitcannotplay."The Dialogue's title is a cross between the word "acrostic" and the word "contrapunctus", a Latin word whichBachusedtodenotethemanyIuguesandcanonsmakinguphisArtoftheFugue.Some explicit reIerences to the Art oI the Fugue are made. The Dialogue itselI conceals some acrostic tricks. Chapter IV: Consistency, Completeness, and Geometry. The preceding Dialogue is explicated to the extent it is possible at this stage. This leads back to the question oI how and when symbols in a Iormal system acquire meaning. The history oI Euclidean and non-Euclidean geometry is given, asanillustrationoItheelusivenotionoI"undeIinedterms".Thisleadstoideasaboutthe consistencyoIdiIIerentandpossibly"rival"geometries.ThroughthisdiscussionthenotionoI undeIinedtermsisclariIied,andtherelationoIundeIinedtermstoperceptionandthought processes is considered. Little Harmonic Labvrinth. This is based on the Bach organ piece by the same name. It is a playIul introduction to the notion oI recursive-i.e., nested structures. It contains stories within stories. The Iramestory,insteadoIIinishingasexpected,isleItopen,sothereaderisleItdanglingwithout resolution.Onenestedstoryconcernsmodulationinmusic-particularlyanorganpiecewhich ends in the wrong key, leaving the listener dangling without resolution. ChapterV:RecursiveStructuresandProcesses.TheideaoIrecursionispresentedinmany diIIerentcontexts:musicalpatterns,linguisticpatterns,geometricstructures,mathematical Iunctions, physical theories, computer programs, and others. CanonbvIntervallicAugmentation.AchillesandtheTortoisetrytoresolvethequestion,"Which containsmoreinIormation-arecord,orthephonographwhichplaysitThisoddquestionarises whentheTortoisedescribesasinglerecordwhich,whenplayedonasetoIdiIIerent phonographs,producestwoquitediIIerentmelodies:B-A-C-HandC-A-G-E.Itturnsout, however, that these melodies are "the same", in a peculiar sense. Chapter VI: The Location of Meaning. A broad discussion oI how meaning is split among coded message,decoder,andreceiver.ExamplespresentedincludestrandsoIDNA,undeciphered inscriptionsonancienttablets,andphonographrecordssailingoutinspace.TherelationshipoI intelligence to "absolute" meaning is postulated. Chromatic Fantasv, And Feud. A short Dialogue bearing hardly any resemblance, except in title, to Bach's Chromatic Fantasv and Fugue. It concerns the proper way to manipulate sentences so as to preserve truth-and in particular the question OverviewVIoI whether there exist rules Ior the usage oI the word "arid". This Dialogue has much in common with the Dialogue by Lewis Carroll. ChapterVII:ThePropositionalCalculus.Itissuggestedhowwordssuchas.,and"canbe governedbyIormalrules.Onceagain,theideasoIisomorphismandautomaticacquisitionoI meaningbysymbolsinsuchasystemarebroughtup.AlltheexamplesinthisChapter, incidentally,are"Zentences"-sentencestakenIromZenkoans.ThisispurposeIullydone, somewhat tongue-in-cheek, since Zen koans are deliberately illogical stories. Crab Canon. A Dialogue based on a piece by the same name Irom the Musical Offering. Both are so named because crabs (supposedly) walk backwards. The Crab makes his Iirst appearance in this Dialogue.ItisperhapsthedensestDialogueinthebookintermsoIIormaltrickeryandlevel-play. Gdel, Escher, and Bach are deeply intertwined in this very short Dialogue. Chapter VIII: Typographical Number Theory. An extension oI the Propositional Calculus called "TNT"ispresented.InTNT,number-theoreticalreasoningcanbedonebyrigidsymbol manipulation. DiIIerences between Iormal reasoning and human thought are considered. AMuOffering.ThisDialogueIoreshadowsseveralnewtopicsinthebook.Ostensiblyconcerned withZenBuddhismandkoans,itisactuallyathinlyveileddiscussionoItheoremhoodand nontheoremhood,truthandIalsity,oIstringsinnumbertheory.ThereareIleetingreIerencesto molecularbiology-particular)theGeneticCode.ThereisnocloseaIIinitytotheMusical Offering, other than in the title and the playing oI selI-reIerential games. ChapterIX:MumonandGdel.AnattemptismadetotalkaboutthestrangeideasoIZen Buddhism.TheZenmonkMumon,whogavewellknowncommentariesonmanykoans,isa central Iigure. In a way, Zen ideas bear a metaphorical resemblance to some contemporary ideas inthephilosophyoImathematics.AIterthis"Zennery",Gdel`sIundamentalideaoIGdel-numbering is introduced, and a Iirst pass through Gdel`s Theorem is made. Part II: EGB Prelude...ThisDialogueattachestothenextone.TheyarebasedonpreludesandIuguesIrom 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 oI the W.T.C.; it is immediately put on.Astheylistentoaprelude,theydiscussthestructureoIpreludesandIugues,whichleads Achilles to ask how to hear a Iugue: as a whole, or as a sum oI parts? This is the debate between holism and reductionism, which is soon taken up in the Ant Fugue. ChapterX:LevelsofDescription,andComputerSystems.VariouslevelsoIseeingpictures, chessboards,andcomputersystemsarediscussed.ThelastoItheseisthenexaminedindetail. This involves describing machine languages, assembly languages, compiler languages, operating systems,andsoIorth.ThenthediscussionturnstocompositesystemsoIothertypes,suchas sportsteams,nuclei,atoms,theweather,andsoIorth.Thequestionarisesastohowman intermediate levels exist-or indeed whether any exist. OverviewVII.AntFugue.AnimitationoIamusicalIugue:eachvoiceenterswiththesamestatement.The theme-holismversusreductionism-isintroducedinarecursivepicturecomposedoIwords composed oI smaller words. etc. The words which appear on the Iour levels oI this strange picture are"HOLISM","REDLCTIONIsM",and"ML".ThediscussionveersoIItoaIriendoIthe Anteater'sAuntHillary,aconsciousantcolony.Thevarious levels oI her thought processes are the topic oI discussion. Many Iugal tricks are ensconced in the Dialogue. As a hint to the reader, reIerences are made to parallel tricks occurring in the Iugue on the record to which the Ioursome islistening.AttheendoItheAntFugue,themesIromthePreludereturn.transIormed considerably. Chapter XI: Brains and Thoughts. "How can thoughts he supported by the hardware oI the brain is the topic oI the Chapter. An overview oI the large scale and small-scale structure oI the brain is Iirstgiven.Thentherelationbetweenconceptsandneuralactivityisspeculativelydiscussedin some detail. EnglishFrenchGermanSuite.AninterludeconsistingoILewisCarroll'snonsensepoem "Jabberwocky`'togetherwithtwotranslations: one into French and one into German, both done last century. Chapter XII: Minds and Thoughts. The preceding poems bring up in a IorceIul way the question oI whether languages, or indeed minds, can be "mapped" onto each other. How is communication possiblebetweentwoseparatephysicalbrains:Whatdoallhumanbrainshaveincommon?A geographicalanalogyisusedtosuggestananswer.Thequestionarises,"Canabrainbe understood, in some objective sense, by an outsider?" Aria with Diverse Jariations. A Dialogue whose Iorm is based on Bach's Goldberg Variations, and whosecontentisrelatedtonumber-theoreticalproblemssuchastheGoldbachconjecture.This hybridhasasitsmainpurposetoshowhownumbertheory'ssubtletystemsIromtheIactthat therearemanydiversevariationsonthethemeoIsearchingthroughaninIinitespace.SomeoI themleadtoinIinitesearches,someoIthemleadtoIinitesearches,whilesomeothershoverin between. ChapterXIII:BlooPandFlooPandGlooP.ThesearethenamesoIthreecomputerlanguages. BlooPprogramscancarryoutonlypredictablyIinitesearches,whileFlooPprogramscancarry out unpredictable or even inIinite searches. The purpose oI this Chapter is to give an intuition Ior the notions oI primitive recursive and general recursive Iunctions in number theory, Ior they are essential in Gdel`s prooI. AironGsString.ADialogueinwhichGdel`sselI-reIerentialconstructionismirroredinwords. The idea is due to W. V. O. Quine. This Dialogue serves as a prototype Ior the next Chapter. ChapterXIV:OnFormallyUndecidablePropositionsofTNTandRelatedSystems.This Chapter'stitleisanadaptationoIthetitleoIGdel`s1931article,inwhichhisIncompleteness Theorem was Iirst published. The two major parts oI Gdel`s prooI are gone through careIully. It isshownhowtheassumptionoIconsistencyoITNTIorcesonetoconcludethatTNT(orany similar system) is incomplete. Relations to Euclidean and non-Euclidean geometry are discussed. Implications Ior the philosophy oI mathematics are gone into with some care. OverviewVIIIBirthday Cantatatata ... In which Achilles cannot convince the wily and skeptical Tortoise that today ishis(Achilles')birthday.HisrepeatedbutunsuccessIultriestodosoIoreshadowthe repeatability oI the Gdel argument. ChapterXV:1umpingoutoftheSystem.TherepeatabilityoIGdel`sargumentisshown,with the implication that TNT is not only incomplete, but "essentially incomplete The Iairly notorious argumentbyJ.R.Lucas,totheeIIectthatGdel`sTheoremdemonstratesthathumanthought cannot in any sense be "mechanical", is analyzed and Iound to be wanting. Edifving Thoughts of a Tobacco Smoker. A Dialogue treating oI many topics, with the thrust being problemsconnectedwithselI-replicationandselI-reIerence.TelevisioncamerasIilming televisionscreens,andvirusesandothersubcellularentitieswhichassemblethemselves,are among the examples used. The title comes Irom a poem by J. S. Bach himselI, which enters in a peculiar way. Chapter XVI: Self-Ref and Self-Rep. This Chapter is about the connection between selI-reIerence initsvariousguises,andselI-reproducingentitiese.g.,computerprogramsorDNAmolecules). The relations between a selI-reproducing entity and the mechanisms external to it which aid it in reproducingitselI(e.g.,acomputerorproteins)arediscussed-particularlytheIuzzinessoIthe distinction. How inIormation travels between various levels oI such systems is the central topic oI this Chapter. TheMagnificrab,Indeed.ThetitleisapunonBach'sMagniIacatinD.ThetaleisabouttheCrab, whogivestheappearanceoIhavingamagicalpoweroIdistinguishingbetweentrueandIalse statementsoInumbertheorybyreadingthemasmusicalpieces,playingthemonhisIlute,and determining whether they are "beautiIul" or not. Chapter XVII: Church, Turing, Tarski, and Others. The Iictional Crab oI the preceding Dialogue isreplacedbyvariousrealpeoplewithamazingmathematicalabilities.TheChurch-Turing Thesis, which relates mental activity to computation, is presented in several versions oI diIIering strengths.Allareanalyzed,particularlyintermsoItheirimplicationsIorsimulatinghuman thoughtmechanically,orprogrammingintoamachineanabilitytosenseorcreatebeauty.The connectionbetweenbrainactivityandcomputationbringsupsomeothertopics:thehalting problem oI Turing, and Tarski's Truth Theorem. SHRDLU,TovofMansDesigning.ThisDialogueisliItedoutoIanarticlebyTerryWinogradon hisprogramSHRDLU:onlyaIewnameshavebeenchanged.Init.aprogramcommunicates withapersonabouttheso-called"blocksworld"inratherimpressiveEnglish.Thecomputer programappearstoexhibitsomerealunderstanding-initslimitedworld.TheDialogue'stitleis based on Jesu, fov of Mans Desiring, one movement oI Bach's Cantata 147. ChapterXVIII:ArtificialIntelligence:Retrospects,ThisChapteropenswithadiscussionoIthe Iamous"Turingtest"-aproposalbythecomputerpioneerAlanTuringIorawaytodetectthe presenceorabsenceoI"thought"inamachine.Fromthere,wegoontoanabridgedhistoryoI ArtiIicialIntelligence.Thiscoversprogramsthatcan-tosomedegree-playgames,prove theorems,solveproblems,composemusic,domathematics,anduse"naturallanguage"(e.g., English). OverviewIXContrafactus.Abouthowweunconsciouslyorganizeourthoughtssothatwecanimagine hypotheticalvariantsontherealworldallthetime.AlsoaboutaberrantvariantsoIthisability-such as possessed by the new character, the Sloth, an avid lover oI French Iries, and rabid hater oI counterIactuals. ChapterXIX:ArtificialIntelligence:Prospects.TheprecedingDialoguetriggersadiscussionoI how knowledge is represented in layers oI contexts. This leads to the modern Al idea oI "Irames". AIrame-likewayoIhandlingasetoIvisualpatternpuzzlesispresented,IorthepurposeoI concreteness.ThenthedeepissueoItheinteractionoIconceptsingeneralisdiscussed,which leadsintosomespeculationsoncreativity.TheChapterconcludeswithasetoIpersonal "Questions and Speculations" on Al and minds in general. SlothCanon.AcanonwhichimitatesaBachcanoninwhichonevoiceplaysthesamemelodyas another, only upside down and twice as slowly, while a third voice is Iree. Here, the Sloth utters thesamelinesastheTortoisedoes,onlynegated(inaliberalsenseoItheterm)andtwiceas slowly, while Achilles is Iree. ChapterXX:StrangeLoops,OrTangledHierarchies.AgrandwindupoImanyoItheideas abouthierarchicalsystemsandselI-reIerence.Itisconcernedwiththesnarlswhicharisewhen systems turn back on themselves-Ior example, science probing science, government investigating governmental wrongdoing, art violating the rules oI art, and Iinally, humans thinking about their ownbrainsandminds.DoesGdel`sTheoremhaveanythingto say about this last "snarl"? Are Iree will and the sensation oI consciousness connected to Gdel`s Theorem? The Chapter ends by tying Gdel, Escher, and Bach together once again. Six-PartRicercar.ThisDialogueisanexuberantgameplayedwithmanyoItheideaswhichhave permeatedthebook.ItisareenactmentoIthestoryoItheMusicalOIIering,whichbeganthe book;itissimultaneouslya"translation"intowordsoIthemostcomplexpieceintheMusical OIIering:theSix-PartRicercar.ThisdualityimbuestheDialoguewithmorelevelsoImeaning than any other in the book. Frederick the Great is replaced by the Crab, pianos by computers, and soon.Manysurprisesarise.TheDialogue'scontentconcernsproblemsoImind,consciousness, Iree will, ArtiIicial Intelligence, the Turing test, and so Iorth, which have been introduced earlier. ItconcludeswithanimplicitreIerencetothebeginningoIthebook,thusmakingthebookinto onebigselI-reIerentialloop,symbolizingatonceBach'smusic,Escher'sdrawings,andGdel`s Theorem. Introduction: A Musico-Logical OIIering10 FIGURE1.JohannSebastianBach,in1748.FromapaintingbyEliasGottlieb Hanssmann. Introduction: A Musico-Logical OIIering11Introduction: A Musico-Logical Offering Author: FREDERICK THE GREAT, King oI Prussia, came to power in 1740. Although he is rememberedinhistorybooksmostlyIorhismilitaryastuteness,hewasalsodevotedto theliIeoIthemindandthespirit.HiscourtinPotsdamwasoneoIthegreatcentersoI intellectualactivityinEuropeintheeighteenthcentury.Thecelebratedmathematician Leonhard Euler spent twenty-Iive years there. Many other mathematicians and scientists came, as well as philosophers-including Voltaire and La Mettrie, who wrote some oI their most inIluential works while there. But music was Frederick's real love. He was an avid Ilutist and composer. Some oI his compositions are occasionally perIormed even to this day. Frederick was one oI the Iirst patronsoItheartstorecognizethevirtuesoIthenewlydeveloped"piano-Iorte"("soIt-loud").ThepianohadbeendevelopedintheIirsthalIoItheeighteenthcenturyasa modiIication oI the harpsichord. The problem with the harpsichord was that pieces could onlybeplayedataratheruniIormloudness-therewasnowaytostrikeonenotemore loudly than its neighbors. The "soIt-loud", as its name implies, provided a remedy to this problem. From Italy, where Bartolommeo CristoIori had made the Iirst one, the soIt-loud idea had spread widely. GottIried Silbermann, the Ioremost German organ builder oI the day, was endeavoring to make a "perIect" piano-Iorte. Undoubtedly King Frederick was thegreatestsupporteroIhiseIIorts-itissaidthattheKingownedasmanyasIiIteen Silbermann pianos! Bach Frederick was an admirer not only oI pianos, but also oI an organist and composer by the nameoIJ.S.Bach.ThisBach'scompositionsweresomewhatnotorious.Somecalled them "turgid and conIused", while others claimed they were incomparable masterpieces. ButnoonedisputedBach'sabilitytoimproviseontheorgan.Inthosedays,beingan organist not only meant being able to play, but also to extemporize, and Bach was known IarandwideIorhisremarkableextemporizations.(ForsomedelightIulanecdotesabout Bach's extemporization, see The Bach Reader, by H. T. David and A. Mendel.) In1747,Bachwassixty-two,andhisIame,aswellasoneoIhissons,hadreached Potsdam: in Iact, Carl Philipp Emanuel Bach was the Capellmeister (choirmaster) at the court oI King Frederick. For years the King had let it be known, through gentle hints to Philipp Emanuel, how Introduction: A Musico-Logical OIIering12pleased he would be to have the elder Bach come and pay him a visit; but this wish had neverbeenrealized.FrederickwasparticularlyeagerIorBachtotryouthisnew Silbermannpianos,whichlie(Frederick)correctlyIoresawasthegreatnewwavein music. ItwasFrederick'scustomtohaveeveningconcertsoIchambermusicinhiscourt. OIten he himselI would be the soloist in a concerto Ior Ilute Here we have reproduced a paintingoIsuchaneveningbytheGermanpainterAdolphvonMenzel,who,inthe 1800's,madeaseriesoIpaintingsillustratingtheliIeoIFredericktheGreat.Atthe cembaloisC.P.E.Bach,andtheIigureIurthesttotherightisJoachimQuantz,the King'sIlutemaster-andtheonlypersonallowedtoIindIaultwiththeKing'sIlute playing.OneMayeveningin1747,anunexpectedguestshowedup.JohannNikolaus Forkel, one oI Bach's earliest biographers, tells the story as Iollows: One evening, just as lie was getting his Ilute ready, and his musicians weressembled, an oIIicer brought him a list oI the strangers who had arrived. With his Ilute in his hand he ran ever the list, but immediately turned to the assembled musicians, and said, with a kind oI agitation, "Gentlemen, old Bach is come." The Hute was now laid aside, and old Bach, who had alighted at his son's lodgings, was immediately summoned to the Palace. WilhelmFriedemann,whoaccompaniedhisIather,toldmethisstory,andImustsay that 1 still think with pleasure on the manner in which lie related it. At that time it was the Iashion to make rather prolix compliments. The Iirst appearance oI J. S. Bach beIore segreataKing,whodidnotevengivehimtimetochangehistravelingdressIora black chanter's gown, must necessarily be attended with many apologies. I will net here dwell en these apologies, but merely observe, that in Wilhelm Friedemann's mouth they made a Iormal Dialogue between the King and the Apologist. ButwhatismereimportantthanthisisthattheKinggaveuphisConcertIorthis evening,andinvitedBach,thenalreadycalledtheOldBach,totryhisIortepianos, madebySilbermann,whichsteedinseveralroomsoIthepalace.|Forkelhereinserts thisIootnote:"ThepianoIortesmanuIacturedbySilbermann,oIFrevberg,pleasedthe King se much, that he resolved to buy them all up. He collected IiIteen. I hear that they all now stand unIit Ior use in various corners oI the Royal Palace."| The musicians went with him Irom room to room, and Bach was invited everywhere to try them and to play unpremeditated compositions. AIter he had gene en Ior some time, he asked the King to givehimasubjectIoraFugue,inordertoexecuteitimmediatelywithoutany preparation.TheKingadmiredthelearnedmannerinwhichhissubjectwasthus executed extempore: and, probably to see hew Iar such art tcouldbecarried,expressed a wish to hear a Fugue with six Obligato parts. But as it is not every subject that is Iit IorsuchIullharmony,BachchoseonehimselI,andimmediatelyexecutedittothe astonishment oI all present in the same magniIicent and learned manner as he had done thatoItheKing.HisMajestydesiredalsotohearhisperIormanceentheorgan.The next day thereIore Bach was taken to all the organs in Potsdam, as lie had beIore been to Silbermann's Iortepianos. AIter his return to Leipzig, he composed the subject, which he had received Irom the King, in three and six parts. added several artiIicial passages instrictcanontoit,andhaditengraved,underthetitleoI"MusikalischesOpIer" |Musical OIIering|, and dedicated it to the Inventor.' Introduction: A Musico-Logical OIIering13

Introduction: A Musico-Logical OIIering14 FIGURE 3. The Royal Theme. When Bach sent a copy oI his Musical OIIering to the King, he included a dedicatory letter,whichisoIinterestIoritsprosestyleiInothingelserathersubmissiveand Ilattersome.Fromamodernperspectiveitseemscomical.Also,itprobablygives something oI the Ilavor oI Bach's apology Ior his appearance.2 MOST GRACIOUS KING! IndeepesthumilityIdedicateherewithtoYourMajestyamusicaloIIering,the noblestpartoIwhichderivesIromYourMajesty'sownaugusthand.Withawesome pleasureIstillremembertheveryspecialRoyalgracewhen,sometimeago,during myvisitinPotsdam,YourMajesty'sSelIdeignedtoplaytomeathemeIoraIugue upontheclavier,andatthesametimechargedmemostgraciouslytocarryitoutin Your Majesty's most august presence. To obey Your Majesty's command was my most humble dim. I noticed very soon, however, that, Ior lack oI necessary preparation, the executionoIthetaskdidnotIareaswellassuchanexcellentthemedemanded.I resoledthereIoreandpromptlypledgedmyselItoworkoutthisrightRoyaltheme moreIully,andthenmakeitknowntotheworld.Thisresolvehasnowbeencarried out as well as possible, and it has none other than this irreproachable intent, to gloriIy, iIonlyinasmallpoint,theIameoIamonarchwhosegreatnessandpower,asinall thesciencesoIwarandpeace,soespeciallyinmusic,everyonemustadmireand revere.Imakeboldtoaddthismosthumblerequest:mayYourMajestydeignto digniIythepresentmodestlaborwithagraciousacceptance,andcontinuetogrant Your Majesty's most august Royal grace to Your Majesty's most humble and obedient servant, THE AUTHOR Leipzig, July 71747 Some twenty-seven years later, when Bach had been dead Ior twentyIour years, a Baron namedGottIriedvanSwieten-towhom,incidentally,ForkeldedicatedhisbiographyoI Bach,andBeethovendedicatedhisFirstSymphony-hadaconversationwithKing Frederick, which he reported as Iollows: He|Frederick|spoketome,amongotherthings,oImusic,andoIagreatorganist namedBach,whohasbeenIorawhileinBerlin.Thisartist|WilhelmFriedemann Bach|isendowedwithatalentsuperior, in depth oI harmonic knowledge and power oIexecution,toany1haveheardorcanimagine,whilethosewhoknewhisIather claim that he, in turn, was even greater. The King Introduction: A Musico-Logical OIIering15is oI this opinion, and to prove it to me he sang aloud a chromatic Iugue subject which he had given this old Bach, who on the spot had made oI it a Iugue in Iour parts, then in Iive parts, and Iinally in eight parts.' OIcoursethereisnowayoIknowingwhetheritwasKingFrederickorBaronvan SwietenwhomagniIiedthestoryintolarger-than-liIeproportions.Butitshowshow powerIul Bach's legend had become by that time. To give an idea oI how extraordinary a six-partIugueis,intheentireWell-TemperedClavierbyBach,containingIorty-eight Preludes and Fugues, only two have as many as Iive parts, and nowhere is there a six-part Iugue! One could probably liken the task oI improvising a six-part Iugue to the playing oI sixtysimultaneousblindIoldgamesoIchess,andwinningthemall.Toimprovisean eight-part Iugue is really beyond human capability. In the copy which Bach sent to King Frederick, on the page preceding the Iirst sheet oI music, was the Iollowing inscription: FIG URE 4. ("AttheKing'sCommand,theSongandtheRemainderResolvedwithCanonicArt.") Here Bach is punning on the word "canonic", since it means not only "with canons" but also "in the best possible way". The initials oI this inscription are R I C E R C A R -anItalianword,meaning"toseek".Andcertainlythereisagreatdealtoseekinthe Musical Offering. It consists oI one three-part Iugue, one six-part Iugue, ten canons, and a triosonata.Musicalscholarshaveconcludedthatthethree-partIuguemustbe,in essence,identicalwiththeonewhichBach improvised Ior King Frederick. The six-part IugueisoneoIBach'smostcomplexcreations,anditsthemeis,oIcourse,theRoyal Theme. That theme, shown in Figure 3, is a very complex one, rhythmically irregular and highlychromatic(thatis,Iilledwithtoneswhichdonotbelongtothekeyitisin).To writeadecentIugueoIeventwovoicesbasedonitwouldnotbeeasyIortheaverage musician! BothoItheIuguesareinscribed"Ricercar",ratherthan"Fuga".Thisisanother meaning oI the word; "ricercar" was, in Iact, the original name Ior the musical Iorm now knownas"Iugue".ByBach'stime,theword"Iugue"(orIuga,inLatinandItalian)had becomestandard,buttheterm"ricercar"hadsurvived,andnowdesignatedanerudite kindoIIugue,perhapstooausterelyintellectualIorthecommonear.Asimilarusage survives in English today: the word "recherche" means, literally, "sought out", but carries the same kind oI implication, namely oI esoteric or highbrow cleverness. ThetriosonataIormsadelightIulrelieIIromtheausterityoItheIuguesandcanons, because it is very melodious and sweet, almost dance- Introduction: A Musico-Logical OIIering16able. Nevertheless, it too is based largely on the King's theme, chromatic and austere as it is.ItisrathermiraculousthatBachcouldusesuchathemetomakesopleasingan interlude. The ten canons in the Musical OIIering are among the most sophisticated canons Bach ever wrote. However, curiously enough, Bach himselI never wrote them out in Iull. This was deliberate. They were posed as puzzles to King Frederick. It was a Iamiliar musical game oI the day to give a single theme, together with some more or less tricky hints, and to let the canon based on that theme be "discovered" by someone else. In order to know how this is possible, you must understand a Iew Iacts about canons. Canons and Fugues TheideaoIacanonisthatonesinglethemeisplayedagainstitselI.Thisisdoneby having"copies"oIthethemeplayedbythevariousparticipatingvoices.Butthereare means'waystodothis.ThemoststraightIorwardoIallcanonsistheround,suchas "Three Blind Mice", "Row, Row, Row Your Boat", or " Frere Jacques". Here, the theme enters in the Iirst voice and, aIter a Iixed time-delay, a "copy" oI it enters, in precisely the samekey.AIterthesameIixedtime-delayinthesecondvoice,thethirdvoiceenters carryingthetheme,andsoon.Mostthemeswillnotharmonizewiththemselvesinthis way.InorderIorathemetoworkasacanontheme,eachoIitsnotesmustbeableto serveinadual(ortriple,orquadruple)role:itmustIirstlybepartoIamelody,and secondlyitmustbepartoIaharmonizationoIthesamemelody.Whentherearethree canonical voices, Ior instance, each note oI the theme must act in two distinct harmonic ways,aswellasmelodically.Thus,eachnoteinacanonhasmorethanonemusical meaning; the listener's ear and brain automatically Iigure out the appropriate meaning, by reIerring to context. TherearemorecomplicatedsortsoIcanons,oIcourse.TheIirstescalationin complexity comes when the "copies" oI the theme are staggered not only in time, but also inpitch;thus,theIirstvoicemightsingthethemestartingonC,andthesecondvoice, overlapping with the Iirst voice, might sing the identical theme starting Iive notes higher, on G. A third voice, starting on the D yet Iive notes higher, might overlap with the Iirst two, and so on. The next escalation in complexity comes when the speeds oI' the diIIerent voicesarenotequal;thus,thesecondvoicemightsingtwiceasquickly,ortwiceas slowly, as the Iirst voice. The Iormer is called diminution, the latter augmentation (since the theme seems to shrink or to expand). We are not yet done! The next stage oI complexity in canon construction is to invert the theme,whichmeanstomakeamelodywhichjumpsdownwherevertheoriginaltheme jumps up, and by exactly the same number oI semitones. This is a rather weird melodic transIormation,butwhenonehasheardmanythemesinverted,itbeginstoseemquite natural. Bach was especially Iond oI inversions, and they show up oIten in his work-and the Musical OIIering is no exception. (For a simple example oI Introduction: A Musico-Logical OIIering17inversion,trythetune"GoodKingWenceslas".Whentheoriginalanditsinversionare sungtogether,startinganoctaveapartandstaggeredwithatime-delayoItwobeats,a pleasingcanonresults.)Finally,themostesotericoI"copies"istheretrogradecopy-wherethethemeisplayedbackwardsintime.Acanonwhichusesthistrickis aIIectionatelyknownasacrabcanon,becauseoIthepeculiaritiesoIcrablocomotion. BachincludedacrabcanonintheMusicalOIIering,needlesstosay.Noticethatevery typeoI"copy"preservesalltheinIormationintheoriginaltheme,inthesensethatthe themeisIullyrecoverableIromanyoIthecopies.SuchaninIormationpreserving transIormationisoItencalledanisomorphism,andwewillhavemuchtraIIicwith isomorphisms in this book. Sometimes it is desirable to relax the tightness oI the canon Iorm. One way is to allow slightdeparturesIromperIectcopying,IorthesakeoImoreIluidharmony.Also,some canonshave"Iree"voices-voiceswhichdonotemploythecanon'stheme,butwhich simply harmonize agreeably with the voices that are in canon with each other. Each oI the canons in the Musical OIIering has Ior its theme a diIIerent variant oI the King'sTheme,andallthedevicesdescribedaboveIormakingcanonsintricateare exploited to the hilt; in Iact, they are occasionally combined. Thus, one three-voice canon is labeled "Canon per Augmentationem, contrario Motu"; its middle voice is Iree (in Iact, itsingstheRoyalTheme),whiletheothertwodancecanonicallyaboveandbelowit, using the devices oI augmentation and inversion. Another bears simply the cryptic label "Quaerendo invenietis" ("By seeking, you will discover"). All oI the canon puzzles have been solved. The canonical solutions were given by one oI Bach's pupils, Johann Philipp Kirnberger. But one might still wonder whether there are more solutions to seek! IshouldalsoexplainbrieIlywhataIugueis.AIugueislikeacanon,inthatitis usually based on one theme which gets played in diIIerent voices and diIIerent keys, and occasionallyatdiIIerentspeedsorupsidedownorbackwards.However,thenotionoI IugueismuchlessrigidthanthatoIcanon,andconsequentlyitallowsIormore emotional and artistic expression. The telltale sign oI a Iugue is the way it begins: with a singlevoicesingingitstheme.Whenitisdone,thenasecondvoiceenters,eitherIive scale-notesup,orIourdown.MeanwhiletheIirstvoicegoeson,singingthe "countersubject": a secondary theme, chosen to provide rhythmic, harmonic, and melodic contrasts to the subject. Each oI the voices enters in turn, singing the theme, oIten to the accompanimentoIthecountersubjectinsomeothervoice,withtheremainingvoices doingwhateverIanciIulthingsenteredthecomposer'smind.Whenallthevoiceshave "arrived", then there are no rules. There are, to be sure, standard kinds oI things to do-but not so standard that one can merely compose a Iugue by Iormula. The two Iugues in the MusicalOIIeringareoutstandingexamplesoIIuguesthatcouldneverhavebeen "composed by Iormula". There is something much deeper in them than mere Iugality. All in all, the Musical OIIering represents one oI Bach's supreme accomplishments in counterpoint. It is itselI one large intellectual Iugue, in Introduction: A Musico-Logical OIIering18whichmanyideasandIormshavebeenwoventogether,andinwhichplayIuldouble meanings and subtle allusions are commonplace. And it is a very beautiIul creation oI the humanintellectwhichwecanappreciateIorever.(TheentireworkiswonderIully described in the book I. S. Bach's Musical OIIering, by H. T. David.) An Endlessly Rising Canon There is one canon in the Musical OIIering which is particularly unusual. Labeled simply "Canon per Tonos", it has three voices. The uppermost voice sings a variant oI the Royal Theme,whileunderneathit,twovoicesprovideacanonicharmonizationbasedona second theme. The lower oI this pair sings its theme in C minor (which is the key oI the canon as a whole), and the upper oI the pair sings the same theme displaced upwards in pitch by an interval oI a IiIth. What makes this canon diIIerent Irom any other, however, isthatwhenitconcludes-or,rather,seemstoconclude-itisnolongerinthekeyoIC minor,butnowisinDminor.SomehowBach has contrived to modulate (change keys) rightunderthelistener'snose.Anditissoconstructedthatthis"ending"tiessmoothly onto the beginning again; thus one can repeat the process and return in the key oI E, only to join again to the beginning. These successive modulations lead the ear to increasingly remoteprovincesoItonality,sothataIterseveraloIthem,onewouldexpecttobe hopelesslyIarawayIromthestartingkey.Andyetmagically,aIterexactlysixsuch modulations, the original key oI C minor has been restored! All the voices are exactly one octave higher than they were at the beginning, and here the piece may be broken oII in a musically agreeable way. Such, one imagines, was Bach's intention; but Bach indubitably also relished the implication that this process could go on ad inIinitum, which is perhaps whyhewroteinthemargin"Asthemodulationrises,somaytheKing'sGlory."To emphasize its potentially inIinite aspect, I like to call this the "Endlessly Rising Canon". In this canon, Bach has given us our Iirst example oI the notion oI Strange Loops. The "StrangeLoop"phenomenonoccurswhenever,bymovingupwards(ordownwards) throughthelevelsoIsomehierarchicalsystem,weunexpectedlyIindourselvesright backwherewestarted.(Here,thesystemisthatoImusicalkeys.)SometimesIusethe term Tangled Hierarchy to describe a system in which a Strange Loop occurs. As we go on, the theme oI Strange Loops will recur again and again. Sometimes it will be hidden, other times it will be out in the open; sometimes it will be right side up, other times it will be upside down, or backwards. "Quaerendo invenietis" is my advice to the reader. Escher To my mind, the most beautiIul and powerIul visual realizations oI this notion oI Strange Loops exist in the work oI the Dutch graphic artist M. C. Escher, who lived Irom 1902 to 1972. Escher was the creator oI some oI the Introduction: A Musico-Logical OIIering19 FIGURE 5. WaterIall, by M. C. Escher (lithograph, 1961). mostintellectuallystimulatingdrawingsoIalltime.ManyoIthemhavetheiroriginin paradox,illusion,ordouble-meaning.MathematicianswereamongtheIirstadmirersoI Escher'sdrawings,andthisisunderstandablebecausetheyoItenarebasedon mathematicalprinciplesoIsymmetryorpattern...Butthereismuchmoretoatypical Escher drawing than just symmetry or pattern; there is oIten an underlying idea, realized in artistic Iorm. And in particular, the Strange Loop is one oI the most recurrent themes in Escher'swork.Look,Iorexample,atthelithographWaterIall(Fig.5),andcompareits six-stependlesslyIallingloopwiththesix-stependlesslyrisingloopoIthe"Canonper Tonos". The similarity oI vision is Introduction: A Musico-Logical OIIering20 FIGURE 6. Ascending and Descending, by M. C. Escher (lithograph, 1960). Introduction: A Musico-Logical OIIering21remarkable. Bach and Escher are playing one single theme in two diIIerent "keys": music and art. EscherrealizedStrangeLoopsinseveraldiIIerentways,andtheycanbearranged according to the tightness oI the loop. The lithograph Ascending and Descending (Fig. 6), in which monks trudge Iorever in loops, is the loosest version, since it involves so many steps beIore the starting point is regained. A tighter loop is contained in WaterIall, which, as we already observed, involves only six discrete steps. You may be thinking that there issomeambiguityinthenotionoIasingle"step"-Iorinstance,couldn'tAscendingand DescendingbeseenjustaseasilyashavingIourlevels(staircases)asIorty-Iivelevels (stairs) It is indeed true that there is an inherent FIGURE 7. Hand with ReIlecting Globe. SelI-portrait In, M. C. Escher (lithograph, 1935). Introduction: A Musico-Logical OIIering22 Introduction: A Musico-Logical OIIering23 hazinessinlevel-counting,notonlyinEscherpictures,butinhierarchical,many-level systems.WewillsharpenourunderstandingoIthishazinesslateron.Butletusnotget toodistractednow'Aswetightenourloop,wecometotheremarkableDrawingHands (Fig.135),inwhicheachoItwohandsdrawstheother:atwo-stepStrangeLoop.And Iinally, the tightest oI all Strange Loops is realized in Print Gallery (Fig. 142): a picture oI a picture which contains itselI. Or is it a picture oI a gallery which contains itselI? Or oI a town which contains itselI? Or a young man who contains himselI'? (Incidentally, the illusion underlying Ascending and Descending and WaterIall was not invented by Escher, butbyRogerPenrose,aBritishmathematician,in1958.However,thethemeoIthe StrangeLoopwasalreadypresentinEscher'sworkin1948,theyearhedrewDrawing Hands. Print Gallery dates Irom 1956.) Implicit in the concept oI Strange Loops is the concept oI inIinity, since what else is a loopbutawayoIrepresentinganendlessprocessinaIiniteway?AndinIinityplaysa largerolenmanyoIEscher'sdrawings.CopiesoIonesinglethemeoItenIitintoeach' other, Iorming visual analogues to the canons oI Bach. Several such patterns can be seen inEscher'sIamousprintMetamorphosis(Fig.8).Itisalittlelikethe"EndlesslyRising Canon": wandering Iurther and Iurther Irom its starting point, it suddenly is back. In the tiledplanesoIMetamorphosisandotherpictures,therearealreadysuggestionsoI inIinity. But wilder visions oI inIinity appear in other drawings by Escher. In some oI his drawings,onesinglethemecanappearondiIIerentlevelsoIreality.Forinstance,one level in a drawing might clearly be recognizable as representing Iantasy or imagination; anotherlevelwouldberecognizableasreality.Thesetwolevelsmightbetheonly explicitly portrayed levels. But the mere presence oI these two levels invites the viewer to look upon himselI as part oI yet another level; and by taking that step, the viewer cannot helpgettingcaughtupinEscher'simpliedchainoIlevels,inwhich,Ioranyonelevel, there is always another level above it oI greater "reality", and likewise, there is always a level below, "more imaginary" than it is. This can be mind-boggling in itselI. However, what happens iI the chain oI levels is not linear, but Iorms a loop? What is real, then, and whatisIantasy?ThegeniusoIEscherwasthathecouldnotonlyconcoct,butactually portray,dozensoIhalI-real,halI-mythicalworlds,worldsIilledwithStrangeLoops, which he seems to be inviting his viewers to enter. Gdel IntheexampleswehaveseenoIStrangeLoopsbyBachandEscher,thereisaconIlict between the Iinite and the inIinite, and hence a strong sense oI paradox. Intuition senses thatthereissomethingmathematicalinvolvedhere.Andindeedinourowncenturya mathematicalcounterpartwasdiscovered,withthemostenormousrepercussions.And, just as the Bach and Escher loops appeal to very simple and ancient intuitions-a musical scale, a staircase-so this discovery, by K. Gdel, oI a Strange Loop in Introduction: A Musico-Logical OIIering24

FIGURE 9. Kurt Godel. Introduction: A Musico-Logical OIIering25mathematicalsystemshasitsoriginsinsimpleandancientintuitions.Initsabsolutely barestIorm,Godel'sdiscoveryinvolvesthetranslationoIanancientparadoxin philosophyintomathematicalterms.Thatparadoxistheso-calledEpimenidesparadox, orliarparadox.EpimenideswasaCretanwhomadeoneimmortalstatement:"All Cretansareliars."AsharperversionoIthestatementissimply"Iamlying";or,"This statementisIalse".ItisthatlastversionwhichIwillusuallymeanwhenIspeakoIthe Epimenidesparadox.Itisastatementwhichrudelyviolatestheusuallyassumed dichotomyoIstatementsintotrueandIalse,becauseiIyoutentativelythinkitistrue, thenitimmediatelybackIiresonyouandmakesyouthinkitisIalse.Butonceyou've decided it is Ialse, a similar backIiring returns you to the idea that it must be true. Try it! TheEpimenidesparadoxisaone-stepStrangeLoop,likeEscher'sPrintGallery.But how does it have to do with mathematics? That is what Godel discovered. His idea was to usemathematicalreasoninginexploringmathematicalreasoningitselI.ThisnotionoI makingmathematics"introspective"provedtobeenormouslypowerIul,andperhapsits richest implication was the one Godel Iound: Godel's Incompleteness Theorem. What the TheoremstatesandhowitisprovedaretwodiIIerentthings.Weshalldiscussbothin quite some detail in this book. The Theorem can De likened to a pearl, and the method oI prooI to an oyster. The pearl is prized Ior its luster and simplicity; the oyster is a complex living beast whose innards give rise to this mysteriously simple gem. Godel'sTheoremappearsasPropositionVIinhis1931paper"OnFormally Undecidable Propositions in Principia Mathematica and Related Systems I." It states: Toeveryw-consistentrecursiveclassKoIIormulaetherecorrespondrecursive class-signs r, such that neither v Gen r nor Neg (v Gen r) belongs to Fig (K) (where v is the Iree variable oI r). Actually,itwasinGerman,andperhapsyouIeelthatitmightaswellbeinGerman anyway. So here is a paraphrase in more normal English: All consistent axiomatic Iormulations oI number theory include undecidable propositions. This is the pearl. In this pearl it is hard to see a Strange Loop. That is because the Strange Loop is buried intheoyster-theprooI.TheprooIoIGodel'sIncompletenessTheoremhingesuponthe writingoIaselI-reIerentialmathematicalstatement,inthesamewayastheEpimenides paradox is a selI-reIerential statement oI language. But whereas it is very simple to talk about language in language, it is not at all easy to see how a statement about numbers can talkaboutitselI.InIact,ittookgeniusmerelytoconnecttheideaoIselI-reIerential statements with number theory. Once Godel had the intuition that such a statement could becreated,hewasoverthemajorhurdle.TheactualcreationoIthestatementwasthe working out oI this one beautiIul spark oI intuition. Introduction: A Musico-Logical OIIering26

We shall examine the Godel construction quite careIully in Chapters to come, but so that you are not leIt completely in the dark, I will sketch here, in a Iew strokes, the core oI the idea, hoping that what you see will trigger ideas in your mind. First oI all, the diIIiculty should be made absolutely clear. Mathematical statements-let us concentrate on number-theoreticalones-areaboutpropertiesoIwholenumbers.Wholenumbersarenot statements,noraretheirproperties.AstatementoInumbertheoryisnotabouta. statement oI number theory; it just is a statement oI number theory. This is the problem; but Godel realized that there was more here than meets the eye. Godel had the insight that a statement oI number theory could be about a statement oI numbertheory(possiblyevenitselI),iIonlynumberscouldsomehowstandIor statements. The idea oI a code, in other words, is at the heart oI his construction. In the Godel Code, usually called "Godel-numbering", numbers are made to stand Ior symbols and sequences oI symbols. That way, each statement oI number theory, being a sequence oI specialized symbols, acquires a Godel number, something like a telephone number or a license plate, by which it can be reIerred to. And this coding trick enables statements oI number theory to be understood on two diIIerent levels: as statements oI number theory, and also as statements about statements oI number theory. OnceGodelhadinventedthiscodingscheme,hehadtoworkoutindetailawayoI transportingtheEpimenidesparadoxintoanumbertheoreticalIormalism.HisIinal transplantoIEpimenidesdidnotsay,"ThisstatementoInumbertheoryisIalse",but rather,"ThisstatementoInumbertheorydoesnothaveanyprooI".AgreatdealoI conIusioncanbecausedbythis,becausepeoplegenerallyunderstandthenotionoI "prooI"rathervaguely.InIact,Godel'sworkwasjustpartoIalongattemptby mathematicians to explicate Ior themselves what prooIs are. The important thing to keep in mind is that prooIs are demonstrations within Iixed systems oI propositions. In the case oIGodel'swork,theIixedsystemoInumbertheoreticalreasoningtowhichtheword "prooI" reIers is that oI Principia Mathematica (P.M.), a giant opus by Bertrand Russell andAlIredNorthWhitehead,publishedbetween1910and1913.ThereIore,theGodel sentence G should more properly be written in English as: This statement oI number theory does not have any prooI in the system oI Principia Mathematica. Incidentally, this Godel sentence G is not Godel's Theorem-no more than the Epimenides sentenceistheobservationthat"TheEpimenidessentenceisaparadox."Wecannow statewhattheeIIectoIdiscoveringGis.WhereastheEpimenidesstatementcreatesa paradoxsinceitisneithertruenorIalse,theGodelsentenceGisunprovable(inside P.M.)buttrue.ThegrandconclusionThatthesystemoIPrincipiaMathematicais "incomplete"-therearetruestatementsoInumbertheorywhichitsmethodsoIprooIare too weak to demonstrate. Introduction: A Musico-Logical OIIering27 But iI Principia Mathematica was the Iirst victim oI this stroke, it was certainly not the last! The phrase "and Related Systems" in the title oI Godel's article is a telling one: Ior iI Godel's result had merely pointed out a deIect in the work oI Russell and Whitehead, then otherscouldhavebeeninspiredtoimproveuponP.M.andtooutwitGodel'sTheorem. Butthiswasnotpossible:Godel'sprooIpertainedtoanyaxiomaticsystemwhich purported to achieve the aims which Whitehead and Russell had set Ior themselves. And IoreachdiIIerentsystem,onebasicmethoddidthetrick.Inshort,Godelshowedthat provability is a weaker notion than truth, no matter what axiomatic system is involved. ThereIoreGodel'sTheoremhadanelectriIyingeIIectuponlogicians,mathematicians, and philosophers interested in the Ioundations oI mathematics, Ior it showed that no Iixed system,nomatterhowcomplicated,couldrepresentthecomplexityoIthewhole numbers:0,1,2,3,...ModernreadersmaynotbeasnonplussedbythisasreadersoI 1931 were, since in the interim our culture has absorbed Godel's Theorem, along with the conceptualrevolutionsoIrelativityandquantummechanics,andtheirphilosophically disorientingmessageshavereachedthepublic,eveniIcushionedbyseverallayersoI translation (and usually obIuscation). There is a general mood oI expectation, these days, oI "limitative" results-but back in 1931, this came as a bolt Irom the blue. Mathematical Logic: A Synopsis A proper appreciation oI Godel's Theorem requires a setting oI context. ThereIore, I will nowattempttosummarizeinashortspacethehistoryoImathematicallogicpriorto 1931-animpossibletask.(SeeDeLong,Kneebone,orNagelandNewman,Iorgood presentationsoIhistory.)Itallbeganwiththeattemptstomechanizethethought processesoIreasoning.NowourabilitytoreasonhasoItenbeenclaimedtobewhat distinguishesusIromotherspecies;so it seems somewhat paradoxical, on Iirst thought, to mechanize that which is most human. Yet even the ancient Greeks knew that reasoning isapatternedprocess,andisatleastpartiallygovernedbystatablelaws.Aristotle codiIied syllogisms, and Euclid codiIied geometry; but thereaIter, many centuries had to pass beIore progress in the study oI axiomatic reasoning would take place again. One oI the signiIicant discoveries oI nineteenth-century mathematics was that there are diIIerent,andequallyvalid,geometries-whereby"ageometry"ismeantatheoryoI properties oI abstract points and lines. It had long been assumed that geometry was what EuclidhadcodiIied,andthat,althoughtheremightbesmallIlawsinEuclid's presentation,theywereunimportantandanyrealprogressingeometrywouldbe achievedbyextendingEuclid.Thisideawasshatteredbytheroughlysimultaneous discoveryoInon-Euclideangeometrybyseveralpeople-adiscoverythatshockedthe mathematics community, because it deeply challenged the idea that mathematics studies the real world. How could there be many diIIer Introduction: A Musico-Logical OIIering28ent kinds oI "points" and "lines" in one single reality? Today, the solution to the dilemma may be apparent, even to some nonmathematicians-but at the time, the dilemma created havoc in mathematical circles. Laterinthenineteenthcentury,theEnglishlogiciansGeorgeBooleandAugustusDe Morgan went considerably Iurther than Aristotle in codiIying strictly deductive reasoning patterns. Boole even called his book "The Laws oI Thought"-surely an exaggeration, but itwasanimportantcontribution.LewisCarrollwasIascinatedbythesemechanized reasoning methods, and invented many puzzles which could be solved with them. Gottlob Frege in Jena and Giuseppe Peano in Turin worked on combining Iormal reasoning with thestudyoIsetsandnumbers.DavidHilbertinGottingenworkedonstricter IormalizationsoIgeometrythanEuclid's.AlloItheseeIIortsweredirectedtowards clariIying what one means by "prooI". In the meantime, interesting developments were taking place in classical mathematics. AtheoryoIdiIIerenttypesoIinIinities,knownasthetheoryoIsets,wasdevelopedby Georg Cantor in the 1880's. The theory was powerIul and beautiIul, but intuition-deIying. BeIore long, a variety oI set-theoretical paradoxes had been unearthed. The situation was verydisturbing,becausejustasmathematicsseemedtoberecoveringIromonesetoI paradoxes-thoserelatedtothetheoryoIlimits,inthecalculusalongcameawholenew set, which looked worse! ThemostIamousisRussell'sparadox.Mostsets,itwouldseem,arenotmembersoI themselves-Ior example, the set oI walruses is not a walrus, the set containing only Joan oI Arc is not Joan oI Arc (a set is not a person)-and so on. In this respect, most sets are rather "run-oI-the-mill". However, some "selI-swallowing" sets do contain themselves as members, such as the set oI all sets, or the set oI all things except Joan oI Arc, and so on. Clearly,everysetiseitherrun-oI-the-millorselI-swallowing,andnosetcanbeboth. Now nothing prevents us Irom inventing R: the set oI all run-o,-the-mill sets. At Iirst, R might seem a rather run-oI-the-mill invention-but that opinion must be revised when you askyourselI,"IsRitselI"arun-oI-the-millsetoraselI-swallowingset?"YouwillIind thattheansweris:"Risneitherrun-oI-the-millnorselI-swallowing,Ioreitherchoice leads to paradox." Try it! ButiIRisneitherrun-oI-the-millnorselI-swallowing,thenwhatisit?Atthevery least,pathological.ButnoonewassatisIiedwithevasiveanswersoIthatsort.Andso people began to dig more deeply into the Ioundations oI set theory. The crucial questions seemed to be: "What is wrong with our intuitive concept oI 'set'? Can we make a rigorous theoryoIsetswhichcorrespondscloselywithourintuitions,butwhichskirtsthe paradoxes?" Here, as in number theory and geometry, the problem is in trying to line up intuition with Iormalized, or axiomatized, reasoning systems. A startling variant oI Russell's paradox, called "Grelling's paradox", can be made using adjectivesinsteadoIsets.DividetheadjectivesinEnglishintotwocategories:those whichareselI-descriptive,suchas"pentasyllabic","awkwardnessIul",and"recherche", and those which are not, such Introduction: A Musico-Logical OIIering29

as "edible", "incomplete", and "bisyllabic". Now iI we admit "non-selIdescriptive" as an adjective,towhichclassdoesitbelong?IIitseemsquestionabletoincludehyphenated words,wecanusetwotermsinventedspeciallyIorthisparadox:autological("selI-descriptive"),andheterological("non-selI-descriptive").Thequestionthenbecomes: "Is 'heterological' heterological?" Try it! Thereseemstoheonecommonculpritintheseparadoxes,namelyselI-reIerence,or "StrangeLoopiness".SoiIthegoalistobanallparadoxes,whynottrybanningselI-reIerenceandanythingthatallowsittoarise?Thisisnotsoeasyasitmightseem, becauseitcanbehardtoIigureoutjustwhereselI-reIerenceisoccurring.Itmaybe spread out over a whole Strange Loop with several steps, as in this "expanded" version oI Epimenides, reminiscent oI Drawing Hands: The Iollowing sentence is Ialse. The preceding sentence is true. Taken together, these sentences have the same eIIect as the original Epimenides paradox: yet separately, they are harmless and even potentially useIul sentences. The "blame" Ior this Strange Loop can't he pinned on either sentence-only on the way they "point" at each other.Inthesameway,eachlocalregionoIAscendingandDescendingisquite legitimate; it is only the way they are globally put together that creates an impossibility. Since there are indirect as well as direct ways oI achieving selI-reIerence, one must Iigure out how to ban both types at once-iI one sees selIreIerence as the root oI all evil. Banishing Strange Loops RussellandWhiteheaddidsubscribetothisview,andaccordingly,Principia MathematicawasamammothexerciseinexorcisingStrangeLoopsIromlogic,set theory,andnumbertheory.TheideaoItheirsystemwasbasicallythis.AsetoIthe lowest "type" could contain only "objects" as membersnot sets. A set oI the next type up couldonlycontainobjects,orsetsoIthelowesttype.Ingeneral,asetoIagiventype couldonlycontainsetsoIlowertype,orobjects.EverysetwouldbelongtoaspeciIic type. Clearly, no set could contain itselI because it would have to belong to a type higher than its own type. Only "run-oI'-the-mill" sets exist in such a system; Iurthermore, old R-the set oI all run-oI-the-mill sets-no longer is considered a set at all, because it does not belong to any Iinite type. To all appearances, then, this theory oI types, which we might also call the "theory oI the abolition oI Strange Loops", successIully rids set theory oI its paradoxes,butonlyatthecostoIintroducinganartiIicial-seeminghierarchy,andoI disallowingtheIormationoIcertainkindsoIsets-suchasthesetoIallrun-oI-the-mill sets. Intuitively, this is not the way we imagine sets. The theory oI types handled Russell's paradox, but it did nothing about the Epimenides paradox or Grelling's paradox. For people whose Introduction: A Musico-Logical OIIering30interest went no Iurther than set theory, this was quite adequate-but Ior people interested intheeliminationoIparadoxesgenerally,somesimilar"hierarchization"seemed necessary,toIorbidloopingbackinsidelanguage.AtthebottomoIsuchahierarchy wouldbeanobjectlanguage.Here,reIerencecouldbemadeonlytoaspeciIicdomain-nottoaspectsoItheobjectlanguageitselI(suchasitsgrammaticalrules,orspeciIic sentences in it). For that purpose there would be a metalanguage. This experience oI two linguisticlevelsisIamiliartoalllearnersoIIoreignlanguages.Thentherewouldbea metametalanguage Ior discussing the metalanguage, and so on. It would be required that everysentenceshouldbelongtosomepreciseleveloIthehierarchy.ThereIore,iIone couldIindnolevelinwhichagivenutteranceIit,thentheutterancewouldbedeemed meaningless, and Iorgotten. Ananalysiscanbeattemptedonthetwo-stepEpimenidesloopgivenabove.TheIirst sentence, since it speaks oI the second, must be on a higher level than the second. But by the same token, the second sentence must be on a higher level than the Iirst. Since this is impossible, the two sentences are "meaningless". More precisely, such sentences simply cannotbeIormulatedatallinasystembasedonastricthierarchyoIlanguages.This prevents all versions oI the Epimenides paradox as well as Grelling's paradox. (To what language level could "heterological" belong?) Nowinsettheory,whichdealswithabstractionsthatwedon'tuseallthetime,a stratiIication like the theory oI types seems acceptable, even iI a little strange-but when it comestolanguage,anall-pervadingpartoIliIe,suchstratiIicationappearsabsurd.We don't think oI ourselves as jumping up and down a hierarchy oI languages when we speak aboutvariousthings.Arathermatter-oI-Iactsentencesuchas,"Inthisbook,Icriticize the theory oI types" would be doubly Iorbidden in the system we are discussing. Firstly, it mentions "this book", which should only be mentionable in a metabook"-andsecondly,itmentionsme-apersonwhomIshouldnotbeallowedto speakoIatall!ThisexamplepointsouthowsillythetheoryoItypesseems,whenyou import it into a Iamiliar context. The remedy it adopts Ior paradoxes-total banishment oI selI-reIerenceinanyIorm-isarealcaseoIoverkill,brandingmanyperIectlygood constructionsasmeaningless.Theadjective"meaningless",bytheway,wouldhaveto applytoalldiscussionsoIthetheoryoIlinguistictypes(suchasthatoIthisvery paragraph) Ior they clearly could not occur on any oI the levels-neither object language, normetalanguage,normetametalanguage,etc.SotheveryactoIdiscussingthetheory would be the most blatant possible violation oI it! NowonecoulddeIendsuchtheoriesbysayingthattheywereonlyintendedtodeal with Iormal languages-not with ordinary, inIormal language. This may be so, but then it showsthatsuchtheoriesareextremelyacademicandhavelittletosayaboutparadoxes except when they crop up in special tailor-made systems. Besides, the drive to eliminate paradoxesatanycost,especiallywhenitrequiresthecreationoIhighlyartiIicial Iormalisms, puts too much stress on bland consistency, and too little on the Introduction: A Musico-Logical OIIering31quirky and bizarre, which make liIe and mathematics interesting. It is oI course important totrytomaintainconsistency,butwhenthiseIIortIorcesyouintoastupendouslyugly theory, you know something is wrong. ThesetypesoIissuesintheIoundationsoImathematicswereresponsibleIorthehigh interest in codiIying human reasoning methods which was present in the early part oI this century.Mathematiciansandphilosophershadbeguntohaveseriousdoubtsabout whether even the most concrete oI theories, such as the study oI whole numbers (number theory),werebuiltonsolidIoundations.IIparadoxescouldpopupsoeasilyinset theory-atheorywhosebasicconcept,thatoIaset,issurelyveryintuitivelyappealing-thenmighttheynotalsoexistinotherbranchesoImathematics?Anotherrelatedworry wasthattheparadoxesoIlogic,suchastheEpimenidesparadox,mightturnouttobe internal to mathematics, and thereby cast in doubt all oI mathematics. This was especially worrisome to those-and there were a good number-who Iirmly believed that mathematics is simply a branch oI logic (or conversely, that logic is simply a branch oI mathematics). InIact,thisveryquestion-"Aremathematicsandlogicdistinct,orseparate"-wasthe source oI much controversy. ThisstudyoImathematicsitselIbecameknownasmetamathematics-oroccasionally, metalogic,sincemathematicsandlogicaresointertwined.ThemosturgentpriorityoI metamathematicians was to determine the true nature oI mathematical reasoning. What is alegalmethodoIprocedure,andwhatisanillegalone?Sincemathematicalreasoning had always been done in "natural language" (e.g., French or Latin or some language Ior normalcommunication),therewasalwaysalotoIpossibleambiguity.Wordshad diIIerentmeaningstodiIIerentpeople,conjuredupdiIIerentimages,andsoIorth.It seemed reasonable and even important to establish a single uniIorm notation in which all mathematicalworkcouldbedone,andwiththeaidoIwhichanytwomathematicians couldresolvedisputesoverwhetherasuggestedprooIwasvalidornot.Thiswould require a complete codiIication oI the universally acceptable modes oI human reasoning, at least as Iar as they applied to mathematics. Consistency, Completeness, Hilbert's Program This was the goal oI Principia Mathematica, which purported to derive all oI mathematics Iromlogic,and,tobesure,withoutcontradictions!Itwaswidelyadmired,butnoone wassureiI(1)alloImathematicsreallywascontainedinthemethodsdelineatedby RussellandWhitehead,or(2)themethodsgivenwereevenselI-consistent.Wasit absolutely clear that contradictory results could never be derived, by any mathematicians whatsoever, Iollowing the methods oI Russell and Whitehead? ThisquestionparticularlybotheredthedistinguishedGermanmathematician(and metamathematician)DavidHilbert,whosetbeIoretheworldcommunityoI mathematicians (and metamathematicians) this chal Introduction: A Musico-Logical OIIering32lenge: to demonstrate rigorously-perhaps Iollowing the very methods outlined by Russell andWhitehead-thatthesystemdeIinedinPrincipiaMathematicawasbothconsistent (contradiction-Iree), and complete (i.e., that every true statement oI, number theory could be derived within the Iramework drawn up in P.M.). This was a tall order, and one could criticizeitonthegroundsthatitwassomewhatcircular:howcanyoujustiIyyour methodsoIreasoningonthebasisoIthosesame methods oI reasoning? It is like liIting yourselIupbyyourownbootstraps.(Wejustdon'tseemtobeabletogetawayIrom these Strange Loops!) HilbertwasIullyawareoIthisdilemma,oIcourse,andthereIoreexpressedthehope that a demonstration oI consistency or completeness could be Iound which depended only on "Iinitistic" modes oI reasoning. "these were a small set oI reasoning methods usually acceptedbymathematicians.Inthisway,Hilberthopedthatmathematicianscould partially liIt themselves by their own bootstraps: the sum total oI mathematical methods might be proved sound, by invoking only a smaller set oI methods. This goal may sound ratheresoteric,butitoccupiedthemindsoImanyoIthegreatestmathematiciansinthe world during the Iirst thirty years oI this century. Inthethirty-Iirstyear,however,Godelpublishedhispaper,whichinsomeways utterlydemolishedHilbert'sprogram.Thispaperrevealednotonlythattherewere irreparable "holes" in the axiomatic system proposed by Russell and Whitehead, but more generally,thatnoaxiomaticsystemwhatsoevercouldproduceallnumber-theoretical truths,unlessitwereaninconsistentsystem!AndIinally,thehopeoIprovingthe consistencyoIasystemsuchasthatpresentedinP.M.wasshowntobevain:iIsucha prooIcouldbeIoundusingonlymethodsinsideP.M.,then-andthisisoneoIthemost mystiIying consequences oI Godel's work-P.M. itselI would be inconsistent! The Iinal irony oI it all is that the prooI oI Gi del's Incompleteness Theorem involved importing the Epimenides paradox right into the heart oIPrincipia Mathematica, a bastion supposedly invulnerable to the attacks oI Strange Loops! Although Godel's Strange Loop didnotdestroyPrincipiaMathematica,itmadeitIarlessinterestingtomathematicians, Ior it showed that Russell and Whitehead's original aims were illusory. Babbage, Computers, Artificial Intelligence ... When Godel's paper came out, the world was on the brink oI developing electronic digital computers. Now the idea oI mechanical calculating engines had been around Ior a while. Intheseventeenthcentury,PascalandLeibnizdesignedmachinestoperIormIixed operations(additionandmultiplication). These machines had no memory, however, and were not, in modern parlance, programmable. The Iirst human to conceive oI the immense computing potential oI machinery was the Londoner Charles Babbage (1792-1871). A character who could almost have stepped out oI the pages oI the Pickwick Papers, Introduction: A Musico-Logical OIIering33 BabbagewasmostIamousduring his liIetime Ior his vigorous campaign to rid London oI "street nuisances"-organ grinders above all. These pests, loving to get his goat, would comeandserenadehimatanytimeoIdayornight,andhewouldIuriouslychasethem downthestreet.Today,werecognizeinBabbageamanahundredyearsaheadoIhis time: not only inventor oI the basic principles oI modern computers, he was also one oI the Iirst to battle noise pollution. HisIirstmachine,the"DiIIerenceEngine",couldgeneratemathematicaltablesoI many kinds by the "method oI diIIerences". But beIore any model oI the "D.E." had been built,Babbagebecameobsessedwithamuchmorerevolutionaryidea:his"Analytical Engine". Rather immodestly, he wrote, "The course through which I arrived at it was the mostentangledandperplexedwhichprobablyeveroccupiedthehumanmind."'Unlike any previously designed machine, the A.E. was to possess both a "store" (memory) and a "mill"(calculatinganddecision-making unit). These units were to be built oI thousands oIintricategearedcylindersinterlockedinincrediblycomplexways.Babbagehada vision oI numbers swirling in and out oI the mill tinder control oI a program contained in punchedcards-anideainspiredbythejacquardloom,acard-controlledloomthatwove amazinglycomplexpatterns.Babbage's brilliant but ill-Iated Countess Iriend, Lady Ada Lovelace(daughteroILordByron),poeticallycommentedthat"theAnalyticalEngine weavesalgebraicpatternsjustastheJacquard-loomweavesIlowersandleaves." UnIortunately,heruseoIthepresenttensewasmisleading,IornoA.E.waseverbuilt, and Babbage died a bitterly disappointed man. Lady Lovelace, no less than Babbage, was proIoundly aware that with the invention oI the Analytical Engine, mankind was Ilirting with mechanized intelligence-particularly iI the Engine were capable oI "eating its own tail" (the way Babbage described the Strange Loop created when a machine reaches in and alters its own stored program). In an 1842 memoir,5shewrotethattheA.E."mightactuponotherthingsbesidesnumber".While BabbagedreamtoIcreatingachessortic-tac-toeautomaton,shesuggestedthathis Engine,withpitchesandharmoniescodedintoitsspinningcylinders,"mightcompose elaborate and scientiIic pieces oI music oI any degree oI complexity or extent." In nearly thesamebreath,however,shecautionsthat"TheAnalyticalEnginehasnopretensions whatever to originate anything. It can do whatever we know how to order it to perIorm." ThoughshewellunderstoodthepoweroIartiIicialcomputation,LadyLovelacewas skepticalabouttheartiIicialcreationoIintelligence.However,couldherkeeninsight allowhertodreamoIthepotentialthatwouldbeopenedupwiththetamingoI electricity? In our century the time was ripe Ior computers-computers beyond the wildest dreams oI Pascal,Leibniz,Babbage,orLadyLovelace.Inthe1930'sand1940's,theIirst"giant electronicbrains"weredesignedandbuilt.TheycatalyzedtheconvergenceoIthree previouslydisparateareas:thetheoryoIaxiomaticreasoning,thestudyoImechanical computation, and the psychology oI intelligence. These same years saw the theory oI computers develop by leaps and Introduction: A Musico-Logical OIIering34bounds. This theory was tightly linked to metamathematics. In Iact, Godel's Theorem has a counterpart in the theory oI computation, discovered by Alan Turing, which reveals the existenceoIinelucPable"holes"ineventhemostpowerIulcomputerimaginable. Ironically,justasthesesomewhateerielimitswerebeingmappedout,realcomputers were being built whose powers seemed to grow and grow beyond their makers' power oI prophecy. Babbage, who once declared he would gladly give up the rest oI his liIe iI he could come back in Iive hundred years and have a three-day guided scientiIic tour oI the newage,wouldprobablyhavebeenthrilledspeechlessamerecenturyaIterhisdeath-both by the new machines, and by their unexpected limitations. Bytheearly1950's,mechanizedintelligenceseemedamerestone'sthrowaway;and yet,Ioreachbarriercrossed,therealwayscroppedupsomenewbarriertotheactual creationoIagenuinethinkingmachine.WastheresomedeepreasonIorthisgoal's mysterious recession? Nooneknowswheretheborderlinebetweennon-intelligentbehaviorandintelligent behaviorlies;inIact,tosuggestthatasharpborderlineexistsisprobablysilly.But essential abilities Ior intelligence are certainly: to respond to situations very Ilexibly; to take advantage oI Iortuitous circumstances; to make sense out oI ambiguous or contradictory messages; to recognize the relative importance oI diIIerent elements oI a situation; to Iind similarities between situations despite diIIerences which may separate them; to draw distinctions between situations despite similarities may link them; tosynthesizenewconceptsbytakingoldthemtogetherinnewways;tocomeup with ideas which are novel. Hereonerunsupagainstaseemingparadox.Computersbytheirverynaturearethe mostinIlexible,desireless,rule-IollowingoIbeasts.Fastthoughtheymaybe,theyare nonethelesstheepitomeoIunconsciousness.How,then,canintelligentbehaviorbe programmed?Isn'tthisthemostblatantoIcontradictionsinterms?OneoIthemajor theses oI this book is that it is not a contradiction at all. One oI the major purposes oI this book is to urge each reader to conIront the apparent contradiction head on, to savor it, to turnitover,totakeitapart,towallowinit,sothatintheendthereadermightemerge withnewinsightsintotheseeminglyunbreathablegulIbetweentheIormalandthe inIormal, the animate and the inanimate, the Ilexible and the inIlexible. This is what ArtiIicial Intelligence (A1) research is all about. And the strange Ilavor oI AI work is that people try to put together long sets oI rules in strict Iormalisms which tell inIlexible machines how to be Ilexible. WhatsortsoI"rules"couldpossiblycapturealloIwhatwethinkoIasintelligent behavior, however? Certainly there must be rules on all sorts oI Introduction: A Musico-Logical OIIering35 diIIerentlevels.Theremustbemany"justplain"rules.Theremustbe"metarules"to modiIythe"justplain"rules;then"metametarules"tomodiIythemetarules,andsoon. TheIlexibilityoIintelligencecomesIromtheenormousnumberoIdiIIerentrules,and levelsoIrules.ThereasonthatsomanyrulesonsomanydiIIerentlevelsmustexistis that in liIe, a creature is Iaced with millions oI situations oI completely diIIerent types. In somesituations,therearestereotypedresponseswhichrequire"justplain"rules.Some situationsaremixturesoIstereotypedsituations-thustheyrequirerulesIordeciding whichoIthe'justplain"rulestoapply.SomesituationscannotbeclassiIied-thusthere must exist rules Ior inventing new rules ... and on and on. Without doubt, Strange Loops involvingrulesthatchangethemselves,directlyorindirectly,areatthecoreoI intelligence.SometimesthecomplexityoIourmindsseemssooverwhelmingthatone Ieels that there can be no solution to the problem oI understanding intelligence-that it is wrong to think that rules oI any sort govern a creature's behavior, even iI one takes "rule" in the multilevel sense described above. ...and Bach In the year 1754, Iour years aIter the death oI J. S. Bach, the Leipzig theologian Johann MichaelSchmidtwrote,inatreatiseonmusicandthesoul,theIollowingnoteworthy passage: NotmanyyearsagoitwasreportedIromFrancethatamanhadmadeastatuethat could play various pieces on the Fleuttraversiere, placed the Ilute to its lips and took it down again, rolled its eyes, etc. But no one has yet invented an image that thinks, or wills, or composes, or even does anything at all similar. Let anyone who wishes to be convinced look careIully at the last Iugal work oI the above-praised Bach, which has appearedincopperengraving,butwhichwasleItunIinishedbecausehisblindness intervened,andlethimobservetheartthatiscontainedtherein;orwhatmuststrike him as even more wonderIul, the Chorale which he dictated in his blindness to the pen oIanother:WennwirinhochstenNothenseen.Iamsurethathewillsoonneedhis soul iI he wishes to observe all the beauties contained therein, let alone wishes to play it to himselI or to Iorm a judgment oI the author. Everything that the champions oI Materialism put Iorward must Iall to the ground in view oI this single example.6 Quitelikely,theIoremostoIthe"championsoIMaterialism"herealludedtowasnone otherthanJulienOIIroydelaMettrie-philosopheratthecourtoIFredericktheGreat, author oI L'homme machine ("Man, the Machine"), and Materialist Par Excellence. It is now more than 200 years later, and the battle is still raging between those who agree with Johann Michael Schmidt, and those who agree with Julien OIIroy de la Mettrie. I hope in this book to give some perspective on the battle. "Godel, Escher, Bach" Thebookisstructuredinanunusualway:asacounterpointbetweenDialoguesand Chapters. The purpose oI this structure is to allow me to Introduction: A Musico-Logical OIIering36presentnewconceptstwice:almosteverynewconceptisIirstpresentedmetaphorically inaDialogue,yieldingasetoIconcrete,visualimages;thentheseserve,duringthe readingoItheIollowing`Chapter,asanintuitivebackgroundIoramoreseriousand abstract presentation oI the same concept. In many oI the Dialogues I appear to be talking about one idea on the surIace, but in reality I am talking about some other idea, in a thinly disguised way. Originally,theonlycharactersinmyDialogueswereAchillesandtheTortoise,who cametomeIromZenooIElea,bywayoILewisCarroll.ZenooIElea,inventoroI paradoxes,livedintheIiIthcenturyB.C.OneoIhisparadoxeswasanallegory,with Achilles and the Tortoise as protagonists. Zeno's invention oI the happy pair is told in my Iirst Dialogue, Three-Part Invention. In 1895, Lewis Carroll reincarnated Achilles and the Tortoise Ior the purpose oI illustrating his own new paradox oI inIinity. Carroll's paradox, whichdeservestobeIarbetterknownthanitis,playsasigniIicantroleinthisbook. Originallytitled"WhattheTortoiseSaidtoAchilles",itisreprintedhereasTwo-Part Invention. When I began writing Dialogues, somehow I connected them up with musical Iorms. I don't remember the moment it happened; I just remember one day writing "Fugue" above an early Dialogue, and Irom then on the idea stuck. Eventually I decided to pattern each DialogueinonewayoranotheronadiIIerentpiecebyBach.Thiswasnotso inappropriate. Old Bach himselI used to remind his pupils that the separate parts in their compositionsshouldbehavelike"personswhoconversedtogetherasiIinaselect company". I have taken that suggestion perhaps rather more literally than Bach intended it;neverthelessIhopetheresultisIaithIultothemeaning.Ihavebeenparticularly inspiredbyaspectsoIBach'scompositionswhichhavestruckmeoverandover,and which are so well described by David and Mendel in The Bach Reader: HisIormingeneralwasbasedonrelationsbetweenseparatesections.Theserelations ranged Irom complete identity oI passages on the one hand to the returnoIasingleprincipleoIelaborationoramerethematicallusionontheother.The resulting patterns were oIten symmetrical, but by no means necessarily so. Sometimes the relations between the various sections make up a maze oI interwoven threads that only detailed analysis can unravel. Usually, however,aIewdominantIeaturesaIIordproper orientation at Iirst sight or hearing, and while in the course oI study one may discover unending sub tleties, one is never at a loss to grasp the unity that holds together every single creation by Bach.' IhavesoughttoweaveanEternalGoldenBraidoutoIthesethreestrands:Godel, Escher, Bach. I began, intending to write an essay at the core oI which would be Godel's Theorem. I imagined it would be a mere pamphlet. But my ideas expanded like a sphere, andsoontouchedBachandEscher.IttooksometimeIormetothinkoImakingthis connectionexplicit,insteadoIjustlettingitbeaprivatemotivatingIorce.ButIinally1 realizedthattome,GodelandEscherandBachwereonlyshadowscastindiIIerent directionsbysomecentralsolidessence.Itriedtoreconstructthecentralobject,and came up with this book. Three-Part Invention 37!"#$$%&'#( *+,$+(-.+ Achilles (a Greek warrior, the fleetest of foot of all mortals) and a Tortoise are standing together on a dustv runwav in the hot sun. Far down the runwav, on a tall flagpole, there hangs a large rectangular flag. The flagis sold red, except where a thin ring-shaped holes has been cut out of it, through which one can see the skv. ACHILLES:WhatisthatstrangeIlagdownattheotherendoIthetrack?Itremindsme somehow oI a print by my Iavourite artists M.C. Escher. TORTOISE:That is Zeno` s IlagACHILLES:Could it be that the hole in it resembles the holes in a Mobian strip Escher once drew? Something is wrong about the Ilag, I can tell. TORTOISE:The ring which has been cut Irom it has the shape oI the numeral Ior zero, which is Zenos Iavourite number. ACHILLES:Theringwhichhasntbeeninventedyet!ItwillonlybeinventedbyaHindu mathematiciansomemillenniahence.Andthus,Mr.T,mtargumentprovesthatsucha Ilag is impossible. TORTOISE:Your argument is persuasive, Achilles, and I must agree that such aIlag is indeed impossible. But it is beautiIul anyway, is it not? ACHILLES:Oh, yes, there is no doubt oI its beauty. TORTOISE:I wonder iI its beauty is related to its impossibility. I dont know, Ive never had the time to analyze Beauty. Its a Capitalized Essence, and I never seem to have time Ior Capitalized Essences. ACHILLES:SpeakingoICapitalizedEssences,Mr.T,haveyoueverwonderedaboutthe Purpose oI LiIe? TORTOISE:Oh, heavens, no; ACHILLES:Haven`t you ever wondered why we are here, or who invented us? TORTOISE:Oh,thatisquiteanothermatter.WeareinventionsoIZeno(asyouwillshortly see) and the reason we are here is to have a Iootrace. ACHILLES:::A Iootrace?How outrageous! Me, the Ileetest oI Ioot oI all mortals, versus you, the ploddingest oI the plodders! There can be no pointto such a race. TORTOISE:You might give me a head start. ACHILLES:It would have to be a huge one. TORTOISE:I don`t object. ACHILLES:But I will catch you, sooner or latermost likely sooner. TORTOISE:NotiIthingsgoaccordingtoZenosparadox,youwon`t.Zenoishopingtouse our Iootrace to show that motion is impossible, you see. It is only in the mind that motion seems possible, according to Zeno. In truth, Motion Is Inherently Impossible.He proves it quite elegantly. Three-Part Invention 38 Figure 10. Mobius strip by M.C.Escher (wood-engraving printed Irom Iour blocks, 1961) ACHILLES:Oh,yes,itcomesbacktomenow:theIamousZenkoanaboutZen Master Zeno. As you say it is very simple indeed. TORTOISE:Zen Koan? Zen Master? What do you mean? ACHILLES:Itgoeslikethis:TwomonkswerearguingaboutaIlag.Onesaid,'The Ilagismoving.Theothersaid,'Thewindismoving.Thesixthpatriarch,Zeno, happenedtobepassingby.Hetoldthem,'Notthewind,nottheIlag,mindis moving. TORTOISE:I am aIraid you are a little beIuddled, Achilles. Zeno is no Zen master, Iar Irom it. He is in Iact, a Greek philosopher Irom the town oI Elea (which lies halIway between points A and B). Centuries hence, he will be celebrated Ior his paradoxes oI motion. In one oI those paradoxes, this very Iootrace between you and me will play a central role. ACHILLES:I`mallconIused.IremembervividlyhowIusedtorepeatoverandover the names oI the six patriarchs oI Zen, and I always said, 'The sixth patriarch is Zeno, The sixth patriarch is Zeno. (Suddenlv a soft warm bree:e picks up.) Oh, look Mr. Tortoiselook at the Ilag waving! How I love to watch the ripples shimmer through it`s soIt Iabric. And the ring cut out oI it is waving, too! Three-Part Invention 39TORTOISE:Dontbesilly.TheIlagisimpossible,henceitcan`tbewaving.Thewindis waving. (At this moment, Zeno happens bv.) Zeno:Hallo! Hulloo! What`s up? What`s new? ACHILLES: The Ilag is moving. TORTOISE:The wind is moving. Zeno:OIriends,Friends!Ceaseyourargumentation!Arrestyourvitriolics!Abandonyour discord! For I shall resolve the issue Ior you Iorthwith. Ho! And on such a Iine day. ACHILLES:This Iellow must be playing the Iool. TORTOISE:No, wait, Achilles. Let us hear what he has to say. Oh Unknown Sir, do impart to us your thoughts on this matter. Zeno:Most willingly. Not thw ind, not the Ilagneither one is moving, nor is anything moving atall.ForIhavediscoveredagreatTheorem,whichstates;'MotionIsInherently Impossible.AndIromthisTheoremIollowsanevengreaterTheoremZeno`s Theorem: 'Motion Unexists. ACHILLES:'Zeno`s Theorem? Are you, sir, by any chance, the philosopher Zeno oI Elea? Zeno:I am indeed, Achilles. ACHILLES: (scratching his head in pu::lement). Now how did he know my name? Zeno:Could I possibly persuade you two to hear me out as to why this is the case? I`ve come allthewaytoEleaIrompointAthisaIternoon,justtryingtoIindsomeonewho`llpay some attention to my closely honed argument. But they`re all hurrying hither and thither, andtheydon`thavetime.You`venoideahowdishearteningitistomeetwithreIusal aIter reIusal. Oh, I`m sorry to burden you with my troubles, I`d just like to ask you one thing:WouldthetwooIyouhumourasilloldphilosopherIoraIewmomentsonlya Iew, I promise youin his eccentric theories. ACHILLES:Oh, by all means! Please do illuminate us! I know I speak Ior both oI us, since my companion, Mr. Tortoise, was only moments ago speaking oI you with great venerationand he mentioned especially your paradoxes. Zeno:Thankyou.Yousee,myMaster,theIiIthpatriarch,taughtmethatrealityisone, immutable,andunchanging,allplurality,change,andmotionaremereillusionsoIthe sense. Some have mocked his views; but I will show the absurdity oI their mockery. My argumentisquitesimple.IwillillustrateitwithtwocharactersoImyownInvention: Achilles )a Greek warrior, the Ileetest oI Ioot oI all mortals), and a Tortoise. In my tale, they are persuaded by a passerby to run a Iootrace down a runway towards a distant Ilag waving in the breeze. Let us assume that, since the Tortoise is a much slowerrunner, he getsaheadstartoI,say,tenrods.Nowtheracebegins.InaIewboundsAchilleshas reached the spot where the Tortoisestarted. Three-Part Invention 40ACHILLES: Hah! Zeno:AndnowtheTortoiseisbutasinglerodaheadoIAchilles.Withinonlyamoment, Achilles has attained that spot. ACHILLES: Ho ho! Zeno:Yet,inthatshortmoment,theTortoisehasmanagedtoadvanceaslightamount.Ina Ilash, Achilles covers that distance too. ACHILLES:Hee hee hee! Zeno:But in that very short Ilash, the Tortoise has managed to inch ahead by ever so little, and soAchillesisstillbehind.NowyouseethatinorderIorAchillestocatchtheTortoise, thisgameoI'try-to-catch-mewillhavetobeplayedanINFINITEnumberoItimes and thereIore Achilles can NEVER catch up with the Tortoise. TORTOISE:Heh heh heh heh! ACHILLES:Hmm.Hmm.Hmm.Hmm.Hmm.Thatargumentsoundswrongtome. And yes, I can`t quite make out what`s wrong with it Zeno:Isn`t it a teaser? It`s my Iavourite paradox. TORTOISE:Excuseme,Zeno,butIbelieveyourtaleillustratesthewrongprinciple,doesit not? You have just told us what will come to known, centuries hence, as Zeno`s 'Achilles paradox , which shows (ahem!) that Achilles will never catch the Tortoise; but the prooI thatMotionIsInherentlyImpossible(andthencethatMotionUnexists)isyour 'dichotomy paradox, isn`t that so? Zeno:Oh, shame on me. OI course, you`re right. That`s the new one about how, in going Irom A to B, one has to go halIway Iirstand oI that stretch one also has to go halIway, and so on and so Iorth. But you see, both those paradoxes really have the same Ilavour. Frankly, I`ve only had one Great IdeaI just exploit it in diIIerent ways. ACHILLES:I swear, these arguments contain a Ilaw. I don`t quite see where, but they cannot be correct. Zeno:YoudoubtthevalidityoImyparadox?Whynotjusttryitout,?YouseethatredIlag waving down here, at the Iar end oI the runway? ACHILLES:The impossible one, based on an Escher print? Zeno:Exactly.WhatdoyousaytoyouandMr.TortoiseracingIorit,allowingMr.TaIair head start oI, well, I don`t knowTORTOISE:How about ten rods? Zeno:Very goodten rods. ACHILLES:Any time. Zeno:Excellent!Howexciting!AnempiricaltestoImyrigorouslyprovenTheorem!Mr. Tortoise, will you position yourselI ten rods upwind? (The Tortoise moves ten rods closer to the flag) Tortoise and Achlles:Ready! Zeno:On your mark! Get set! Go! The MU-puzzle41Chapter 1 The MU-puzzle Formal Systems ONEOFTHEmostcentralnotionsinthisbookisthatoIaformalsvstem.ThetypeoI Iormal system I use was invented by the American logician Emil Post in the 1920's, and isoItencalleda"Postproductionsystem".ThisChapterintroducesyoutoaIormal systemandmoreover,itismyhopethatyouwillwanttoexplorethisIormalsystemat least a little; so to provoke your curiosity, I have posed a little puzzle. "Can you produce MU?" is the puzzle. To begin with, you will be supplied with a string (which means a string oI letters).* Not to keep you in suspense, that string will be MI. Then you will be told some rules, with which you can change one string into another. IIoneoIthoserulesisapplicableatsomepoint,andyouwanttouseit,youmay,but-thereisnothingthatwilldictatewhichruleyoushoulduse,incasethereareseveral applicablerules.ThatisleItuptoyou-andoIcourse,thatiswhereplayingthegameoI anyIormalsystemcanbecomesomethingoIanart.Themajorpoint,whichalmost doesn'tneedstating,isthatyoumustnotdoanythingwhichisoutsidetherules.We mightcallthisrestrictionthe"RequirementoIFormality".InthepresentChapter,it probably won't need to be stressed at all. Strange though it may sound, though, I predict thatwhenyouplayaroundwithsomeoItheIormalsystemsoIChapterstocome,you will Iind yourselI violating the Requirement oI Formality over and over again, unless you have worked with Iormal systems beIore. TheIirstthingtosayaboutourIormalsystem-the/*0-svstem-isthatitutilizes only three letters oI the alphabet: M, I, U. That means that the only strings oI the MIU-system are strings which are composed oI those three letters. Below are some strings oI the MIU-system: MU UIM MUUMUU UIIUMIUUIMUIIUMIUUIMUIIU *Inthisbook,weshallemploytheIollowingconventionswhenwereIertostrings.Whenthe stringisinthesametypeIaceasthetext,thenitwillbeenclosedinsingleordoublequotes. Punctuation which belongs to the sentence and not to the string under discussion will go outside oI the quotes, as logic dictates. For example, the Iirst letter oI this sentence is 'F', while the Iirst letteroI'thissentence`.is't'.WhenthestringisinQuadrataRoman,however,quoteswill usually be leIt oII, unless clarity demands them. For example, the Iirst letter oI Quadrata is Q. The MU-puzzle42ButalthoughalloIthesearelegitimatestrings,theyarenotstringswhichare"inyour possession".InIact,theonlystringinyourpossessionsoIarisMI.Onlybyusingthe rules,abouttobeintroduced,canyouenlargeyourprivatecollection.HereistheIirst rule: RULE I: II you possess a string whose last letter is I, you can add on a U at the end. By the way, iI up to this point you had not guessed it, a Iact about the meaning oI "string" is that the letters are in a Iixed order. For example, MI and IM are two diIIerent strings. A string oI symbols is not just a "bag" oI symbols, in which the order doesn't make any diIIerence. Here is the second rule: RULE II: Suppose you have Mx. Then you may add Mxx to your collection. What I mean by this is shown below, in a Iew examples. From MIU, you may get MIUIU.From MUM, you may get MUMUM.From MU, you may get MUU. So the letter `x' in the rule simply stands Ior any string; but once you have decided which stringitstandsIor,youhavetostickwithyourchoice(untilyouusetheruleagain,at which point you may make a new choice). Notice the third