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Chapter26
TheUniverseislawlessor
“Pantˆonchrˆematˆonmetronanthrˆoponeinai”⇤
CristianS.Calude1 ,F.WalterMeyerstein2 &ArtoSalomaa3
1 ComputerScienceDepartment,TheUniversityof, Auckland,NewZealand.
2 Barcelona,Spain.
3 TurkuCentreforComputerScience,TUCS,Turku,Finland.
ThebeliefthatthephysicalUniverseisa knowable systemgovernedby ruleswhichdetermineitsfuture uniquely and completely hasdominated theWesterncivilisationinthelasttwoandahalfmillennia.Thegoal ofthispaperistoprovidenewargumentsinfavourofthehypothesis thattheUniverseislawless,ahypothesisproposedanddiscussedinour papers.7,9,11,14,15,18
1.Introduction
Theendeavourtodiscoveranddeterminethelawspresumedtogovernthe physicalUniverseisasoldasWesterncivilisationitself,asarethedi culties herewithassociated.WitnesstheanecdotetransmittedbyPlato(inhis dialogue Theaetetus)concerningThalesofMiletus,thefirstmathematician toaccuratelypredictasolareclipse(forthe28thMay585BC):
WhileThaleswasstudyingthestarsandlookingupwards,he fellintoapit,andaneat,wittyThracianservantgirljeeredat him,becausehewassoeagertoknowthethingsintheskythat hecouldnotseewhatwastherebeforehimathisveryfeet.
Nowadayswecontinue“tolookupwards”,albeitwiththehelpofthe latesttechnologyanditsfabulousinstruments.Thisprocessisunavoidably markedbythehuman“measure”whichbiasesthelawswepresumetohold intheentireUniverse.
Inwhatfollowsweprovidenewargumentsinfavourofthehypothesis thattheUniverseislawless,ahypothesisproposedanddiscussedinourpa⇤“Manisthemeasureofallthings”,Protagoras,5thcenturyBC.
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C.S.Calude,F.W.Meyerstein&ASalomaapers.7,9,11,14,15,18 Westartbydescribingthenotionsof(physical)Universe andlawoftheUniverse(sometimescallednaturallaworthelawofnature), thenwediscussthelawfulnesshypothesisandlawlessnesshypothesis.We continuebyarguinginfavourofthelawlessnesshypothesisinvarioustypes ofUniverses.Finallywediscusstheprovabilityofthelawlessnesshypothesis.
2.TheUniverse
Thedictionarydefinitionoftheterm,“allthatexists”,isatautology.For thescientificendeavour(“lookingupwards”)thetermcoverstwoquiteseparatedomainsoftherealityaccessibletohumans:I)theSolarSystem andII)theelectromagneticradiationsignalsfrombeyondtheSolarSystem capturedbytheantennasofourinstruments.
TheSunandtheplethoraofplanets,moons,comets,asteroids, andotherdirectlydetectableobjectshavebeenintenselyscrutinisedby humans—fromtheverydawnoftheirhistory—bymeansoftheirinnate radiation-detectionantenna:theretina.Remarkably,beyondthenaked eye,nofurtherlight-amplifyinginstrumentwasavailabletoThales,orto Ptolemy,untilGalileo’sinventionofthetelescoperadicallychangedthis wayof“lookingupwards”.ThetelescopeandCopernicuspresentedhumanswithadi↵erentUniverse,aUniversegiganticbutstillreasonably comprehensiblebyminusculehumans,althoughdefinitelyremovingthem fromthecentralposition.Inthefollowingcenturies,greatphysicistsdiscoveredthefirst“laws”ofnature,indomainssodi↵erentasoptics,electricity, movement,gravity,chemistry,etc.,givingrisetotheideathattheremight existrulesof“universal”validity.But,aswithGalileo’stelescope,advances intechnologyagainchangedtheentireoutlook.Nowa“large”bracketof theelectromagneticspectrum,notjustthenarrowvisible-lightwindow,is availableforscrutiny.
Contemporaneouswiththesetechnologicaladvances,theoreticalphysicistsdevelopedthetwofundamentalexplanatorymodelsofphysical“reality”,thestandardmodelofquantummechanicsandgeneralrelativityof gravitation.Theseadvancesagainfundamentallychangedwhatwasmeant bytheterm“Universe”.
Inthefirstplace,thehuman“measure”vanished:therealityencompassedbythistermisenormous,bothintimeasinspace.Justonetiny example:thedistancefromtheSolarSystemtotheneareststar,AlphaCentaurii,isapproximately40, 000, 000, 000, 000, 000m.Further,iftheSolar
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Systeminspectedwiththemoderninstrumentsrevealeditselfasbeingofan unsuspectedcomplexity,theelectromagneticsignalsnowdetectedinallfrequenciesofthespectrumshowed—notjustanextraordinarycomplexity— butalsowhatcouldonlybeinterpretedinthelightofpresentlyadmitted theoriesasincrediblygiganticphenomena,forwhichevennewnameshad tobecoined:super-massiveblackholes,pulsars,quasars,neutronstars, andavariegatedcatalogueofsupernovae,tonameonlyafew.
Nevertheless,thesearchforauniversalexplanatorytheory—“the laws“—wenton.Ithadtoincorporatethetwofundamentaltheories:quantummechanicsandgravitation.Butthefirstisaprobabilistictheory,the secondadeterministictheory,andtheirmarriagehassofarresistedall e↵orts.Thatistosay,thesee↵ortsnowtakeoutlandishforms:inthem theUniversehasmoredimensionsthanthetraditionalfour,eleven,forinstance.Worse:thereisnotjustoneUniverse—“ours”—butmanyofthem, althoughcompletelydetachedandunreachableforus.
Iftheseconundrumswerenotenough,furtherobservationshavecreated evenmoreproblems.Severaldecadesagoitwasdiscoveredthatthemovementofthestarsinagalaxy,includingourMilkyWay,donotcomplywith thespeedvaluesassignedtothembyNewton’sorEinstein’sgravitation laws.Neitherdogroupsofgalaxies.Theremedy:“blackmatter”,anundetectable(“black”)gravitatingcomponentoftheUniverse.Thenrecently itwasfoundthattheUniverseexpandsfasterthanwhatwasallowedby thelatesttheories.Theremedy:“blackenergy”,aconceptoriginallyput forwardbyEinsteinalbeitinadi↵erentcontext.Whatarethesemysterious matter-energyforms?Untiltodaynobodyknowsandofcoursenothingof thatkindhassofarbeendetected.However,basedonmoreandmoreexact measurementsofthecosmicmicrowavebackground,initiallypredictedto existasafossilremnantoftheBigBang† itself,thefollowingcompositionof theUniverseispresentlyputforwardbycosmologists:darkmatter23.3%, darkenergy72.1%,ordinarymatter,ofwhichstars,planetsandpeopleare made:4.6%.
Inaquiteabbreviatedform,thisiswhatthetermUniversestandsfor nowadays.Clearly,notawell-definedconceptbutapatchworkofobservationsnotyetunderstood,theoriesandprejudices.
† ToarriveatacosmologyoftheBigBangtype,manyadditionalpostulatesarerequired, see,forexample,.5
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530 C.S.Calude,F.W.Meyerstein&ASalomaa3.Thelaws
WhatarethelawsoftheUniverse?Aretheyjustmetaphors(cf.Zilsel36 ) or“likeveinsofgold,[...]thatscientistsareextractingtheore,”(cf. Johnson24 )?
AccordingtoFeynman22 andDavies,21 thephysicallawsorthelaws oftheUniverse—shortly,thelaws,areexpressedinsimplemathematical terms;furtheron,thelawsareuniversal(theyapplyeverywhereinthe Universe),infinite,absolute,stable,omnipotent(everythingintheUniverse mustcomplywiththem).
Whentheadjective“lawful”ispredicatedfromthetermUniverse,what istherebymeant?Herealltheusualhumanprejudicesimpinge.Todeterminethelawsthatrulethechangesanddevelopmentofsaidobjectis equatedwithacquiring knowledge aboutthisentity.Inotherwords,one jumpsfromthe how tothe why.Buttheselaws,assumingwewilleverfind them,arenot causal lawsatall.ItistruethatAristotlehasdefinedknowledgeofsomethingas knowledge ofthecause(orcauses) why thatthingis asitis.Probablybecausetheassumptionofcausalityisaninnate—fitness enhancing—traitofhumans,causalityisinmostcasesimmediatelyassociatedwithknowledge.Butcausalitycanonlybeobservedbyhumansinthe formofshortcausalchains,shortasmeasuredfromaparticularhereand now(hicetnunc)andbasicallyonlyinthepast-timedirection.
Wegiveatrivialexample.Amanpassesunderabalconyfromwhich aflowerpotfallskillinghim.Thecauseofhisdeath?Theflowerpot,of course.Butalsotheredtra clight:haditbeengreenhewouldhave passedearlierunderthebalcony...etc.Itisclearthatfromevery hicet nunc sproutexponentiallymanyinterconnected“causalchains”,andthe wholeideabecomesmeaninglessatashortpastdistancefromany nunc.In theoppositedirection,towardsthefuture,causalitychangesintoprediction, alwaysaprobabilistica↵airinthebestcase.Finally,letusrepeatagain thatthedimensionoftheUniversemakesanyreasonablereferencetothe humanmeasure,asrequiredbyProtagoras,ifnotdirectlyabsurd,atleast untenable.
Consequently,itseemsthatpredicating“lawful”,inanycommon-use senseofthatterm,withthenounUniverse,cannotbereasonablymade. Infact,itisprecisely not inthecommon-usesensethatthetermisappliedinmostcases.Loadedwithcenturiesofreligiousbelief,thesearchis notfor laws butfora design,oratleasta designprinciple,ofthatUniverse.Thisisagainaveryoldidea.Divinitieswerealwayscreditedwith
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TheUniverseisLawlessor“Pantˆonchrˆematˆonmetronanthrˆoponeinai” 531 asuperiorknowledge,includingthepossessionoftheultimateaccountand justificationoftheworld.
InPlato’s Timaeus wehaveoneofthemostfamousexamples.The Universeinthe Timaeus isfashionedbyadivinecraftsman,thedemiurge, ofwhomitisrepeatedlystatedthathewas“good”andthathedesigned theCosmoswiththeviewtomakeitas“good”ashepossiblycould(the platonicdemiurgeisnotomnipotent).Notehowever,thatnowhereinthe Timaeus (orelsewhere)doesPlatoneatlydefinethe“good”.Buttheidea isclearlyexpressed:thedi↵erencebetweenachaotic,lawlessUniverseand aUniversethatcanbeclaimedtobe“lawful”,i.e.tobeaCosmos,isthe existenceofsomeoverlying,unifyingconceptpresumablyof“divine”origin (suchas“theGood”, t’agathon,forPlato).
Plato’sideasdirectlyinfluencedBrahe,Kepler,Galileowhoputforward whatbecametheo cialprogramofscience:Findthelawfulpartofthe Universe,and,iflucky,trytoformulatethe(mathematical)lawsdescribing its“kinesis”(change).Inthisspiritsomeaudaciouspresent-dayphysicists andcosmologistsarelookingfora“theoryofeverything”orwhatevername theymaychoose.
4.Thelawfulnesshypothesis
Frommillennia-oldaspirations“toknowmore”comestheideathatthe Universeislawful.Thishypothesisseemstobesupportedbyourdailyobservations:therhythmofdayandnight,thepatternofplanetarymotion, theregulartickingofclocks.Thestageissetatthebeginningandeverythingfollows“mechanistically”withouttheinterventionofGod,without theoccurrenceof“miracles”.Thefutureisdeterminedfromthepastby universal,infiniteandeternallaws:
[The] entirehistoryoftheUniverseisfixed,accordingtosome precisemathematicalscheme, foralltime,cf.Penrose,28 p. 558–559.
Mostimportantly,thelawsare knowable bymeansofobservations/measurementsandreason/logic.Itisuptoustodiscoverthem. Thegreatlaw,thelawofcauseande↵ect—athingcannotoccurwithout acausewhichproducesit inLaplace’swords—transcendsallknownlaws andiseveratworkwithchainsofcausationsande↵ectsgoverningallof manifestedmatterandlife.
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C.S.Calude,F.W.Meyerstein&ASalomaa5.Thelawlessnesshypothesis
Itisasimplematterofreflectiontopointoutsomelimitsofthelawfulness hypothesis:thevagariesofweather,thedevastationofearthquakesorthe fallofmeteoritesare“perceived”asfortuitous.
Thegreatlawofcauseande↵ectisillusiveandcouldnotbeproven, justobserved.Infact,itispossibletodisproveitsuniversalityasweshall seesoon.
Thelawlessnesshypothesis—accordingtowhichtherearenolawsof theUniverse—doesnotexcludetheexistenceof localrules functioningon large,butfinitescales.Localregularitiesarenotonlycompatiblewith randomness,butinfactaconsequenceofrandomness.Following 15 wewill illustrateourargumentsforaUniversecrudelyrepresentedbyaninfinite binarysequence.
Forexample,everyMartin-Lofrandomsequence‡ containseverypossiblestring(ofanylength)andeverysuchstringmustappearinfinitely manytimes.6 ThefactthatthefirstbilliondigitsofaMartin-Lofrandom sequenceareperfectlylawful,forinstancebybeingexactlythefirstdigitsof thebinaryexpansionof ⇡ ,doesnotmodifyinanywaytheglobalproperty ofrandomnessofthe(infinite)sequence.
Thesefactsareconsistentwithourcommonexperience.Spacescientists canpinpointandpredictplanetarylocationsandvelocities“wellenough” toplanmissionsmonthsinadvance,astronomerscanpredictsolarorlunar eclipsescenturiesbeforetheiroccurrences,etc.Alltheseresults—asimpressiveastheymaybe—areonlytrue locally andwithinacertain degree ofprecision. Theyarenot“lawsoftheUniverse”.
ThehypothesisthattheUniverseislawlessisnotanewidea.Twentyfourcenturiesago,Platointhe Timaeus inventedacosmology(seemore in14 )whichstatesthatinthebeginningthedemiurgefindsacompletely chaoticsubstrate,“Chora”,whichhasonlyoneproperty:itisthematerial substrateoftheUniverseinaprimordialstate,astatewhichwewouldcall todayrandom.Faithfultothelawofcauseande↵ect,Platoproposesan actingprincipleofdisorder,acauseofrandomness,whichhecalls“Anagke” (necessity).Thedemiurgeistryingto“persuade”Anagketoacceptamathematicalorder.Ifsuccessful,onearrivesatafinitesetofpurelymathematicalelementarybuildingblocks—Plato’sperfectpolyhedra—which,when combinedbysimplemathematicalrules,constitutetheorderedUniverse, ‡ AsequenceisMartin-Lofrandomifthereisaconstant c suchthatallitsfiniteprefixes are c-incompressiblewithrespecttoaself-delimitinguniversalTuringmachine.
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TheUniverseisLawlessor“Pantˆonchrˆematˆonmetronanthrˆoponeinai” 533 the“cosmos”(order).Butonlythepartwherethedemiurgesucceededin persuadingAnagkeisordered.Infact,thedemiurgeisnotall-powerful, henceinPlato’sUniverse,orderisonlypartial.Andanirreducibledisorder,chaos,randomnessremains,soirreduciblethatnothingcanbe said aboutit.PlatodoesnotindicateanywherewhatpartoftheUniverseis lawful,andwhatpartisentirelyrandom.
WenotethatthedemiurgeisnottheGodofGenesis,aslaterinterpretershopedtoprove.Infact,thedemiurgedoesnot“create”anythingat all,itisonlythesu cientcauseoforder,wheresuchorderexists.Insteadof saying,“thereisalawwhichunderpinstheorderdetectedinthiscontext”, Platosays,“thedemiurgecaused...”,andthenheaddsthemathematical expressiondescribinginrigorousterms,thispartialorder.
Twentyfourcenturieslater,Poincar´ealsosuspectedthechaotic,random natureoftheUniversewhenhewrote:§
Ifweknewexactlythelawsofnatureandthesituationofthe universeattheinitialmoment,wecouldpredictexactlythe situationofthatuniverseatasucceedingmoment.Butevenif itwerethecasethatthenaturallawnolongerhadanysecretfor us,wecouldstillonlyknowtheinitialsituationapproximately. Ifthatenabledustopredictthesucceedingsituationwiththe sameapproximation,thatisallwerequire,that[it]isgoverned bythelaws.Butitisnotalwaysso;itmayhappenthatsmall di↵erencesintheinitialconditionsproduceverygreatonesin thefinalphenomena.Asmallerrorintheformerwillproduce anenormouserrorinthelatter.Predictionbecomesimpossible, andwehavethefortuitousphenomenon.
InourtimeBarrow 2 hasproventhatEinstein’sequationsexhibita formalchaoticbehaviour,whichmeansthattheevolutionoftheUniverse becomesunpredictableafteratimeshortincosmologicalscales.Hawking’s views(see23 p.26)areevenstronger:
Theintrinsicentropymeansthatgravityintroducesanextra levelofunpredictabilityoverandabovetheuncertaintyusually associatedwithquantumtheory.¶ SoEinsteinwaswrongwhen hesaid,“Goddoesnotplaydice.”Considerationofblackholes suggests,notonlyGoddoesplaydice,butthathesometimes confusesusbythrowingthemwheretheycan’tbeseen.
§ QuotedfromPeterson,29 p.216.
¶ Amassivestar,whichhasexhausteditssuppliesofnuclearenergy,collapsesgravitationallyanddisappearsleavingbehindonlyanintensegravitationalfieldtomarkits presence.Thestarremainsinastateofcontinuousfreefall,collapsingendlesslyinward intothegravitationalpitwithoutreachingthebottom.
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C.S.Calude,F.W.Meyerstein&ASalomaaAdetailedaccountofunknowablesinphysicsisgivenbySvozil.34
AretherebetterwaystodescribetheUniversethanthemathematical one?Inretrospect,mathematicalformalismsseemtobeinevitable,Inany case,thereisnothingtoindicatebettercandidates.Thegrowingpreference tomovefromanalyticaldescriptionsofphysicallawstoalgorithmicones (seeforexampleWolfram35 orthediscussionsin9,11 )isnotaparadigmshift asprogramsarefundamentallymathematicalentities.
6.Argumentsinfavourofthelawlessnesshypothesis
WeconcentrateoncontinuousmodelsfortheUniverse.Firstwewillargue thateveniftheUniverseislawfulthenwewon’tbeabletoknowthis; secondly,weshalldiscussreasonswhytheUniversecannotbelawful.
Asthetoolstounderstandthelawsaremathematicalandmuchofthe elementaryintuitionaboutnumbersderivesfromourlinguisticabilitiesto assignnamestoobjectsk itisnotsurprisingthatourargumentswillfocus onnumbers.ThispointofviewisconsistentwithLandauer’s26
Thelawsofphysicsareessentiallyalgorithmsforcalculation. Thesealgorithmsaresignificantonlytotheextentthattheyare executableinourrealphysicalworld.Ourusuallawsdepend onthemathematician’srealnumbersystem.
Towhatextentisthesystemofrealnumberscontaminatedby“chaoticity”and“randomness”?Arealnumberinbase b is disjunctive (cf.J ¨ urgensen andThierrin25 )incaseits b-expansionsequencecontainsallpossiblestrings overthatalphabet {0, 1,...,b 1}.A lexicon isarealnumberwhichisdisjunctiveinanybase.Alexiconcontainsallwritings,whichhavebeenor willbeeverwritten,inanypossiblelanguage.Alexiconexpressesastrong qualitativeideaofrandomness.
Accordingtothelawoflargenumbers,ineverybinaryexpansionof almosteveryrealnumberintheunitintervaleverystringappearswith its“natural”probability.Forexample1appearswithprobability1/2,0 appearswithprobability1/2,00appearswithprobability1/4,andsoon. Thishappensforalmostall,butnotexactlyallofthem:thelawoflarge numbersisfalseinthesenseofBairecategorywithrespecttothenatural topologyoftheunitinterval,27 butitisstilltrueforasmallmodificationofthistopology.13 Lexiconsformresiduals19 forthenaturaltopology, k AccordingtoBarrow(,3 p.4),“linguisticabilitiesarefarmoreimpressivethanour mathematicalabilities,bothintheircomplexityandtheiruniversalityamonghumansof allraces.”
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TheUniverseisLawlessor“Pantˆonchrˆematˆonmetronanthrˆoponeinai” 535 hence mostrealsdonotobeyanyprobabilitylaws.Thisshowsthatthe systemofrealnumbers,ourverybasiclanguageofexpressinglaws,isfully contaminatedbyrandomness.
Martin-Lofrandomness,astrongerquantitativeformofrandomness, while“lesspervasive”thandisjunctivity,isstillomnipresentamongreal numbers:withprobabilityoneeveryrealnumberisMartin-Lofrandom.6 Evenmore,Martin-Lofrandomrealsareinasensethe“bricks”ofthewhole setofreals:byG´acstheoremimprovedbyHertling(see,6 p.155–165)every realise↵ectivelyreducibletoaMartin-Lofrandomone.
Thelawofcauseande↵ectbreaksdownwiththeadventofalgorithmic informationtheory:mathematics,evenelementarynumbertheory,isfull offactstruefornoformalreasonasChaitinhasproved:6,20,30
Godnotonlyplaysdiceinphysicsbutalsoinpuremathematics.
Randomnessnotonlyexists,itiseverywhere.10
Thelawlessnessidentifiedinthesystemofrealsappearsinquantum mechanics.Thisisnonews,exceptthatnowonecangobeyondthemere postulationofquantumrandomness:onecanprovesomemathematical factsaboutthequalityofquantumrandomness.Consideraquantumrandomnumbergeneratorgeneratingbitsproducedbysuccessivepreparation andmeasurementofastateinwhicheachoutcomehasprobabilityone-half. Byenvisagingthisdevicerunningadinfinitum,wecanconsidertheinfinite sequence x itproduces.Ifweassume:a)astandardpictureofquantum mechanics,i.e.aCopenhagen-likeinterpretationinwhichmeasurementirreversiblyaltersthequantumstate,b)the“many-worlds”interpretationand other“exotic”possibilitiesincludingcontextualhiddencounterfactualobservablesareexcluded,andc)theexperimenterhasfreedominthechoiceof measurementbasis(the“free-willassumption”),then x isincomputable,17 thatisnoTuringmachinecanreproduceexactlythebitsofthesequence x Forexample, x canstartwithabillionof0’s,butcannotconsistsofonly0’s. Infact,onecanproveastrongerproperty:thesequence x isbi-immune, i.e.onlyfinitelymanybitsof x arecomputable.Everybi-immunesequence isincomputable,buttheconverseisnottrue.Experimentalconfirmation ofthistheoreticalresultwasobtainedin.12
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C.S.Calude,F.W.Meyerstein&ASalomaa7.Digitaluniversesarealsolawless
Asthe“free-willassumption”usedintheprevioussectionexcludesadigital Universe⇤⇤ ,itisnaturaltoaskwhethersuchaUniverseislawfulornot?
Digitalphysicsdistinguishesthreepossiblescenarios:a)theUniverse†† is(maybe)continuous,butourmodelisdigital,sayauniversal(prefix) Turingmachineworkingwithadiscreteinfinitetime,b)theUniverseis auniversal(prefix)Turingmachineworkingwithadiscreteinfinitetime, c)theUniverseisauniversal(prefix)Turingmachineworkingforafinite, albeithuge,timeonly.
AlawofaUniverseincasesa)andb)canbeexpressedbyaninfinite sequencewhileforc)thelawhastobeexpressedbyafinitestring.All resultsregardingqualitativeandquantitativerandomnessdescribedinthe previoussectionapplyforthescenariosa)andb).Thestatusofa“law”in thescenarioc)isnotsoclear.ThelawlessnessofsuchaUniversecomesfrom thefactthatstringscodingprogramsexpressinglawsofsuchaUniverse cannotbedistinguishedfromalgorithmicrandomstrings.6
TheinfluentialNKSprogrammeinitiatedbyWolfram’sbook35 —the systematic,empiricalinvestigationofcomputationalsystemsfortheirown sake—isrelevantforunderstandingtheUniverseinallthreepossiblescenariosdescribedabove,irrespectiveoftheparticularphilosophicalviewsof researchersinNKS.Proposeddigitalversionsofvariouspartsofcontinuousphysicshaveconsistentlyrevealedvariousformsofrandomness;seefor exampletheworkindigitalstatisticalmechanicsin.1,16,31–33
8.Canthelawlessnesshypothesisbeproved?
Inspiteofmanyunknowablesinphysics,34 therelevanceofincompleteness ofmathematicsforphysicsisstillunclear.4 Itisunlikelythataformalproof forthelawlessnesshypothesiscanbefound.Ofcourse,thehypothesis canbeexperimentallyillustratedandtested(see12 andthediscussionin Zenil37 ).
InagreementwithHawking(,23 p.3–4):
Itakethepositivistviewpointthataphysicaltheoryisjusta mathematicalmodelandthatitismeaninglesstoaskwhether
⇤⇤ Inatrulydeterministictheory—sometimescalledsuper-determinism—theexperimentermighthavetheillusionofexercisingherindependentfreechoice,butinreality shejustobeystherulesofthetheory.
†† NotethatthetermUniverse,asdescribedinSection2,isamodelitself.
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TheUniverseisLawlessor“Pantˆonchrˆematˆonmetronanthrˆoponeinai” 537 itcorrespondstoreality.Allthatonecanaskisthatitspredictionsshouldbeinagreementwithobservation.
onecansaythatourpartialandprovisionalunderstandingoftheUniversecomesthroughmeasurements,soultimatelythroughnumbers.With extremelyrareexceptions,therealnumbersrepresentingtheoutcomeof measurementsarelexicons,sotheyaredevoidofanyorderorlaw.Can suchasystemexpressany“laws”ofthephysicalUniverse?
Finally,doesthelawlessnesshypothesismeantheendofscience?Should onedefinitelyabandonthehopeoffindingsenseandmeaningintheUniverse?Theanswerstobothquestionsarenegative.Withthelawfulness hypothesisweleaveinadreamofglobal,universalorderandlaw,when,accordingtothelawlessnesshypothesis,thereisonlyChora(chaos)withlocal lawsonly.Thesearejusthypothesesandtheirmeritsshouldbepragmaticallyjudgedonly.IfonefeelselatedtodiscoverthelawsoftheUniverse, thenthetraditionalassumptionfitsbetter;thealternativehypothesisis preferableforthemorerealisticandhumbleminds.Scienceisandwill bealive,andprogressinansweringfundamentalquestionsanddeveloping applicationswillcontinue.
Acknowledgement
WethankE.Calude,J.Casti,G.Chaitin,B.Doran,S.Marcus,B.Pavlov, M.Stay,K.SvozilandH.Zenilformanyilluminatingdiscussionsonthese issues.
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