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This is a draft of the article to be published in Springer book. The final publication will be available at www.springerlink.com

GordanaDodig‐Crnkovic

MälardalenUniversity,ComputerScienceLaboratory,SchoolofInnovation,Designand

Engineering,Västerås,Sweden;E‐mail:gordana.dodig‐crnkovic@mdh.se

Abstract.StephenWolfram’swork,andespeciallyhisNewKindofScience,presentsasmucha

newscienceasanewnaturalphilosophy‐naturalcomputationalism.Inthesamewayas

AndrewHodges,basedonAlanTuring’spioneeringworkoncomputabilityandhisideason

morphologicalcomputingandartificialintelligence,arguesthatTuringisbestviewedasa

naturalphilosopherwecanalsoassertthatWolfram’sworkconstitutesnaturalphilosophy.Itis

evidentthroughnaturalandformalcomputationalphenomenastudiedindifferentmedia,from

thebookwithrelatedmaterialstoprogramsanddemonstrationsandcomputational

knowledgeengine.Wolfram’stheoreticalstudiesandpracticalcomputationalconstructs

includingMathematicaandWolframAlpharevealaresearchprogramreminiscentofLeibniz’

Mathesisuniversalis,theprojectofauniversalsciencesupportedbyalogicalcalculation

framework.Wolfram’snewkindofsciencemaybeseeninthesenseofNewton’sPhilosophiæ

NaturalisPrincipiaMathematicabeingbothnaturalphilosophyandscience,notonlybecauseof

thenewmethodologyofexperimentalcomputerscienceandsimulation,orbecauseof

particularcontributionsaddressingvarietyofphenomena,butinthefirstplaceasanewunified

scientificframeworkforallofknowledge.Itisnotonlyaboutexplainingspecialpatternsseenin

natureandmodelsofcomplexbehaviors;itisaboutthecomputationalnaturederivedfromthe

firstcomputationalprinciples.Wolfram’saswellasTuring’snaturalphilosophydiffersfrom

Galileo’sviewofnature.Computationusedinmodelingismorethanalanguage.Itproduces

realtimebehaviorsofphysicalsystems:computationisthewaynatureis.Cellularautomataas

exploredbyWolframareawholefascinatingcomputationaluniverse.Dotheyexhaustall

possiblecomputationalbehaviorsthatourphysicaluniverseexhibit?Ifweunderstandphysical

processesascomputationsinamoregeneralsensethanthecomputationsperformedby

symbolmanipulationdonebyourcurrentcomputers,thenuniversalTuringmachinesand

universalcellularautomataexhibitonlyasubsetofallpossibleinformation‐processing

behaviorsfoundinnature.Eventhoughmathematically,thereisaprincipleofcomputational

equivalence,inphysicalnatureexistsahierarchyofemergentprocessesonmanylevelsof

organizationthatexhibitsdifferentphysicalbehaviorandthuscanbesaidcomputewith

differentexpressivepower.Thisarticlearguesthat,basedonthenotionofcomputingnature,

wherecomputingstandsforallkindsofinformationprocessing,thedevelopmentofnatural

computationalismhaveapotentialtoenrichcomputationalstudiesinthesamewayasthe

explorationsinthecomputationaluniverseholdapromisetoprovidecomputationalmodels

applicabletothephysicaluniverse.

EvolvingIdeasofSystèmesduMonde

Cosmogoniesasaccountsoftheoriginandthenatureoftheuniverseevolvewithgrowthof

humanknowledgethroughallegories,myths,models,theoriesandparadigms.This

developmentgoesinparallelwiththeincreaseinthesizeoftheknownuniverse–from

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immediatesurroundingsintheageofgreatmyths,totheearth,solarsystem,MilkyWay,to

astonishing500billiongalaxies‐accordingtocurrentstateofknowledge.Afteralonghistoryof

mythopoethicandallegoricaccountsoftheoriginsandfunctioningoftheuniverse,Antiquity

formulatedfirstnaturalphilosophicalandscientifictheories.ForPythagoras,numberswerethe

essenceandtheprincipleoftheuniverse,whileforPlatogeometrywasfundamental.Plutarch

(Convivialiumdisputationum,liber8,2)reports:"PlatosaidGodgeometrizescontinually".This

wasinmoderntimesre‐interpretedbyGaussas“otheosarithmetizei,”or“Godcomputes”,

Svozil(2005).Irrespectiveofthechoiceofarithmeticorgeometry,thelawsoftheuniverseare

governedbymathematicalprinciples,eventhoughoneisdiscreteandtheothercontinuous.

Leibniz(1646‐1716)withhisphilosophyofMonadologyholdsaspecialplacewhenitcomesto

theSystèmesduMonde.Monadsweredefinedaselementaryautomataconstitutingthe

complexworldthroughcommunicatingnetworks,Mainzer(1994).IntheSection18of

Monadology,Leibnizdepictsamonadasfollows:“AllsimplesubstancesorcreatedMonads

mightbecalledEntelechies,fortheyhaveinthemcertainperfection(echousitoenteles);anda

certainself‐sufficiency(autarkeia)whichmakesthemthesourcesoftheirinternalactivitiesand,

sotospeak,incorporealautomata.”Leibnizhadvisionaryideasaboutcalculatingmachines,he

introducedbinarynotationandarguedfortheessentialroleofformallanguages,Davis(2000).

Wiener,inTheHumanUseofHumanBeings,describesLeibnizasaforerunnerofcybernetics

“Leibniz,dominatedbyideasofcommunication,isinmorethanonewaytheintellectual

ancestoroftheideasofthisbookforhewasalsointerestedinmachinecomputationand

automata.”,(Wiener1988,p.19).Accordingtocontemporaryinformationalinterpretationof

Uchii(2009),Leibniz’smonadscanbeinterpretedasinformationcarriersprogrammedbydivine

codetochangeinformationalcontentsoftheirinternalstates.Thedivinecodingguaranteed

correspondencebetweentheactivitiesofmonadsandtheworldofphenomena.

SystèmeduMondeoftheClockwork(mechanistic)universeisanexampleofaflawlesslylawful

scientifically‐baseduniverse,intheformofaperfectmachine,governedbythelawsofphysics.

Laplace(1749‐1827)believedthataSupremeIntelligence,basedonthelawsofnatureandon

knowledgeofthepositionsandvelocitiesofallparticlesintheuniverseatanymomentcould

inferthestateoftheuniverseatanyfutureorpasttimeaccordingtothelawsofmechanics

discoveredbyNewton(1642‐1727).Eventhoughtheuniverse‐automatonisaphysicalsystem,

Galileo(1564‐1642)inhisbookTheAssayer‐IlSaggiatore,pointstovitalconnectionbetween

physicsandmathematics,claimingthatthewaytounderstandnatureisthroughmathematics.

Themechanisticworldisbasedonthefollowingprinciples,DodigCrnkovicandMüller(2011):

(M1) Theontologicallyfundamentalentitiesofthephysicalrealityarephysicalstructures

(space‐time&matter‐energy)andchangeofphysicalstructures(dynamics).

(M2)Allthepropertiesofanycomplexphysicalsystemcanbederivedfromtheproperties

ofitscomponents.

(M3)Changeofphysicalstructuresisgovernedbylaws.

(M4)Theobserverisoutsideofthesystemobserved.

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Mechanisticmodelsassumethatthesystemisclosed,isolatedfromtheenvironment,andlaws

ofconservation(energy,mass,momentum,etc.)thushold.Environment,ifmodeledatall,is

treatedasaperturbationforthesteadystateofthesystem.

Thelimitsofamechanisticuniverseanddeterminismwereuncoveredbytheincreasinguseof

computersastoolsofexploration,especiallyinthebiologicalworld.Whatbeginstoemerge

nowadaysisafundamentallynewparadigmofnotonlysciencesbutevenamoregeneral

paradigmoftheuniverse,comparableinitsradicallynovelapproachwithitshistorical

predecessorstheMytho‐poeticalUniverse,theUniverseofIdealMathematicalPrinciplesand

theMechanisticUniverse.ThisnewparadigmisdubbedInfo‐ComputationalUniverse;forthe

details,seeDodig–Crnkovic(2006).

Ourcurrentunderstandingofthefundamentalityofinformationandcomputationforthe

structureanddynamicsofthenaturalworld,hasledtoanarticulationoftheuniverseasa

computer,anetworkofcomputationalprocessesoninformationalstructures.

TheComputingUniverse–NaturalistComputationalism

“Willwefindthewholeofphysics?Idon’tknowforsure.ButIthinkatthispointit’ssortof

almostembarrassingnottoatleasttry.”Wolframtalkfromthe2010TEDConference

Theideaofcomputingnature(naturalcomputationalism,pancomputationalism)isold,andina

generalsensecanbetracedbacktoLeibniz.Amongthefirstcontemporaryresearcherssharing

computationalviewofnatureareKonradZuse,EdwardFredkin,TommasoToffoliandStephen

Wolfram,togetherwithJürgenSchmidhuber,SethLloyd,CharlesSeife,andGregoryChaitin.

KonradZusewasthefirsttosuggestin1967thatthephysicalbehaviouroftheentireuniverse

isbeingcomputedonabasiclevel,possiblyoncellularautomata,bytheuniverseitself,which

hereferredtoas"RechnenderRaum"orComputingSpace,Zuse(1969;1970).“Theideathat

spacemightbedefinedbysomesortofcausalnetworkofdiscreteelementaryquantumevents

aroseinvariousformsinworkbyCarlvonWeizsäcker(ur‐theory),JohnWheeler

(pregeometry),DavidFinkelstein(spacetimecode),DavidBohm(topochronology)andRoger

Penrose(spinnetworks).Generalargumentsfordiscretespacewerealsosometimesmade‐‐

notablybyEdwardFredkin,MarvinMinskyandtosomeextentRichardFeynman‐‐onthebasis

ofanalogiestocomputersandinparticulartheideathatagivenregionofspaceshouldcontain

onlyafiniteamountofinformation.”,Wolfram(2002,p.1026).Zusehadtheideaof“going

beyondquantummechanicsindiscretizingphysics,avisionhesharedwiththelateEinsteinand

manyresearchers,amongothersFredkin,Toffoli,Margolus,andWolfram.”,Svozil(2005).

Wolfram(2002),basedonextensivestudiesofcellularautomata,advocatesfora

pancomputationalistviewasanewdynamickindofreductionisminwhichthecomplexityof

behaviorsandstructuresfoundinnaturearederived(generated)fromafewbasicstructures

andprocesses:

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“Istronglysuspectthatthevastmajorityofphysicallawsdiscoveredsofararenottruly

fundamental,butareinsteadmerelyemergentfeaturesofthelarge‐scalebehaviorofsome

ultimateunderlyingrule.Andwhatthismeansisthatanysimplicityobservedinknownphysical

lawsmayhavelittleconnectionwithsimplicityintheunderlyingrule.Soperhapsintheend

thereistheleasttoexplainifIamcorrectthattheuniversejustfollowsasingle,simple,

underlyingrule.”(Wolfram,2002,p.471)

WolframandFredkin(1990),inthesimilarveinasZuse,assumethattheuniverseis,ona

fundamentallevel,adiscretesystem.Followingtheprinciplethat“theultimatemodelof

physicsistobeassimpleaspossible”Wolfram(2002,p.475)expectsthefeaturesofthe

universetoemerge”purelyfrompropertiesofspace”.Thispresupposesthatspaceisthe

independentfirstprinciple.Itishoweveralsopossiblethatspace‐timeandmatter‐energy

emergeatonce;thatthereisnospacewithoutmatter‐energy.Butinthiscontext,thisisa

detail.Themostimportantistheexpressivepower,productivityandinternalcoherenceof

models,andmodelscandiffer.

Moreover,eventhoughdiscretemodelspossessmanyattractivefeatures,physicsregularly

usesboth.(Lesne,2007)arguesforthenecessityofcontinuuminphysicalmodelingofthe

world.Hereisthesummary:

“Thispaperpresentsasampleofthedeepandmultipleinterplaybetweendiscreteand

continuousbehavioursandthecorrespondingmodellingsinphysics.Theaimofthisoverviewis

toshowthatdiscreteandcontinuousfeaturescoexistinanynaturalphenomenon,depending

onthescalesofobservation.Accordingly,differentmodels,eitherdiscreteorcontinuousintime,

space,phasespaceorconjugatespacecanbeconsidered.”(Lesne,2007,p.185)

Howeverthecomputinguniverse(naturalcomputationalism)doesnotcriticallydependonthe

discretenessofthemodelsofthephysicalworld.Therearedigitalaswellasanalog,discrete

andcontinuous‐statemodelsaswellascomputers.Onaquantum‐mechanicallevel,the

universeperforms,oncharacteristicallydualwave‐particleobjects,bothcontinuousand

discretecomputation,Lloyd(2006).

TuringandtheComputingNature

NotonlyLeibnizcanbeseenasapredecessorofnaturalcomputationalism,Turingcanbe

addedtothelistaswell,basedonhisconvictionthatmachines(canbemadethat)canthink

andonhisworkonunorganizedmachines(neuralnetworks)andmorphogenesis.

Turingiswellknowninthefirstplaceforhiscontributionstothetheoryofcomputation,

computerscience,(Turingmachinemodel)Turing(1936;1950),andartificialintelligence

(Turingtest),butforhisbiographerHodges,Turingisultimatelyanaturalphilosopher:

“Hethoughtandlivedagenerationaheadofhistime,andyetthefeaturesofhisthoughtthat

bursttheboundariesofthe1940sarebetterdescribedbytheantiquewords:natural

philosophy.”(Hodges,1997,p.3)

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ItisimportanttonoticethatTuring’snaturalphilosophygoesfurtherthanGalileo’sviewabout

thelanguageofnature:

“Philosophy[i.e.physics]iswritteninthisgrandbook—Imeantheuniverse—whichstands

continuallyopentoourgaze,butitcannotbeunderstoodunlessonefirstlearnstocomprehend

thelanguageandinterpretthecharactersinwhichitiswritten.Itiswritteninthelanguageof

mathematics,anditscharactersaretriangles,circles,andothergeometricalfigures,without

whichitishumanlyimpossibletounderstandasinglewordofit;withoutthese,oneis

wanderingaroundinadarklabyrinth.”(Galileo,1623,p.237)

Computingdiffersfrommathematicsinthatcomputersnotonlycalculatenumbers,butmore

importantlyproducerealtimebehaviors.Turingstudiedavarietyofnaturalphenomenaand

proposedtheircomputationalmodeling.Hemadeapioneeringcontributionintheelucidation

ofconnectionsbetweencomputationandintelligenceandhisworkonmorphogenesisprovides

evidencefornaturalphilosophers’approach.

Turing’spaperonmorphogenesisproposedachemicalmodelasthebasisofthedevelopment

ofbiologicalpatterns,Turing(1952).Hedidnotoriginallyclaimthatthephysicalsystem

producingpatternsactuallyperformscomputationthroughmorphogenesis.Nevertheless,from

theperspectiveofcontemporarynaturalcomputationalismandparticularlyinfo‐

computationalismwecanarguethatmorphogenesisisaprocessofmorphologicalcomputing,

Dodig‐Crnkovic(2012).

Physicalprocess,thoughnotcomputationalinthetraditionalsense,presentsnatural

(unconventional),physical,morphologicalcomputation.Anessentialelementinthisprocessis

theinterplaybetweentheinformationalstructureandthecomputationalprocess–information

self‐structuring.

Theprocessofcomputationimplementsphysicallawswhichactoninformationalstructures.

Throughtheprocessofcomputation,structureschangetheirforms,asarguedinDodig‐

Crnkovic(2011).Allcomputationonsomelevelofabstractioncanbeviewedasmorphological

computation–aform‐changing/form‐generatingprocessoninformationalstructures,Dodig‐

Crnkovic(2012).

GenerationofFormbyMorphogeneticandMorphologicalComputing

“Withthisbackgrounditbecomesunderstandablethatweneednointelligentdesignof

complexstructures,butonlyverysimplerulesforlocalelementsthatgenerateglobalstructures

duringtheirevolution.”(MainzerandChua,2011,p.9)

Generationofformcanbestudiedbycellularautomatabasedonrulesdefiningupdated

configurationsofagridofcells,andequivalentrulesforothersimpleprograms,butitcanalso

bestudiedinphysicalsystemsundergoingmorphogenesisormetamorphoses.Insuchsystems

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underlyingphysicallawsexpressthemselvesasacomputationcausingchangesofexisting

forms.Thisprocesshasbeenstudiedinroboticsandnano‐systemsMacLennan(2010),and

recentlyevenonthemacroscopicscalesinmaterialsandarchitectureasmaterialcomputation,

computingmatterandmaterialcomputation,Menges(2012),butitdeservesmoreattentionas

abasicphenomenonofformgenerationinphysicalmatter,especiallyintricateinliving

systems.

Despitethemathematicalprincipleofcomputationalequivalence1,inphysicalnaturethereisa

hierarchyofemergentprocessesonmanylevelsoforganizationexhibitingdifferentphysical

behaviors.Basedonthenotionofcomputingnature,wherecomputingstandsforallkindsof

informationprocessing,thedevelopmentofnaturalcomputationalismenrichesour

understandingofcomputationbycomputationalstudiesofphysicalsystems,inthesimilarway

astheexplorationsinthecomputationaluniverseprovidenewmodelsapplicabletothe

physicaluniverse.Theprocessgoesinbothdirections–fromthephysicaltothemodelsandthe

otherwayround,RozenbergandKari(2008).

CriticismsoftheComputationalViewsoftheUniverse

InhisarticleonPhysicalComputationforTheStanfordEncyclopediaofPhilosophy,Gualtiero

Piccinini(2010)presentsseveralcriticalargumentsagainstPancomputationalism(Naturalist

computationalism).TheunlimitedPancomputationalism,themostradicalversionof

Pancomputationalism,accordingtoPiccininiassertsthat“everyphysicalsystemperformsevery

computation—oratleast,everysufficientlycomplexsystemimplementsalargenumberofnon‐

equivalentcomputations”.Iarguethesetobetwosubstantiallydifferentclaims.Thefirstone,

thateverysystemexecuteseverycomputation,haslittlesupportinphysicsandothernatural

sciences.Differentsortsofsystemsperformdifferentsortsofdynamicalbehaviors.Thesecond

claim,thatasufficientlycomplexsystemsimplementalargenumberofdifferentcomputations,

isinaccordancewithnaturalsciencesandessentiallydifferentfromtheclaimthateverysystem

performseverycomputation,Dodig‐CrnkovicinDodig‐CrnkovicandMüller(2011).

AsforthesourcesofNaturalistcomputationalism,Piccininiidentifiesseveral:

Onesourceis“amatterofrelativelyfreeinterpretation”whichcomputationasystem

performs.Thismaywellbetrueofhumancomputationaldeviceslikefingers,pebbles,

abacuses,andcomputerseventhoughinterpretationsoncechosenarekeptconstant(thusno

longerfree),inordertoallowsocialcommunicationofresults.

AnothersourceofPancomputationalismisthecausalstructureofthephysicalworld.Thatclaim

goesonestepfurtherthanthefirstone,actuallysearchingforthebasisof“free

1“Almostallprocessesthatarenotobviouslysimplecanbeviewedascomputationsofequivalentsophistication.”(Wolfram

2002,pp.5and716‐717).“Almostanydynamicalsystemthatdoesn'tleadtorandomortransparentlyfixedoroscillatory

behavior,islikelytobeauniversalcomputer.”(Goertzel,DynamicalPsychology,2002)

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interpretation”.Wecanfreelychosesystemsusedforcalculation/computation,butthe

computationaloperationsperformedarepredictablebecauseofthelawsofphysicswhich

guaranteethatphysicalobjectsbehaveinthesamewayandaccordingtophysicallawssothat

wecanpredictandusetheirbehaviourforcomputation.

Info‐computationalismisinthePiccininischemebasedonthethirdsource:

“Athirdallegedsourceofpancomputationalismisthateveryphysicalstatecarriesinformation,

incombinationwithaninformation‐basedsemanticsplusaliberalversionofthesemanticview

ofcomputation.Accordingtothesemanticviewofcomputation,computationisthe

manipulationofrepresentations.Accordingtoinformation‐basedsemantics,arepresentationis

anythingthatcarriesinformation.Assumingthateveryphysicalstatecarriesinformation,it

followsthateveryphysicalsystemperformsthecomputationsconstitutedbythemanipulation

ofitsinformation‐carryingstates(cf.Shagrir2006).Bothinformation‐basedsemanticsandthe

assumptionthateveryphysicalstatecarriesinformation(intherelevantsense)remain

controversial.”

Theuseoftheword“manipulation”seemstosuggestaconsciousintervention,while

computationingeneral,asunderstoodwithintheframeworkofComputingNature/Natural

Computationalism/Pancomputationalism,isanaturaldynamicalprocessthatdrives(through

thephysicalinteractionmechanisms)changesininformationalstructures.Notwithstanding

Piccinini’sskepticism,therearewellestablishedtheoriesincomputersciencewhichdoexactly

thejobofconnectingcomputationalprocessesandinformationalstructuresassuggestedby

info‐computationalism,Dodig‐Crnkovic(2011).

Recently,PiccininimadeasubstantialmoveinthedirectionofNaturalComputationalismby

advocating,whathecallsthemodestviewofthephysicalChurch‐TuringthesisPiccinini(2011).

HerehisclaiminshortisthatnotallofphysicalcomputationisTuringcomputable.Thisview

agreeswithourbestknowledgeaboutNaturalComputationtodayanditalsobringsuscloser

backtotheTuring’sworkconcerningunorganizedmachineswithoracles(advice,learning).

YetanotherinterestingsourceofcriticismtowardsNaturalComputationalismandinparticlar

Info‐ComputationalismisexpressedbyVincentMüllerinDodig‐CrnkovicandMüller(2011):

”TheremightbeasetofcomputingproceduresthatislargerthantheonedefinedbyChurch‐

Turing–andthereiscertainlyamathematicalsetofcomputablefunctionslargerthanthat

computablebyTuringmachine(e.g.thatcomputablebyTuring’sideaofhismachineplus

“oracle”).(…)Myunderstandingof‘computer’,assuggestedby[Turing,1936],isthatsuch

machinescharacteristicallygobeyondmerecalculators(likethosealreadyinventedbyLeibniz

andPascal)inthattheyareuniversal;theycan,inprinciple,computeanyalgorithm,because

theyareprogrammable–inthissense,Zuse’sZ3wasthefirstcomputer(1941).Ifthisfeatureof

universalityisacriterionforbeingacomputer,thenanalogmachinesdonotqualifybecause

theycanonlybeprogrammedinaverylimitedsense.(…)First,howcanyouguaranteethatthe

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notionof‘computing’youareusinghereisinanysenseunified,i.e.onenotion?”(Dodig-

Crnkovic and Müller, 2011, p.162.)

Soinwhatwayisphysicalcomputation/naturalcomputationimportant?Oneofthecentral

questionswithincomputing,cognitivescience,AIandotherrelatedfieldsisabout

computationalmodeling(andsimulating)ofintelligentbehaviour.Whatcanbecomputedand

how?Ithasbecomeobviousthatwemusthaverichermodelsofcomputation,beyondTuring

machines,ifwearetoefficientlymodelandsimulatebiologicalsystems.Whatexactlycanwe

learnfromnatureandespeciallyfromintelligentorganisms?

IthastakenamorethansixtyyearsfromthefirstproposalofTuringtesthecalledthe

“ImitationGame”,describedinTuring(1950)p.442,totherecent(2011)IBM’sWatson

machinewinningJeopardybypurelycomputationalmeans.Thatisjustthebeginningofwhat

Turingbelievedonedaywillbepossible–aconstructionofcomputationalmachinescapableof

generallyintelligentbehavioraswellastheaccuratecomputationalmodelingofnaturalworld.

Computationvs.UniversalComputation

Computationisaprocessthataphysicalsystemundergoeswhenprocessinginformation

(computing).Computationasaphenomenonisstudiedwithinseveralresearchfields:theoryof

computation,includingcomputabilitytheory,physics,biology,logic,andsoon.Itisworth

noticingthattheGerman,FrenchandItalianlanguagesusetherespectiveterms"Informatik",

"Informatique"and“Informatica”(InformaticsinEnglish)todenoteComputing,indicatingclose

relationshipsbetweencomputationandinformation.InDodig‐Crnkovic(2006)itisarguedthat

informationconstitutethestructure,thefabricoftheuniverse,whilecomputationis

synonymouswithphysicalprocessthat,implementingphysicallaws,incessantlychanges

informationalstructures.

Theabilityofacomputertoperformuniversalcomputation(i.e.toprocessnotonlyinputdata

butalsothecodedescribinganyothercomputingmachine)isconsideredcentral.Hereisthe

explanationgivenbySvozil(2005):“Thenotionofuniversalcomputationisrobustinthesense

thatanyuniversalcomputercanemulateanyotheruniversalcomputer(regardlessofefficiency

andoverhead),sothatitdoesnotreallymatterwhichoneisactuallyimplemented.(…)So,

whenitcomestotheirgenericproperties,itisnotreallyimportantwhetherautomaton

universesaremodeledtobeCellularAutomata,TuringMachines,collidingbilliardballs[8],or

biologicalsubstrates.”Fredkin(1991)notices:”Thisisalsothedisadvantage.Itishardtothink

aboutthepropertiesofthemembersofaclasswheneachmembercandoeverything.Thefield

ofComputerSciencehasveryfewexamplesofusefulormeaningfulanalyticsolutionsasto

whatsomedigitalsystemwillorwon'tdo.Onthecontrary,thereisacelebratedproofthat,in

general,therearenoanalyticalshortcutsthatcantellthefuturestateofsomegeneral

computationanyquickerthandoingthecomputationstepbystep(thisisthesocalled”halting

problem”forTuringMachines,Turing1936).Therearenormallynosolutionsinclosedform.

Thereisnotyetanygoodhierarchyofconceptsthatexpresscomplexbehaviorintermsof

simplerbehavior,asisdoneinphysics.”(Emphasisadded)

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Thisisthecoreoftheproblem:Thereisnohierarchy.Inphysicsthereisnaturalencapsulation,

soinprincipleseparationbetweendifferentlevelsoforganization.Ameta‐languageis

informationcompressionofthelevelbelow.However,discreteautomataareallonthesame

organizationallevel,eventhoughtheyshowtemporaldevelopment.Thatiswhyauniversal

automaton,whichasaninputtakesarbitrarymachineandexecutesitsalgorithmcannotdoany

betterthanthemachineitself,asitoperatesonthesameinformation.Thewaytomakeit

possibleforauniversalmachinetobemorepowerfulistofirstseparatelevelsofabstraction

betweenmetalevel(universal)andobjectlevel(particularalgorithm).Thatiswhatisdonein

physics“forfree”byself‐organizationbasedonnaturallawsondifferentorganizationlevels

(spatialscales).

Ingeneralitisnotnecessaryforcomputationbelongingtodifferentclassesofprocessestobe

universal.Physicalprocessesinquantummechanicsaredifferentfromprocessesintheclassical

clockworkuniverseanditisnotabigproblemiftheyaremodeleddifferently,bydifferent

classesofcomputers.Thatiswhatpresentdayphysicsdoes–itproducesdifferenttheoretical

frameworksfordifferentlevelsoforganization–fromquarkstogalaxies.Wehavedifferent

frameworksexecutedonthesamesortofcomputer.Inthefuturewecanhavethesame

frameworkexecutedondifferentsortsofcomputers.

QuestionsBeyondPresentComputationalExperiments

Evenwithintheworldofcellularautomata,therearenumberofinterestingquestionsfor

futureinvestigations.MainzerandChua(2011)propose:

”GoingbeyondthenumericalexperimentsofStevenWolfram,itisarguedthatcellular

automatamustbeconsideredcomplexdynamicalsystemsintheirownright,requiring

appropriateanalyticalmodelsinordertofindpreciseanswersandpredictionsintheuniverseof

cellularautomata.Indeed,eventuallywehavetoaskwhethercellularautomatacanbe

consideredmodelsoftherealworldand,conversely,whethertherearelimitstoourmodern

approachofattributingtheworld,andtheuniverseforthatmatter,essentiallyadigitalreality.”

Insteadofexploringcellularpatternsfromaphenomenologicalpointofview,Mainzer&Chua

(2011)applyanalyticalmethodologyliketheoneusedinmathematicalphysics.

IwouldaddsomeofquestionsthatcametomymindwhenreadingWolfram’sbook.Hereare

someofthem.

Cellularautomatagetupdatedsynchronously.Howaboutdiachronicprocesses?Iftheyare

modelingphysicalworldasweknowit,itshouldbepossibletomodelaneventoriginatedin

thepast(likeaphotoncreatedintheBigBang)tointeractwiththeinformationalstructurein

thecontemporaryuniverse(triggeradetectortoday).Howaboutnon‐localsystems?

Cellularautomataandsimpleprogramshavedemonstratedsurprisinglyrichexpressivepower

inmodelingself‐organizationandemergentpropertiesinsystemsconsistingofsimilarunitsbut

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therearephenomenainnaturethatseemtoberadicallydifferent:Howaboutinteractionsin

totallyheterogeneoussystems?Cantheybereducedtothepropertiesoftheunderlyinggrid?

Howaboutevolution?Howcouldevolutionanddevelopmentbeimplementedintheworldof

cellularautomata?Deacon(2011)forexampleproposesconstructivemechanismstoexplain

(theunavoidable)evolutionfromthermodynamictoanticipatory(teleological)systemsthat

areinagreementwithnaturalcomputation(physicalcomputation).

Itispossiblethatcomputationonamathematicallevelis“allornothing”(computational

equivalence),butifwewanttoascribecomputationalcharacteristicstothephysicalworldand

explainitsfullcomplexity,wemustadmitthattherearehierarchicalstructuresinphysical

systemsthathavecomplexsystemicproperties.Howcouldthearchitecturebebuildupoutof

form‐generatingalgorithms?Innaturethereisahierarchicalsuccessionoflevelsof

organizationandeveryhigherlevelcanbedescribedbymeta‐languagewithrespecttoprevious

level.Howaboutsecondorderalgorithms,oralgorithmschangingalgorithms?

Couldthatbethattheunderlyingcellularautomatonoftheuniverseexpandsproducing

expandinguniversewhichweobserve?Whatwouldthatmeanforthepropertiesofthe

automaton?

Conclusion.TheDreamofLeibnizComingTrue?

“Coulditbethatsomeplaceoutthereinthecomputationaluniversewemightfindourphysical

universe?Willwefindthewholeofphysics?...Ithinkcomputationisdestinedtobethedefining

ideaofourfuture.”StephenWolfram,TEDtalk,filmedFeb.2010

Wolfram’sNewKindofScienceisoneofhiscloselyinterconnectedprojectsthatcanbe

understoodinrelationtotheLeibniz’squestforautomationofreasoninauniversalscience,

Mathesisuniversalis(1695).Leibniz’scharacteristicauniversaliswasenvisagedasalgebra

expressingconceptualthoughtbyaformalsystembasedontherulesforsymbolicmanipulation

ofcalculusratiocinator.TherearetwoopposedinterpretationsofLeibniz’scalculus

ratiocinator:thefirstisanalyticviewrelatingcalculustosoftwareand”algebraoflogic",and

thesecond,syntheticview,foundincybernetics,understandscalculusratiocinatorasreferring

toa"calculatingmachine".Thisdualitymaybeseenasreflectingthedichotomybetween

mathematicalandphysicalviewofcomputation.

Thedevelopmentofformalsystems,Hilbert’sprogramandthedevelopmentofprogrammable

computationalmachineryallcontributedtothegradualrealizationoftheformalizationproject

ofLeibniz.However,atthesametimethedevelopmentofhumanknowledgeruninto

increasingfragmentationandspecializationwhichhasreachedalarmingproportions.So,for

example,atpresentnoindividualcanhavegeneralknowledgeofphysicsbroadenoughto

coverallitsdifferentfields–fromstringtheorytoastrophysics.

11

Wolfram’sproject,contrarytothegeneraldivisionintodisparateknowledgecompartments,

runstowardscommonsyntheticframeworkusingtoolsofformalreasoningandMathematica

ascalculusratiocinator,achievingawide‐rangingsynthesisofknowledge.AddingWolfram

Alpha’scapabilitytoaccumulateandcomputegeneralknowledge,thisprojectbearsa

resemblancetotheambitionsofMathesisuniversalis,andbringsrenewedrenaissance

optimismaboutthehumancapabilitytoknowtheworldbasedonnaturallaws,with

computationasanorganizingprincipleofallknowledge.

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