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Abstract

Stephen Wolfram’s work, and especially his New Kind of Science, presents as much a new science as a new natural philosophy-natural computationalism. In the same way as Andrew Hodges, based on Alan Turing’s pioneering work on computability and his ideas on morphological computing and artificial intelligence, argues that Turing is best viewed as a natural philosopher we can also assert that Wolfram’s work constitutes natural philosophy. It is evident through natural and formal computational phenomena studied in different media, from the book with related materials to programs and demonstrations and computational knowledge engine. Wolfram’s theoretical studies and practical computational constructs including Mathematica and Wolfram|Alpha reveal a research program reminiscent of Leibniz’ Mathesis universalis, the project of a universal science supported by a logical calculation framework. Wolfram’s new kind of science may be seen in the sense of Newton’s Philosophiæ Naturalis Principia Mathematica being both natural philosophy and science, not only because of the new methodology of experimental computer science and simulation, or because of particular contributions addressing variety of phenomena, but in the first place as a new unified scientific framework for all of knowledge. It is not only about explaining special patterns seen in nature and models of complex behaviors; it is about the computational nature derived from the first computational principles. Wolfram’s as well as Turing’s natural philosophy differs from Galileo’s view of nature. Computation used in modeling is more than a language. It produces real time behaviors of physical systems: computation is the way nature is. Cellular automata as explored by Wolfram are a whole fascinating computational universe. Do they exhaust all possible computational behaviors that our physical universe exhibit? If we understand physical processes as computations in a more general sense than the computations performed by symbol manipulation done by our current computers, then universal Turing machines and universal cellular automata exhibit only a subset of all possible information processing behaviors found in nature. Even though mathematically, there is a principle of computational equivalence, in physical nature exists a hierarchy of emergent processes on many levels of organization that exhibits different physical behavior and thus can be said compute with different expressive power. This article argues that, based on the notion of computing nature, where computing stands for all kinds of information processing, the development of natural computationalism have a potential to enrich computational studies in the same way as the explorations in the computational universe hold a promise to provide computational models applicable to the physical universe.
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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
GordanaDodigCrnkovic
MälardalenUniversity,ComputerScienceLaboratory,SchoolofInnovation,Designand
Engineering,Västerås,Sweden;Email:gordana.dodigcrnkovic@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
universalcellularautomataexhibitonlyasubsetofallpossibleinformationprocessing
behaviorsfoundinnature.Eventhoughmathematically,thereisaprincipleofcomputational
equivalence,inphysicalnatureexistsahierarchyofemergentprocessesonmanylevelsof
organizationthatexhibitsdifferentphysicalbehaviorandthuscanbesaidcomputewith
differentexpressivepower.Thisarticlearguesthat,basedonthenotionofcomputingnature,
wherecomputingstandsforallkindsofinformationprocessing,thedevelopmentofnatural
computationalismhaveapotentialtoenrichcomputationalstudiesinthesamewayasthe
explorationsinthecomputationaluniverseholdapromisetoprovidecomputationalmodels
applicabletothephysicaluniverse.
EvolvingIdeasofSystèmesduMonde
Cosmogoniesasaccountsoftheoriginandthenatureoftheuniverseevolvewithgrowthof
humanknowledgethroughallegories,myths,models,theoriesandparadigms.This
developmentgoesinparallelwiththeincreaseinthesizeoftheknownuniversefrom
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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
wasinmoderntimesreinterpretedbyGaussas“otheosarithmetizei,”or“Godcomputes”,
Svozil(2005).Irrespectiveofthechoiceofarithmeticorgeometry,thelawsoftheuniverseare
governedbymathematicalprinciples,eventhoughoneisdiscreteandtheothercontinuous.
Leibniz(16461716)withhisphilosophyofMonadologyholdsaspecialplacewhenitcomesto
theSystèmesduMonde.Monadsweredefinedaselementaryautomataconstitutingthe
complexworldthroughcommunicatingnetworks,Mainzer(1994).IntheSection18of
Monadology,Leibnizdepictsamonadasfollows:“AllsimplesubstancesorcreatedMonads
mightbecalledEntelechies,fortheyhaveinthemcertainperfection(echousitoenteles);anda
certainselfsufficiency(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
scientificallybaseduniverse,intheformofaperfectmachine,governedbythelawsofphysics.
Laplace(17491827)believedthataSupremeIntelligence,basedonthelawsofnatureandon
knowledgeofthepositionsandvelocitiesofallparticlesintheuniverseatanymomentcould
inferthestateoftheuniverseatanyfutureorpasttimeaccordingtothelawsofmechanics
discoveredbyNewton(16421727).Eventhoughtheuniverseautomatonisaphysicalsystem,
Galileo(15641642)inhisbookTheAssayer‐IlSaggiatore,pointstovitalconnectionbetween
physicsandmathematics,claimingthatthewaytounderstandnatureisthroughmathematics.
Themechanisticworldisbasedonthefollowingprinciples,DodigCrnkovicandMüller(2011):
(M1) Theontologicallyfundamentalentitiesofthephysicalrealityarephysicalstructures
(spacetime&matterenergy)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
predecessorstheMythopoeticalUniverse,theUniverseofIdealMathematicalPrinciplesand
theMechanisticUniverse.ThisnewparadigmisdubbedInfoComputationalUniverse;forthe
details,seeDodig–Crnkovic(2006).
Ourcurrentunderstandingofthefundamentalityofinformationandcomputationforthe
structureanddynamicsofthenaturalworld,hasledtoanarticulationoftheuniverseasa
computer,anetworkofcomputationalprocessesoninformationalstructures.
TheComputingUniverseNaturalistComputationalism
“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(urtheory),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,butareinsteadmerelyemergentfeaturesofthelargescalebehaviorofsome
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.Itishoweveralsopossiblethatspacetimeandmatterenergy
emergeatonce;thatthereisnospacewithoutmatterenergy.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
andcontinuousstatemodelsaswellascomputers.Onaquantummechanicallevel,the
universeperforms,oncharacteristicallydualwaveparticleobjects,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]iswritteninthisgrandbookImeantheuniversewhichstands
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,
DodigCrnkovic(2012).
Physicalprocess,thoughnotcomputationalinthetraditionalsense,presentsnatural
(unconventional),physical,morphologicalcomputation.Anessentialelementinthisprocessis
theinterplaybetweentheinformationalstructureandthecomputationalprocessinformation
selfstructuring.
Theprocessofcomputationimplementsphysicallawswhichactoninformationalstructures.
Throughtheprocessofcomputation,structureschangetheirforms,asarguedinDodig
Crnkovic(2011).Allcomputationonsomelevelofabstractioncanbeviewedasmorphological
computationaformchanging/formgeneratingprocessoninformationalstructures,Dodig
Crnkovic(2012).
GenerationofFormbyMorphogeneticandMorphologicalComputing
“Withthisbackgrounditbecomesunderstandablethatweneednointelligentdesignof
complexstructures,butonlyverysimplerulesforlocalelementsthatgenerateglobalstructures
duringtheirevolution.”(MainzerandChua,2011,p.9)
Generationofformcanbestudiedbycellularautomatabasedonrulesdefiningupdated
configurationsofagridofcells,andequivalentrulesforothersimpleprograms,butitcanalso
bestudiedinphysicalsystemsundergoingmorphogenesisormetamorphoses.Insuchsystems
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underlyingphysicallawsexpressthemselvesasacomputationcausingchangesofexisting
forms.ThisprocesshasbeenstudiedinroboticsandnanosystemsMacLennan(2010),and
recentlyevenonthemacroscopicscalesinmaterialsandarchitectureasmaterialcomputation,
computingmatterandmaterialcomputation,Menges(2012),butitdeservesmoreattentionas
abasicphenomenonofformgenerationinphysicalmatter,especiallyintricateinliving
systems.
Despitethemathematicalprincipleofcomputationalequivalence1,inphysicalnaturethereisa
hierarchyofemergentprocessesonmanylevelsoforganizationexhibitingdifferentphysical
behaviors.Basedonthenotionofcomputingnature,wherecomputingstandsforallkindsof
informationprocessing,thedevelopmentofnaturalcomputationalismenrichesour
understandingofcomputationbycomputationalstudiesofphysicalsystems,inthesimilarway
astheexplorationsinthecomputationaluniverseprovidenewmodelsapplicabletothe
physicaluniverse.Theprocessgoesinbothdirectionsfromthephysicaltothemodelsandthe
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,DodigCrnkovicinDodigCrnkovicandMüller(2011).
AsforthesourcesofNaturalistcomputationalism,Piccininiidentifiesseveral:
Onesourceis“amatterofrelativelyfreeinterpretation”whichcomputationasystem
performs.Thismaywellbetrueofhumancomputationaldeviceslikefingers,pebbles,
abacuses,andcomputerseventhoughinterpretationsoncechosenarekeptconstant(thusno
longerfree),inordertoallowsocialcommunicationofresults.
AnothersourceofPancomputationalismisthecausalstructureofthephysicalworld.Thatclaim
goesonestepfurtherthanthefirstone,actuallysearchingforthebasisof“free

1“Almostallprocessesthatarenotobviouslysimplecanbeviewedascomputationsofequivalentsophistication.”(Wolfram
2002,pp.5and716717).“Almostanydynamicalsystemthatdoesn'tleadtorandomortransparentlyfixedoroscillatory
behavior,islikelytobeauniversalcomputer.”(Goertzel,DynamicalPsychology,2002)
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interpretation”.Wecanfreelychosesystemsusedforcalculation/computation,butthe
computationaloperationsperformedarepredictablebecauseofthelawsofphysicswhich
guaranteethatphysicalobjectsbehaveinthesamewayandaccordingtophysicallawssothat
wecanpredictandusetheirbehaviourforcomputation.
InfocomputationalismisinthePiccininischemebasedonthethirdsource:
“Athirdallegedsourceofpancomputationalismisthateveryphysicalstatecarriesinformation,
incombinationwithaninformationbasedsemanticsplusaliberalversionofthesemanticview
ofcomputation.Accordingtothesemanticviewofcomputation,computationisthe
manipulationofrepresentations.Accordingtoinformationbasedsemantics,arepresentationis
anythingthatcarriesinformation.Assumingthateveryphysicalstatecarriesinformation,it
followsthateveryphysicalsystemperformsthecomputationsconstitutedbythemanipulation
ofitsinformationcarryingstates(cf.Shagrir2006).Bothinformationbasedsemanticsandthe
assumptionthateveryphysicalstatecarriesinformation(intherelevantsense)remain
controversial.”
Theuseoftheword“manipulation”seemstosuggestaconsciousintervention,while
computationingeneral,asunderstoodwithintheframeworkofComputingNature/Natural
Computationalism/Pancomputationalism,isanaturaldynamicalprocessthatdrives(through
thephysicalinteractionmechanisms)changesininformationalstructures.Notwithstanding
Piccinini’sskepticism,therearewellestablishedtheoriesincomputersciencewhichdoexactly
thejobofconnectingcomputationalprocessesandinformationalstructuresassuggestedby
infocomputationalism,DodigCrnkovic(2011).
Recently,PiccininimadeasubstantialmoveinthedirectionofNaturalComputationalismby
advocating,whathecallsthemodestviewofthephysicalChurchTuringthesisPiccinini(2011).
HerehisclaiminshortisthatnotallofphysicalcomputationisTuringcomputable.Thisview
agreeswithourbestknowledgeaboutNaturalComputationtodayanditalsobringsuscloser
backtotheTuring’sworkconcerningunorganizedmachineswithoracles(advice,learning).
YetanotherinterestingsourceofcriticismtowardsNaturalComputationalismandinparticlar
InfoComputationalismisexpressedbyVincentMüllerinDodigCrnkovicandMüller(2011):
”TheremightbeasetofcomputingproceduresthatislargerthantheonedefinedbyChurch
Turingandthereiscertainlyamathematicalsetofcomputablefunctionslargerthanthat
computablebyTuringmachine(e.g.thatcomputablebyTuring’sideaofhismachineplus
“oracle”).(…)Myunderstandingof‘computer’,assuggestedby[Turing,1936],isthatsuch
machinescharacteristicallygobeyondmerecalculators(likethosealreadyinventedbyLeibniz
andPascal)inthattheyareuniversal;theycan,inprinciple,computeanyalgorithm,because
theyareprogrammableinthissense,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
Turingbelievedonedaywillbepossibleaconstructionofcomputationalmachinescapableof
generallyintelligentbehavioraswellastheaccuratecomputationalmodelingofnaturalworld.
Computationvs.UniversalComputation
Computationisaprocessthataphysicalsystemundergoeswhenprocessinginformation
(computing).Computationasaphenomenonisstudiedwithinseveralresearchfields:theoryof
computation,includingcomputabilitytheory,physics,biology,logic,andsoon.Itisworth
noticingthattheGerman,FrenchandItalianlanguagesusetherespectiveterms"Informatik",
"Informatique"and“Informatica”(InformaticsinEnglish)todenoteComputing,indicatingclose
relationshipsbetweencomputationandinformation.InDodigCrnkovic(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.Ametalanguageis
informationcompressionofthelevelbelow.However,discreteautomataareallonthesame
organizationallevel,eventhoughtheyshowtemporaldevelopment.Thatiswhyauniversal
automaton,whichasaninputtakesarbitrarymachineandexecutesitsalgorithmcannotdoany
betterthanthemachineitself,asitoperatesonthesameinformation.Thewaytomakeit
possibleforauniversalmachinetobemorepowerfulistofirstseparatelevelsofabstraction
betweenmetalevel(universal)andobjectlevel(particularalgorithm).Thatiswhatisdonein
physics“forfree”byselforganizationbasedonnaturallawsondifferentorganizationlevels
(spatialscales).
Ingeneralitisnotnecessaryforcomputationbelongingtodifferentclassesofprocessestobe
universal.Physicalprocessesinquantummechanicsaredifferentfromprocessesintheclassical
clockworkuniverseanditisnotabigproblemiftheyaremodeleddifferently,bydifferent
classesofcomputers.Thatiswhatpresentdayphysicsdoesitproducesdifferenttheoretical
frameworksfordifferentlevelsoforganizationfromquarkstogalaxies.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).Howaboutnonlocalsystems?
Cellularautomataandsimpleprogramshavedemonstratedsurprisinglyrichexpressivepower
inmodelingselforganizationandemergentpropertiesinsystemsconsistingofsimilarunitsbut
10
therearephenomenainnaturethatseemtoberadicallydifferent:Howaboutinteractionsin
totallyheterogeneoussystems?Cantheybereducedtothepropertiesoftheunderlyinggrid?
Howaboutevolution?Howcouldevolutionanddevelopmentbeimplementedintheworldof
cellularautomata?Deacon(2011)forexampleproposesconstructivemechanismstoexplain
(theunavoidable)evolutionfromthermodynamictoanticipatory(teleological)systemsthat
areinagreementwithnaturalcomputation(physicalcomputation).
Itispossiblethatcomputationonamathematicallevelis“allornothing”(computational
equivalence),butifwewanttoascribecomputationalcharacteristicstothephysicalworldand
explainitsfullcomplexity,wemustadmitthattherearehierarchicalstructuresinphysical
systemsthathavecomplexsystemicproperties.Howcouldthearchitecturebebuildupoutof
formgeneratingalgorithms?Innaturethereisahierarchicalsuccessionoflevelsof
organizationandeveryhigherlevelcanbedescribedbymetalanguagewithrespecttoprevious
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
coverallitsdifferentfieldsfromstringtheorytoastrophysics.
11
Wolfram’sproject,contrarytothegeneraldivisionintodisparateknowledgecompartments,
runstowardscommonsyntheticframeworkusingtoolsofformalreasoningandMathematica
ascalculusratiocinator,achievingawiderangingsynthesisofknowledge.AddingWolfram
Alpha’scapabilitytoaccumulateandcomputegeneralknowledge,thisprojectbearsa
resemblancetotheambitionsofMathesisuniversalis,andbringsrenewedrenaissance
optimismaboutthehumancapabilitytoknowtheworldbasedonnaturallaws,with
computationasanorganizingprincipleofallknowledge.
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