On the Relation of Computing to the World by demandside

OntheRelationofComputingtotheWorld

WilliamJ.Rapaport

DepartmentofComputerScienceandEngineering, DepartmentofPhilosophy,DepartmentofLinguistics, andCenterforCognitiveScience

UniversityatBuffalo,TheStateUniversityofNewYork, Buffalo,NY14260-2500

rapaport@buffalo.edu

http://www.cse.buffalo.edu/∼rapaport/

April21,2015

Abstract

Isurveyacommonthemethatpervadesthephilosophyofcomputerscience (andphilosophymoregenerally):therelationofcomputingtotheworld. Arealgorithmsmerelycertainproceduresentirelycharacterizableinan“indigenous”,“internal’,“intrinsic”,“local”,“narrow”,“syntactic”(moregenerally:“intra-system”)purelyTuring-machinelanguage?Ormusttheyinteractwiththerealworld,withapurposethatisexpressibleonlyinalanguage withan“external”,“extrinsic”,“global”,“wide”,“inherited”(moregenerally:“extra-”or“inter-”sytem)semantics?

1Preface

IfyoubeginwithComputerScience,youwillendwithPhilosophy.1

Iwassimultaneouslysurprisedanddeeplyhonoredtoreceivethe2015Covey AwardfromtheInternationalAssociationforComputingandPhilosophy.2 The honorisdueinparttolinkingmetotheillustriouspredecessorswhohavereceived thisaward,butalsotoitshavingbeennamedforPrestonCovey,3 whomIknew andwhoinspiredmeasIbeganmytwinjourneysinphilosophyandcomputing.

1.1FromPhilosophytoComputerScience,andBackAgain

Contrarytothemottoabove,Ibeganwithphilosophy,foundmywaytocomputer science,andhavereturnedtoamixtureofthetwo.InspiredbyDouglasHofstadter’sreview[Hofstatder,1980]ofAaronSloman’s TheComputerRevolutionin Philosophy [Sloman,1978],whichquotedSlomantotheeffectthataphilosopher ofmindwhoknewnoAIwaslikeaphilosopherofphysicswhoknewnoquantum mechanics,4 myphilosophicalinterestsinphilosophyofmindledmetostudyAI atSUNYBuffalowithStuartC.Shapiro.5 ThiseventuallyledtoafacultyappointmentincomputerscienceatBuffalo.(Alongtheway,myphilosophycolleagues andIatSUNYFredoniapublishedoneoftheﬁrstintroductorylogictextbooksto useacomputationalapproach[Schagrinetal.,1985].)

AtBuffalo,Iwasamazedtodiscoverthatmyrelativelyarcanephilosophy dissertationonAlexiusMeinongwasdirectlyrelevanttoShapiro’sworkinAI, providinganintensionalsemanticsforhisSNePSsemantic-networkprocessing system(see,e.g.,[ShapiroandRapaport,1987],[ShapiroandRapaport,1991]).6 AndthenIrealizedthatthediscoveryofquasi-indexicals(‘hehimself’,‘sheherself’,etc.;[Castaneda,1966])bymydissertationadvisor,Hector-NeriCastaneda

1“ClickingontheﬁrstlinkinthemaintextofaWikipediaarticle,andthenrepeatingthe processforsubsequentarticles,usuallyeventuallygetsyoutothePhilosophyarticle.Asof May26,2011,94.52%ofallarticlesinWikipedialeadeventuallytothearticlePhilosophy” (http://en.wikipedia.org/wiki/Wikipedia:Getting to Philosophy).Ifyoubeginwith“ComputerScience”,youwillendwith“Philosophy”(in12links).

2http://www.iacap.org/awards/

3http://en.wikipedia.org/wiki/Covey Award

4“Iampreparedtogosofarastosaythatwithinafewyears,ifthereremainanyphilosopherswho arenotfamiliarwithsomeofthemaindevelopmentsinartiﬁcialintelligence,itwillbefairtoaccuse themofprofessionalincompetence,andthattoteachcoursesinphilosophyofmind,epistemology, aesthetics,philosophyofscience,philosophyoflanguage,ethics,metaphysics,andothermainareas ofphilosophy,withoutdiscussingtherelevantaspectsofartiﬁcialintelligencewillbeasirresponsible asgivingadegreecourseinphysicswhichincludesnoquantumtheory”[Sloman,1978,p.5].

5http://www.cse.buffalo.edu/∼shapiro/

6IpresentedsomeofthisworkatCAP1987.

couldrepaira“bug”inaknowledge-representationtheorythatShapirohaddevelopedwithanotherconverttocomputerscience(frompsychology),AnthonyS. Maida[MaidaandShapiro,1982].(Thisworkwasitselfdebuggedwiththehelpof yetanotherconvert(fromEnglish),mydoctoralstudentJanyceM.Wiebe [Rapaportetal.,1997].)

MyworkwithShapiroandourSNePSResearchGroupatBuffaloenabledme torebutmyCoveyAwardpredecessorJohnR.Searle’sChinese-RoomArgument [Searle,1980]usingaphilosophicaltheoryofwhatIcall“syntacticsemantics” [Rapaport,1986b],[Rapaport,1988],[Rapaport,1995],[Rapaport,2012].7 And bothoftheseprojects,aswellasoneofmyearlyMeinongpapers[Rapaport,1981], ledme,togetherwithanotherdoctoralstudent(KarenEhrlich)and(later)acolleaguefromBuffalo’sDepartmentofLearningandInstruction(MichaelW.Kibby) todevelopacomputationalandpedagogicaltheoryofvocabularyacquisitionfrom context[RapaportandKibby,2007],[RapaportandKibby,2014].8

1.2ThePhilosophyofComputerScience

Allofthisinspiredmetocreateandteachacourseonthephilosophyofcomputer science[Rapaport,2005b]9 and,nowinretirement,towriteupmylecturenotesas atextbook[Rapaport,2015].

Thecourseandthetextbeginwithasinglequestion:Whatiscomputerscience?10 Toanswerthis,weneedtoconsideraseriesofquestions,eachofwhich leadstoanother:

• Iscomputerscienceascience?(Andwhatisscience?)Orisitabranchof engineering?(Whatisengineering?)(Orisitacombinationofthese?Or perhapssomethingcompletelydifferent,new, suigeneris?)

• Ifitisascience,whatisitascienceof?Ofcomputers?(Inthatcase,what isacomputer?)Orofcomputation?Thatquestionleadstomanyothers:

• Whatiscomputation?Whatisanalgorithm?Doalgorithmsdifferfromprocedures?Fromrecipes?(Andwhatarethey?)Whatisthe(Church-Turing) ComputabilityThesis?11 Whatishypercomputation?Whatisacomputer

7IpresentedsomeofthisworkatIACAP2009andNACAP2010.

8IpresentedsomeofthisworkatNACAP2006.

9PresentedatNACAP2006inmyHerbertA.SimonKeynoteAddress, http://www.hass.rpi.edu/streaming/conferences/cap2006/nacp 8 11 2006 9 1010.asx

10Thisistheﬁrstquestion,but,becausetherearetwointendedaudiences—philosophystudents andcomputer-sciencestudents—Iactuallybeginwithazerothquestion:Whatisphilosophy?

11Ipreferthisnameover“Church’sThesis”,“Turing’sThesis”,orthe“Church-TuringThesis”for reasonsgivenin[Soare,2009, §§3.5,12].

program?Thisquestionalsogivesrisetoahostofothers:

• Whatisanimplementation?Whatistherelationofaprogramtothatwhich itmodelsorsimulates?(Forthatmatter,whatissimulation?Andcanprogramsbeconsideredtobe(scientiﬁc)theories?)Whatissoftware,andhow doesitrelatetohardware?(Andcan,orshould,oneorbothofthosebe copyrightedorpatented?)Cancomputerprogramsbe(logically)veriﬁed?

• Thereare,ofcourse,issuesinthephilosophyofAI(WhatisAI?Whatisthe relationofcomputationtocognition?Cancomputersthink?Whatarethe TuringTestandtheChinese-RoomArgument?).

• Finally,thereareissuesincomputerethics,butIonlytouchontwothatI thinkarenotwidelydealtwithinthealreadyvoluminouscomputer-ethics literature:Shouldwetrustdecisionsmadebycomputers?Shouldwebuild “intelligent”computers?

TherearemanyissuesthatIdon’tdealwith:Thenatureofinformation,theroleof theInternetinsociety,theroleofcomputingineducation,andsoon.ButIhaveto stopsomewhere!

Mygoalinthebookisnottobecomprehensive,buttoprovidebackground onsomeofthemajorissuesandaguidetosomeofthemajorpapers,andtoraise questionsforreaderstothinkabout(togetherwithaguidetohowtothinkabout them—thetextcontainsabriefintroductiontocriticalthinkingandlogicalevaluationofarguments).Foraphilosophytextbook,raisingquestionsismoreimportant thanansweringthem.Mygoalistogivereaderstheopportunityandthemeansto joinalong-standingconversationandtodevisetheirownanswerstosomeofthese questions.

2ACommonThread:ComputingandtheWorld

Inthecourseofwritingthebook,Ihavenoticedathemethatpervadesitstopics.In linewithmygoalsforthebook,Ihavenotyetcommittedmyselftoaposition;Iam stillaskingquestionsandexploring.Inthisessay,Iwanttosharethosequestions andexplorationswithyou.

Thecommonthreadthatrunsthroughmost,ifnotall,ofthesetopics:the relationofcomputingtotheworld:

Iscomputingaboutthe world?Isit“external”,“global”,“wide”,or “semantic”?

Orisitabout descriptions oftheworld?Isit,instead,“internal”,“local”,“narrow”,or“syntactic”?

AndIwillquicklyagreethatitmightbeboth!Inthatcase,thequestionishowthe answerstothesequestionsarerelated.

Thisthemeshouldbefamiliartomostofyou;Iamnotannouncingsomenewly discoveredphilosophicalpuzzle.Butitisn’tnecessarilyfamiliarorobviousto mybook’sintendedaudience,andIdowanttorecommenditasatopicworth ponderingandworthdiscussingwithstudents.

Inthissection,we’llsurveytheseissuesastheyappearinsomeofthephilosophyofcomputersciencequestionsof §1.2.Insubsequentsections,we’llgointo moredetail.ButIonlypromisetoraisequestions,nottoanswerthem(inthecourse andthetext:tochallengemystudents’thinking,nottotellthemwhattothink).

LeslieLamportrecentlysaid(quotingcartoonistRichardGuindon),“‘Writingis nature’swayoflettingyouknowhowsloppyyourthinkingis.’Wethinkinorder tounderstandwhatwearedoing.Ifweunderstandsomething,wecanexplainit clearlyinwriting.Ifwehavenotexplaineditinwriting,thenwedonotknowif wereallyunderstandit”[Lamport,2015,38].12

2.1SomeThoughtExperiments

Castanedausedtosaythatphilosophizingmustbeginwithdata.Solet’sbeginwith somedataintheformofrealandimaginedcomputerprograms.

2.1.1Rey’sandFodor’sChessandWarPrograms

JerryFodor,takingupasuggestionbyGeorgesRey,asksustoconsideracomputer thatsimulatestheSixDayWarandacomputerthatsimulates(oractuallyplays?) agameofchess,butwhichare(are?)suchthat“theinternalcareerofamachine runningoneprogramwouldbeidentical,stepbystep,tothatofamachinerunning theother”[Fodor,1978,p.232].

Arealexampleofthesamekindis“amethodforanalyzingx-raydiffraction datathat,withafewmodiﬁcations,alsosolvesSudokupuzzles”.13 Orconsidera computerversionofthemurder-mysterygameCluethatexclusivelyusestheResolutionruleofinference,andsocouldbeageneral-purposepropositionaltheorem proverinstead.14

12AsE.M.Forsterisallegedtohavesaid,“HowcanIknowwhatIthinktillIseewhatIsay?” https://rjheeks.wordpress.com/2011/04/13/discovery-writing-and-the-so-called-forster-quote/

13CitedinthebiographicalsketchforabookreviewbyVeitElser[Elser,2012].

14ThankstoRobinHill,personalcommunication,forthisexample.

Similarexamplesabound,notablyinapplicationsofmathematicstoscience, andthesecanbesuitably“computationalized”byimaginingcomputerprograms foreach.Forexample,

NicolaasdeBruijnoncetoldmeroughlythefollowinganecdote:Some chemistsweretalkingaboutacertainmolecularstructure,expressing difﬁcultyinunderstandingit.DeBruijn,overhearingthem,thought theyweretalkingaboutmathematicallatticetheory,sinceeverything theysaidcouldbe—andwas—interpretedbyhimasbeingaboutthe mathematical,ratherthanthechemical,domain.Hetoldthemthe solutionoftheirproblemintermsoflatticetheory.They,ofcourse, understooditintermsofchemistry.WeredeBruijnandthechemists talkingaboutthesamething?[Rapaport,1995, §2.5.1,p.63].

Otherexamples,concerning,e.g.,theapplicabilityofgrouptheorytophysics arediscussedin[Frenkel,2013].And[Halmos,1973,p.384]pointsoutthatonce aspectsofHilbertspacesareseentobestructurallyidenticaltoaspectsofquantummechanicalsystems,“thedifferencebetweenaquantumphysicistandamathematicaloperator-theoristbecomesoneoflanguageandemphasisonly”.Clearly,EugeneWigner’sclassicpaper[Wigner,1960]isrelevanthere.Theissueshereare alsoakinto,ifnotthesameas,thosethatmotivatestructuralisminthephilosophy ofmathematics(we’llreturnbrieﬂytothisin §5.2).

Intheseexamples,dowehaveonealgorithm,ortwo?15

2.1.2Cleland’sRecipeforHollandaiseSauce

CarolClelandoffersanexampleofarecipeforhollandaisesauce[Cleland,1993], [Cleland,2002].Let’ssupposethatwehaveanalgorithm(arecipe)thattellsusto mixeggsandoil,andthatoutputshollandaisesauce.16 Supposethat,onEarth,the resultofmixingtheeggandoilisanemulsionthatis,infact,hollandaisesauce. Andletussupposethat,ontheMoon,mixingeggsandoildoesnotresultinan emulsion,sothatnohollandaisesauceisoutput(instead,theoutputisamessy mixtureofeggsandoil).

CanaTuringmachinemakehollandaisesauce?Ismakinghollandaisesauce computable?

15Comparethisremark:“Recoveringmotivesandintentionsisaprincipaljobofthehistorian. Forwithoutsomeattrributionofmentalattitudes,actionscannotbecharacterizedanddecisionsassessed. Thesameovertbehavior,afterall,mightbedescribedas‘mailingaletter’or‘fomentinga revolution.’ ”[Richards,2009,415].

16Callingarecipean“algorithm”,despiteitsubiquityinintroductorypresentationsofcomputerscience,iscontroversial;see[Preston,2013,Ch.1]forsomeusefuldiscussionofthe nonalgorithmic,improvisationalnatureofrecipes.

2.1.3ABlocks-WorldRobot

Considerablocks-worldcomputerprogramthatinstructsarobothowtopickup blocksandmovethemontooroffofotherblocks[Winston,1977].Ioncesawa livedemoofsuchaprogram.Unfortunately,therobotfailedtopickuponeofthe blocks,becauseitwasnotcorrectlyplaced,yettheprogramcontinuedtoexecute “perfectly”eventhoughtheoutputwasnotwhatwasintended.[Rapaport,1995, §2.5.1].

Did theprogrambehaveasintended?

2.1.4AGCDProgram

MichaelRescorla[Rescorla,2013, §4]offersanexamplethatisreminiscentofCleland’s,butless“physical”.HereisaSchemeprogramforcomputingthegreatest commondivisor(GCD)oftwonumbers:

(define(gcdab) (if(=b0) a (gcdb(remainderab))))

Implementthisprogramontwocomputers,one(M10)usingbase-10notation andone(M13)usingbase-13notation.Rescorlaarguesthatonly M10 executesthe Schemeprogram forcomputingGCDs,eventhough,ina“narrow”sense,both computersareexecutingthe“same”program.Whenthe numerals ‘115’and‘20’ areinputto M10,itoutputsthenumeral‘5’;“ittherebycalculatesthegreatest commondivisorofthecorresponding numbers”[Rescorla,2013,p.688].Butthe numbers expressedinbase-13by‘115’and‘20’are18710 and2610,respectively, andtheirGCDis110,not510.So,ina“wide”sense,thetwomachinesaredoing “differentthings”.

Are thesemachinesdoingdifferentthings?

2.1.5ASpreadsheet

IvividlyremembertheﬁrstsemesterthatItaughta“GreatIdeasinComputer Science”courseaimedatcomputer-phobicstudents.Weweregoingtoteachthe studentshowtouseaspreadsheetprogram,somethingthatIhadneverused!So, withrespecttothis,Iwasasnaiveasanyofmystudents.MyTA,whohadused spreadsheetsbefore,gavemesomethinglikethefollowinginstructions:

enteranumberincell 1; enteranumberincell 2; enter‘= clickoncell 1 clickoncell 2 ’incell 3

IhadnoideawhatIwasdoing.Iwasblindlyfollowingherinstructions andhad noideathatIwasaddingtwointegers.OnceshetoldmethatthatwaswhatIwas doing,myinitialreactionwas“Whydidn’tyoutellmethatbeforewebegan?”.

WhenIenteredthosedataintothespreadsheet,wasIaddingtwonumbers?

2.2WhatIsComputerScience?

Science,nomatterhowconceived,isgenerallyagreedtobeawayof understanding theworld.So,ifcomputerscienceisascience—and,ofcourse,justbecauseitis called‘computer science’doesnotimplythatit is ascience!—thenitshouldbea wayofunderstandingtheworld computationally. Engineering,nomatterhowconceived,isgenerallyagreedtobeawayof changing theworld(preferablybyimprovingit).17 So,ifcomputerscienceis anengineeringdiscipline,thenitshouldbeawayofchangingtheworld(forthe better?)—byimplementingalgorithmsincomputerprogramsthatcanhavephysicaleffects.

Computersciencetriestodoboth:tounderstandtheworldcomputationally, andtochangetheworldbybuildingcomputationalartifacts.Inintroductoryclasses, Iofferthefollowingdeﬁnition:

Computerscienceisthescientiﬁcstudy18 of:

• what canbecomputed(narrowly,whichmathematicalfunctions arecomputable;widely,whichreal-worldtasksareautomatable [Forsythe,1968]),

• how tocomputesuchfunctionsortasks(howtorepresentthe necessaryinformationandhowtoconstructappropriatealgorithms),

• howtocomputethem efﬁciently,

17“[S]sciencetriestounderstandtheworld,whereasengineeringtriestochangeit”[Staples,2015, §1],paraphrasingMarx:“Thephilosophershaveonlyinterpretedtheworld,invariousways;the pointistochangeit”[Marx,1845,Thesis11].WasMarxproposingadisciplineof“philosophical engineering”?

18Usingtheadjective‘scientific’insteadofthenoun‘science’neatlyputsthescience-vs.engineeringdisputeinthebackground.Afterall,surelyengineeringis“scientific”,evenifitisn’ta “science”.And,whetherornotcomputerscienceisa“science”,itissurelyasystematic, scientific fieldofstudy.

• andhowto engineer computersthatcanimplementthosecomputationsintherealworld.

Andhereweglimpsetheﬁrststrandofourthread:Iscomputationconcerned with(a)theinternalworkingsofacomputer(bothabstractlyintermsofthetheory ofcomputation—e.g.,thewayinwhichaTuringmachineworks—aswellasmore concretelyintermsoftheinternalphysicalworkingsofaphysicalcomputer)?Or (butnotnecessarilyexclusively“or”!)with(b)howthoseinternalworkingscan reachouttotheworldinwhichtheyareembedded?

And,here,itisalsoimportanttopointoutsimilaritieswithaclassicissuein thephilosophyofmind:Howdoesthemind(or,morematerialistically,thebrain) reachoutto“harpoon”theworld?19

2.3WhatIsaComputer?

Computersciencemightbethestudyofcomputers orofcomputation.Ifitisthe scienceoreventheengineeringofcomputers,thenweneedtoaskwhatacomputeris.Isacomputeranabstract,mathematicalentity,inparticular,aTuring machine(orauniversalTuringmachine)?(Or,forthatmatter,anythinglogically equivalenttoaTuringmachine,suchasa λ-“calculator”orarecursive-function “machine”?)Orisitanyphysical,real-worldobjectthatimplementsa(universal)Turingmachine?(Orboth,ofcourse.)Ifitisphysical,whichthingsinthe worldarecomputers(besidestheobvioussuspects,suchasMacs,PCs,iPhones, etc.)?Notably,isthebrainacomputer?Isthesolarsystemacomputer(computing Kepler’slaws)?Istheuniverseacomputer?20

Here,weseeanotherstrandofourthread:Whereshouldwelookforananswer towhatacomputeris?Shouldwelooknarrowlytomathematics,ormorewidely totherealworld?(AndIwon’tevenmentiontherelatedissueofmathematicalPlatonism,whichmightrespondthatlooking“narrowly”tomathis,indeed,looking evenmorewidelythanthemerelyrealworld.)

2.4WhatIsComputation?

Whenweaskwhatcomputationis,thingsgetmorecomplex.Iscomputation“narrow”,focusingonlyontheoperationsofaTuringmachine(print,move)oronbasic recursivefunctions(successor,predecessor,projection)?Or“wide”,involving,say,

19See,e.g.,[Castaneda,1989,p.114,“TheHarpooningModel”].

20Onthebrain,see,e.g.,[Searle,1990];onthesolarsystem,see,e.g.,[Copeland,1996, §2],[PerruchetandVinter,2002, §1.3.4],[Shagrir,2006,p.394];ontheuniverse,see,e.g., [Weinberg,2002],[Wolfram,2002],[LloydandNg,2004].

chesspiecesandachessboard(forachessprogram)orsoldiersandabattleﬁeld (forawargamesimulator)?Isitindependentoftheworld,orisitworld-involving?

Arealgorithmspurelylogical?Orarethey“intentional”[Hill,2015]and“teleological”[Anderson,2015]?Whichofthesetwoformsdotheytake?:

Do P (where‘P’iseitheraprimitivecomputation,orasetofcomputations recursivelystructuredbysequence,selection,andrepetition,i.e.,a “procedure”).

Inordertoaccomplishgoal G,do P

Iftheformer,thencomputationis“narrow”;ifthelatter,then“wide”.(We’llreturn tothisin §4.)

What is aprocedure?Arerecipesprocedures?Is“Makehollandaisesauce”a high-levelprocedurecall?Ifso,thencomputationiswide.(We’llreturntothisin §4.4.)

IsmakinghollandaisesauceTuring-machinecomputable?Orisitataskthat goesbeyondTuringmachines?Howdoesinteractive(ororacle)computation,relatetoTuring-machinecomputation?Turing-machinecomputationseemstobe “narrow”;theothers,“wide”.(We’llreturntothisin §6.)

2.5WhatIsaComputerProgram?

Isanalgorithmanimplementationofafunction,acomputerprogramanimplementationofanalgorithm,andaprocess(i.e.,aprogrambeingexecutedonacomputer) animplementationofaprogram?Implementationisarelationbetweensomething moreorless abstract andsomethingelsethatismoreorless concrete (atleast,that islessabstract).Itisthecentralrelationbetweenabstractionandreality(aswellas wheresciencemeetsengineering).

Elsewhere,Ihavearguedthatimplementationismostusefullyunderstood as(external)semanticinterpretation.Moreprecisely, I isanimplementation,in medium M,of“Abstraction” A iff I isasemanticinterpretationormodelof A, where A issomesyntacticdomainand M isthesemanticdomain[Rapaport,1999, 128],[Rapaport,2005a].Typically(butnotnecessarily), I isareal-worldsystem insomephysicalmedium M,and A isanabstractorformalsystem(butboth I and A couldbeabstract.Thethemeoftherelationofcomputingtotherealworldis obviouslyrelatedtothisissue.

Ithasbeenclaimedthat(atleastsome)computerprogramsaretheories [SimonandNewell,1962,p.97],[Johnson-Laird,1981,pp.185–186]

[Pylyshyn,1984,p.76].(Forthecontrastingview,see[Moor,1978, §4], [Thagard,1984].)Howdotheoriesrelatetotheworld?Docomputerprograms simulatetheworld?(We’llreturntothisalittlebitin §7.)

Arecomputerprogramssoftwareorhardware?Herewehaveacomputational versionofthemind-bodyproblem.Andithaslegalramiﬁcationsintermsofwhat canbecopyrightedandwhatcanbepatented.

Cancomputerprogramsbeveriﬁed?Thisquestionconcerns,atleastinpart, thecorrelationofasyntacticdescription(oftheworld)withadomainofsemantic interpretation(i.e.,theworldbeingdescribed).(We’llreturntothisin §7.)

Itistimetodelveintosomeofthese.

3Inputs,TuringMachines,andOutputs

Anymachineisaprisonerofitsinputandoutputdomains. [Newell,1980,148]

Let’sbeginatthebeginning,withTuringmachines.ThetapeofaTuringmachine recordssymbolsinits“cells”,usually‘0’or‘1’.Isthetapetheinput-outputdevice oftheTuringmachine?Orisitthemachine’sinternalmemorydevice?

GivenaTuringmachineforcomputingacertainmathematicalfunction,itis certainlytruethatthefunction’sinputswillbeinscribedonthetapeatthebeginning ofthecomputation,andtheoutputswillbeprintedontothetapebythetimethatthe computationhalts.Moreover,theinscriptionsonthetapewillbeusedandmodiﬁed bythemachineduringthecomputation,inthesamewaythataphysicalcomputer usesitsinternalmemoryforstoringintermediateresultsofacomputation.Soit certainlylooksliketheanswertoourquestionsis:both.

But,althoughTuring’s a-machinesweredesignedtosimulatehumancomputers,21 Turingdoesn’ttalkaboutthehumanswhowould use them.ATuringmachine doesn’t acceptuser-suppliedinputfromtheexternalworld!Itbeginswith alldatapre-storedonitstapeandthensimplydoesitsownthing,computingthe outputofafunctionandleavingtheresultonthetape.Turingmachinesdon’t“tell” anyoneintheexternalworldwhattheanswersare,thoughtheanswersarethere foranyonetoreadbecausethe“internalmemory”ofthemachineisvisibletothe externalworld.Ofcourse,auserwouldhavetobeableto interpret thesymbolson thetape,andthereonhangsatale.

21Thatis,humanswhocompute.Insomerecentwritings(see §4.2),theyarecalled‘computors’ [Sieg,1994],soastobeabletouseawordthatsoundslike‘computer’withoutthe21st-century implicationthatitissomethinglikeaMacoraPC.Iprefertocallthem‘clerks’.

ThequestionIwanttolookathereiswhetherthesymbolsonthetapearereally inputsandoutputsinthesenseofcomingfrom,andbeingreportedto,theexternal world.

Areinputsandoutputsanessentialpartofanalgorithm?Afterall,theinputoutputinterface“merely”connectsthealgorithmwiththeworld.Itmayseem outrageoustodenythattheyareessential,butit’sbeendone!

3.1AreInputsNeeded?

It’soutrageous,ofcourse,becausealgorithmsaresupposedtobewaysofcomputingmathematicalfunctions,andmathematicalfunctions,bydeﬁnition,haveinputs andoutputs.Theyare,afterall,certainsetsoforderedpairsofinputsandoutputs, andyoucan’tverywellhaveanordered pair thatismissingoneorbothofthose.

A.A.Markov’sinformalcharacterizationofalgorithmhasan“applicability”conditionstatingthatalgorithmsmusthave“Thepossibilityofstartingfromoriginal givenobjectswhichcanvarywithinknownlimits”[Markov,1954,p.1].Those “originalgivenobjects”are,presumably,theinput.

ButDonaldKnuth’sinformalcharacterizationofthenotionofalgorithmhasan “input”conditionstatingthat“Analgorithmhas zeroormore inputs”[Knuth,1973, p.5;myitalics]!Henotonlydoesn’texplainthis,buthegoesontocharacterize outputsas“quantitieswhichhaveaspeciﬁedrelationtotheinputs”[Knuth,1973, p.5].The“relation”wouldnodoubtbethefunctionalrelationbetweeninputs andoutputs,but,ifthereisnoinput,whatkindofarelationwouldtheoutputbe in?22 Knuthisnotaloneinthis:JurisHartmanisandR.E.Stearns’sclassicpaper oncomputationalcomplexityallowstheirmulti-tapeTuringmachinestohaveat mostonetape,whichisanoutput-onlytape;thereneednotbeanyinputtapes [HartmanisandStearns,1965,p.288].

Onewaytounderstandthisisthatsomeprogramsmerelyoutputinformation, suchasprime-numbergenerators.Incasessuchasthis,althoughtheremaynot beany explicit input,thereisan implicit input(roughly,ordinals:thealgorithm outputsthe nthprime,withoutexplicitlyrequestingan n tobeinput).Another kindoffunctionthatmightseemnottohaveany(explicit)inputsisaconstant function,but,again,itsimplicitinputcouldbeanything(oranythingofacertain type,“varyingwithinknownlimits”,asMarkovmighthavesaid).

So,whatconstitutesinput?Isitsimplytheinitialdataforacomputation?Or isitinformationsuppliedtothecomputerfromtheexternalworld(andinterpreted ortranslatedintoarepresentationofthatinformationthatthecomputercan“understand”andmanipulate)?

22Isthisarelationtoanon-existententityinaMeinongiansense?See[Grossmann,1974,p.109], [Rapaport,1986a, §4.5].

3.2AreOutputsNeeded?

Whataboutoutput?Markov,Knuth,andHartmanis&Stearnsallrequireatleast oneoutput.Markov,forexample,hasan“effectiveness”condition(usingthatterm slightlydifferentlyfromothers,suchasChurchorKnuth)statingthatanalgorithm must“obtainacertainresult”.

ButB.JackCopelandandOrenShagrirsuggestthataTuringmachine’soutput mightbeunreadable[CopelandandShagrir,2011,pp.230–231].Imagine,nota Turingmachinewithatape,butaphysicalcomputerthatliterallyprintsoutits results.Supposethattheprinterisbrokenorthatithasrunoutofink.Orsuppose thattheprogrammerfailedtoincludea‘print’commandintheprogram.The computer’sprogramwouldcomputearesultbutnotbeabletotelltheuserwhat itis.Considerthisalgorithmfrom[ChaterandOaksford,2013,p.1172,citingan examplefrom[Pearl,2000]]:

1.input P

2.multiply P by2;storein Y

3.add1to Y ;storein Z

Thisalgorithmdoesnotappeartohaveanoutput.Thecomputerhascomputed 2X + 1andstoreditawayin Z forsafekeeping,butdoesn’ttellyouitsanswer. There is ananswer,butitisn’toutput.(“Iknowsomethingthatyoudon’t!”?) So,whatconstitutes“output”?Isitsimplytheﬁnalresultofacomputation?Or isitsomekindoftranslationorinterpretationoftheﬁnalresultthatisphysically outputandimplementedintherealworld?Intheformercase,wouldn’tbothof §2.1.4’sbase-10andbase-13GCDcomputersbedoingthesamething?Aproblemwouldariseonlyiftheytolduswhatresultstheygot,andwe—readingthose results—wouldinterpretthem,possiblyincorrectly.

3.3WhenAreInputsandOutputsNeeded?

HereisGualtieroPiccininionthistopic:

Docomputationshavetohaveinputsandoutputs?Themathematical resourcesofcomputabilitytheorycanbeusedtodeﬁne‘computations’ thatlackinputs,outputs,orboth. Butthecomputationsthataregenerallyrelevantforapplicationsarecomputationswithbothinputsand outputs. [Piccinini,2011,p.741,n.11;myitalics]

Or,asAllenNewellputit:

Machinesliveintherealworldandhaveonlyalimitedcontactwithit. Anymachine,nomatterhowuniversal,thathasnoears(sotospeak) willnothear;thathasnowings,willnotﬂy.[Newell,1980,148]23

Thereyouhaveit:Narrowlyconceived,algorithmsmightnotneedinputsand outputs.Widelyconceived,theydo.Anyinputfromtheexternalworldwouldhave tobe encoded byauserintoalanguage“understandable”bytheTuringmachine (orelsetheTuringmachinewouldneedtobeableto decode suchexternal-world input).Andanyoutput from theTuringmachinetobereported to theexternal world(e.g.,auser)wouldhavetobe encoded bytheTuringmachine(or decoded bytheuser).Suchcodingswould,themselves,havetobealgorithmic.

Infact,thekeytodeterminingwhichreal-worldtasksarecomputable—one ofthemainquestionsthatcomputerscienceisconcernedwith(§2.2)—isﬁnding codingschemesthatallowthesequenceof‘0’sand‘1’s(i.e.,anaturalnumberin binarynotation)onaTuringmachine’stapetobe interpreted asasymbol,apixel, asound,etc.Amathematicalfunctiononthenaturalnumbersiscomputableiffitis computablebyaTuringmachine(accordingtotheChurch-TuringComputability Thesis);thus,areal-worldproblemiscomputableiffitcanbeencodedassucha computablemathematicalfunction.

Butit’sthatwideconception,requiringalgorithmic,semanticinterpretations oftheinputsandoutputs,thatleadstovariousdebates.

3.4MustInputsandOutputsBeInterpretedAlike?

Thereisanotherinput-outputissue,whichhasnotbeendiscussedmuchinthe literaturebutwhichisrelevanttoourtheme.[Rescorla,2007,p.254]notesthat

DifferenttextbooksemploydifferentcorrelationsbetweenTuringmachinesyntaxandthenaturalnumbers.Thefollowingthreecorrelations areamongthemostpopular:24

d1(n)= n

d2(n + 1)= n

d3(n + 1)= n, asaninput.

d3(n)= n, asanoutput.

23‘Universal’,asNewellusesithere,meansbeingableto“produceanarbitraryinput-output function”[Newell,1980,147].

24[Thesymbol‘x’representsasequenceof x strokes,where x isanaturalnumber.–WJR]

Amachinethatdoublesthenumberofstrokescomputes f (n)= 2n under d1, g(n)= 2n + 1under d2,and h(n)= 2n + 2under d3.Thus,the sameTuringmachinecomputesdifferentnumericalfunctionsrelative todifferentcorrelationsbetweensymbolsandnumbers.

Let’sfocusoninterpretationslike d3 (we’lllookat d1 and d2 in §5.2).This ideaofhavingdifferentinputandoutputinterpretationsoccursallthetimeinthe realworld.(Idon’tknowhowoftenitisconsideredinthemorerareﬁedatmosphereofcomputationtheory.)Forexample,machine-translationsystemsthatuse an“interlingua”workthisway:Chineseinputisencodedintoan“interlingual” representationlanguage(oftenthoughtofasaninternal,“meaning”-representation languagethatencodesthe“proposition”expressedbytheChineseinput),andEnglishoutputisgeneratedfromthatinterlingua(re-expressinginEnglishthepropositionthatwasoriginallyexpressedinChinese).25 Cognition(assumingthatit iscomputable!)alsoworksthisway:Perceptualencodingsintothe“language” ofthebiologicalneuralnetworkofourbrainsurelydifferfrommotordecodings. (Newell’sabove-quotedexamplesofhearingandﬂyingaresurelydifferent.)

Consider,again,Rescorla’sGCDprogram.Andletusconsider,notScheme, butaCommonLispversion.TheCommonLispversionwilllookidenticaltothe Schemeversion(thelanguagessharemostoftheirsyntax),buttheCommonLisp versionhastwoglobalvariables—*read-base* and *print-base*—that tellthecomputerhowtointerpretinputandhowtodisplayoutput.Bydefault, *read-base* issetto10.SotheCommonLispread-procedureseesthethreecharactersequence‘115’(forexample);decidesthatitsatisﬁesthesyntaxofan integer;convertsthatsequenceofcharacterstoaninternalrepresentationoftype integer whichisrepresentedinternallyasabinarynumeralimplementedas bitsorswitch-settings—doesthesamewith(say)‘20’;andcomputestheirGCDusingthealgorithmfrom §2.1.4onthebinaryrepresentation.Ifthephysicalcomputer hadbeenanoldIBMmachine,thecomputationmighthaveusedbinary-codeddecimalnumeralsinstead,thuscomputinginbase10.If *read-base* hadbeenset to13,theinputcharacterswouldhavebeeninterpretedasbase-13numerals,and theverysameCommonLisp(orScheme)codewouldhavecorrectlycomputed theGCDof18710 and2610.Onecouldeithersaythatthealgorithmcomputes with numbers—notnumerals—orwith base-2numeralsasacanonicalrepresentationofnumbers,dependingonone’sviewconcerningsuchthingsasPlatonism ornominalism.Andsimilarlyforoutput:Theswitch-settingscontainingtheGCD oftheinputarethenoutputasbase-10orbase-13numerals,aspixelsonascreen orinkonpaper,dependingonthevalueofsuchthingsas *print-base*.The 25MyformerstudentMin-HungLiaousedSNePSforthispurposein[Liao,1998].Formoreon thehistoryofinterlinguasincomputerscience,see[Daylight,2013, §2].

point,onceagain,withrespecttoRescorla’sexample,isthatasingleCommonLisp (orScheme)algorithmisbeingexecutedbyboth M10 and M13.Thosemachines are different;theydo not “havethesamelocal,intrinsic,physicalproperties” [Rescorla,2013,687],because M10 has *read-base* and *print-base* set to10,whereas M13 has *read-base* and *print-base* setto13.26

TheaspectofthissituationthatIwanttoremindyouofiswhetherthetapeis the external inputandoutputdevice,oris,rather,themachine’s internal memory.27 Ifitisthemachine’sinternalmemory,then,insomesense,thereisno(visibleor user-accessible)inputoroutput(§3).Ifitisanexternalinput-outputdevice,then themarksonitarefor our convenienceonly.Intheformercase,theonlyaccurate descriptionoftheTuringmachine’sbehaviorissyntacticallyintermsofstrokeappending.Inthelattercase,wecanusethatsyntacticdescriptionbutwecanalso embellishitwithoneintermsofourinterpretationofwhatitisdoing.(We’llreturn tothisin §5.2.)

4AreAlgorithmsTeleological(Intentional)?

Let’sbeginuntanglingourthreadwiththequestionofwhethertheproperwayto characterizeanalgorithmmustincludean“intentionalorteleologicalpreface”,so tospeak.

4.1WhatIsanAlgorithm?

Thehistoryofcomputationtheoryis,inpart,anattempttomakemathematically precisetheinformalnotionofanalgorithm.Turingmoreorless“won”thecompetition.(Atleast,hetiedwithChurch.G¨odel,alsointherace,placedhisbeton Turing[Soare,2009]).Manyinformalcharacterizationsof“algorithm”exist(such asKnuth’sandMarkov’s,mentionedin §3.1);theycanbesummarizedasfollows [Rapaport,2012,Appendix,pp.69–71]:

Analgorithm(forexecutor E)[toaccomplishgoal G]is:

1.aprocedure P,i.e.,afiniteset(orsequence)ofstatements(or rules,orinstructions),suchthateachstatement S is: (a)composedofafinitenumberofsymbols(better:uninterpretedmarks)fromafinitealphabet (b)andunambiguous(for E—i.e.,

26IamindebtedtoStuartC.Shapiro,personalcommunication,fortheideasinthisparagraph.

27[Dresner,2003]and[Dresner,2012]discussthis.

i. E “knowshow”todo S, ii. E cando S, iii. S canbedoneinaﬁniteamountoftime iv.and,afterdoing S, E “knows”whattodonext—), 2. P takesaﬁniteamountoftime,i.e.,halts, 3.[and P endswith G accomplished].28

4.2DoAlgorithmsNeedaPurpose?

Ithinkthatthenotionofanalgorithmisbestunderstoodwithrespecttoanexecutor. Onemachine’salgorithmmightbeanother’sungrammaticalinput [Suber,1988].Wecanprobablyrephrasetheabovecharacterizationwithoutreferenceto E,albeitmoreawkwardly,hencethe parentheses aroundreferencesto E.

Butthepresentissueiswhetherthe bracketed commentsaboutthetasktobe accomplishedareessential.MyformerstudentRobinK.Hillhasrecentlyargued infavorofincluding G,roughlyonthegroundsthata“prospectiveuser”needs “someunderstandingofthetaskinquestion”overandabovethemereinstructions ([Hill,2015, §5].Algorithms,accordingtoHill,mustbeexpressedintheform“In ordertoaccomplish G,do P”,notmerely“Do P”.

StephenC.Kleene’sinformalcharacterizationof“algorithm”[Kleene,1995, p.18]beginsasfollows:

[A]methodforansweringanyoneofagiveninﬁniteclassofquestions...isgivenbyasetofrulesorinstructions,describingaprocedurethatworksasfollows. After theprocedurehasbeendescribed, [then]ifweselect any questionfromtheclass,theprocedurewillthen tellushowtoperformsuccessivesteps,...

28“[N]ote...thatthemoreonetriestomakeprecisethese informal requirementsforsomething tobeanalgorithm,themoreonerecapitulatesTuring’smotivationfortheformulationofaTuring machine”[Rapaport,2012,p.71].Afewotherthingstonote:Thecharacterizationofaprocedureas a set of statements (withtheparentheticalalternativesofsequence,rules,andinstructions)isintended toabstractawayfromissuesaboutimperative/proceduralvs.declarativepresentations.Whetherthe “letters”ofthealphabetinwhichthealgorithmiswrittenareconsideredtobeeithersymbolsinthe classicalsenseofmark-plus-semantic-interpretationorelseuninterpretedmarks(symbolsthatare not“symbolicof”anyting)isanotheraspectofourcommonthread.Talkof“knowinghow”does notpresumethat E isacognitiveagent(itmightbeaCPU),butIdowanttodistinguishbetween E’s (i)beingable inprinciple (i.e.,“knowinghow”)toexecuteastatementand(ii)beingable inpractice to(i.e.,“can”)executeit.Similarly,“knowing”whattodonextdoesnotpresumecognition,but merelyhavingadeteministicwaytoproceedfromonestatementtothe“next”.Iamusing‘know’ anditscognateshereinthesame(notnecessarilycognitive)waythatAIresearchersuseitinthe phrase‘knowledgebase’.

Notethattheprocedurehasapurpose:“answeringanyoneofagiveninﬁniteclass ofquestions”.Andtheproceduredependsonthatclass:Givenaclassofquestions, thereisaproceduresuchthat,given“any questionfromtheclass,”theprocedure “enable[s]ustorecognizethatnowwehavetheanswerbeforeusandtoreadit off”.

DavidMarranalyzedinformationprocessingintothreelevels:computational (what asystemdoes),algorithmic(how itdoesit),andphysical(howitis implemented)[Marr,1982].Ihaveneverlikedtheseterms,preferring‘functional’, ‘computational’,and‘implementational’,respectively:Certainly,whenoneisdoingmathematicalcomputation(thekindthatTuringwasconcernedwithin [Turing,1936],onebeginswitha mathematicalfunction (i.e.,acertainsetoforderedpairs),asksforanalgorithmto compute it,andthenseeksan implementation ofit, possibly ina physical systemsuchasacomputerorthebrain(orperhapseven [Searle,1982]’sbeercansandleverspoweredbywindmills),but notnecessarily (e.g.,thefunctionalityofanabstractdatatypesuchasastackcanbe abstractly implementedusingalist.29

In non-mathematicalﬁelds(e.g.,cognition),thesetoforderedpairsofinputoutput behavior isexpressedinproblem-speciﬁclanguage;30 thealgorithmiclevel willalsobeexpressedinthatlanguage;andtheimplementationlevelmightbethe brainoracomputer.Arecipeforhollandaisesaucedevelopedinthiswayneeds tosaymorethanjustsomethingalongthelinesof“mixtheseingredientsinthis way”;itmusttaketheexternalenvironmentintoaccount.(Wewillreturntothisin §4.4,andwewillseehowitcantaketheexternalworldintoaccountin §5.4.)

Marr’s“computational”levelisrathermurky.FrancesEgantakesthemathematicalfunctionalviewjustoutlined.31 Onthatview,Marr’s“computational”level ismathematical.

BartonAnderson[Anderson,2015, §1],ontheotherhand,saysthatMarr’s “computational”level

concern[s]thepresumed goal or purpose ofamapping,32 thatis,the speciﬁcationofthe‘task’thataparticularcomputation‘solved.’Algorithmiclevelquestionsinvolvespecifyinghowthismappingwas achievedcomputationally,thatis,theformalprocedurethattransforms aninputrepresentationintoanoutputrepresentation.

Onthisview,Marr’s“computational”levelis teleological.Intheformulation“To

29ThisisonereasonthatIhavearguedthatimplementationissemanticinterpretation [Rapaport,1999],[Rapaport,2005a].

30Usingan“external”or“inherited”semantics;thesetermswillbeexplainedbelow.

31[Egan,1991,pp.196–107],[Egan,1995,p.185];cf.[ShagrirandBechtel,2015, §2.2].

32Ofamathematicalfunction?

accomplish G,do P”,the“To G”prefaceexpressestheteleologicalaspectofMarr’s “computational”level;the“do P”seemstoexpressMarr’s“algorithm”level.

Accordingto[Bickle,2015],Marrwastryingtocounterthethen-prevailing methodologyoftryingto describe whatneuronsweredoing(a“narrow”,internal, implementation-leveldescription)withouthavinga“wide”,external,“computational”-level purpose (a“function”intheteleological,notmathematical,sense). Suchateleologicaldescriptionwouldtellus“why”[Marr,1982,p.15,asquoted in[Bickle,2015]]neuronsbehaveastheydo.

ShagrirandWilliamBechtelsuggestthatMarr’s“computational”levelconﬂatestwoseparate,albeitrelated,questions:notonly“why”,butalso“what”.On thisview,Eganisfocusingonthe“what”,whereasAndersonisfocusingonthe “why”[ShagrirandBechtel,2015, §2.2].Wewillreturntothisinamoment.

Certainly,knowingwhatthegoalofanalgorithmismakesiteasierfora cognitiveagent executortofollowthealgorithmandtohaveafullerunderstandingof whats/heisdoing.Ididn’tunderstandthatIwasaddingwhenmyTAtoldmeto entercertaindataintothecellsofthespreadsheet.Itwasonlywhenshetoldme thatthatwashowIcouldaddtwonumberswithaspreadsheetthatIunderstood.

Now,(Iliketothinkthat)Iamacognitiveagentwhocancometounderstand thatenteringdataintoaspreadsheetcanbeawayofadding.ATuringmachinethat addsoraMacrunningExcelisnotsuchacognitiveagent.Itdoesnotunderstand whatadditionisorthatthatiswhatitisdoing.Anditdoesnothaveto.However,an AIprogramrunningonarobotthatpassestheTuringtestwouldbeaverydifferent matter;IhavearguedelsewherethatsuchanAIprogramcould,would,andshould (cometo)understandwhatitwasdoing.33

Theimportantpointisthat—despitethefactthatunderstandingwhattaskan algorithmisaccomplishingmakesiteasiertounderstandthe algorithm itself— “blind”followingofthealgorithmisallthatisnecessaryto accomplish thetask. Understandingthetask—thegoalofthealgorithm—isexpressedbytheintentional/teleologicalpreface.Thisisakintodubbingitwithanamethatismeaningfultotheuser,aswewilldiscussin §5.3.

Thatcomputationcanbe“blind”inthiswayiswhat[Fodor,1980]expressed byhis“formalitycondition”andwhatDanielC.Dennetthascalled Turing’s...strangeinversionofreasoning.ThePre-Turingworldwas oneinwhichcomputerswerepeople,whohadtounderstandmathematicsinordertodotheirjobs.Turingrealisedthatthiswasjust notnecessary:youcouldtakethetaskstheyperformedandsqueeze outthelasttinysmidgensofunderstanding,leavingnothingbutbrute, 33[Rapaport,1988],[Rapaport,2012].SeemyformerdoctoralstudentAlbertGoldfain’sworkon howtogetAIcomputersystemstounderstandmathematics[Goldfain,2006],[Goldfain,2008].

mechanicalactions.INORDERTOBEAPERFECTANDBEAUTIFULCOMPUTINGMACHINEITISNOTREQUISITETOKNOW WHATARITHMETICIS.[Dennett,2013,p.570,capsinoriginal]34

AsIreadit,thepointisthataTuringmachineneednot“know”thatitisadding, butagentswhodounderstandaddingcanusethatmachinetoadd.

Orcanthey?Inordertodoso,themachine’sinputsandoutputshavetobe interpreted—understood—bytheuserasrepresentingthenumberstobeadded. Andthatseemstorequireanappropriaterelationshipwiththeexternalworld.It seemstorequirea“usermanual”thattellstheuserwhatthealgorithmdoesinthe waythatHillprescribes,notinthewaythatmyTAexplainedwhataspreadsheet does.Andsucha“usermanual”—anintentionorapurposeforthealgorithm—in turnrequiresaninterpretationofthemachine’sinputsandoutputs.

Butbeforepursuingthislineofthought,let’stakeafewmoreminutestoconsider“Turing’sinversion”,theideathataTuringmachinecanbedoingsomething veryparticularbyexecutinganalgorithmwithoutanyspeciﬁcationofwhatthatalgorithmis“doing”intermsoftheexternalworld.Algorithms,onthisview,seem nottohavetobeintentionalorteleological,yettheyremainalgorithms.

BrianHayesofferstwoexpressionsofanalgorithmthatantsexecute [Hayes,2004]:

teleologicalversion:

Tocreateanantgraveyard,gatherallyourdeadinoneplace.35 non-teleologicalversion:

1.Ifyouseeadeadant,36 andifyouarenotcarryingone,thenpickitup.

2.Ifyouseeadeadant,andifyouarecarryingone,thenputyoursdown neartheotherone.

AsHayesnotes,theteleologicalversionrequiresplanningandorganizationskills farbeyondthosethatanantmighthave,nottomentionconceptualunderstanding thatwemightverywellbeunwillingtoascribetoants.Moretothepoint,the

34Seealsothemoreeasilyaccessible[Dennett,2009,p.10061].

35Thisisa“fully”teleologicalversion,withahigh-level,teleologicallyformulatedexecutionstatement.A“partially”teleologicalversionwouldsimplypreﬁx“Tocreateanantgraveyard”tothe followingnon-teleologicalversion.

36Notethattestingthisconditiondoesnotrequiretheanttohavea dedicto conceptofdeath; itissufﬁcientfortheanttosense—eithervisiblyorperhapschemically—what we woulddescribe, dere,asadeadant.Foracomputationaltheoryofthe dere/dedicto distinction,see [WiebeandRapaport,1986],[Rapaportetal.,1997].

antneedsnoneofthat.Theteleologicaldescriptionhelps us describeandperhaps understandtheant’sbehavior;itdoesn’thelptheant.

Thesameistrueinmyspreadsheetexample.KnowingthatIamaddinghelps me understandwhatIamdoingwhenIﬁllthespreadsheetcellswithcertainvalues orformulas.Butthespreadsheetdoesitsthingwithoutneedingthatknowledge.

AnditistrueforSearleintheChineseRoom[Searle,1980]:Searle-in-theroommightnotunderstandwhatheisdoing,buthe is understandingChinese.37 WasSearle-in-the-roomsimplytold,“Followtherulebook!”?Orwashetold, “TounderstandChinese,followtherulebook!”?Ifhewastoldtheformer(which seemstobewhatSearle-the-authorhadinmind),then,(a)fromanarrow,internal, first-personpointofview,Searle-in-the-roomcantruthfullysaythathedoesn’t knowwhatheisdoing(inthewidesense).Inthenarrowsense,hedoesknowthat heisfollowingtherulebook,justasIdidn’tknowthatIwasusingaspreadsheet toadd,eventhoughIknewthatIwasfillingcertaincellswithcertainvalues.And (b)fromthewide,external,third-personpointofview,thenative-Chinese-speaking interrogatorcantruthfullytellSearle-in-the-roomthathe is understandingChinese. WhenSearle-in-the-roomistoldthathehaspassedaTuringtestforunderstanding Chinese,hecan—paraphrasingMolière’sbourgeoisgentleman—truthfullyadmit thathewasspeakingChinesebutdidn’tknowit.38

HereisanicedescriptionofcomputationthatmatchestheChinese-Roomsetup:

ConsideragainthescenariodescribedbyTuring:anidealizedhumancomputor39 manipulatessymbolsinscribedonpaper.Thecomputormanipulatesthesesymbols becausehe[sic]wantstocalculate thevaluesomenumber-theoreticfunctionassumesonsomeinput. Thecomputorstartswithasymbolicrepresentationfortheinput,performsaseriesofsyntacticoperations,andarrivesatasymbolicrepresentationfortheoutput. Thisproceduresucceedsonlywhenthe computorcan understand thesymbolicrepresentationshemanipulates. Thecomputorneednotknowinadvancewhichnumberagiven symbolrepresents, buthemustbecapable,inprinciple,ofdeter-

37ToomuchhasbeenwrittenontheChinese-RoomArgumenttocitehere,but[Cole,1991],my responsetoColein[Rapaport,1990],[Rapaport,2000],and[Rapaport,2006,390–397]touchon thisparticularpoint.

38“Parmafoi!ilyaplusdequaranteansquejedisdelaprosesansquej’ensusserien,etje voussuisleplusoblig´edumondedem’avoirappriscela.”“Uponmyword!Ithasbeenmorethan fortyyearsthatIhavebeenspeakingprosewithoutmyknowinganythingaboutit,andIammost obligatedtoyouintheworldforhavingapprisedmeofthat.”(mytranslation) (http://en.wikipedia.org/wiki/Le Bourgeois gentilhomme).

39Seeexplanationofthistermin §3.

miningwhichnumberthesymbolrepresents.Onlythendoeshis syntacticactivityconstituteacomputationoftherelevantnumbertheoreticfunction. Ifthecomputorlacksanypotentialunderstanding oftherelevantsyntacticitems,thenhisactivitycountsasmeremanipulationofsyntax,ratherthancalculationofonenumberfromanother. [Rescorla,2007,pp.261–262;myboldface,Rescorla’sitalics]

Withouttheboldfacedclauses,thisisanicedescriptionoftheChineseRoom.The differenceisthat,intheChineseRoom,Searle-in-the-roomdoesnot“wantto” communicateinChinese;hedoesn’tknowwhathe’sdoing,inthatsenseofthe phrase.Still,he’sdoingit,accordingtotheinterpretationofthenativespeaker outsidetheroom.Ialsoclaimthattheboldfacedclausesareirrelevanttothecomputationitself;thatisthelessonofTuring’s“strangeinversion”. Theseexamplessuggestthattheuser-manual/externalworldinterpretationis notnecessary.Algorithms can beteleological,andtheirbeingsocanhelpcognitive agentswhoexecutethemtomorefullyunderstandwhattheyaredoing.Butthey don’t haveto beteleological.40

4.3CanAlgorithmsHaveMorethanOnePurpose?

Inadditiontobeingteleological,algorithmsseemtobeabletobe multiply teleological,asinthechess-warexampleanditskin.Thatis,therecanbealgorithmsof theform:

Toaccomplish G1,do P andalgorithmsoftheform:

Toaccomplish G2,do P.

40Thistouchesonanotherphilosophicalissueincognitivescience:theroleof“embodiedcognition”.As[Adams,2010,p.619]describesit,

Onanon-embodiedapproach,thesensorysysteminformsthecognitivesystemand themotorsystemdoesthecognitivesystem’sbidding.Therearecausalrelations betweenthesystemsbutthesensoryandmotorsystemsarenotconstitutiveofcognition.Forembodiedviews,therelationtothesensori-motorsystemtocognitionis constitutive,notjustcausal.

Comparethisparaphrase:Onanon-teleologicalapproach,inputsfromtheexternalworldinforma TuringmachineandoutputstotheexternalworlddotheTuringmachine’sbidding.Therearecausal orsemanticrelationsbetweenthesystemsbutthesemanticinterpretationsoftheinputsandoutputs arenotconstitutiveofcomputation.Forteleologicalviews,therelationtothesemanticinterpretations oftheinputsandoutputsisconstitutive.

where G1 = G2,andwhere G2 doesnotsubsume G1 (orviceversa),althoughthe Psarethesame.Inotherwords,whatifdoing P accomplishesboth G1 and G2? Howmanyalgorithmsdowehaveinthatcase?Two?(Onethataccomplishes G1, andanotherthataccomplishes G2,countingteleologically,or“widely”?)Orjust one?(Asinglealgorithmthatdoes P,countingmorenarrowly?)

WeredeBruijnandthechemiststalkingaboutthesamething?Ontheteleological(orwide)view,theyweren’t;onthenarrowview,theywere.Multipleteleologiesaremultiplerealizationsofanalgorithmnarrowlyconstrued:‘Do P’can beseenasawaytoalgorithmicallyimplementthehigher-level“function”(mathematical or teleological)ofaccomplishing G1 aswellasG2.E.g.,executinga particularsubroutineinagivenprogrammightresultincheckmateorwinninga battle.Viewingmultipleteleologiesasmultiplerealizations(multipleimplementations)canalsoaccountforhollandaise-saucefailuresontheMoon,whichcould betheresultofan“implementation-leveldetail”[Rapaport,1999]thatisirrelevant totheabstract,underlyingcomputation.

4.4WhatIf G and X ComeApart?

Whatif“successfully”executing Pfails toaccomplishgoal G?Thiscouldhappen forexternal,environmentalreasons(hencemyuseof‘wide’,above).Doesthis meanthat G mightnotbeacomputabletaskeventhough P is?

Theblocks-worldcomputer’smodeloftheworldwasanincomplete,partial model;itassumedthatitsactionswerealwayssuccessful.I’llhavemoretosay aboutpartialmodelsin §7.Fornow,thepointisthatthisprogramlackedfeedback fromtheexternalworld.Therewasnothingwrongwiththeenvironment,asthere isinthelunarhollandaise-saucecase;rather,therewasincompleteinformation abouttheenvironment.

Rescorla’sGCDcomputersdo“differentthings” by doingthe“samething”. Thedifferenceisnotin how theyaredoingwhattheyaredoing,butintheinterpretationsthat we usersofthemachinesgivetotheirinputsandoutputs.Would [Hill,2015]saythattheprocedureencodedinthatSchemeprogramwastherefore notanalgorithm?

Whatismorecentraltothenotionof“algorithm”?:Allofparts1–3inour informalcharacterizationin §4(“To G,do P”),orjustparts1–2,i.e.,withoutthe bracketedgoals(just“Do P”)?Isthealgorithmthenarrow,non-teleological,“purposeless”(ornon-purposed)entity?Oristhealgorithmthewide,intentional,teleological(i.e.,goal-directed)entity?Onthenarrowview,thewargameandchess algorithmsarejust one algorithm,thehollandaise-saucerecipe does workonthe Moon(itscomputerprogrammightbelogicallyveriﬁableevenifitfailstomake hollandaisesauce),andthe“two”GCDprogramsarealsojust one algorithmthat

doesitsthingcorrectly(butonlywebase-10folkscan use ittocomputeGCDs).

Onthewideview,thewargameandchessprogramsare two,distinctalgorithms, thehollandaise-saucerecipe fails ontheMoon(despitethefactthattheprogram mighthavebeenveriﬁed—shadesoftheFetzercontroversythatwewilldiscuss in §7!),andtheSchemeprogramwhenfedbase-13numeralsisdoingsomething wrong (inparticular,its“remainder”subroutineisincorrect).41

Theseexamplessuggestthatthewide,goal-directednatureofalgorithmsteleologicallyconceivedisduetotheinterpretationoftheirinputandoutput.AsShagrir &Bechtelputit,Marr’s“algorithmiclevel...isdirectedtothe innerworking of themechanism....Thecomputationallevellooks outside,toidentifyingthefunctioncomputedandrelatingittotheenvironmentinwhichthemechanismoperates” [ShagrirandBechtel,2015, §2.3].

Thereisawaytocombinetheseinsights:Hill’sformulationoftheteleological orintentionalnatureofalgorithmshadtwoparts,ateleological“preface”specifyingthetasktobeaccomplished,andastatmentofthealgorithmthataccomplishes it.OnewaytoclarifythenatureofMarr’s“computational”levelistosplititinto its“why”andits“what”parts.The“why”partisthetasktobeaccomplished. The“what”partcanbeexpressed“computationally”(Iwouldsay“functionally”) asamathematicalfunction(possibly,butnotnecessarily,expressedin“why”terminology),butitcanalsobeexpressed algorithmically.Finally,thealgorithmcan beimplemented.So,wecandistinguishthefollowing four Marr-likelevelsof analysis:

“Computational”-WhatLevel

Do f (i)= o

“Computational”-WhyLevel

Toaccomplish G,do f (i)= o

AlgorithmicLevel

Toaccomplish G,do A f (i)= o

ImplementationLevel

Toaccomplish G,do I(A f )(i)= o

where:

• G isthetasktobeaccomplishedorexplained,expressedinthelanguageof theexternalworld,sotospeak;

41AtleastasRescorladescribesit;itdoesthe right thingontheShapiro-Rapaportinterpretation discussedin §3.4.

• f isan input-outputfunctionthataccomplishes G,expressedeitherinthe samelanguageorperhapsexpressedinpurelymathematicallanguage;

• A f isanalgorithmthatimplements f (i.e.,itisanalgorithmthathasthesame input-outputbehavioras f );and

• and I isanimplementation(perhapsinthebrainoronsomecomputer)of A).42

[ShagrirandBechtel,2015, §4]saythat“The what aspect[ofthe“computational”level]providesadescriptionofthemathematicalfunctionthatisbeing computed.The why aspectemploysthecontextualconstraintsinordertoshow howthisfunctionmatcheswiththeenvironment.”Theseseemtometonicely describethetwoclausesofwhatIcallthe“computational-why”levelabove.

Andthisraisesanotherinterestingquestion:

5DoWeComputewithSymbolsorwithMeanings?

5.1WhereDoesContextBelong?

Analogous(ifnotidentical)problemsariseelsewhere:Is“3isanintegerpowerof 2”true?[Stewart,2000,p.98](cf.[Rescorla,2013]).Notinbase-10arithmetic. Butitistrueintheintegersmodulo5.43

Onecantaketwoattitudestowardssituationslikethis.First,onecansaythat thetruth-valuesdifferbecausetheexternalcontextisdifferentinthetwocases, inmuchthesamewaythat‘Ilikewater’istrueonEarth(where‘water’refersto H2O)but(say)falseonTwinEarth(where‘water’referstoXYZ).Alternatively, onecansaythattheoriginalpropositionwasnotstatedcarefullyenough:Itshould havebeensomethinglike:“310 isanintegerpowerof210”or,perhaps,“Inbase10,3isanintegerpowerof2”(withacorrespondingpairofpropositionsforthe Z/5case).44 Weshouldhaveincorporatedtheinformationabouttheexternalcon-

42[Egan,1995,p.187,n.8],citingColinMcGinn,notesthatevenwhatIamcallingthe “computational”-whatlevelcanbephrasedintentionallyas,e.g.,“TocomputetheLaplaceanofa Gaussian,do f (i, o).”Soperhapsthereisalevelintermediatebetweenthewhat-andwhy-levels, somethingalongtheselines:“Toaccomplish A f (i, o),do A f (i, o)”,where A isexpressedinpure Turing-machinelanguage.Note,too,that both clausescanvaryindependently:Notonlycan f implementmanydifferent Gs(asinthechess-wargameexample),but G canbeimplementedbymany different A f s.

43Because (210)(110) = 210 < 310 < (210)(210) = 410.But23 = 8 = 3Z/5

44Therearemanyotheroptions:“The number representedbythe numeral ‘3’inbase-10isan integerpowerofthe number representedbythe numeral ‘2’inbase-10”.Or“Thenumbercanonically representableas {{{ / 0}}} (etc.)”.

text into theproposition.Inotherwords,the“externalcontext”shouldhavebeen “internalized”.45 (Wewillreturntothisin §5.4.)

5.2WhatIsThisTuringMachineDoing?

Hereisaquestiontoconsider:Aresuchpropositionsaboutnumbers?Orarethey aboutnumerals?WhatdoTuringmachinescomputewith?Forthatmatter,whatdo we computewith?Thisisnottheplaceforustogetintoadiscussionofnominalism inmathematics,thoughitisworthpointingoutthatourcommonthreadseemsto leadusthere.

BothChristopherPeacockeandRescorlaremindusthat

ATuringmachinemanipulatessyntacticentities:stringsconsistingof strokesandblanks....Ourmaininterestisnotstring-theoreticfunctionsbutnumber-theoreticfunctions.Wewanttoinvestigatecomputablefunctionsfromthenaturalnumberstothenaturalnumbers.To doso,wemustcorrelatestringsofstrokeswithnumbers.46 [Rescorla,2007,p.253]

Onceagain,weseethatitisnecessarytointerpretthestrokes.

Hereis[Peacocke,1999]’sexample:SupposethatwehaveaTuringmachine thatoutputsacopyoftheinputappendedtoitself(thusdoublingthenumberof inputstrokes):input‘/’,output‘//’;input‘//’,output‘////’,andsoon.Whatisour Turingmachinedoing?Isn’t“outputtingacopyoftheinputappendedtoitself”the mostneutraldescription?Afterall,thatdescribes exactly whattheTuringmachine isdoing,leavingtheinterpretationuptotheobserver.Ifwehadcomeacrossthat Turingmachineinthemiddleofthedesertandweretryingtoﬁgureoutwhatit does,somethinglikethatwouldbethemostreasonableanswer. Why someone mightwantacopy-appendingTuringmachineisadifferentmatterthatprobably would requireaninterpretationofthestrokes.Butthatgoesfarbeyondwhatthe Turingmachineisdoing.

Aswesawin §3.4,Rescorlaofferedthreeinterpretationsofthestrokes.Do wereallyhaveonemachinethatdoesthreedifferentthings?Whatitdoes(inone senseofthatphrase)dependsonhowitsinputandoutputareinterpreted,thatis,on theenvironmentinwhichitisworking.Indifferentenvironments,itdoesdifferent things;atleast,that’swhatClelandsaidaboutthehollandaisesaucerecipe.

45ThisissomethingthatIhavearguedisnecessaryforcomputationalcognition;see,e.g., [Rapaport,2006,pp.385–387],[Rapaport,2012, §3].

46Here,bytheway,wehaveaninterestingdifferencebetweenTuringmachinesandtheirlogical equivalentsintheComputabilityThesis,suchasthe λ-calculusorrecursive-functiontheory,because thelatterdealwithfunctionsandnumbers,notsymbolsforthem.

Piccininisaysmuchthesamething:

Incomputabilitytheory,symbolsaretypicallymarksonpaperindividuatedbytheirgeometricalshape(asopposedtotheirsemanticproperties).Symbolsandstringsofsymbolsmayormaynotbeassigned aninterpretation;iftheyareinterpreted,thesamestringmaybeinterpreteddifferently....Inthesecomputationaldescriptions,theidentity ofthecomputingmechanismdoesnothingeonhowthestringsare interpreted.[Piccinini,2006, §2]

By‘individuated’,Piccininiistalkingabouthowonedecideswhetherwhatappear tobetwoprograms(say,oneforawargamebattleandoneforachessmatch)are, infact,twodistinctprogramsorreallyjustoneprogram(perhapsbeingdescribed differently).Here,hesuggeststhatitisnothowtheinputsandoutputsareinterpreted(theirsemantics)thatmatters,butwhattheinputsandoutputslooklike (theirsyntax).So,forPiccinini,thewargameandchessprogramsarethesame. ForCleland,theywouldbedifferent.ForPiccinini,thehollandaise-sauceprogram runningontheMoonworksjustaswellastheonerunningonEarth;forCleland, onlythelatterdoeswhatitissupposedtodo.

So,thequestion“WhichTuringmachineisthis?”hasonlyoneanswer,given intermsofitssyntax(“determinedby[its]instructions,notby[its]interpretations” [Piccinini,2006, §2]).Butthequestion“WhatdoesthisTuringmachinedo?”has n + 1answers:onesyntacticanswerand n semanticanswers(oneforeachof n differentsemanticinterpretations).

Hereisarelatedissue,showingourthreadrunningthroughactiontheory: GivenacalculatorthatIusetoaddtwonumbers,howwouldyoudescribemy behavior?AmIpushingcertainbuttonsinacertainsequence?(A“syntactic”, narrow,internalanswer:Iam“doing P”.)OramIaddingtwonumbers?(A teleological,“semantic”,wide,externalanswer:Iamaccomplishing G.)Oram Iaddingtwonumbers by pushingthosebuttonsinthatsequence?(Ateleological (etc.)answer,togetherwithasyntacticdescriptionofhowIamdoingit:Iamaccomplishing G,bydoing P.)[Rapaport,1990],[Rapaport,1993].Thisisthesame situationthatwesawinthespreadsheetexample(andwewillseeitagainwhenwe discussnamedsubroutinesin §5.3).

Clearly,insomesense,alloftheseanswersarecorrect,merely(?)focusingon differentaspectsofthesituation.Butthereisafurtherquestion: Why (orhow) doesaTuringmachine’sprintingandmovingthusandso,ormypushingcertain calculatorbuttonsthusandso,resultinaddingtwonumbers?Andtheanswerto thatseemstorequireasemanticinterpretation.Thisisthekindofquestionthat Marr’s“computational”levelissupposedtorespondto.

IfIwanttoknowwhichTuringmachinethisis,Ishouldlookattheinternal mechanism(roughly,Dennett’s“design”stance[Dennett,1971])fortheanswer [Piccinini,2006].ButifI’minterestedinbuyingachessprogram(ratherthana wargamesimulator),thenIneedtolookattheexternal/inherited/widesemantics [Cleland,1993].

Sincewecanarbitrarilyvaryinheritedmeaningsrelativetosyntactic machinations,inheritedmeaningsdonot makeadifference tothose machinations.Theyareimposeduponanunderlyingcausalstructure. [Rescorla,2014,p.181]

Onthisview,thehollandaise-sauce-makingcomputerdoesitsthingwhetherit’s onEarthortheMoon(whetheritsoutputishollandaisesauceornot).Perhaps itsoutputissomekindofgeneralized,abstract,hollandaise-sauce type,whoseimplementations/instantiations/tokensontheMoonaresomesortofgoop,butwhose implementations/instantiations/tokensonEartharewhatarenormallyconsidered tobe(successful)hollandaisesauce.(Ihavebeentoldthatpackagesofcakemix thataresoldinmile-highDenvercomewithalternativedirections).

Hereisanotherniceexample:

aloomprogrammedtoweaveacertainpatternwillweavethatpattern regardlessofwhatkindsofthreaditisweaving.Thepropertiesof thethreadsmakenodifferencetothepatternbeingwoven.Inother words,theweavingprocessisinsensitivetothepropertiesoftheinput.

[Piccinini,2008,p.39]

AsPiccininigoesontopointout,theoutputmighthavedifferentcolorsdepending onthecolorsoftheinputthreads,butthe pattern willremainthesame.Thepattern isinternal;thecolorsareexternal,tousetheterminologyabove.Ifyouwantto weaveanAmericanﬂag,youhadbetterusered,white,andbluethreadsinthe appropriateways.Butevenifyouusepuce,purple,andplumthreads,youwill weaveanAmerican-ﬂag pattern.Whichismoreimportant:thepatternorthe colors?That’sprobablynotexactlytherightquestion.Rather,theissueisthis:If youwantacertainpattern,thisprogramwillgiveittoyou;ifyouwantacertain patternwithcertaincolors,thenyouneedtohavetherightinputs(youneedtouse theprogramintherightenvironment).(Thisaspectofourthreadreappearsinthe philosophyofmathematicsconcerning“structuralism”:Isthepattern,orstructure, ofthenaturalnumbersallthatmatters?Ordoesitalsomatterwhatthenatural numbersinthepattern“really”are?Forasurvey,see[Horsten,2015, §4].)

5.3SyntacticSemantics

‘Syntax’isusuallyusedinthenarrowsenseofthegrammarofalanguage,and ‘semantics’isusuallyunderstoodasthemeaningsofthemorphemes,words,and sentencesofalanguage.Following[Morris,1938,pp.6–7],Iprefertounderstand ‘syntax’verybroadlytobethestudyofthepropertiesof,andrelationsamong, theelementsofasingleset(orformalsystem),and‘semantics’verybroadlytobe thestudyoftherelationsbetweenanytwosetswhatsoever(eachwithitsownsyntax).47 Syntaxisconcernedwith“intra-system”propertiesandrelations;semanticsisconcernedwith“extra-system”relations(wherethe“system”inquestionis the“syntactic”domain),or,viewed“subspecieaeternitatis”,itisconcernedwith “inter-system”relations(i.e.,relationsbetweentwodomains,oneofwhichistaken asthesyntacticdomainandtheotherasasemanticdomain).

So,onewaytoanswerthequestionsattheendof §5.2isbyusinganexternal semanticinterpretation:TheseTuring-machineoperationsorthosebuttonpresses (consideredasbeinglocatedinaformal,syntacticsystemofTuring-machineoperationsorbuttonpressings)canbeassociatedwithnumbersandarithmeticaloperationsonthem(consideredasbeinglocatedinadistinct,Platonic(oratleast external)realmofmathematicalentities).48 Intheformulation“To G,do P”, Pcan be identiﬁed syntactically(atthe“computational-what”level),but Gneeds tobe identiﬁedsemantically—andthen P canbe interpreted semanticallyin G’sterms (atthe“computational-why”level).

Butthereisanotherwaytoanswerthesequestions,byusingan“internal”kind ofsemantics,thekindthatIhavecalled“syntacticsemantics”.Syntacticsemantics ariseswhenthesemanticdomainofanexternalsemanticinterpretationis“internalized”intothesyntacticdomain.Inthatway,theprevioussemanticrelations betweenthetwopreviouslyindependentdomainshavebecomerelationswithinthe newuniﬁeddomain,turningthemintosyntacticrelations.49 Syntacticsemantics isakinto(ifnotidenticalwith)whatRescorlahascalled“indigenoussemantics” [Rescorla,2012],[Rescorla,2014].Onepossibledifferencebetweenthemisthat

47So,thegrammarofalanguage—syntaxinthenarrowsense—isthestudyoftheproperties of,andrelationsamong,itswordsandsentences.Andtheirreferentialmeaningsaregivenbya semanticinterpretationrelatingthelinguisticitemstoconceptsorobjects.Formoredetails,see [Rapaport,2012, §3.2].

48Whichrealm,incidentally,hasitsownorganizationintermsofpropertiesandrelationsamong itsentities,i.e.,itsontology,whichcanbethoughtofasitsown“syntax”[Rapaport,2006,p.392]. Similarly,theorganizationofthesyntacticrealm,intermsofthepropertiesandrelationsamong itsentities,canbeconsidered,atleastinthementalcase,asan“epistemologicalontology” [Rapaport,1986a].Ontologyissyntax(bythedeﬁnitionof‘syntax’givenhere).Relationsbetween two domains,eachwithitsownsyntax(orontology)issemantics.

49[Rapaport,1988],[Rapaport,1995],[Rapaport,2006],[Rapaport,2012];seealso[Kay,2001].

myversionemphasizestheimportanceof conceptual-role semantics(tobedistinguishedfrom,butincluding, inferential-role semantics)[Rapaport,2002],whereas Rescorla’sversionemphasizes causal relations.50

Withoutgoingintodetails(someofwhicharespelledoutinthecitedpapers), letmenoteherethatonewaytogivethiskindofsemanticsisintermsof(named) subroutines(whichaccomplishsubtasksoftheoverallalgorithm).Wecanidentify collectionsofstatementsinaprogramthat“worktogether”,thenpackagethemup, namethepackage,andthusidentifysubtasks.

E.g.,thefollowingLogoprogramdrawsaunitsquarebymovingforward1 unit,thenturning90degreesright,anddoingthat4times: repeat4[forward1right90]

ButLogowon’t“know”whatitmeanstodrawasquareunlesswetellitthat tosquare repeat4[forward1right90] end

Ofcourse,theLogoprogramstillhasnounderstandingintheway we doofwhat asquareis.Itisnowcapableonlyofassociatingthatnewlydeﬁnedsymbol (‘square’)withacertainprocedure.Thesymbol’smeaningfor us isits external semantics;theword’smeaning(or“meaning”?)fortheLogoprogramisitsinternal “syntacticsemantics”duetoitsrelationshipwiththebodyofthatprogram.Anotherexampleisthesequenceofinstructions“turnleft;turnleft;turnleft”,in Karel theRobot [Pattisetal.,1995],whichcanbepackagedupandnamed“turnright”:

DEFINE-NEW-INSTRUCTIONturnrightAS

BEGIN

turnleft;turnleft;turnleft END

NoticeherethatKarelstillcan’t“turn right”(i.e.,90degclockwise);itcanonly turnleftthreetimes(i.e.,270degcounterclockwise).

Thereisacaveat:Merely naming asubroutinedoesnotautomaticallyendow itwiththemeaningofthatname,as[McDermott,1980]makesclear.Butthe ideathatconnections(whetherconceptual,inferential,orcausal)canbe“packaged”togetherisawayofproviding“syntactic”or“indigenous”semantics.Ifthe nameisassociatedwithobjectsthatare external totheprogram,thenwehave external/wide/inherited/extra-systemsemantics.Ifitisassociatedwithobjects 50[Egan,1995,p.181]’s“structuralproperties”and[Bickle,2015,esp. §5]’sdescriptionof “causal-mechanisticexplanations”inneurosciencemayalsobe“syntactic/indigenous”semantics.

internal totheprogram,thenwehaveinternal/narrow/syntactic/indigenous/intrasystemsemantics. Identifying subroutinesissyntactic; naming themleadstosemantics:Ifthenameisexternallymeaningfultoauser,becausetheusercanassociatethenamewithotherexternalconcepts,thenwehavesemanticsintheordinary sense(subjecttothecaveatinthepreviousfootnote);ifitisinternallymeaningful tothecomputer,becausethecomputercanassociatethenamewithotherinternal names,thenwehave“syntactic”or“indigenous”semantics.

5.4Internalization

Anotherwaytogetsyntacticsemanticsisby“internalization”.Externalsemantic relationsbetweentheelementsoftwodomains(a“syntactic”domaindescribed syntacticallyanda“semantic”domainalsodescribedsyntactically(or,ifyouprefer,“ontologically”)canbeturnedintointernalsyntacticrelations(“syntacticsemantics”)byinternalizingthesemanticdomainintothesyntacticdomain.After all,ifyoutaketheunionofthesyntacticandsemanticdomains,thenallformerly externalsemanticrelationsarenowinternalsyntacticones.

Andonewaythatthishappensforuscognitive(andarguablycomputational) agentsisbysensoryperception,whichisaformofinputencoding.Foranimal brains,perceptioninterpretssignalsfromtheexternalworldintoabiologicalneuralnetwork.Foracomputerthatacceptsinputfromtheexternalworld,theinterpretationofexternaloruserinputasitsinternalswitchsettings(orinscriptionson aTuring-machinetape)constitutesaformofperception.Bothareformsofwhat Iamcalling“internalization”.Asaresult,theinterpretationbecomespartofthe computer’sorthebrain’sintra-system,syntactic/indigenoussemantics.(See,e.g., [Rapaport,2012]forfurtherdiscussionandrefereces.)

MycolleagueStuartC.Shapiroadvocatesinternalizationinthefollowingform:51

Shapiro’sInternalizationTactic

Algorithms do taketheteleologicalform,“Toaccomplish G,do P”, but G mustinclude everything thatisrelevant:

• Tomakehollandaisesauce onEarth,do P.

• ToﬁndtheGCDof2integers inbase-10,do Q

• Toplaychess,do R, whereR’svariablesrangeoverchesspiecesand achessboard

51Personalcommunication.Asimilarpointwasmadein[Smith,1985,p.24]:“aswellasmodellingtheartifactitself,youhavetomodeltherelevantpartoftheworldinwhichitwillbeembedded.”

• Tosimulateawargamebattle,do R, whereR’svariablesrangeover soldiersandabattleﬁeld

Andtheproperlocationfortheseteleologicalclausesisinthepreconditionsand postconditionsoftheprogram.Oncetheyarelocatedthere,theycanbeusedinthe formalverificationoftheprogram,whichproceedsbyprovingthat, ifthepreconditionsaresatisfied,thentheprogramwillaccomplishitsgoal asarticulatedinthe postconditions.Thisbuildstheexternalworld(andanyattendantexternalsemantics) into thealgorithm.AsLamporthassaid,“Thereisnoeasywaytoensurea blueprintstayswithabuilding,butaspecificationcanandshouldbeembeddedas acommentwithinthecodeitisspecifying”[Lamport,2015,41].Theseparabilityofblueprintfrombuildingisakintotheseparabiiityof G from P;embedding aspecificationintocodeas(atleast)acommentistointernalizeitasapre-or postcondition.52

AsIsuggestedin §4.1,wecanavoidhavingCleland’shollandaise-saucerecipe outputamessygoopbylimitingitsexecutiontoonelocation(Earth,say)without guaranteeingthatitwillworkelsewhere(ontheMoon,say).Thisisnodifferent fromapartialmathematicalfunctionthatissilentaboutwhattodowithinputfrom outsideitsdomain,orfromanalgorithmforaddingtwointegersthatspeciﬁesno particularbehaviorfornon-numericalinput.53 Anotherwayistousethe“Denver cakemix”strategy:Therecipeoralgorithmshouldbeexpressedconditionally:If location=Earth,thendo P;iflocation=Moon,thendo Q (where Q mightbethe outputofanerrormessage).

Athirdwayistowriteanalgorithm(arecipe)thattellsustomixeggsandoil untilanemulsionisproduced,andthatoutputshollandaisesauce.OnEarth,an emulsionisindeedproduced,andhollandaisesauceisoutput.ButontheMoon, thisalgorithmgoesintoaninﬁniteloop;nothing(and,inparticular,nohollandaise sauce)iseveroutput.Oneproblemwiththisisthatthe“until”clause(“untilan emulsionisproduced”)isnotclearlyalgorithmic(cf.[Cooper,2012,p.78]).How wouldthe computer tellifanemulsionhasbeenproduced?Thisisnotaclearly algorithmic,Booleanconditionwhosetruthvaluecanbedeterminedbythecomputersimplybycheckingoneofitsswitchsettings(thatis,avaluestoredinsome variable).Itwouldneedsensorstodeterminewhattheexternalworldislike.But thatisaformof interactive computing,towhichwenowturn.

52Notethesimilarityof(a)internalizingexternal/inheritedsemanticsintointernal/syntacticsemanticsto(b)theDeductionTheoreminlogic,whichcanbethoughtofassayingthatapremiseof anargumentcanbe“internalized”asanantecedentoftheargument’sconclusion: P ⊢ C ⇔⊢ (P → C)

53“Crashing”isawell-deﬁnedbehavioriftheprogramissilentaboutillegalinput.More“wellbehaved”behaviorrequiressomekindoferrorhandling.

6Interactive(Hyper?)computation

Therearemanyvarietiesofhypercomputation,i.e.,“thecomputationoffunctions thatcannotbe”computedbyaTuringmachine[Copeland,2002,p.461].ThenotionisputforthasachallengetotheComputabilityThesis.ManyvarietiesofhypercomputationinvolvesucharacanaascomputersoperatinginMalament-Hogarth spacetime.54 Here,Iwanttofocusononeformofhypercomputationthatismore downtoearth.Itgoesundervariousnames(thoughwhetherthereisasingle it withmultiplenames,ormultiple itsisanissuethatIwillignorehere):‘interactive computation’[Wegner,1995,p.45],‘reactivecomputation’(AmirPneuli’sterm; see[Hoffmann,2010]),or‘oraclecomputation’(Turing’sterm;usefulexpositions arein[Feferman,1992],[Davis,2006],[Soare,2009],and[Soare,2013]).

RememberthatTuringmachinesdonotreally accept inputfromtheexternalworld;theinputtoafunctioncomputedbyaTuringmachineis pre-stored alreadyprintedonitstape;Turingmachineswork“ofﬂine”.Givensuchatape, theTuringmachinecomputes(and,hopefully,halts).Astudentinanintroductory programmingcoursewhoisaskedtowritean interactive programthattakesasinputtwointegers chosenrandomlybyauser andthatproducesasoutputtheirGCD hasnotwrittenaTuring-machineprogram.Thestudent’sprogrambeginswitha Turingmachinethatprintsaqueryonitstapeandhalts;theuserthendoesthe equivalentofsupplyinganewtapewiththeuser’sinputpre-storedonit;andthen a(nother)TuringmachineusesthattapetocomputeaGCD,queryanotherinput, and(temporarily)halt.

Eachrunofthestudent’sprogram,however,couldbeconsideredtobethe runofaTuringmachine.Buttheread-loopinwhichtheGCDcomputationis embeddedinthatstudent’sprogramtakesitoutoftherealmofaTuringmachine, strictlyspeaking.

Thathardlymeansthatourfreshmanstudenthascreatedahypercomputerthat computessomethingthataTuringmachinecannotcompute.Suchinteractivecomputations,whichareattheheartofmodern-daycomputing,weremathematically modeledbyTuringusinghisconceptofanoracle.Ourfreshman’scomputerprogram’squerytotheusertoinputanotherpairofintegersisnothingbutacalltoan oraclethatprovidesunpredictable,andpossiblyuncomputable,values.(Computer userswhosupplyinputareoracles!)

Now,manyinteractivecomputationscanbe simulated byTuringmachines, simplybysupplyingalltheactualinputsonthetapeatthestart.TheKleeneSubstitutionProperty55 statesthatdatacanbestoredeffectively(i.e.,algorithmically)

54http://en.wikipedia.org/wiki/Hypercomputation

55AlsocalledtheKleeneRecursionTheorem[Case,nd].

inprograms;thedataneednotbeinputfromtheexternalworld.AtypicalinteractivecomputermightbeanATMatthebank.Noonecanpredictwhatkindofinput willbegiventothatATMonanygivenday;but,attheendoftheday,allofthe day’sinputsareknown,andthatATMcanbesimulatedbyaTM.

ButthatisofnohelptoanyonewhowantstouseanATMonadailybasis. Computationinthewildmustallowforinputfromtheexternalworld(including oracles).

Andthatiswhereourthreadre-appears:Computationmustinteractwiththe world.Acomputerwithoutphysicaltransducersthatcoupleittotheenvironment [Sloman,2002, §5,#F6,pp.17–18]wouldbesolipsistic.Thetransducersallowfor perceptionoftheexternalworld(andtherebyforinteractivecomputing),andthey allowformanipulationoftheexternalworld(andtherebyforcomputers—robots, includingcomputationalchefs—thatcanmakehollandaisesauce).Butthecomputationandtheexternal-worldinteraction(theinterpretationofthecomputer’s outputintheexternalworld)areseparableanddistinct.Andtherecan,therefore, beslippagebetweenthem(leadingtosuchthingsasblocks-worldandhollandaisesaucefailures),mulitipleinterpretations(chessvs.wargame),etc.

7ProgramVeriﬁcationandtheLimitsofComputation

Let’sconsiderprogramsthatspecifyphysicalbehaviorsabitfurther.Inaclassic andcontroversialpaper,JamesH.Fetzer[Fetzer,1988]arguedtotheeffectthat, givenacomputerprogram,youmightbeabletologicallyprovethatits(primitive) instructionRINGBELLwillbeexecuted,butyoucannotlogicallyprovethatthe physicalbellwillactuallyring(awireconnectingcomputerandbellmightbe broken,thebellmightnothaveaclapper,etc.).Similarly,wemightbeableto logicallyprovethatthehollandaise-sauceprogramwillexecutecorrectly,butnot thathollandaisesaucewillactuallybeproduced.ForFetzerandCleland,it’sthe bellsandthesaucethatmatterintherealworld:Computingisabout theworld.

DonaldMackenzieagrees[MacKenzie,1992,p.1066]:

...mathematicalreasoningalonecanneverestablishthe“correctness” ofaprogramorhardwaredesigninanabsolutesense,butonlyrelative tosomeformalspeciﬁcationofitsdesiredbehavior”.

Similarly,aformalproofofatheoremonlyestablishesitstruth relativeto thetruth ofitsaxioms,notits“absolute”truth[Rapaport,1984,p.613].

MacKenziecontinues:

Mathematicalargumentcanestablishthataprogramordesignisacorrectimplementationofthatspeciﬁcation,butnotthatimplementation

ofthespeciﬁcationmeansacomputersystemthatis“safe”,“secure”, orwhatever.[MacKenzie,1992,p.1066].

Therearetwopointstonoticehere.First,amathematicalargumentcanestablish thecorrectnessofaprogramrelativetoitsspecification,thatis,whethertheprogramsatisfiesthespecification.Inpart,thisisthefirstpoint,above.But,second, notonlydoesthisnotnecessarilymeanthatthecomputersystemissafe(orwhatever),italsodoesnotmeanthatthe specification isitself“correct”.Presumably,a specificationisarelativelyabstractoutlineofthesolutiontoaproblem.Proving thatacomputerprogramiscorrectrelativeto—thatis,satisfies—thespecification doesnotguaranteethatthespecificationactuallysolvestheproblem! Lamporthasrecentlysaidmuchthesamething:

Aspecificationisanabstraction.Itshoulddescribetheimportantaspectsandomittheunimportantones.Abstractionisanartthatis learnedonlythroughpractice....[A]specificationofwhatapiece ofcodedoesshoulddescribeeverythingoneneedstoknowtousethe code.Itshouldneverbenecessarytoreadthecodetofindoutwhatit does.[Lamport,2015,39]

Thegoalofanalgorithmcanbeexpressedinitsspecification.Thisiswhyyou wouldn’t have toreadthecodetofindoutwhatitdoes.Ofcourse,ifthespecificationhasbeeninternalizedintothecode,youmightbeableto.Butit’salsowhy youcanhaveachessprogramthatisalsoawargamesimulator:Theymighthave differentspecificationsbutthesamecode.

...whatcanbeprovencorrectisnotaphysicalpieceofhardware, orprogramrunningonaphysicalmachine,butonlyamathematical modelofthathardwareorprogram.[MacKenzie,1992,1066].

ThisisthecaseforreasonsthatarerelatedtotheChurch-TuringComputability

Thesis:Youcan’tshowthattwosystemsarethesame,insomesense,unlessyou cantalkaboutbothsystemsinthesamelanguage.InthecaseoftheComputability

Thesis,theproblemconcernstheinformalityofthelanguageofalgorithmsversus theformalityofthelanguageofTuringmachines.Inthepresentcase,theproblemconcernsthemathematicallanguageofprogramsversusthenon-linguistic, physicalnatureofthehardware.Onlyby describing thehardwareina(formal, mathematical)language,canaproofofequivalencebeattempted.Butthenwe alsoneedaproofthatthatformaldescriptionofthehardwareiscorrect;andthat can’tbehad.

Itcan’tbehad,because,tohaveit,wewouldneed another formaldescription ofthehardwaretocomparewiththeformaldescriptionthatweweretryingto

verify.AndthatleadstoaZeno-orBradley-likeinﬁniteregress.(Wecancome “close,butnocigar”.)

BrianCantwellSmithhasarticulatedthisproblemmostclearly[Smith,1985]. Forhim,computingisabouta modelof theworld.AccordingtoSmith,todesign acomputersystemtosolveareal-worldproblem,wemustdotwothings:

1.Createa model ofthereal-worldproblem.

2. Representthe model inthecomputer.

Themodelthatwecreatehasnochoicebuttobe“delimited”,thatis,itmust beabstract—itmustomitsomedetailsofthereal-worldsituation.Abstractionis theoppositeofimplementation.Itistheremovalof“irrelevant”implementation details.Hispointisthatcomputersonlydealwith theirrepresentationsof these abstractmodelsof therealworld.Theyare twice removedfromreality.56

Allmodelsarenecessarily“partial”,henceabstract.Butactionis not abstract: You and thecomputermustact in thecomplex,realworld,andinrealtime.Yet suchreal-worldactionmustbebasedon partialmodels oftherealworldandinferencesbasedonincompleteandnoisyinformation(cf.HerbertSimon’snotion of“boundedrationality”andtheneedfor“satisﬁcing”[Simon,1996]).Moreover, thereisnoguaranteethatthe models arecorrect.

Actioncanhelp:Itcanprovidefeedbacktothecomputersystem,sothatthe systemwon’tbeisolatedfromtherealworld.Recallourblocks-worldprogram thatdidn’t“know”thatithaddroppedablock,but“blindly”continuedexecuting itsprogramtoputtheblockonanother).Ifithadhadsomesensorydevicethat wouldhaveletitknowthatitnolongerwasholdingtheblockthatitwassupposed tomove,andiftheprogramhadhadsomekindoferror-handlingprocedureinit, thenitmighthaveworkedmuchbetter(itmighthaveworked“asintended”).

Theproblem,onSmith’sview,isthatmathematicalmodeltheoryonlydiscussestherelationbetweentwo descriptions:themodelitself(whichisapartial descriptionoftheworld)andadescriptionofthemodel.Itdoesnotdiscussthe relationbetweenthemodelandtheworld;thereisanunbridgeablegap.InKantianfashion,amodelislikeeyeglassesforthecomputer,throughwhichitseesthe world,anditcannotseetheworldwithoutthoseglasses.Themodel is theworld asfarasthecomputercansee.Themodelistheworld as thecomputerseesit.

BothSmithandFetzeragreethattheprogram-veriﬁcationprojectfails,butfor slightlydifferentreasons:ForFetzer(andCleland),computingisaboutthe world; itisexternalandcontextual.Thus,computerprogramscan’tbeveriﬁed,because 56“Humanfallibilitymeanssomeofthemoresubtle,dangerousbugsturnouttobeerrorsin design;thecodefaithfullyimplementstheintendeddesign,butthedesignfailstocorrectlyhandlea particular‘rare’scenario”[Newcombeetal.,2015,67].

theworldmaynotbeconduciveto“correct”behavior:Aphysicalpartmightbreak; theenvironmentmightpreventanotherwise-perfectly-running,“correct”program fromaccomplishingitstask(suchasmakinghollandaisesauceontheMoonusing anEarthrecipe);etc.

ForSmith,computingisabouta model oftheworld;itisinternalandnarrow. Thus,computerprogramscan’tbeveriﬁed,butforadifferentreason,namely,the modelmightnotmatchtheworld.57 NotethatSmithalsobelievesthatcomputers mustact in therealworld,butitistheirabstractnarrownessthatisolatesthemfrom theconcrete,realworldatthesametimethattheymustactinit.

Thedebateoverwhethercomputationconcernstheinternal,syntacticmanipulationofsymbolsortheexternal,semanticinterpretationofthemisreminiscentof Smith’sgap.As[Rescorla,2007,p.265]observes,weneedacomputabletheory ofthesemanticinterpretationfunction,but,asSmithobserves,wedon’t(can’t?) haveone,forreasonsakintotheComputabilityThesisproblem.

Smith’sgapisdue,inpart,tothefactthatspeciﬁcationsareabstractions.How doesoneknowifsomethingthathasbeenomittedfromthespeciﬁcationisimportantornot?Thisiswhy“abstractionisanart”,asLamportsaid,andwhythere’s noguaranteethatthemodeliscorrect(inthesensethatitmatchesreality).

8Conclusion

Wheredowestand?First,Ihavenotattemptedinthisoverviewtoresolvethese issues.Iamstillstrugglingwiththem,andmygoalwastoconvinceyouthatthey areinteresting,andperhapsimportant,issuesthatarewidespreadthroughoutthe philosophyofcomputerscienceandbeyond,toissuesinthephilosophyofmind, philosophyoflanguage,andtheethicsandpracticalusesofcomputers.ButIthink wecanseeopportunitiesforsomepossibleresolutions.

WecandistinguishbetweenthequestionofwhichTuringmachineacertain computationisandthequestionofwhatgoalthatcomputationistryingtoaccomplish.Bothquestionsareimportant,andtheycanhaveverydifferentanswers. TwocomputationsmightimplementthesameTuringmachine,butbedesignedto accomplishdifferentgoals.

Andwecandistinguishbetweentwokindsofsemantics:wide/external/extrinsic/inheritedandnarrow/internal/intrinsic/“syntactic”/indigenous.Bothkindsexist, haveinterestingrelationshipsandplaydifferent,albeitcomplementary,roles.

57Perhapsabetterwayoflookingatthingsistosaythattherearetwodifferentnotionsof“veriﬁcation”:aninternalandanexternalone.Forrelateddistinctions,see[TedreandSutinen,2008, pp.163–164].

Algorithmsnarrowlyconstrued(minustheteleologicalpreface)iswhatisstudiedinthemathematicaltheoryofcomputation.Todecidewhetherataskiscomputable,weneedtoﬁndanalgorithmthatcanaccomplishit.Thus,wehavetwo separatethings:analgorithm(narrowlyconstrued,ifyouprefer)andatask.Some algorithmscanaccomplishmorethanonetask(dependingonhowtheirinputs andoutputsareinterpretedbyexternal/inheritedsemantics).Somealgorithmsmay fail,notbecauseofabuggy,narrowalgorithm,butbecauseofaproblematthe real-worldinterface.Thatinterfaceisthe(algorithmic)codingofthealgorithm’s inputsandoutputs,typicallythrougha sequence oftransducersatthereal-world end(cf.[Smith,1987]).Physicalsignalsfromtheexternalworldmustbetransduced(encoded)intothecomputer’sswitch-settings(thephysicalanaloguesofa Turingmachine’s‘0’sand‘1’s),andtheoutputswitch-settingshavetobetransduced(decoded)intosuchreal-worldthingsasdisplaysonascreenorphysical movementsbyarobot.

Atthereal-worldendofthiscontinuum,werunintoSmith’sgap.Fromthe narrowalgorithm’spointofview,sotospeak,itmightbeabletoasymptotically approachtherealworld,inZeno-likefashion,withoutclosingthegap.But,justas someonetryingtocrossaroombyonlygoinghalftheremainingdistanceateach step will eventuallycrosstheroom(thoughnotbecauseofdoingitthatway),so thenarrowalgorithmimplementedinaphysicalcomputer will dosomethinginthe realworld.Whetherwhatitaccomplisheswaswhatitsprogrammerintendedis anothermatter.(Intherealworld,thereareno“partialfunctions”!)

Onewaytomaketeleologicalalgorithmsmorelikelytobesuccessfulisby Shapiro’sstrategy:Internalizingtheexternal,teleologicalaspectsintothepre-and post-conditionsofthe(narrow)algorithm,therebyturningtheexternal/inherited semanticinterpretationofthealgorithmintoaninternal/indigenoussyntacticsemantics.

WhatSmithshowsisthattheexternalsemanticsforanalgorithmisneverarelationdirectlywiththerealworld,butonlytoa model oftherealworld.Thatis,the real-worldsemanticshasbeeninternalized.Butthatinternalizationisnecessarily partialandincomplete.

Therearealgorithms simpliciter,andtherearealgorithms foraccomplishing aparticulartask.Alternatively, all algorithmsaccomplishaparticulartask,but sometasksormore“interesting”thanothers.Thealgorithmswhosetasksare notcurrentlyofinterestmayultimately become interestingwhenanapplication isfoundforthem,asinthecaseofnon-Euclideangeometry.Putotherwise,the algorithmsthatdonotaccomplishtasksmayultimatelybeusedtoaccomplisha task.

Algorithmsthat explicitly accomplisha(ninteresting)taskcanbeconverted intoalgorithmswhosetasksare not explicitinthatmannerbyinternalizingthe

taskintothealgorithmnarrowlyconstrued.Thiscanbedonebyinternalizingthe task,perhapsintheformofpre-andpostconditions,perhapsintheformofnamed subroutines(modules).Bothinvolvesyntacticorindigenoussemantics. Aspromised,IhaveraisedmorequestionsthanIhaveanswered.Butthat’s whatphilosophersaresupposedtodo!58

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