GödelandPhysics JohnD.Barrow DAMTP CentreforMathematicalSciences CambridgeUniversity WilberforceRd., CambridgeCB30WA UK Summary Weintroducesomeearlyconsiderationsofphysicalandmathematicalimpossibilityas preludestoGödel’sincompletenesstheorems.Weconsidersomeinformalaspectsof thesetheoremsandtheirunderlyingassumptionsanddiscusssometheresponsestothese theoremsbythoseseekingtodrawconclusionsfromthemaboutthecompletabilityof theoriesofphysics.WearguethatthereisnoreasontoexpectGödelincompletenessto handicapthesearchforadescriptionofthelawsofNature,butwedoexpectittolimit whatwecanpredictabouttheoutcomesofthoselaws,andexamplesaregiven.We discussthe'Gödeluniverse',asolutionofEinstein'sequationsdescribingarotating universewheretimetravelispossible,whichwasdiscoveredbyGödelin1949,andthe roleitplayedinexposingthefullspectrumofpossibilitiesthataglobalunderstandingof space-timewouldreveal.Finally,weshowhowrecentstudiesofsupertaskshaveshown howglobalspace-timestructuredeterminestheultimatecapabilityofcomputational deviceswithinthem. 2 1: SomeHistorical Background PhysicalImpossibilities Thereisalonghistoryofscientificandphilosophicalconsiderationofphysical impossibilities 1.TheAristotelianworldviewoutlawedthepossibilitythatphysical infinitiesorlocalphysicalvacuacouldbecreatedorobserved 2.DuringtheMiddleAges, physicistsdevisedingeniousthoughtexperimentstotrytoimaginehowNaturecouldbe ‘tricked’intoallowinganinstantaneousvacuumtoform,andthenarguingabouthowthis possibilitywasstoppedfromoccurringbynaturalprocessesor,ifthatfailed,bythe invocationofaCosmicCensor,topreventitsappearance 3.Chemistryhaditsown ongoingalchemicaldebateaboutthepossibilityorimpossibilityofmakinggoldfrom basermetals,andengineeringmaintainedanenduringattachmenttothequestfora perpetualmotionmachinesthatonlyfullyabatedwhentheconsequencesoflawsof thermodynamicsweresystematicallyunderstoodduringthenineteenthcentury.Subtle examples,likeMaxwell’ssortingdemon,stillremaineduntiltheywereeventuallyfully exorcisedbytheapplicationofthemodernthermodynamictheoryofcomputationin 1961 4. MathematicalImpossibilities Mathematiciansalsooccasionallyconsideredthequestionofimpossibilityinthe contextofaseveralfundamentalproblemsofarithmetic,geometry,andalgebra. Supposedly,inabout550BC,thePythagoreansfirstencounteredthe‘irrationality’of numberslike √2whichcannotbeexpressedastheratiooftwointegers(‘irrational’ originallymeaningsimply'notaratio',ratherthanbeyondreason,asmightbesuggested today).5.Legendhasitthatthisdiscoverywassuchascandal,thatthediscoverer, Hippasos,wasdrownedbythemembersofthePythagoreanbrotherhoodforhistrouble. Thisgivesusthefirstglimpseofoperationsandquestionswhichhavenoanswersgivena particularsetofrules.Inthefirstquarterofthe19 th century,theproblemoffindingan explicitformforthesolutionofageneralquinticalgebraicequationintermsofits coefficientswasprovedtohavenosolutioninvolvingordinaryarithmeticoperationsand radicalsbytheyoungNorwegianmathematician,HenrikAbel 6.Unlikethecaseof quadratic,cubic,orquarticequations,thegeneralquinticcannotbesolvedbyanyexact formula.Justafewyearslater,in1837,rigorousproofsweregiventhatanangleof60 degreescouldnotbetrisectedjustbyuseofastraightedgeandpairofcompasses.These examplesrevealedforthefirsttime,tothosewholookedatthemintherightway,some hintsastothelimitationsofparticularaxiomaticsystems. InthelightoftheongoingimpactofGödel’sworkonspeculationsaboutthe limitationsofthehumanmind,itisinterestingtoreflectbrieflyonthesociologicaland psychologicaleffectsofsomeoftheseearlyresults.Theexistenceofirrationalnumbers wasofthedeepestconcerntothePythagoreans;however,asfaraswecanjudge,there werenodeepphilosophicalquestionsaboutthelimitationsofmathematicalreasoning raisedbythedemonstrationthatthequinticcouldnotbesolved.Yet,therewasachange. Previously,thereweremanythingsthoughtimpossiblethatcouldnotbesoproven despitemanyeffortstodoso.Butnowtherewereproofsthatsomethingcouldnotbe done. 3 Axiomatics Thedevelopmentofunderstandingofwhatconstructionsandproofscouldbecarried outbylimitedmeans,suchasrulerandcompassconstruction,orusingonlyarithmetic operationsandradicals,showedthataxiomsmattered.Thepowerandscopeofasystem ofaxiomsdeterminedwhatitsallowedrulesofreasoningcouldencompass. Untilthe19 th century,thearchetypalaxiomaticsystemwasthatofEuclidean geometry.Butitimportanttoappreciatethatthissystemwasnotthenviewed,asitis today,asjustoneamongmanyaxiomaticpossibilities.Euclideangeometrywashowthe worldreallywas.ItwaspartoftheabsolutetruthabouttheUniverse.Thisgaveita specialstatusanditsconstructionsandelucidation,largelyunchangedformorethan2000 years,providedastylethatwasapedbymanyworksofphilosophyandtheology.The widespreadbeliefinitsabsolutetruthprovidedanimportantcornerstoneforthebeliefsof theologiansandphilosophersthathumanreasoncouldgraspsomethingoftheultimate natureofthings.Ifchallengedthatthiswasbeyondthepowerofourmindstopenetrate, theycouldalwayspointtoEuclideangeometryasaconcreteexampleofhowandwhere thistypeofinsightintotheultimatenatureofthingshadalreadybeenpossible.Asa result,thediscoveries,byBolyai,Lobachevskii,Gauss,andRiemann,thatother geometriesexisted,butinwhichEuclid’sparallelpostulatewasnotincluded,hadamajor impactoutsideofmathematics 7.Theexistenceofotherlogicallyconsistentgeometries meantthatEuclid’sgeometrywasnot the truth:itwassimplyamodelforsomepartsof thetruth.Asaresult,newformsofrelativismsprangup,nourishedbythedemonstration thatevenEuclid’sancientfoundationalsystemwasmerelyoneofmanypossible geometries–andindeedoneofthesealternativeswasafarmoreappropriatemodelfor describingthegeometryoftheEarth’ssurfacethanEuclid’s.Curiousbooksappeared aboutnon-Euclideanmodelsofgovernmentandeconomics.‘Non-Euclidean’becamea bywordfornewandrelativetruth,theverylatestintellectualfashion 8.Later,newlogics wouldbecreatedaswell,bychangingtheaxiomsoftheclassicallogicalsystemthat Aristotlehaddefined. Outofthesestudiesemergedadeeperappreciationoftheneedforaxiomstobe consistentlydefinedandclearlystated.Thetraditionalrealistviewofmathematicsasa descriptionofhowtheworld‘was’hadtobesupersededbyamoresophisticatedview thatrecognisedmathematicstobeanunlimitedsystemofpatternswhicharisefromthe infinitenumberofpossibleaxiomaticsystemsthatcanbedefined.Someofthose patternsappeartobemadeuseofinNature,butmostarenot.Mathematicalsystemslike Euclideangeometryhadbeenassumedtobepartoftheabsolutetruthabouttheworld anduniquelyrelatedtoreality.Butthedevelopmentofnon-Euclideangeometriesand non-standardlogicsmeantthatmathematicalexistencenowmeantnothingmorethan logicalself-consistency(ieitmustnotbepossibletoprovethat0=1).Itnolongerhad anynecessaryrequirementofphysicalexistence. Hilbert’sprogramme ThecarefulstudyofaxiomaticsystemsrevealedthatevenEuclid’sbeautiful developmentofplanegeometrymadeuseofunstatedaxioms.In1882,MoritzPasch gaveaverysimpleexampleofanintuitively‘obvious’propertyofpointsandlinesthat 4 couldnotbeprovedfromEuclid’sclassicalaxioms 9.IfthepointsA,B,C,andDlieona straightlinesuchthatBliesbetweenAandCandCliesbetweenBandDthenitisnot possibletoprovethatBliesbetweenAandD.Thepictureofthesetupmadeitappear inevitablebutthatisnotasubstituteforaproof. _____________________________ A BCD Paschwantedtodistinguishbetweenthelogicalconsequencesoftheaxiomsofgeometry andthosepropertiesthatwejustassumedwereintuitivelytrue.Forhim,mathematical argumentationshouldnotdependonanyphysicalinterpretationorvisualisationofthe quantitiesinvolved.Hewasconcernedthataxiomaticsystemsshouldbecompleteand hasbeendescribedas'thefatherofrigoringeometry'byFreudenthal 10 . DavidHilbert,thegreatestmathematicianoftheday,felttheinfluence ofPasch’s writingsbothdirectlyandthroughtheireffectsonPeano'swork 11 from1882to1899,and beganasystematicprogrammein1899toplacemathematicsuponaformalaxiomatic footing 12 .Thiswasanewemphasis,conveyedbyHilbert'sremarkthatinmathematics 'Onemustbeabletosay..--insteadofpoints,straightlinesandplanes--tables,chairs, andbeermugs' 13 .Hebelievedthatitwouldbepossibletodeterminetheaxioms underlyingeachpartofmathematics(andhenceofthewhole),demonstratethatthese axiomsareself-consistent,andthenshowthattheresultingsystemofstatementsand deductionsformedfromtheseaxiomsisbothcompleteanddecidable.Moreprecisely,a systemis consistent ifwecannotprovethatastatementSanditsnegation, ∼S,areboth truetheorems.Itis complete ifforeverystatementSwecanforminitslanguage,eitherS oritsnegation, ∼Sisatruetheorem.Itis decidable ifforeverystatementSthatcanbe formedinitslanguage,wecanprovewhetherSistrueorfalse.Thus,ifasystemis decidableitmustbecomplete. Hilbert’sformalisticvisionofmathematicswasofatightwebofdeductionsspreading outwithimpeccablelogicalconnectionsfromthedefiningaxioms.Indeed,mathematics was defined tobethecollectionofallthosedeductions.Hilbertsetouttocompletethis formalisationofmathematicswiththehelpofothers,andbelievedthatitwould ultimatelybepossibletoextenditsscopetoincludescienceslikephysics 14 whichwere builtuponappliedmathematics.HebeganwithEuclideangeometryandsucceededin placingitonarigorousaxiomaticbasis.Hisprogrammethenimaginedstrengtheningthe systembyaddingadditionalaxioms,showingateachstepthatconsistencyand decidabilityremained,untileventuallythesystemhadbecomelargeenoughto encompassthewholeofarithmetic. Hilbert’sprogrammebeganconfidentlyandhebelievedthatitwouldjustbeamatter oftimebeforeallofmathematicswascorralledwithinitsformalisticweb.Alas,the worldwassoonturneduponitsheadbytheyoungKurtGödel.Gödelhadcompletedone oftheearlystepsinHilbert’sprogrammeaspartofhisdoctoralthesis,byprovingthe consistencyandcompletenessof1st -orderlogic(laterAlonzoChurchandAlanTuring wouldshowthatitwasnotdecidable).Butthenextstepsthathetookhaveensuredhis