PhysicalComputation
AMechanisticAccount
GualtieroPiccinini
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# GualtieroPiccinini2015
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Acknowledgements
Iencounteredthedebateaboutthefoundationsofcognitivesciencewhen,asan undergraduateattheUniversityofTurin,Itookanintroductorycourseincognitive psychologytaughtbyBrunoBara.Thoseearlywonderingseventuallyledtomy Ph.D.dissertationattheUniversityofPittsburgh,andthentoastreamofpaperson computationandrelatedtopics.Thisbookorganizes,streamlines,revises,and systematizesmostofwhatI’vethoughtandpublishedaboutconcretecomputation. (Itleavesoutmyworkoncomputationaltheoriesofcognition.)
IcouldnothavegottenwhereIamwithoutthehelpofmyteachers,beginning withmyparentsandcontinuingallthewaytomycollegeandgraduateschool teachers.Incollege,thefollowingwereespeciallyinfluential:inphilosophy,Elisabetta Galeotti,DiegoMarconi,GianniVattimo,andCarloAugustoViano;inpsychology, BrunoBaraandGiulianoGeminiani.Ingraduateschool,thefollowingplayed importantrolesinmythinkingaboutcomputation:myadvisorPeterMachameras wellasNuelBelnap,RobertDaley(ComputabilityTheory),BardErmentrout(ComputationalNeuroscience),JohnEarman,ClarkGlymour,PaulGriffiths,RickGrush, KenManders,JohnNorton,andMerrileeSalmon.Otherimportantteachersduring graduateschoolwereRobertBrandom,JoeCamp,AdolfGrünbaum,SusieJohnson (TheoryofMind),PatLevitt(SystemsNeuroscience),JayMcClelland(Cognitive Neuroscience),JohnMcDowell,TedMcGuire,RobertOlby,CarlOlson(Cognitive Neuroscience),FritzRinger,LauraRuetsche,andRichmondThomason.
Manypeople’sworkshapedmine.Oncomputation,theclassicworkofAlonzo Church,KurtGodel,StephenKleene,EmilPost,andAlanTuringwasparamount. Onneuralnetworksandtheirconnectiontodigitalcomputers,Ilearnedthemost fromWarrenMcCulloch,WalterPitts,JohnvonNeumann,andNorbertWiener.On analogcomputers,theworkofClaudeShannon,MarianPour-El,andLeeRubelwas crucial.Onartificialintelligenceandcomputationalcognitivescience,AllenNewell andHerbertSimonhadthelargestimpact.Thecanonicaltextbookoncomputer organizationanddesignbyDavidPattersonandJohnHennessyconfirmedmyearly hunchthatcomputationalexplanationismechanisticandhelpedmearticulateit.On thephilosophyofcomputation,IfoundmosthelpfultheworksofJackCopeland, RobertCummins,DanielDennett,FrancesEgan,JerryFodor,JustinGarson,Gilbert Harman,JohnHaugeland,ZenonPylyshyn,OronShagrir,WilfriedSieg,andStephenStich.Oninformation,FredDrestkeandClaudeShannonweremostsignificant.Onfunctions,ChristopherBoorseandRuthMillikanmadethebiggest difference.Onmechanisms,IwasmostinfluencedbyCarlCraver,LindleyDarden, PeterMachamer,andWilliamWimsatt.Ongeneralmetaphysics,JohnHeilandthe NEHSummerSeminarhedirectedhelpedmeanenormousamount.
Theworkthatgoesintothisbookhasbenefitedfromsomanyinteractionswithso manypeopleoversuchalongtime includingaudiencesatmanytalks that Icannothopetoremembereveryonewhoplayedarole.Iapologizeinadvanceto thoseIforget.HerearethoseIrememberasmosthelpful:DarrenAbramson,Fred Adams,NealAnderson,KenAizawa,SonyaBahar,MarkBalaguer,BillBechtel, KennyBoyce,DaveChalmers,MarkCollier,CarlCraver,JackCopeland,Martin Davis,DanielDennett,MichaelDickson,EliDresner,FrancesEgan,ChrisEliasmith, IlkeErcan,CarrieFigdor,JerryFodor,FlorentFranchette,NirFresco,JustinGarson, CarlGillett,RobertGordon,SabrinaHaimovici,GilbertHarman,JohnHeil,Alberto Herna ´ ndez,DavidM.Kaplan,SaulKripke,FredKroon,BillLycan,CoreyMaley, MarcinMilkowski,JonathanMills,AlexMorgan,TobyOrd,AnyaPlutynski,Tom Polger,HilaryPutnam,MichaelRabin,MichaelRescorla,BrendanRitchie,Brad Rives,MartinRoth,Anna-MariRusanen,DanRyder,AndreaScarantino,Matthias Scheutz,SusanSchneider,SamScott,LarryShapiro,RebeccaSkloot,MarkSprevak, OronShagrir,StuartShapiro,DoronSwade,WilfriedSieg,EricThomson,Brandon Towl,RayTurner,DanWeiskopf,andJulieYoo.
SpecialthankstoMarkSprevak,themembersofthereadinggroupheranona previousversionofthemanuscript,andananonymousrefereeforOUPfortheir extremelyhelpfulfeedback.ThankstoPeterMomtchiloff,myeditoratOUP,forhis help,patience,andsupport.
Chapters1–4areindebtedtoG.Piccinini, “ComputationinPhysicalSystems,” TheStanfordEncyclopediaofPhilosophy (Fall2010Edition),EdwardN.Zalta(ed.), URL=http://plato.stanford.edu/archives/fall2010/entries/computation-physicalsystems/ Chapter1isindebtedtoG.Piccinini, “ComputingMechanisms,” Philosophyof Science,74.4(2007),501–26.
Chapter3isindebtedtoG.Piccinini, “ComputationwithoutRepresentation,” PhilosophicalStudies,137.2(2008),205–41andG.Piccinini, “Functionalism,Computationalism,andMentalContents,” CanadianJournalofPhilosophy,34.3(2004), 375–410.
Chapter4isindebtedtoG.Piccinini, “ComputationalModelingvs.ComputationalExplanation:IsEverythingaTuringMachine,andDoesItMattertothe PhilosophyofMind?” AustralasianJournalofPhilosophy,85.1(2007),93–115.
Chapter5isindebtedtoG.Piccinini, “Functionalism,Computationalism,and MentalStates,” StudiesintheHistoryandPhilosophyofScience,35.4(2004), 811–33andG.PiccininiandC.Craver, “IntegratingPsychologyandNeuroscience: FunctionalAnalysesasMechanismSketches,” Synthese,183.3(2011),283–311.
Chapter6isindebtedtoC.MaleyandG.Piccinini, “AUnifiedMechanistic AccountofTeleologicalFunctionsforPsychologyandNeuroscience,” forthcoming inDavidKaplan,ed., IntegratingPsychologyandNeuroscience:ProspectsandProblems.Oxford:OxfordUniversityPress.
Chapter7isindebtedtoG.Piccinini, “ComputingMechanisms,” Philosophyof Science,74.4(2007),501–26andG.Piccinini, “ComputationwithoutRepresentation,” PhilosophicalStudies,137.2(2008),205–41.
Chapters10–13areindebtedtoG.Piccinini, “Computers,” PacificPhilosophical Quarterly,89.1(2008),32–73.
Chapter13isindebtedtoG.Piccinini, “SomeNeuralNetworksCompute,Others Don’t,” NeuralNetworks,21.2–3(2008),311–21.
Chapter14isindebtedtoG.PiccininiandA.Scarantino, “InformationProcessing, Computation,andCognition,” JournalofBiologicalPhysics,37.1(2011),1–38.
Chapters15and16areindebtedto “ThePhysicalChurch-TuringThesis:Modest orBold?” TheBritishJournalforthePhilosophyofScience,62.4(2011),733–69.
Myco-authorsforsomeoftheabovearticles CarlCraver,CoreyMaley,and AndreaScarantino helpedmedevelopimportantaspectsoftheviewIdefendand Iamdeeplygratefultothem.
Thearticlesfromwhichthebookborrowswereusuallyrefereedanonymously. Thankstothemanyanonymousrefereesfortheirhelpfulfeedback.
Thankstomyresearchassistant,FrankFaries,especiallyfordrawingthe figures forChapters8and9.
ThismaterialisbasedonworksupportedbytheCollegeofArtsandSciencesatthe UniversityofMissouri St.Louis,theCharlesBabbageInstitute,theInstitutefor AdvancedStudiesattheHebrewUniversityofJerusalem,theNationalEndowment fortheHumanities,theNationalScienceFoundationundergrantsno.SES-0216981 andSES-0924527,theUniversityofMissouri,theUniversityofMissouri St.Louis, theMellonFoundation,andtheRegioneSardegna.Anyopinions, findings,conclusions,andrecommendationsexpressedinthisbookarethoseoftheauthoranddo notnecessarilyreflecttheviewsofthesefundinginstitutions.
Deepthankstomyfriends,partners,andfamily myparents,sisters,brothers-inlaw,andnieces fortheirloveandsupport,especiallyduringmymostchallenging times.Thankstomydaughters Violet,Brie,andMartine forbringingsomuch meaningandjoytomylife.
Introduction
Thisbookisaboutthenatureofconcretecomputation thephysicalsystemsthat performcomputationsandthecomputationstheyperform.Iarguethatconcrete computingsystemsareakindoffunctionalmechanism.Afunctionalmechanismisa systemofcomponentpartswithcausalpowersthatareorganizedtoperforma function.Computingmechanismsaredifferentfrom non-computingmechanisms becausetheyhaveaspecialfunction:tomanipulatevehiclesbasedsolelyondifferencesbetweendifferentportionsofthevehiclesinaccordancewitharulethatis definedoverthevehiclesand,possibly,certaininternalstatesofthemechanism.Icall thisthe mechanisticaccount ofcomputation.
WhenIbeganarticulatingandpresentingthemechanisticaccountofcomputationtophilosophicalaudiencesovertenyearsago,Ioftenencounteredoneof twodismissiveresponses. Responseone:yourviewisobvious,wellknown,and uncontroversial utterlydull. Responsetwo:yourviewiscounterintuitive,implausible,anduntenable totallyworthless.Thesewerenottheonlyresponses.Plentyof peopleengagedthesubstanceofthemechanisticaccountofcomputationand discusseditsprosandcons.Buttheseradicalresponsesweresufficientlycommon thattheydeservetobeaddressedupfront.
Ifthemechanisticaccountelicitedeitheroneoftheseresponsesbutnottheother, perhapsthemechanisticaccountwouldbeatfault.Butthepresenceofboth responsesisencouragingbecausetheycanceleachotherout,asitwere.Thosewho responddismissivelyappeartobeunawarethattheoppositedismissiveresponseis equallycommon.Iftheyknewthis,presumablytheywouldtoneitdown.For althoughreasonablepeoplemaydisagreeaboutwhetheraviewistrueorfalse,itis unreasonabletodisagreeonwhethersomethingis obviously trueor obviously false.If it’ssoobvious,howcantherebeequallyinformedpeoplewhothinkthe opposite is obvious?
The firstdismissiveresponse thatthemechanisticaccountissoobviousthatit’ s dull seemstobemotivatedbysomethinglikethefollowingreasoning.Forthesake oftheargument,let’sassumealongwithmanyphilosophersthatcomputationisa kindofsymbolmanipulation.Thereisanimportantdistinctionbetweenthesyntax ofsymbols(and,moregenerally,theirformalproperties)andtheirsemantics.Toa firstapproximation,syntactic(moregenerally,formal)propertiesarethosethat determinewhetherasymbolicstructureiswellformed theymakethedifference
between ‘thepuzzleissolvable’ and ‘puzzletheissolvable’;semanticpropertiesare thosethatdeterminewhatsymbolsmean theymakethedifferencebetween ‘ivitelli deiromanisonobelli’ inmostlanguages,whereitmeansnothing;inLatin,whereit means go,Vitellus,attheRomangod’ swarcry;andinItalian,whereitmeans the calvesoftheRomansaregood-looking.Mostpeople finditintuitivelycompellingthat computationsoperateonsymbolsbasedontheirformalorsyntacticpropertiesalone andnotatallbasedontheirsemanticproperties.Furthermore,manyphilosophers assimilatecomputationalexplanationandfunctionalanalysis:computationalstates areoftensaidtobeindividuatedbytheirfunctionalrelationstoothercomputational states,inputs,andoutputs.Therefore,computationalstatesandprocessesareindividuatedfunctionally,i.e.,formallyorsyntactically.Sayingthatcomputationis mechanistic,asmyaccountdoes,isjustarelabelingofthisstandardview.Therefore, themechanisticaccountofcomputationisnothingnew.Somethinglikethisreasoningisbehindthe firstdismissiveresponse.Itisdeceptivelypersuasivebut,alas,itgoes waytoofast.
A firstproblemisthatphysicalsystemsdon’tweartheirsyntactic(orformal) propertiesontheirsleeves.Ifthemechanisticaccountwerebasedonsyntactic properties,itshouldbeginwithanaccountofsyntacticpropertiesthatdoesnot presupposethenotionofcomputation.Idon’tknowofanysuchaccount,and fortunatelyIdon’tneedone.Forthemechanisticaccountofcomputationispainstakinglybuiltbyspecifyingwhichpropertiesofwhichmechanismsarecomputational,withouteverinvokingthenotionofsyntax(orformalproperty).Thus,the mechanisticaccountmayprovideingredientsforanaccountofsyntax notvice versa(Chapter3,Section4).
Asecondproblemistheimplicitassimilationoffunctionalanalysisandcomputationalexplanation,whichispervasiveintheliterature.Irejectsuchanassimilation andarguethatfunctionalanalysisprovidesapartialsketchofamechanism (Chapter5),defendateleologicalaccountoffunctionalmechanisms(Chapter6), andarguethatcomputationalexplanationisaspecifickindofmechanisticexplanation(Chapter7).
Anadditionalissueisthatcomputationsareoftenindividuatedsemantically in termsoffunctionsfromwhatisdenotedbytheirinputstowhatisdenotedbytheir outputs.Andphilosophersinterestedincomputationareofteninterestedinhow computationcanexplaincognition,whichisusuallyassumedtodealinrepresentations.Afterall,cognitivestatesandprocessesaretypicallyindividuatedatleastin partbytheirsemanticcontent.Thus,manyphilosophersinterestedincomputation believethatcomputationalstatesandprocessesareindividuatedbytheircontentin suchawaythatatleastpartoftheiressenceissemantic.Icallthisthe semantic account ofcomputation.Thereinliesthemotivationfortheseconddismissive response:sincecomputationisessentiallysemanticandthemechanisticaccountof computationdeniesthis,themechanisticaccountisobviouslyandhorriblywrong.
Butthesemanticaccountofcomputationhasitsownproblems.Forstarters,the notionofsemanticpropertyisevenmoreobscureandmoreinneedofnaturalistic explicationthanthatofsyntacticproperty.Inaddition,Iarguethatindividuating computationssemanticallyalwayspresupposestheirnon-semanticindividuation, andthatsomecomputationsareindividuatedpurelynon-semantically.Therefore, contrarytotheseconddismissiveresponse,computationdoesnotpresuppose representation(Chapter3).
Butifwerejecttheviewthatcomputationpresupposesrepresentation,werisk fallingintotheviewthateverythingperformscomputations pancomputationalism (Chapter4).Thisisnotonlycounterintuitive italsorisksunderminingthefoundationsofcomputerscienceandcognitivescience.Itisalsoasurprisinglypopular view.Yet,Iarguethatpancomputationalismismisguidedandwecanavoiditbya judicioususeofmechanisticexplanation(Chapter4).
Themechanisticaccountbeginsbyadaptingamechanisticframeworkfromthe philosophyofscience.Thisgivesusidentityconditionsformechanismsintermsof theircomponents,theirfunctions,andtheirorganization,withoutinvokingthe notionofcomputation.Tothisgeneralframework,amechanisticaccountofcomputationmustaddcriteriaforwhatcountsascomputationallyrelevantmechanistic properties.Idothisbyadaptingthenotionofastringofletters,takenfromlogicand computabilitytheory,andgeneralizingittothenotionofasystemofvehiclesthatare definedsolelybasedondifferencesbetweendifferentportionsofthevehicles.Any systemwhosefunctionistomanipulatesuchvehiclesinaccordancewitharule, wheretheruleisdefinedintermsofthevehiclesthemselves,isacomputingsystem. Iexplainhowasystemofappropriatevehiclescanbefoundinthenatural(concrete) world,yieldingarobust(nontrivial)notionofcomputation(Chapter7).
Afterthat,Idevelopthisgeneralmechanisticaccountbyexplicatingspecific computingsystemsandtheirpropertiesinmechanisticterms.Iexplicatethenotion ofprimitivecomputingcomponents(Chapter8),complexcomputingcomponents (Chapter9),digitalcalculators(Chapter10),digitalcomputers(Chapter11),analog computers(Chapter12),andneuralnetworks(Chapter13).Afterthat,Ireturnto semanticpropertiesundertheguiseofinformationprocessing(inseveralsensesof theterm);Iarguethatprocessinginformationisaformofcomputationbutcomputationneednotbeaformofinformationprocessing(Chapter14).Iconcludethe bookwiththelimitsofphysicalcomputation.Oncetherelevantquestionisclarified (Chapter15),theevidencesuggeststhatanyfunctionthatisphysicallycomputableis computablebyTuringmachines(Chapter16).
TowardsanAccountofPhysical Computation
1.AbstractComputationandConcreteComputation
Computationmaybestudiedmathematicallybyformallydefiningcomputingsystems,suchasalgorithmsandTuringmachines,andprovingtheoremsabouttheir properties.Themathematicaltheoryofcomputationisawell-establishedbranchof mathematics.Itstudieswhichfunctionsdefinedoveradenumerabledomain,suchas thenaturalnumbersorstringsoflettersfroma finitealphabet,arecomputableby algorithmorbysomerestrictedclassofcomputationalsystems.Italsostudieshow muchtimeorspace(i.e.,memory)ittakesforacomputationalsystemtocompute certainfunctions,withoutworryingmuchabouttheparticularunitsoftimeinvolved orhowthememorycellsarephysicallyimplemented.
Bycontrast,mostusesofcomputationinscienceandeverydaylifeinvolve concrete computation:computationinconcretephysicalsystemssuchascomputersand brains.Concretecomputationiscloselyrelatedtomathematicalcomputation:we speakofphysicalsystemsasrunninganalgorithmorasimplementingaTuring machine,forexample.Buttherelationshipbetweenconcretecomputationand mathematicalcomputationisnotpartofthemathematicaltheoryofcomputation perseandrequiresfurtherinvestigation.Thisbookisaboutconcretecomputation. Wewillseeinduecoursethatquestionsaboutconcretecomputationarenotneatly separablefrommathematicalresultsaboutcomputation.Thefollowingmathematicalresultsareespeciallycrucialtooursubsequentinvestigation.
Themostimportantnotionofcomputationisthatofdigitalcomputation,which AlanTuring,KurtGodel,AlonzoChurch,EmilPost,andStephenKleeneformalized inthe1930s(seetheAppendixforasketchofthemostrelevantbackgroundand results).Theirworkinvestigatedthefoundationsofmathematics.Onecrucialquestionwaswhether firstorderlogicisdecidable whetherthereisanalgorithmthat determineswhetheranygiven firstorderlogicalformulaisatheorem.
Turing(1936–7)andChurch(1936)provedthattheanswerisnegative:thereisno suchalgorithm.Toshowthis,theyofferedprecisecharacterizationsoftheinformal notionofalgorithmicallycomputablefunction.Turingdidsointermsofso-called Turingmachines devicesthatmanipulatediscretesymbolswrittenonatapein
accordancewith finitelymanyinstructions.Otherlogiciansdidthesamething they formalizedthenotionofalgorithmicallycomputablefunction intermsofother notions,suchas º-definablefunctionsandgeneralrecursivefunctions.
Totheirsurprise,allsuchnotionsturnedouttobeextensionallyequivalent,thatis, anyfunctioncomputablewithinanyoftheseformalismsiscomputablewithinanyof theothers.Theytookthisasevidencethattheirquestforaprecisedefinitionof ‘algorithm,’ or ‘effectiveprocedure,’ or ‘algorithmicallycomputablefunction,’ had beensuccessful.Theyhadfoundaprecise,mathematicallydefinedcounterparttothe informalnotionofcomputationbyalgorithm amathematicalnotionthatcouldbe usedtostudyinarigorouswaywhichfunctionscanandcannotbecomputedby algorithm,andthereforewhichfunctionscanandcannotbecomputedbymachines thatfollowalgorithms.Theresultingview thatTuringmachinesandotherequivalentformalismscapturetheinformalnotionofalgorithm isnowknownasthe Church-Turingthesis (moreonthisinChapter15).Itprovidesthefoundationforthe mathematicaltheoryofcomputationaswellasmainstreamcomputerscience.
ThetheoreticalsignificanceofTuringetal.’sformalizationofcomputationcan hardlybeoverstated.AsGo ¨ delpointedout(inalecturefollowingonebyTarski): Tarskihasstressedinhislecture(andIthinkjustly)thegreatimportanceoftheconceptof generalrecursiveness(orTuring’scomputability).Itseemstomethatthisimportanceislargely duetothefactthatwiththisconceptonehasforthe firsttimesucceededingivinganabsolute definitionofaninterestingepistemologicalnotion,i.e.,onenotdependingontheformalism chosen(Godel1946,84).
AstandardTuringmachinecomputesonlyonefunction.Turingalsoshowedthat thereare universal Turingmachines machinesthatcancomputeanyfunction computablebyanyotherTuringmachine.Universalmachinesdothisbyexecuting instructionsthatencodethebehaviorofthemachinetheysimulate.Assumingthe Church-Turingthesis,universalTuringmachinescancomputeanyfunctioncomputablebyalgorithm.Thisresultissignificantforcomputerscience:youdon’tneed tobuilddifferentcomputersfordifferentfunctions;oneuniversalcomputerwill sufficetocomputeanycomputablefunction.Moderndigitalcomputersapproximate universalmachinesinTuring’ssense:digitalcomputerscancomputeanyfunction computablebyalgorithmforaslongastheyhavetimeandmemory.(Strictly speaking,auniversalmachinehasanunboundedmemory,whereasdigitalcomputer memoriescanbeextendedbutnotindefinitely,sotheyarenotquiteunboundedin thesameway.)
Theaboveresultshouldnotbeconfusedwiththecommonclaimthatcomputers cancompute anything.Nothingcouldbefurtherfromthetruth:anotherimportant resultofcomputabilitytheoryisthatmostfunctionsarenotcomputablebyTuring machines(thus,bydigitalcomputers).Turingmachinescomputefunctionsdefined overdenumerabledomains,suchasstringsoflettersfroma finitealphabet.There areuncountablymanysuchfunctions.ButthereareonlycountablymanyTuring
machines;youcanenumerateTuringmachinesbyenumeratingalllistsofinstructions.Sinceanuncountableinfinityismuchlargerthanacountableone,itfollows thatTuringmachines(andhencedigitalcomputers)cancomputeonlyatinyportion ofallfunctions(overdenumerabledomains,suchasnaturalnumbersorstringsof letters).
Turingmachinesandmostmoderncomputersareknownas(classical) digital computers,thatis,computersthatmanipulatestringsofdiscrete,unambiguously distinguishablestates.Digitalcomputersaresometimescontrastedwith analog computers,thatis,machinesthatmanipulatecontinuousvariables.Continuous variablesarevariablesthatcanchangetheirvaluecontinuouslyovertimewhile takinganyvaluewithinacertaininterval.Analogcomputersareusedprimarilyto solvecertainsystemsofdifferentialequations(Pour-El1974;Rubel1993).
Classicaldigitalcomputersmayalsobecontrastedwith quantum computers (NielsenandChuang2000).Quantumcomputersmanipulatequantumstatescalled qudits (mostcommonly binary qudits,or qubits).Unlikethecomputationalstatesof digitalcomputers,quditsarenotunambiguouslydistinguishablefromoneanother. Thisbookwillfocusonclassical(i.e.,non-quantum)computation.
Thesameentitiesstudiedinthemathematicaltheoryofcomputation Turing machines,algorithms,andsoon aresaidtobeimplementedbyconcretephysical systems.Thisposesaproblem:underwhatconditionsdoesaconcrete,physical systemperformacomputationwhencomputationisdefinedbyanabstractmathematicalformalism?Thismaybecalledthe problemofcomputationalimplementation
Theproblemofcomputationalimplementationmaybeformulatedinacoupleof differentways,dependingonourontologyofmathematics.Somepeopleinterpret theformalismsofcomputabilitytheory,aswellasotherportionsofmathematics,as definingandreferringtoabstractobjects.Accordingtothisinterpretation,Turing machines,algorithms,andthelikeareabstractobjects.1
Abstractobjectsinthissenseshouldnotbeconfusedwithabstractioninthesense offocusingononeaspectofsomethingattheexpenseofotheraspects.Forinstance, wetalkabouttheeconomyofacountryanddiscusswhetheritisgrowingor contracting;wedosobyabstractingawayfrommanyotheraspectsoftheobjects andpropertiesthatconstitutethatcountry.Iwilldiscussthisnotionofabstraction (partialconsideration)later.Nowlet’sdealwithabstractobjects.
Abstractobjectsareputativeentitiesthataresupposedtobenon-spatial,nontemporal,andnon-causal.Inotherwords,abstractobjectshavenospatiallocation, donotexistthroughtime,andarecausallyinert.Theviewthatthereareabstract mathematicalobjectsandthatourmathematicaltruthsdescribesuchobjectstrulyis called platonism (Balaguer2009;Linnebo2011;Rodriguez-Pereyra2011;Rosen2012;
1 E.g.: ‘Computationalmodelsare abstractentities.Theyarenotlocatedinspaceandtime,andtheydo notparticipateincausalinteractions’ (Rescorla2014b,1277,emphasisadded).
Swoyer2008).Accordingtoplatonism,mathematicalentitiessuchasthenumber2 areabstractobjects:thenumber2isinnoparticularplace,existsatemporally,andis causallyinert.Yet,whenwesaythat2isthesquarerootof4,wesaysomethingtrue aboutthatabstractobject.
Ifyoubelievethatcomputabilitytheoryisaboutabstractobjectsandtherearesuch things thatis,ifyouareaplatonist youmayask:whatmustbethecaseforagiven concretephysicalsystemtoimplementagivenabstractcomputationalobject(as opposedtoanotherabstractobject,ornoneatall)?Thisishowplatonistsformulate theproblemofcomputationalimplementation.2
Non-platoniststreattheformalismsofcomputabilitytheorysimplyasabstract computational descriptions,withoutpositinganyabstractobjectsastheirreferents. Themaincontemporaryalternativetomathematicalplatonismistheviewthat mathematicsisputativelyaboutabstractobjectsbuttherearenosuchthings. Platonistsarerightthatmathematicalobjectssuchasnumbers,sets,functions,and Turingmachineswouldbeabstractobjects,iftherewereany.Butthereareno abstractobjects.Strictlyspeaking,therearenonumbers,nosets,nofunctions,no Turingmachines,etc.Therefore,strictlyspeaking,existentialmathematicalstatementsarenottrue;universalmathematicalstatementsarevacuouslytrue(forlackof referents).Thisviewgoesbythenameof fictionalism (Field1980,1989;Balaguer 2013;Paseau2013;Bueno2009;Leng2010).
Ifyoubelievetherearenoabstractobjects,youmayask:whatdoesittakefora givenconcretephysicalsystemtosatisfyagivenabstractcomputationaldescription (asopposedtoanother,ornoneatall)?Thisishowanon-platonistformulatesthe problemofcomputationalimplementation.Regardlessofhowtheproblemofcomputationalimplementationisformulated,solvingitrequiresanaccountofconcrete computation anaccountofwhatittakesforagivenphysicalsystemtoperforma givencomputation.
Acloselyrelatedproblemisthatofdistinguishingbetweenphysicalsystemssuch asdigitalcomputers,whichappeartocompute,andphysicalsystemssuchasrocks, whichappearnottocompute.Unlikecomputers,ordinaryrocksarenotsoldin computerstoresandareusuallynottakentoperformcomputations.Why?Whatdo computershavethatrockslack,suchthatcomputerscomputeandrocksdon’t? (Ifindeedtheydon’t?)
Questionsonthenatureofconcretecomputationshouldnotbeconfusedwith questionsaboutcomputationalmodeling.Thedynamicalevolutionofmanyphysical systemsmaybedescribedbycomputationalmodels.Computationalmodels describethedynamicsofasystemthatarewritteninto,andrunby,acomputer. Thebehaviorofrocks aswellasrivers,ecosystems,andplanetarysystems,among
2 E.g.: “weneedatheoryofimplementation:therelationthatholdsbetweenan abstractcomputational object... andaphysicalsystem” (Chalmers2011,325,emphasisadded).
manyothers maywellbemodeledcomputationally.Fromthis,itdoesn’tfollow thatthemodeledsystemsarecomputingdevices thattheythemselvesperform computations.Primafacie,onlyrelativelyfewandquitespecialsystemscompute. Explainingwhatmakesthemspecial orexplainingawayourfeelingthattheyare special isthejobofanaccountofconcretecomputation.
Thisbookoffersanaccountofconcretecomputation.Ihavereservationsabout bothplatonismand fictionalismbutthisisnottheplacetoairthem.Iwillremain officiallyneutralabouttheontologyofmathematics;myaccountworksunderany viewaboutmathematicalontology.
2.AbstractDescriptions
Beforedevelopinganaccountofconcretecomputation,weshouldintroduceanotion ofabstractionthathaslittletodowithabstractobjects.Concretephysicalsystems satisfyanumberofdifferentdescriptions.Forexample,oneandthesamephysical systemmaybeajuicypineapple,acollectionofcellsthatareboundtogetherand releaseacertainamountofliquidunderacertainamountofpressure,anorganized systemofmoleculesthatgiveoutcertainchemicalsundercertainconditions,abunch ofatomssomeofwhosebondsbreakundercertainconditions,etc.Bythesame token,thesamephysicalsystemmaybeaDellLatitudecomputer,aprocessor connectedtoamemoryinsideacase,agroupofwell-functioningdigitalcircuits, anorganizedcollectionofelectroniccomponents,avastsetofmoleculesarrangedin specificways,etc.
A firstobservationabouttheaboveexamplesisthatsomedescriptionsinvolve smallerentities(andtheirproperties)thanothers.Atomsaregenerallysmallerthan molecules,whicharesmallerthancellsandelectroniccomponents,whichinturnare smallerthanpineapplesandprocessors.
Asecondobservationisthat,withinacomplexsystem,usuallylargerthingsare madeoutofsmallerthings.Atomscomposemolecules,whichcomposecellsand circuits,whichcomposepineapplesandprocessors.
Athirdobservationisthatinordertodescribethesamepropertyof,oractivityby, thesamesystem,descriptionsintermsoflargerthingsandtheirpropertiesareoften moreeconomicalthandescriptionsintermsoftheircomponentsandtheirproperties.Forexample,itonlytakesacoupleofpredicatestosaythatsomethingisajuicy pineapple.Ifwehadtoprovidethesameinformationbydescribingthesystem’scells andtheirproperties,wewouldneedtodescribethekindsofcell,howtheyare arranged,andhowmuchliquidtheyreleasewhentheyaresquished(andtherelevant amountofsquishing).Todothisinanadequatewaywouldtakealotofwork.Todo thisintermsofthesystem’smoleculesandtheirpropertieswouldrequiregivingeven moreinformation,whichwouldtakeevenmorework.Doingthisintermsofthe system’satomsandtheirpropertieswouldbeevenworse.Thesamepointappliesto computingsystems.ItonlytakesonepredicatetosaythatsomethingisaDell
Latitude.Itwouldtakeconsiderablymoretosayenoughaboutthesystem’ sprocessor,memory,andcasetodistinguishthemfromnon-Dell-Latitudeprocessors, memories,andcases.Itwouldtakeevenmoretosaythesamethingintermsof thesystem’sdigitalcircuits,evenmoreintermsofthesystem’selectroniccomponents,andevenmoreintermsofthesystem’satoms.
Descriptionsofthesamesystemmayberankedaccordingtohowmuchinformationtheyrequireinordertoexpressthesamepropertyorpropertiesofthesystem. Descriptionsthataremoreeconomicalthanothersachievetheireconomyby expressingselectinformationaboutthesystemattheexpenseofotherinformation. Inotherwords,economicaldescriptionsomitmanydetailsaboutasystem.They capturesomeaspectsofasystemwhileignoringothers.Followingtradition,Iwillcall adescriptionmoreorless abstract dependingonhowmanydetailsitomitsinorder toexpressthatasystempossessesacertainproperty.Themoreeconomicallya descriptionaccomplishesthis,themoreabstractitis.Inourexample, ‘beingaDell Latitude’ ismoreabstractthan ‘beingacertaintypeofprocessorconnectedtoa certaintypeofmemoryinsideacertainkindofcase,’ whichinturnismoreabstract than ‘beingsuchandsuchasystemoforganizeddigitalcircuits.’
Mathematicaldescriptionsofconcretephysicalsystemsareabstractinthissense. Theyexpresscertainproperties(e.g.,shape,size,quantity)whileignoringothers(e.g., color,chemicalcomposition).Inthisbook,Iwillgiveanaccountofcomputing systemsaccordingtowhichcomputationaldescriptionsofconcretephysicalsystems aremathematicalandthusabstractinthesamesense.Theyexpresscertainproperties (e.g.,whichfunctionasystemcomputes)inaneconomicalwaywhileignoring others.
Aplatonistmayobjectasfollows.WhataboutTuringmachinesandother computationalentitiesdefinedpurelymathematically,whosepropertiesarestudied bycomputabilitytheory?Surelythosearenotconcretephysicalsystems.Surely mathematiciansarefreetode finecomputationalentities,studytheirproperties, anddiscovertruthsaboutthem,withouttheirdescriptionsneedingtobetrueof anyphysicalsystem.Forinstance,Turingmachineshaveunboundedtapes,whereas thereisalimittohowmuchthememoryofaconcretecomputercanbeextended. Therefore,Turingmachinesarenotconcretephysicalsystems.Therefore,Turing machinesmustbeabstractobjects.Therefore,westillneedabstractcomputational objectstoactastruthmakersforourcomputationaldescriptions.
MyreplyissimplythatIamafteranaccountofcomputationinthephysicalworld. IfTuringmachinesandothermathematicallydefinedcomputationalentitiesare abstractobjects andtheirdescriptionsarenottrueofanyphysicalsystem theyfall outsidethescopeofmyaccount.Therefore,evenifweconcludedthat,ifcomputabilitytheoryrefers,itreferstoabstractobjects,thiswouldnotchangemyaccountof concretecomputation.
3.TowardsanAccountofConcreteComputation
Hereiswherewe’vegottensofar:theproblemofcomputationalimplementationis whatdistinguishesphysicalprocessesthatcountascomputationsfromphysical processesthatdonot,andwhichcomputationstheycountas.Equivalently,itis thequestionofwhatdistinguishesphysicalsystemsthatcountascomputingsystems fromphysicalsystemsthatdonot,andwhichcomputationstheyperform.
Theproblemofcomputationalimplementationtakesdifferentformsdepending onhowwethinkaboutmathematicallydefinedcomputation.InthisbookIwill assumethatmathematicallyde finedcomputationisanabstract(mathematical) descriptionofahypotheticalprocess,whichmaybeidealizedinvariouswaysand mayormaynotbephysicallyrealizable.Themathematicalstudyofcomputation employsabstractdescriptionswithoutmuchconcernforphysicalimplementation. Concretecomputationisakindofphysicalprocess.Itmaybedescribedinvery concretetermsormoreabstractterms.Ifthedescriptionisabstractenoughinways thatIwillarticulate,itisacomputationaldescription.
If,onthecontrary,youtakeabstractcomputationtobethedescriptionofabstract objects,thenthequestionbecomesoneofidentifyingarelationshipbetweenconcrete objectsandtheirabstractcounterpartssuchthattheformercountasimplementationsofthelatter.EverythingIsayinthisbookmaybereinterpretedwithinthis platonistframeworksothatitanswerstheplatonistquestion.(Mutatismutandis for the fictionalist.)ForbythetimeIshowhowcertainconcreteobjectssatisfythe relevantcomputationaldescriptions,Ialsosolvetheplatonisticallyformulatedproblemofimplementation.Alltheplatonisthaslefttodoistointerpretthecomputationaldescriptionsasreferringtoabstractobjects,andpositanappropriaterelation betweentheabstractobjectsthusreferredtoandtheconcreteobjectsthatputatively implementthem.
Intherestofthebook,Iwilldefendthefollowingsolutiontotheproblemof computationalimplementation.
TheMechanisticAccountofComputation:Aphysicalcomputingsystemisa mechanismwhoseteleologicalfunctionisperformingaphysicalcomputation. Aphysicalcomputationisthemanipulation(byafunctionalmechanism)ofa medium-independentvehicleaccordingtoarule.Amedium-independentvehicle isaphysicalvariabledefinedsolelyintermsofitsdegreesoffreedom(e.g.,whether itsvalueis1or0duringagiventimeinterval),asopposedtoitsspeci ficphysical composition(e.g.,whetherit’savoltageandwhatvoltagevaluescorrespondto1or 0duringagiventimeinterval).Aruleisamappingfrominputsand/orinternal statestointernalstatesand/oroutputs.
AllofthiswillbeunpackedinmoredetailinChapter7,afterlayingappropriate groundwork.Theoverarchingargumentforthismechanisticaccountisthatit satisfiesanumberofdesideratabetterthanthecompetition.
4.Desiderata
Anaccountofconcretecomputationthatdoesjusticetothesciencesofcomputation shouldhavethefollowingfeatures:1)objectivity;2)explanation;3)therightthings compute;4)thewrongthingsdon’tcompute;5)miscomputationisexplained;and 6)anadequatetaxonomyofcomputingsystemsisprovided.
1Objectivity
Anaccountwithobjectivityissuchthatwhetherasystemperformsaparticular computationisamatteroffact.Contrarytoobjectivity,someauthorshavesuggested thatcomputationaldescriptionsarevacuous amatteroffreeinterpretationrather thanfact.Theallegedreasonisthatanysystemmaybedescribedasperformingjust about any computation,andthereisnofurtherfactofthematterastowhetherone computationaldescriptionismoreaccuratethananother(Putnam1988;Searle1992; cf.Chapter4,Section2).Thisconclusionmaybeinformallyderivedasfollows. Assumeasimplemappingaccountofcomputation(Chapter2):asystemperforms computation C ifandonlyifthereisamappingbetweenasequenceofstates individuatedby C andasequenceofstatesindividuatedbyaphysicaldescription ofthesystem(Putnam1960,1967b,1988).Assume,assomehave,thatthereareno constraintsonwhichmappingsareacceptable,sothatanysequenceofcomputationalstatesmaymapontoanysequenceofphysicalstatesofthesamecardinality.If thesequenceofphysicalstateshaslargercardinality,thecomputationalstatesmay mapontoeitherequivalentclassesorasubsetofthephysicalstates.Sincephysical variablescangenerallytakerealnumbersasvaluesandthereareuncountablymany ofthose,physicaldescriptionsgenerallygiverisetouncountablymanystatesand statetransitions.Butordinarycomputationaldescriptionsgiveriseonlytocountably manystatesandstatetransitions.Therefore,thereisamappingfromany(countable) sequenceofcomputationalstatetransitionsontoeitherequivalentclassesorasubset ofphysicalstatesbelongingtoany(uncountable)sequenceofphysicalstatetransitions.Therefore,generally,anyphysicalsystemperformsanycomputation.
Ifthisresultissound,thenempiricalfactsaboutconcretesystemsmakeno differencetowhatcomputationstheyperform.BothPutnam(1988)andSearle (1992)takeresultsofthissorttotrivializetheempiricalimportofcomputational descriptions.Bothconcludethatcomputationalism theviewthatthebrainisa computingsystem isvacuous.But,asusual,oneperson’ s modusponens isanother person ’ s modustollens.ItakePutnamandSearle’sresulttorefutetheiraccountof concretecomputation.
Computerscientistsandengineersappealtoempiricalfactsaboutthesystemsthey studytodeterminewhichcomputationsareperformedbywhichsystems.Theyapply computationaldescriptionstoconcretesystemsinawayentirelyanalogoustoother bona fidescientificdescriptions.Inaddition,manypsychologistsandneuroscientists areinthebusinessofdiscoveringwhichcomputationsareperformedbymindsand
brains.Whentheydisagree,theyaddresstheiropponentsbymusteringempirical evidenceaboutthesystemstheystudy.Unlesstheprimafacielegitimacyofthose scientificpracticescanbeexplainedaway,agoodaccountofconcretecomputation shouldentailthatthereisafactofthematterastowhichcomputationsare performedbywhichsystems.
2Explanation
Computationsperformedbyasystemmayexplainitscapacities.Ordinarydigital computersaresaidtoexecuteprograms,andtheircapacitiesarenormallyexplained byappealingtotheprogramstheyexecute.Theliteratureoncomputationaltheories ofcognitioncontainsexplanationsthatappealtothecomputationsperformedbythe brain.Thesameliteraturealsocontainsclaimsthatcognitivecapacitiesoughttobe explainedcomputationally,andmorespeci fically,byprogramexecution(e.g.,Fodor 1968b;Cummins1977;cf.Chapter5).Agoodaccountofcomputingmechanisms shouldsayhowappealstoprogramexecution,andmoregenerallytocomputation, explainthebehaviorofcomputingsystems.Itshouldalsosayhowprogramexecutionrelatestothegeneralnotionofcomputation:whethertheyarethesameand,if not,howtheyarerelated.
3Therightthingscompute
Agoodaccountofcomputingmechanismsshouldentailthatparadigmaticexamples ofcomputingmechanisms,suchasdigitalcomputers,calculators,bothuniversaland non-universalTuringmachines,and finitestateautomata,compute.
4Thewrongthingsdon’tcompute
Agoodaccountofcomputingmechanismsshouldentailthatallparadigmatic examplesofnon-computingmechanismsandsystems,suchasplanetarysystems, hurricanes,anddigestivesystems,don’tperformcomputations.
Contrarytodesideratum4,manyauthorsmaintainthateverythingperforms computations(e.g.,Chalmers1996b,331;Scheutz1999,191;Shagrir2006b;cf. Chapter4).Butcontrarytotheirviewaswellasdesideratum3,thereareaccounts ofcomputationsorestrictivethatunderthem,evenmanyparadigmaticexamplesof computingmechanismsturnout not tocompute.Forinstance,accordingtoJerry FodorandZenonPylyshyn,anecessaryconditionforsomethingtoperformcomputationsisthatthestepsitfollowsbecausedbyinternalrepresentationsofrulesfor thosesteps(Fodor1968b,1975,1998,10–11;Pylyshyn1984).Butnon-universal Turingmachinesand finitestateautomatadonotrepresentrulesforthestepsthey follow.Thus,accordingtoFodorandPylyshyn’saccount,theydonotcompute.
Theaccountsjustmentionedfaildesideratum3or4.Why,what’swrongwiththat? Therearedebatablecases,suchaslook-uptablesandanalogcomputers.Whether thosereallycomputemaybeopentodebate,andinsomecasesitmaybeopento stipulation.Butthereareplentyofclearcases.Digitalcomputers,calculators,Turing
machines,and finitestateautomataareparadigmaticcomputingsystems.They constitutethesubjectmatterofcomputerscienceandcomputabilitytheory.Planetary systems,theweather,anddigestivesystemsareparadigmaticnon-computingsystems3;attheveryleast,it’snotobvioushowtoexplaintheirbehaviorcomputationally. Ifwecan findanaccountthatworksfortheclearcases,theunclearonesmaybeleftto fallwherevertheaccountsaystheydo “spoilstothevictor” (Lewis1986,203;cf. Collins,Hall,andPaul2004,32).
Insofarastheassumptionsofcomputerscientistsandcomputabilitytheorists groundthesuccessoftheirscienceaswellastheappealoftheirnotionofcomputationtopractitionersofotherdisciplines,theyoughttoberespected.Bysatisfying desiderata3and4,agoodaccountofphysicalcomputationdrawsaprincipled distinctionbetweensystemsthatcomputeandsystemsthatdon’t,anditdrawsit inaplacethat fitsthepresuppositionsofgoodscience.
5Miscomputationisexplained
Computationscangowrong.Toa firstapproximation,asystem M miscomputesjust incase M iscomputingfunction f oninput i,f(i)= o1,M outputs o2,and o2 ¼ o1.Here o1 and o2 representanypossibleoutcomeofacomputation,includingthepossibility thatthefunctionisundefinedforagiveninput,whichcorrespondstoanon-halting computation.Miscomputationisanalogoustomisrepresentation(Dretske1986),but it’snotthesame.Something(e.g.,asorter)maycomputecorrectlyorincorrectly regardlessofwhetheritrepresentsormisrepresentsanything.Something(e.g.,a painting)mayrepresentcorrectlyorincorrectlyregardlessofwhetheritcomputesor miscomputesanything.
Fresco(2013,Chapter2)pointsoutthattheabovedefinitionofmiscomputationis toopermissive.Considerasystem M thatiscomputingfunction f oninput i,with f(i)= o1,andoutputs o1.Eventhoughtheoutputiscorrect, M mayhavefailedto compute f(i)correctly. M outputting o1 maybeduenottoacorrectcomputationof f(i)buttorandomnoisein M’soutputdevice,orto M followingacomputational paththathasnothingtodowith f(i)butstillresultsin o1,orto M switchingfrom computing f(i)tocomputing g(i),where g ¼ f,but g(i)= o1.Inpractice,theseare unlikelyscenarios.Nevertheless,wemaystillwishtosaythatamiscomputation occurred.
Toinsurethis,wemayrestrictourdefinitionofmiscomputationasfollows.As before,letMbeasystemcomputingfunction f(i)= o1.Let P betheprocedure M issupposedtofollowincomputing f(i),with P consistingofcomputationalsteps s1, s2,... sn.Bydefinition,theoutcomeof sn is o1.Let si bethe ithstepinthesequence s1, s2,... sn–1.
3 Forevidence,seeFodor1968b,632;Fodor1975,74;Dreyfus1979,68,101–2;Searle1980,37–8;Searle 1992,208.
Mmiscomputesf(i)justincase,foranystep si,after M takesstep si,either M takes acomputationalstepotherthan si+1 or M failstocontinuethecomputation.
Inotherwords, M miscomputesjustincaseitfailstofolloweverystepofthe procedureit’ssupposedtofollowallthewayuntilproducingthecorrectoutput. Althoughmiscomputationhasbeenignoredbyphilosophersuntilveryrecently,a goodaccountofcomputingmechanismsshouldexplainhowit’spossiblefora physicalsystemtomiscompute.Thisisdesirablebecausemiscomputation,ormore informally,makingcomputationalmistakes,playsanimportantroleincomputer scienceanditsapplications.Thosewhodesignandusecomputingmechanisms devotealargeportionoftheireffortstoavoidingmiscomputationsanddevising techniquesforpreventingthem.Totheextentthatanaccountofcomputingmechanismsmakesnosenseofthateffort,itisunsatisfactory.
6Taxonomy
Differentclassesofcomputingmechanismshavedifferentcapacities.Logicgatescan performonlytrivialoperationsonpairsofbits.Non-programmablecalculatorscan computea finitebutconsiderablenumberoffunctionsforinputsofboundedsize. Ordinarydigitalcomputerscancomputeanycomputablefunctiononanyinputuntil theyrunoutofmemoryortime.Differentcapacitiesrelevanttocomputingplayan importantroleincomputerscienceandcomputingapplications.Anyaccountof computingsystemswhoseconceptualresourcesexplainorshedlightonthose differencesispreferabletoanaccountthatisblindtothosedifferences.
Toillustrate,considerRobertCummins’saccount.AccordingtoCummins(1983), forsomethingtocompute,itmustexecuteaprogram.Healsomaintainsthat executingaprogramamountstofollowingthestepsdescribedbytheprogram. Thisleadstoparadoxicalconsequences.Considerthatmanyparadigmaticcomputingmechanisms(suchasnon-universalTuringmachinesand finitestateautomata) arenotcharacterizedbycomputerscientistsasexecutingprograms,andtheyare considerablylesspowerfulthanthesystemsthataresocharacterized(i.e.,universal Turingmachinesandidealizeddigitalcomputers).Accordingly,wemightconclude thatnon-universalTuringmachines, finitestateautomata,etc.,donotreallycompute.Butthisviolatesdesideratum3(therightthingscompute).Alternatively,we mightobservealongwithCumminsthatnon-universalTuringmachinesand finite statesautomatadofollowthestepsdescribedbyaprogram.Therefore,byCummins’ s light,theyexecuteaprogram,andhencetheycompute.Butnowwe finditdifficultto explainwhytheyarelesspowerfulthanordinarydigitalcomputers. 4 Forunder Cummins’saccount,wecannotsaythat,unlikedigitalcomputers,thoseother
4 Ifwemakethesimplifyingassumptionthatordinarydigitalcomputershaveamemoryof fixedsize, theyareequivalentto(veryspecial) finitestateautomata.HereIamcontrastingordinarydigitalcomputers withordinary finitestateautomata.
systemslackthe flexibilitythatcomeswiththecapacitytoexecuteprograms.The differencebetweencomputingsystemsthatexecuteprogramsandthosethatdon’tis importanttocomputerscienceandcomputingapplications,anditshouldmakea differencetotheoriesofcognition.Weshouldpreferanaccountthathonorsthat kindofdifferencetoonethatisblindtoit.
Withthesedesiderataaslandmarks,Iwillproceedtoformulateandevaluate differentaccountsofconcretecomputation.Iwillarguethattheaccountthatbest satisfiesthesedesiderataisthemechanisticone.
