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

Contents

Overview

viii

ListofIllustrations

xiv

WordsofThanks

xix

PartI:GEB

Introduction:AMusico-LogicalOffering

3

Three-PartInvention

29

ChapterI:TheMU-puzzle

33

Two-PartInvention

43

ChapterII:MeaningandForminMathematics

46

SonataforUnaccompaniedAchilles

61

ChapterIII:FigureandGround

64

Contracrostipunctus

75

ChapterIV:Consistency,Completeness,andGeometry

82

LittleHarmonicLabyrinth

103

ChapterV:RecursiveStructuresandProcesses

127

CanonbyIntervallicAugmentation

153

ChapterVI:TheLocationofMeaning

158

ChromaticFantasy,AndFeud

177

ChapterVII:ThePropositionalCalculus

181

CrabCanon

199

ChapterVIII:TypographicalNumberTheory

204

AMuOffering

231

ChapterIX:MumonandGödel

246

Contents II

PartIIEGB

Prelude...

275

ChapterX:LevelsofDescription,andComputerSystems

285

AntFugue

311

ChapterXI:BrainsandThoughts

337

EnglishFrenchGermanSuit

366

ChapterXII:MindsandThoughts

369

AriawithDiverseVariations

391

ChapterXIII:BlooPandFlooPandGlooP

406

AironG'sString

431

ChapterXIV:OnFormallyUndecidablePropositionsofTNT

andRelatedSystems

438

BirthdayCantatatata...

461

ChapterXV:JumpingoutoftheSystem

465

EdifyingThoughtsofaTobaccoSmoker

480

ChapterXVI:Self-RefandSelf-Rep

495

TheMagnfierab,Indeed

549

ChapterXVII:Church,Turing,Tarski,andOthers

559

SHRDLU,ToyofMan'sDesigning

586

ChapterXVIII:ArtificialIntelligence:Retrospects

594

Contrafactus

633

ChapterXIX:ArtificialIntelligence:Prospects

641

SlothCanon

681

ChapterXX:StrangeLoops,OrTangledHierarchies

684

Six-PartRicercar

720

Notes

743

Bibliography

746

Credits

757

Index

759

Contents III

Overview

PartI:GEB

Introduction:AMusico-LogicalOffering.ThebookopenswiththestoryofBach'sMusicalOffering.BachmadeanimpromptuvisittoKingFredericktheGreatofPrussia,andwasrequestedtoimproviseuponathemepresentedbytheKing.Hisimprovisationsformedthebasisofthatgreatwork.TheMusicalOfferinganditsstoryformathemeuponwhichI"improvise"throughoutthebook,thusmakingasortof"MetamusicalOffering".Self-referenceandtheinterplaybetweendifferentlevelsinBacharediscussed:thisleadstoadiscussionofparallelideasinEscher'sdrawingsandthenGödel’sTheorem.AbriefpresentationofthehistoryoflogicandparadoxesisgivenasbackgroundforGödel’sTheorem.Thisleadstomechanicalreasoningandcomputers,andthedebateaboutwhetherArtificialIntelligenceispossible.Iclosewithanexplanationoftheoriginsofthebook-particularlythewhyandwhereforeoftheDialogues.

Three-PartInvention.Bachwrotefifteenthree-partinventions.Inthisthree-partDialogue,theTortoiseandAchilles-themainfictionalprotagonistsintheDialogues-are"invented"byZeno(asinfacttheywere,toillustrateZeno'sparadoxesofmotion).Veryshort,itsimplygivestheflavoroftheDialoguestocome.

ChapterI:TheMU-puzzle.Asimpleformalsystem(theMIL'-system)ispresented,andthereaderisurgedtoworkoutapuzzletogainfamiliaritywithformalsystemsingeneral.Anumberoffundamentalnotionsareintroduced:string,theorem,axiom,ruleofinference,derivation,formalsystem,decisionprocedure,workinginside/outsidethesystem.

Two-PartInvention.Bachalsowrotefifteentwo-partinventions.Thistwo-partDialoguewaswrittennotbyme,butbyLewisCarrollin1895.CarrollborrowedAchillesandtheTortoisefromZeno,andIinturnborrowedthemfromCarroll.Thetopicistherelationbetweenreasoning,reasoningaboutreasoning,reasoningaboutreasoningaboutreasoning,andsoon.Itparallels,inaway,Zeno'sparadoxesabouttheimpossibilityofmotion,seemingtoshow,byusinginfiniteregress,thatreasoningisimpossible.Itisabeautifulparadox,andisreferredtoseveraltimeslaterinthebook.

ChapterII:MeaningandForminMathematics.Anewformalsystem(thepq-system)ispresented,evensimplerthantheMIU-systemofChapterI.Apparentlymeaninglessatfirst,itssymbolsaresuddenlyrevealedtopossessmeaningbyvirtueoftheformofthetheoremstheyappearin.Thisrevelationisthefirstimportantinsightintomeaning:itsdeepconnectiontoisomorphism.Variousissuesrelatedtomeaningarethendiscussed,suchastruth,proof,symbolmanipulation,andtheelusiveconcept,"form".

SonataforUnaccompaniedAchilles.ADialoguewhichimitatestheBachSonatasforunaccompaniedviolin.Inparticular,Achillesistheonlyspeaker,sinceitisatranscriptofoneendofatelephonecall,atthefarendofwhichistheTortoise.Theirconversationconcernstheconceptsof"figure"and"ground"invarious

Overview IV

contexts-e.g.,Escher'sart.TheDialogueitselfformsanexampleofthedistinction,sinceAchilles'linesforma"figure",andtheTortoise'slines-implicitinAchilles'lines-forma"ground".

ChapterIII:FigureandGround.Thedistinctionbetweenfigureandgroundinartiscomparedtothedistinctionbetweentheoremsandnontheoremsinformalsystems.Thequestion"Doesafigurenecessarilycontainthesameinformationasitsground%"leadstothedistinctionbetweenrecursivelyenumerablesetsandrecursivesets.

Contracrostipunctus.ThisDialogueiscentraltothebook,foritcontainsasetofparaphrasesofGödel’sself-referentialconstructionandofhisIncompletenessTheorem.OneoftheparaphrasesoftheTheoremsays,"Foreachrecordplayerthereisarecordwhichitcannotplay."TheDialogue'stitleisacrossbetweentheword"acrostic"andtheword"contrapunctus",aLatinwordwhichBachusedtodenotethemanyfuguesandcanonsmakinguphisArtoftheFugue.SomeexplicitreferencestotheArtoftheFuguearemade.TheDialogueitselfconcealssomeacrostictricks.

ChapterIV:Consistency,Completeness,andGeometry.TheprecedingDialogueisexplicatedtotheextentitispossibleatthisstage.Thisleadsbacktothequestionofhowandwhensymbolsinaformalsystemacquiremeaning.ThehistoryofEuclideanandnon-Euclideangeometryisgiven,asanillustrationoftheelusivenotionof"undefinedterms".Thisleadstoideasabouttheconsistencyofdifferentandpossibly"rival"geometries.Throughthisdiscussionthenotionofundefinedtermsisclarified,andtherelationofundefinedtermstoperceptionandthoughtprocessesisconsidered.

LittleHarmonicLabyrinth.ThisisbasedontheBachorganpiecebythesamename.Itisaplayfulintroductiontothenotionofrecursive-i.e.,nestedstructures.Itcontainsstorieswithinstories.Theframestory,insteadoffinishingasexpected,isleftopen,sothereaderisleftdanglingwithoutresolution.Onenestedstoryconcernsmodulationinmusic-particularlyanorganpiecewhichendsinthewrongkey,leavingthelistenerdanglingwithoutresolution.

ChapterV:RecursiveStructuresandProcesses.Theideaofrecursionispresentedinmanydifferentcontexts:musicalpatterns,linguisticpatterns,geometricstructures,mathematicalfunctions,physicaltheories,computerprograms,andothers.

CanonbyIntervallicAugmentation.AchillesandtheTortoisetrytoresolvethequestion,"Whichcontainsmoreinformation-arecord,orthephonographwhichplaysitThisoddquestionariseswhentheTortoisedescribesasinglerecordwhich,whenplayedonasetofdifferentphonographs,producestwoquitedifferentmelodies:B-A-C-HandC-A-G-E.Itturnsout,however,thatthesemelodiesare"thesame",inapeculiarsense.

ChapterVI:TheLocationofMeaning.Abroaddiscussionofhowmeaningissplitamongcodedmessage,decoder,andreceiver.ExamplespresentedincludestrandsofDNA,undecipheredinscriptionsonancienttablets,andphonographrecordssailingoutinspace.Therelationshipofintelligenceto"absolute"meaningispostulated.

ChromaticFantasy,AndFeud.AshortDialoguebearinghardlyanyresemblance,exceptintitle,toBach'sChromaticFantasyandFugue.Itconcernstheproperwaytomanipulatesentencessoastopreservetruth-andinparticularthequestion

Overview V

ofwhetherthereexistrulesfortheusageoftheword"arid".ThisDialoguehasmuchincommonwiththeDialoguebyLewisCarroll.

ChapterVII:ThePropositionalCalculus.Itissuggestedhowwordssuchas.,and"canbegovernedbyformalrules.Onceagain,theideasofisomorphismandautomaticacquisitionofmeaningbysymbolsinsuchasystemarebroughtup.AlltheexamplesinthisChapter,incidentally,are"Zentences"-sentencestakenfromZenkoans.Thisispurposefullydone,somewhattongue-in-cheek,sinceZenkoansaredeliberatelyillogicalstories.

CrabCanon.ADialoguebasedonapiecebythesamenamefromtheMusicalOffering.Botharesonamedbecausecrabs(supposedly)walkbackwards.TheCrabmakeshisfirstappearanceinthisDialogue.ItisperhapsthedensestDialogueinthebookintermsofformaltrickeryandlevel-play.Gödel,Escher,andBacharedeeplyintertwinedinthisveryshortDialogue.

ChapterVIII:TypographicalNumberTheory.AnextensionofthePropositionalCalculuscalled"TNT"ispresented.InTNT,number-theoreticalreasoningcanbedonebyrigidsymbolmanipulation.Differencesbetweenformalreasoningandhumanthoughtareconsidered.

AMuOffering.ThisDialogueforeshadowsseveralnewtopicsinthebook.OstensiblyconcernedwithZenBuddhismandkoans,itisactuallyathinlyveileddiscussionoftheoremhoodandnontheoremhood,truthandfalsity,ofstringsinnumbertheory.Therearefleetingreferencestomolecularbiology-particular)theGeneticCode.ThereisnocloseaffinitytotheMusicalOffering,otherthaninthetitleandtheplayingofself-referentialgames.

ChapterIX:MumonandGödel.AnattemptismadetotalkaboutthestrangeideasofZenBuddhism.TheZenmonkMumon,whogavewellknowncommentariesonmanykoans,isacentralfigure.Inaway,Zenideasbearametaphoricalresemblancetosomecontemporaryideasinthephilosophyofmathematics.Afterthis"Zennery",Gödel’sfundamentalideaofGödel-numberingisintroduced,andafirstpassthroughGödel’sTheoremismade.

PartII:EGB

Prelude...ThisDialogueattachestothenextone.TheyarebasedonpreludesandfuguesfromBach'sWell-TemperedClavier.AchillesandtheTortoisebringapresenttotheCrab,whohasaguest:theAnteater.ThepresentturnsouttobearecordingoftheW.T.C.;itisimmediatelyputon.Astheylistentoaprelude,theydiscussthestructureofpreludesandfugues,whichleadsAchillestoaskhowtohearafugue:asawhole,orasasumofparts?Thisisthedebatebetweenholismandreductionism,whichissoontakenupintheAntFugue.

ChapterX:LevelsofDescription,andComputerSystems.Variouslevelsofseeingpictures,chessboards,andcomputersystemsarediscussed.Thelastoftheseisthenexaminedindetail.Thisinvolvesdescribingmachinelanguages,assemblylanguages,compilerlanguages,operatingsystems,andsoforth.Thenthediscussionturnstocompositesystemsofothertypes,suchassportsteams,nuclei,atoms,theweather,andsoforth.Thequestionarisesastohowmanintermediatelevelsexist-orindeedwhetheranyexist.

Overview VI

…AntFugue.Animitationofamusicalfugue:eachvoiceenterswiththesamestatement.Thetheme-holismversusreductionism-isintroducedinarecursivepicturecomposedofwordscomposedofsmallerwords.etc.Thewordswhichappearonthefourlevelsofthisstrangepictureare"HOLISM","REDLCTIONIsM",and"ML".ThediscussionveersofftoafriendoftheAnteater'sAuntHillary,aconsciousantcolony.Thevariouslevelsofherthoughtprocessesarethetopicofdiscussion.ManyfugaltricksareensconcedintheDialogue.Asahinttothereader,referencesaremadetoparalleltricksoccurringinthefugueontherecordtowhichthefoursomeislistening.AttheendoftheAntFugue,themesfromthePreludereturn.transformedconsiderably.

ChapterXI:BrainsandThoughts."HowcanthoughtshesupportedbythehardwareofthebrainisthetopicoftheChapter.Anoverviewofthelargescaleandsmall-scalestructureofthebrainisfirstgiven.Thentherelationbetweenconceptsandneuralactivityisspeculativelydiscussedinsomedetail.

EnglishFrenchGermanSuite.AninterludeconsistingofLewisCarroll'snonsensepoem"Jabberwocky`'togetherwithtwotranslations:oneintoFrenchandoneintoGerman,bothdonelastcentury.

ChapterXII:MindsandThoughts.Theprecedingpoemsbringupinaforcefulwaythequestionofwhetherlanguages,orindeedminds,canbe"mapped"ontoeachother.Howiscommunicationpossiblebetweentwoseparatephysicalbrains:Whatdoallhumanbrainshaveincommon?Ageographicalanalogyisusedtosuggestananswer.Thequestionarises,"Canabrainbeunderstood,insomeobjectivesense,byanoutsider?"

AriawithDiverseVariations.ADialoguewhoseformisbasedonBach'sGoldbergVariations,andwhosecontentisrelatedtonumber-theoreticalproblemssuchastheGoldbachconjecture.Thishybridhasasitsmainpurposetoshowhownumbertheory'ssubtletystemsfromthefactthattherearemanydiversevariationsonthethemeofsearchingthroughaninfinitespace.Someofthemleadtoinfinitesearches,someofthemleadtofinitesearches,whilesomeothershoverinbetween.

ChapterXIII:BlooPandFlooPandGlooP.Thesearethenamesofthreecomputerlanguages.BlooPprogramscancarryoutonlypredictablyfinitesearches,whileFlooPprogramscancarryoutunpredictableoreveninfinitesearches.ThepurposeofthisChapteristogiveanintuitionforthenotionsofprimitiverecursiveandgeneralrecursivefunctionsinnumbertheory,fortheyareessentialinGödel’sproof.

AironG'sString.ADialogueinwhichGödel’sself-referentialconstructionismirroredinwords.

TheideaisduetoW.V.O.Quine.ThisDialogueservesasaprototypeforthenextChapter.

ChapterXIV:OnFormallyUndecidablePropositionsofTNTandRelatedSystems.ThisChapter'stitleisanadaptationofthetitleofGödel’s1931article,inwhichhisIncompletenessTheoremwasfirstpublished.ThetwomajorpartsofGödel’sproofaregonethroughcarefully.ItisshownhowtheassumptionofconsistencyofTNTforcesonetoconcludethatTNT(oranysimilarsystem)isincomplete.RelationstoEuclideanandnon-Euclideangeometryarediscussed.Implicationsforthephilosophyofmathematicsaregoneintowithsomecare.

Overview VII

BirthdayCantatatata...InwhichAchillescannotconvincethewilyandskepticalTortoisethattodayishis(Achilles')birthday.HisrepeatedbutunsuccessfultriestodosoforeshadowtherepeatabilityoftheGödelargument.

ChapterXV:JumpingoutoftheSystem.TherepeatabilityofGödel’sargumentisshown,withtheimplicationthatTNTisnotonlyincomplete,but"essentiallyincompleteThefairlynotoriousargumentbyJ.R.Lucas,totheeffectthatGödel’sTheoremdemonstratesthathumanthoughtcannotinanysensebe"mechanical",isanalyzedandfoundtobewanting.

EdifyingThoughtsofaTobaccoSmoker.ADialoguetreatingofmanytopics,withthethrustbeingproblemsconnectedwithself-replicationandself-reference.Televisioncamerasfilmingtelevisionscreens,andvirusesandothersubcellularentitieswhichassemblethemselves,areamongtheexamplesused.ThetitlecomesfromapoembyJ.S.Bachhimself,whichentersinapeculiarway.

ChapterXVI:Self-RefandSelf-Rep.ThisChapterisabouttheconnectionbetweenself-referenceinitsvariousguises,andself-reproducingentitiese.g.,computerprogramsorDNAmolecules).Therelationsbetweenaself-reproducingentityandthemechanismsexternaltoitwhichaiditinreproducingitself(e.g.,acomputerorproteins)arediscussed-particularlythefuzzinessofthedistinction.HowinformationtravelsbetweenvariouslevelsofsuchsystemsisthecentraltopicofthisChapter.

TheMagnificrab,Indeed.ThetitleisapunonBach'sMagnifacatinD.ThetaleisabouttheCrab,whogivestheappearanceofhavingamagicalpowerofdistinguishingbetweentrueandfalsestatementsofnumbertheorybyreadingthemasmusicalpieces,playingthemonhisflute,anddeterminingwhethertheyare"beautiful"ornot.

ChapterXVII:Church,Turing,Tarski,andOthers.ThefictionalCraboftheprecedingDialogueisreplacedbyvariousrealpeoplewithamazingmathematicalabilities.TheChurch-TuringThesis,whichrelatesmentalactivitytocomputation,ispresentedinseveralversionsofdifferingstrengths.Allareanalyzed,particularlyintermsoftheirimplicationsforsimulatinghumanthoughtmechanically,orprogrammingintoamachineanabilitytosenseorcreatebeauty.Theconnectionbetweenbrainactivityandcomputationbringsupsomeothertopics:thehaltingproblemofTuring,andTarski'sTruthTheorem.

SHRDLU,ToyofMan'sDesigning.ThisDialogueisliftedoutofanarticlebyTerryWinogradonhisprogramSHRDLU:onlyafewnameshavebeenchanged.Init.aprogramcommunicateswithapersonabouttheso-called"blocksworld"inratherimpressiveEnglish.Thecomputerprogramappearstoexhibitsomerealunderstanding-initslimitedworld.TheDialogue'stitleisbasedonJesu,joyofMansDesiring,onemovementofBach'sCantata147.

ChapterXVIII:ArtificialIntelligence:Retrospects,ThisChapteropenswithadiscussionofthefamous"Turingtest"-aproposalbythecomputerpioneerAlanTuringforawaytodetectthepresenceorabsenceof"thought"inamachine.Fromthere,wegoontoanabridgedhistoryofArtificialIntelligence.Thiscoversprogramsthatcan-tosomedegree-playgames,provetheorems,solveproblems,composemusic,domathematics,anduse"naturallanguage"(e.g.,English).

Overview VIII

Contrafactus.Abouthowweunconsciouslyorganizeourthoughtssothatwecanimaginehypotheticalvariantsontherealworldallthetime.Alsoaboutaberrantvariantsofthisability-suchaspossessedbythenewcharacter,theSloth,anavidloverofFrenchfries,andrabidhaterofcounterfactuals.

ChapterXIX:ArtificialIntelligence:Prospects.TheprecedingDialoguetriggersadiscussionofhowknowledgeisrepresentedinlayersofcontexts.ThisleadstothemodernAlideaof"frames".Aframe-likewayofhandlingasetofvisualpatternpuzzlesispresented,forthepurposeofconcreteness.Thenthedeepissueoftheinteractionofconceptsingeneralisdiscussed,whichleadsintosomespeculationsoncreativity.TheChapterconcludeswithasetofpersonal"QuestionsandSpeculations"onAlandmindsingeneral.

SlothCanon.AcanonwhichimitatesaBachcanoninwhichonevoiceplaysthesamemelodyasanother,onlyupsidedownandtwiceasslowly,whileathirdvoiceisfree.Here,theSlothuttersthesamelinesastheTortoisedoes,onlynegated(inaliberalsenseoftheterm)andtwiceasslowly,whileAchillesisfree.

ChapterXX:StrangeLoops,OrTangledHierarchies.Agrandwindupofmanyoftheideasabouthierarchicalsystemsandself-reference.Itisconcernedwiththesnarlswhicharisewhensystemsturnbackonthemselves-forexample,scienceprobingscience,governmentinvestigatinggovernmentalwrongdoing,artviolatingtherulesofart,andfinally,humansthinkingabouttheirownbrainsandminds.DoesGödel’sTheoremhaveanythingtosayaboutthislast"snarl"?ArefreewillandthesensationofconsciousnessconnectedtoGödel’sTheorem?TheChapterendsbytyingGödel,Escher,andBachtogetheronceagain.

Six-PartRicercar.ThisDialogueisanexuberantgameplayedwithmanyoftheideaswhichhavepermeatedthebook.ItisareenactmentofthestoryoftheMusicalOffering,whichbeganthebook;itissimultaneouslya"translation"intowordsofthemostcomplexpieceintheMusicalOffering:theSix-PartRicercar.ThisdualityimbuestheDialoguewithmorelevelsofmeaningthananyotherinthebook.FredericktheGreatisreplacedbytheCrab,pianosbycomputers,andsoon.Manysurprisesarise.TheDialogue'scontentconcernsproblemsofmind,consciousness,freewill,ArtificialIntelligence,theTuringtest,andsoforth,whichhavebeenintroducedearlier.Itconcludeswithanimplicitreferencetothebeginningofthebook,thusmakingthebookintoonebigself-referentialloop,symbolizingatonceBach'smusic,Escher'sdrawings,andGödel’sTheorem.

Overview IX

FIGURE1.JohannSebastianBach,in1748.FromapaintingbyEliasGottliebHanssmann.

Introduction:AMusico-LogicalOffering

10

Introduction:

AMusico-LogicalOffering

Author:

FREDERICKTHEGREAT,KingofPrussia,cametopowerin1740.Althoughheisrememberedinhistorybooksmostlyforhismilitaryastuteness,hewasalsodevotedtothelifeofthemindandthespirit.HiscourtinPotsdamwasoneofthegreatcentersofintellectualactivityinEuropeintheeighteenthcentury.ThecelebratedmathematicianLeonhardEulerspenttwenty-fiveyearsthere.Manyothermathematiciansandscientistscame,aswellasphilosophers-includingVoltaireandLaMettrie,whowrotesomeoftheirmostinfluentialworkswhilethere.

ButmusicwasFrederick'sreallove.Hewasanavidflutistandcomposer.Someofhiscompositionsareoccasionallyperformedeventothisday.Frederickwasoneofthefirstpatronsoftheartstorecognizethevirtuesofthenewlydeveloped"piano-forte"("soft-loud").Thepianohadbeendevelopedinthefirsthalfoftheeighteenthcenturyasamodificationoftheharpsichord.Theproblemwiththeharpsichordwasthatpiecescouldonlybeplayedataratheruniformloudness-therewasnowaytostrikeonenotemoreloudlythanitsneighbors.The"soft-loud",asitsnameimplies,providedaremedytothisproblem.FromItaly,whereBartolommeoCristoforihadmadethefirstone,thesoft-loudideahadspreadwidely.GottfriedSilbermann,theforemostGermanorganbuilderoftheday,wasendeavoringtomakea"perfect"piano-forte.UndoubtedlyKingFrederickwasthegreatestsupporterofhisefforts-itissaidthattheKingownedasmanyasfifteenSilbermannpianos!

Bach

Frederickwasanadmirernotonlyofpianos,butalsoofanorganistandcomposerbythenameofJ.S.Bach.ThisBach'scompositionsweresomewhatnotorious.Somecalledthem"turgidandconfused",whileothersclaimedtheywereincomparablemasterpieces.ButnoonedisputedBach'sabilitytoimproviseontheorgan.Inthosedays,beinganorganistnotonlymeantbeingabletoplay,butalsotoextemporize,andBachwasknownfarandwideforhisremarkableextemporizations.(ForsomedelightfulanecdotesaboutBach'sextemporization,seeTheBachReader,byH.T.DavidandA.Mendel.)

In1747,Bachwassixty-two,andhisfame,aswellasoneofhissons,hadreachedPotsdam:infact,CarlPhilippEmanuelBachwastheCapellmeister(choirmaster)atthecourtofKingFrederick.ForyearstheKinghadletitbeknown,throughgentlehintstoPhilippEmanuel,how

Introduction:AMusico-LogicalOffering 11

pleasedhewouldbetohavetheelderBachcomeandpayhimavisit;butthiswishhadneverbeenrealized.FrederickwasparticularlyeagerforBachtotryouthisnewSilbermannpianos,whichlie(Frederick)correctlyforesawasthegreatnewwaveinmusic.

ItwasFrederick'scustomtohaveeveningconcertsofchambermusicinhiscourt.OftenhehimselfwouldbethesoloistinaconcertoforfluteHerewehavereproducedapaintingofsuchaneveningbytheGermanpainterAdolphvonMenzel,who,inthe1800's,madeaseriesofpaintingsillustratingthelifeofFredericktheGreat.AtthecembaloisC.P.E.Bach,andthefigurefurthesttotherightisJoachimQuantz,theKing'sflutemaster-andtheonlypersonallowedtofindfaultwiththeKing'sfluteplaying.OneMayeveningin1747,anunexpectedguestshowedup.JohannNikolausForkel,oneofBach'searliestbiographers,tellsthestoryasfollows:

Oneevening,justasliewasgettinghisfluteready,andhismusiciansweressembled,anofficerbroughthimalistofthestrangerswhohadarrived.Withhisfluteinhishandheraneverthelist,butimmediatelyturnedtotheassembledmusicians,andsaid,withakindofagitation,"Gentlemen,oldBachiscome."TheHutewasnowlaidaside,andoldBach,whohadalightedathisson'slodgings,wasimmediatelysummonedtothePalace.WilhelmFriedemann,whoaccompaniedhisfather,toldmethisstory,andImustsaythat1stillthinkwithpleasureonthemannerinwhichlierelatedit.Atthattimeitwasthefashiontomakeratherprolixcompliments.ThefirstappearanceofJ.S.BachbeforesegreataKing,whodidnotevengivehimtimetochangehistravelingdressforablackchanter'sgown,mustnecessarilybeattendedwithmanyapologies.Iwillnetheredwellentheseapologies,butmerelyobserve,thatinWilhelmFriedemann'smouththeymadeaformalDialoguebetweentheKingandtheApologist.

ButwhatismereimportantthanthisisthattheKinggaveuphisConcertforthisevening,andinvitedBach,thenalreadycalledtheOldBach,totryhisfortepianos,madebySilbermann,whichsteedinseveralroomsofthepalace.[Forkelhereinsertsthisfootnote:"ThepianofortesmanufacturedbySilbermann,ofFrevberg,pleasedtheKingsemuch,thatheresolvedtobuythemallup.Hecollectedfifteen.IhearthattheyallnowstandunfitforuseinvariouscornersoftheRoyalPalace."]Themusicianswentwithhimfromroomtoroom,andBachwasinvitedeverywheretotrythemandtoplayunpremeditatedcompositions.Afterhehadgeneenforsometime,heaskedtheKingtogivehimasubjectforaFugue,inordertoexecuteitimmediatelywithoutanypreparation.TheKingadmiredthelearnedmannerinwhichhissubjectwasthusexecutedextempore:and,probablytoseehewfarsucharttcouldbecarried,expressedawishtohearaFuguewithsixObligatoparts.Butasitisnoteverysubjectthatisfitforsuchfullharmony,Bachchoseonehimself,andimmediatelyexecutedittotheastonishmentofallpresentinthesamemagnificentandlearnedmannerashehaddonethatoftheKing.HisMajestydesiredalsotohearhisperformanceentheorgan.ThenextdaythereforeBachwastakentoalltheorgansinPotsdam,asliehadbeforebeentoSilbermann'sfortepianos.AfterhisreturntoLeipzig,hecomposedthesubject,whichhehadreceivedfromtheKing,inthreeandsixparts.addedseveralartificialpassagesinstrictcanontoit,andhaditengraved,underthetitleof"MusikalischesOpfer"[MusicalOffering],anddedicatedittotheInventor.'