what is recursion? - princeton universityassets.press.princeton.edu/chapters/s9424.pdf · 1 what is...
TRANSCRIPT
1
What Is Recursion
In1637theFrenchphilosopherReacuteneacuteDescarteswrotetheimmor-tallineldquoJepensedoncjesuisrdquoCuriouslythisisusuallyrenderedinLatinasCogito ergo sumand is translated inEnglishasldquoIthinkthereforeIamrdquoInmakingthisstatementDescarteswasnotmerelythinkinghewasthinkingaboutthinkingwhichledhimtotheconclusionthatheexistedTherecursivenatureofDescartesrsquosinsight isperhapsbetter rendered in theversionofferedbyAm-broseBierceinThe Devilrsquos DictionaryCogito cogito ergo cogito summdashldquoIthinkIthinkthereforeIthinkIamrdquoDescarteshimselfthoughwasmorepronetodoubtandexpandedhisdictumasldquoJedoutedonc jepensedonc jesuisrdquomdashldquoIdoubtthereforeI thinkthereforeIamrdquoHethusconcludedthatevenifhedoubtedsome-oneorsomethingmustbedoingthedoubtingsotheveryfactthathedoubtedprovedhisexistenceThisprobablycameasarelieftohisfriends
In thisbookI examine themoregeneral roleof recursion inour mental livesand argue that it is the primary characteristicthatdistinguishesthehumanmindfromthatofotheranimalsItunderliesourabilitynotonlytoreflectuponourownmindsbutalsotosimulatethemindsofothersItallowsustotravelmentallyintimeinsertingconsciousnessofthepastorfutureintopresentconsciousnessRecursionisalsothemainingredientdistinguishinghumanlanguagefromallotherformsofanimalcommunication
Recursion though is a fairly elusive concept often used inslightlydifferentways1 BeforeIdelveintosomeofthecomplexi-tiesletrsquosconsidersomefurtherexamplestogivethegeneralideaFirstthenanot-too-seriousdictionarydefinition
Recursion(rĭ-kucircrrsquo-zhən)nounSeerecursion
Oneproblemhereofcourseisthatthisimpliesaninfiniteloop
chapteR 1
Figure1Thethinkerthinksofthinkingofthinking(authorrsquosdrawing)
fromwhichyoumayneverescapeinordertoreadtheotherstuffinthisbookThefollowingvariantsuggestsawayout
Recursion(rĭ-kucircrrsquo-zhən)nounIfyoustilldonrsquotgetitseerecursion
ThisbanksonthepossibilitythatifyoudogetitafteraroundortwoyoucanescapeandmoveonIfyoudonrsquotwellIrsquomsorry
ThepostmodernnovelistJohnBarthconcoctedwhatisprobablyboththeshortestandthelongeststoryeverwrittencalledFrameshyTaleItcanbereproducedasfollowsWritethesentenceONCE UPON A TIME THERE ononesideofastripofpaperandWAS A STORY THAT BEGAN ontheothersideThentwistoneendonceandattachittotheotherendtoformaMobiusstripAsyouworkyourwayroundthestripthestorygoesonforever
AsimilarexamplecomesfromananonymousparodyofthefirstlineofBulwer-LyttonrsquosinfamousnovelPaul Clifford
ItwasadarkandstormynightandwesaidtothecaptainldquoTellusa storyrdquoAnd this is the story thecaptain toldldquoItwasadarkandstormynightandwesaidtothecaptainlsquoTellusastoryrsquoAndthisisthestorythecaptaintoldlsquoItwasadarkrsquordquo
Anotheramusingexample isprovidedbyacompetitionrunbyThe Spectator magazinewhichaskedreaderstostatewhatthey
What Is RecuRsIon
wouldmostliketoreadonopeningthemorningpaperThewin-ningentryreadasfollows
OurSecondCompetition
TheFirstPrize inthesecondof thisyearrsquoscompetitionsgoes toMrArthurRobinsonwhosewittyentrywaseasilythebestofthosewereceivedHischoiceofwhathewouldliketoreadwhenopeningthepaperwasheadedldquoOurSecondCompetitionrdquoandwasas followsldquoTheFirstprizeinthesecondofthisyearrsquoscompetitionsgoestoMrArthurRobinsonwhosewittyentrywaseasilythebestofthosewereceivedHischoiceofwhathewouldliketoreadwhenopeningthepaperwasheadedlsquoOurSecondCompetitionrsquobutowingtopaperre-strictionswecannotprintallofitrdquo2
Taking a different tack John Barthrsquos story Autobiography A Selfshyrecorded Fiction is a recursive tale inwhich thenarrator isostensiblythestoryitselfwritingaboutitself3Itendsrecursivelyinitsownend
NonsenseIrsquollmutter to the endonewordafter anotherstring therascalsoutmadornotheardornotmylastwordswillbemylastwords
Tomyknowledgenostoryhasyetattemptedtowriteastoryofastorythatwritesaboutitself
AndthenthereistherecurringproblemoffleasaspennedbytheVictorianmathematicianAugustusdeMorgan
GreatfleashavelittlefleasupontheirbackstobitersquoemAndlittlefleashavesmallerfleasandsoad infinitumAndthegreatfleasthemselvesinturnhavegreaterfleastogoonWhiletheseagainhavegreaterstillandgreaterstillandsoon4
Thisnotionof insertingprogressivelysmallerentities into largeronesad infinitum canalsogiverisetointerestingvisualeffectsasintheexamplesshowninfigure2
chapteR 1
Apollonian gasket Sierpinski triangle
Figure2Twofiguresshowingrecursiveinsertionsofcircles(left)andtriangles(right)TheApolloniangasketderivesfromApolloniusaGreekgeometerfromthethirdcenturyBCwhostudiedtheproblemofhowtodrawacirclethatistangentialtothreecirclesStartingwiththreecirclesthataretangentialtooneanotheronecancontinuetheprocessofconstructingcirclestangentialtoalltripletsad infinitumTheresultingfigureservesasamathematicalmodelforfoam(seeMackenzie2009formoreinformation)(2DApolloniangasketwithfourinitialcirclescourtesyofGuillaumeJacquenot)
TheuseofrecursiontocreateinfinitesequencesisalsoexploitedbymathematicsOnesuchsequenceisthesetofnatural(iewhole)numberswhich IrsquollwriteasNThuswecangenerateallof thepositivenaturalnumbersbythedefinitions
1isinNIfnisinNthen(n+1)isinN
ThisseconddefinitionisrecursivebecauseNappearsinthecondi-tionthatneedstobesatisfiedforN
YoumayrememberpossiblyfromschooldayswhatfactorialsareAsaschoolboy I foundthemamusing inachildishkindofwaybecausetheyweresignaledwithexclamationmarksthusfac-torial3usuallywritten3is321andequals65Similarlywecancomputethefollowing
5=54321=1208=87654321=40320
What Is RecuRsIon
Clearlythiscangoonforeverbutwecancapturetheentiresetbyusingjusttwodefiningequations
0=1n=n(nminus1) [wherengt0]
ThissecondequationisrecursiveinthatafactorialisdefinedintermsofafactorialWeneedthefirstequationtokickthethingoff
ThenextexampleisforrabbitsandiscalledtheFibonaccise-riesdefinedbythefollowingthreeequations
fibonacci(0)=1fibonacci(1)=1fibonacci(n)=fibonacci(n minus1)+fibonacci(n minus2)[wherengt1]
Ifyouarefollowingmeyoushouldbeabletocomputetheserieswhichgoes11235813WhatthedefinitionsaysthenisthateachnumberintheseriesisthesumofthetwopreviousonesWhyrabbitsFibonacci(c1170ndash1250)wasanItalianmathemati-cianwhousedtheseriestopredictthegrowthofahypotheticalpopulationofrabbits6
ForafinalinformalexampleItakeyoutoKyotoJapanwhereIoncehappeneduponasignonagatethatwaswritteninKanjiscriptIaskedwhatitmeantandmyguidetoldmeIhopecorrectlythatitmeantPost no billsThereisaparadoxhereinthatthesignwas itselfabilltherebycontravening itsownpresencePerhapsthereneededtobeanothersignthatsaidPost no ldquoPost no billsrdquo billsButofcoursethisisitselfinviolationofitsownmessagesowemightenvisageanothersignthatreadsPost no lsquoPost ldquoPost no billsrdquo billsrsquo billsThereisnoendtothisprocesssoitmighthavebeenmoresensible tohaveallowedbillson thegate in thefirstplaceInpracticethoughlimitationsoftimespaceormemorywillpreventarecursivesequenceofstructurecontinuingforever
toward a Working Definition
Oneof thecharacteristicsofrecursionthenis that itcantakesits own output as the next input a loop that can be extended
chapteR 1
indefinitelytocreatesequencesorstructuresofunboundedlengthorcomplexityInpracticeofcoursewedonotgetcaughtupininfiniteloopsmdashlifeissimplytooshortforthatForthepurposesofthisbookthenweshallnotbeinterestedsomuchinthegenera-tionofinfinitesequencesasinadefinitionthatmightapplyuse-fullytohumanthoughtAdefinitionthatmeetsthisrequirementissuggestedbyStevenPinkerandRayJackendoffwhodefinere-cursionasldquoaprocedurethatcallsitselforaconstituentthatcontainsaconstituentofthesamekindrdquo7
Thesecondpartofthisdefinitionisimportantespeciallyinlan-guagebecauseitallowsthatrecursiveconstructionsneednotin-volvetheembeddingofthesame constituentsasintheexampleof the gate inKyotobutmay contain constituents of the samekindmdasha process sometimes known asldquoself-similar embeddingrdquoForexamplenounphrasescanbebuiltfromnounphrasesinre-cursivefashionTecumsehFitchgivestheexampleofsimplenounphrasessuchasthe dogthe catthe treethe lakeandonecanthencreatenewnounphrasesbyplacingthewordbeside betweenanypairthe dog beside the treethe cat beside the lake8OronemighthavetwosentencesJane loves John andJane flies airplanesandembedone in theother (withappropriatemodification)asJane who flies airplanes loves JohnThesecanbeextendedrecursivelytowhateverlevelofcomplexityisdesiredForexamplewecouldextendthenounphrasetothe dog beside the tree beside the lakeor thesentenceaboutJaneandJohntoJane who flies airplanes that exceed the sound barrier loves John who is prone to selfshydoubtMost languagesmakeuseof recursiveoperationsof thissortmdashalthoughweshallseeinthenextchapterthattheremaybeafewlanguagesthatdonrsquotoperateinthisway
AlthoughitiscommontoprovideillustrationsfromlanguagethemainthemeofthisbookisthatitisinthoughtratherthaninlanguagethatrecursionoriginatesAsPinkerandJackendoffputitldquoTheonlyreasonlanguageneedstoberecursiveisbecauseitsfunction is to express recursive thoughts If there were not anyrecursivethoughtsthemeansofexpressionwouldnotneedrecur-sioneitherrdquo9 Inrememberingepisodesfromthepastforinstanceweessentiallyinsertsequencesofpastconsciousnessintopresent
What Is RecuRsIon
consciousnessor inour interactionswithotherpeoplewemayinsertwhattheyarethinkingintoourownthinkingThesethemesareexploredinlaterchapters
Process and Structure
AssuggestedbyPinkerandJackendoffrsquosdualdefinitionrecursioncanbeunderstoodeitherasaprocess orasastructureThedistinc-tioncanbeimportantArecursiveprocessmayleadtoastructurethatneednotbeseenasitselfrecursiveForexamplesupposeweconstructasequenceofmusicalnoteswithanembeddingroutinebypairingpairsofnoteseachconsistingofa randomlychosennoteplayedonapianowitharandomlychosennoteplayedonaviolinThefirstpairisembeddedinanotherpairandthefour-noteoutputthenembeddedinanotherpairThisprocesscanbecontin-uedindefinitelytocreateasequenceofnotesAsillustratedinfig-ure3thoughthesequencecanbeinterpretednotasarecursivelyembeddedstructurebutasasequenceofpianonotesfollowedbyanequallylongsequenceofviolinnotesThisfailuretodistinguishrecursiveembeddingfromrecursivestructurehasledtosomecon-fusionespeciallyinclaimsaboutrecursioninnonhumanspecies10
Again in his most recent theory on the nature of languageknownastheMinimalistProgram11 NoamChomskyhasarguedthathumanthoughtisgeneratedbyaMergeoperationappliedre-cursivelyThatisunitsaremergedtoformlargerentitiesandthemergedentitiescanbethemselvesmergedtoformstilllargerenti-tiesandsoonThisoperationunderliestheembeddedstructureofhumanlanguagealthoughinChomskyrsquostheoryitappliesstrictlytowhathecallsIshylanguagewhichisthethoughtprocesspreced-ingEshylanguagetheexternal language that isactually spokenorsignedMergecanproducestringsofelementsbetheywordsorelementsofthoughtandalthoughitmaybeappliedrecursivelytoproducehierarchicalstructurethatstructuremaynotbeevidentinthefinaloutputForinstanceevensentencesmightberegardedsimplyaswordsallmergedinunstructuredsequenceasinritual-izedsongsorprayersEverydaylanguagetoomayincludemen-tally undifferentiated clicheacutes and slogansor sequences that are
chapteR 1
P P P P V V V V P P P P V V V V
Figure3ThesequenceofPsandVscanbecreatedeitherbyrecursivelynest-ingPVpairsinPVpairs(left)orbyarrangingasequenceofPsfollowedbyasequenceofanequalnumberofVs(right)Thesequencemightbegeneratedasintheleftpanelbutinterpretedasintherightpanel
highlyautomatedPoliticiansmaybeespeciallypronetothiskindoftalk
Asnotedaboverecursiveprocessesandstructurescaninprin-cipleextendwithoutlimitbutarelimitedinpracticeNeverthelessrecursiondoes give rise to theconcept of infinity itself perhapslimited to the human imaginationAfter all only humans haveacquired theability to count indefinitelyand tounderstand thenatureofinfiniteserieswhereasotherspeciescanatbestmerelyestimatequantityandareaccurateonlyup to somesmallfinitenumber12EveninlanguageweunderstandthatasentencecaninprinciplebeextendedindefinitelyeventhoughinpracticeitcannotbemdashalthoughthenovelistHenryJameshadadamngoodtrySuchunderstandingsareindeedpartofhumanmentalachievementanddependonahumancapacityforrecursivethoughtNeverthelesstheyarenottheprimaryconcernsofthisbook
The appealing aspect of recursion is precisely that it can in principle extend indefinitely to create thoughts (and sentences)ofwhatevercomplexityisrequiredTheideahasanelegantsim-plicitygivingrisetowhatChomskycalledldquodiscreteinfinityrdquo13orWilhelmHumboldt(1767ndash1835)famouslycalledldquotheinfiniteuseoffinitemeansrdquoAndalthoughrecursionislimitedinpracticewecanneverthelessachieveconsiderabledepthsofrecursivethoughtarguablyunsurpassedinanyotherspeciesInchessforexampleaplayermaybeabletothinkrecursivelythreeorfourstepsahead
What Is RecuRsIon
examiningpossiblemovesandcountermovesbutthenumberofpossibilitiessoonmultipliesbeyondthecapacityofthemindtoholdthem
Deeperlevelsofrecursionmaybepossiblewiththeaidofwrit-ingorsimplywithextendedtimeforrehearsalandcontemplationorextendedmemorycapacitythroughartificialmeansTheslowdevelopmentofacomplexmathematicalproofforexamplemayrequiresubtheoremswithinsubtheoremsPlaysornovelsmayin-volverecursiveloopsthatbuildslowlymdashinShakespearersquosTwelfth NightforexampleMariaforeseesthatSirTobywilleagerlyan-ticipate thatOliviawill judgeMalvolio absurdly impertinent tosuppose that she wishes him to regard himself as her preferredsuitor14Asinfictionsoinlifeweallliveinawebofcomplexre-cursiverelationshipsandplanningadinnerpartymayneedcare-fulattentiontowhothinkswhatofwhom
ThestructuresresultingfromrecursiveprocessesneednotrevealthenatureofthoseprocessesjustasaloafofbreadmaynotrevealtheprocessesofkneadingthatwentintothemakingofthebreadorthetasteofwinethepickingandtramplingofthegrapesOftenthoughthestructureofasentenceorstreamofthoughtmayrevealrecursiveembeddingmdashinterpretationofasentencemayrequiretheunderstandingofphrasesembeddedinphrasesregardlessofhowtheembeddingwasactuallyaccomplishedandaninternalunder-standingofastreamofthoughtmayrequirethesegmentationofepisodeswithinepisodes
What Recursion Is not
Recursionisnottheonlydeviceforcreatingsequencesorstruc-tures of potentially infinite length or size I now consider someexamplesthatdonotmeetthecriteriaforrecursion
Repetition
Simple repetition can lead to sequences of potentially infinitelengthbutdoesnot classifyas true recursionForexamplethesentence thatopenschapter9ofAAMilnersquosWinnie the Pooh
10 chapteR 1
goesIt rained and it rained and it rained Thiscouldgoonfor-evermdashoratleastuntilPigletisdrownedmdashbuttherepetitionsimplyconveys the information that it rained rathera lotcausingPig-letsomeennuiItisnotrecursivebecauseeachadditionofand it rained isnotdrivenbythepreviousoneitissimplyaddedatthediscretionofthewriter
InanyeventrepetitiondoesnotdistinguishhumanactivityfromthatofnonhumananimalsBirdsongforexampleisrelentlesslyrepetitivebuteachrepeatedthemedoesnotembellishorqualifythepreviousoneAtmosttherepetitionmightsignalurgencyorsimplysignalcontinuingpresenceasonemightrepeatedlyknockonadoorinthehopeofarousingsomeoneinsideRepetitionisubiquitousinhumanandanimallifeinactivitiesrangingfromtherepeatedjawmovementsineatingtothecuriouslyrepetitivena-tureofsexualactivityThespidernolessiscapableofrepetitionasinWaltWhitmanrsquosLeaves of Grass15
ANOISELESSpatientspiderImarkrsquodwhereonalittlepromontoryitstoodisolatedMarkrsquodhowtoexplorethevacantvastsurroundingItlaunchrsquodforthfilamentfilamentfilamentoutofitselfEverunreelingthemmdashevertirelesslyspeedingthem
Informationcanalsobeaggregatedinnonrecursivefashionaswhentheshort-storywriterSaki(HHMunro)wroteldquoHungerfatigueanddespairinghopelessnesshadnumbedhisbrainrdquo16Ag-gregationofdifferentphrases similarlycompoundsmeaningad-ditivelyaswhenthehistorianPeterHennessywrote
ThemodelofamodernPrimeMinisterwouldbeakindofgrotesquecompositefreakmdashsomeonewiththededicationtodutyofaPeelthephysicalenergyofaGladstonethedetachmentofaSalisburytheballsofaLloydGeorgetheword-powerofaChurchilltheadministrativegiftsof anAttleethe styleof aMacmillanthemanagerialismof aHeathandthesleeprequirementsofaThatcher17
ThesentenceitselfhasrecursiveelementsbuttheaggregationofphrasestodescribethefreakishcompositeisnotrecursiveinthateachdoesnotcallthenextInsteadtheyareeffectivelyelements
What Is RecuRsIon 11
inalistinsertedtoaddinformationNonhumanspeciesmaywellhaveasimilarabilitytoaccumulateinformationaswhenunder-standingapredatoraslargefierceandwithsharpteethandclaws
Iteration
AslightlymoresubtlevariantonrepetitionandaggregationisitshyerationwhereaprocessisrepeatedbutinthiscasethereisinputfromthepreviousapplicationoftheprocessInthisrespect it islikerecursionandindeedconsideredbymathematicianstobelongtotheclassofldquogeneralrecursivefunctionsrdquoForthemainpurposesofthisbookthoughitdoesnotqualifyastruerecursionbecauseeachoutputisdiscardedonceithasbeenenteredintothenextap-plicationThedictionarydefinitionofrecursionthatIgaveearlierinthischapterwasalsoreallyanexampleofiterationratherthanrecursionbecauseyoujustkeepgoingroundandroundtheloopwithoutanyaddedstructureTheiterationsthereforedonotleadtoaddedcomplexity18
Iterativeproceduresareusedincomputationalmathematicstoarriveat increasinglyaccuratesolutionstoaproblemThebasicideaistostartwithapreliminarysolutionmdashperhapsaguessmdashandthenuseaproceduretocomputeanewsolutionThissolutionisthenusedasthestartingpointforthenextcomputationandthenew solution is then the starting point for the next roundThecycle is repeateduntil the solutions stabilize to someacceptablecriterion19Feedbacksystemsoperateinmuchthesamewaytypi-callyasameansofmaintaininghomeostasisForexampleather-mostatmayinvolveasystemforraisingorloweringtemperatureandthegoalistoachievesomegiventemperatureTheactualtem-peratureisfedintothesystemwhichoperatestoraiseorlowerthe temperatureuntil thedesired is reachedThebody isawashwith feedback systems to maintain homeostasis of temperatureironenergybloodcompositionandsoonThemainregulatoristhehypothalamusinthelimbicsystemofthebrainSuchsystemsagaindonotdifferentiatehumansfromotheranimals
Sometimesthedistinctionbetweenrecursionanditerationmaybeamatterof interpretationIn the infinite loopcreatedby the
1 chapteR 1
parodyofPaul CliffordonemightsaythateachbeginningofthestoryisinitiatedbythepreviousonewhichisthenforgottenTheparodyisbestappreciatedthoughifthestoryisseenasanendlessever deepening whirlpoolwith each segment remaining as partofitIrsquomtoldthestoryworksbestifeachsegmentisspokenwithadifferentaccent
Considertoothislinefromawell-knownchildrenrsquosverse
ThisisthedogthatworriedthecatthatkilledtheratthatatethemaltthatlayinthehousethatJackbuilt
TounderstandthissentenceastrulyrecursiveonemustappreciatethatitdescribesastateofaffairsasacomplexwholeandreferstoparticularcasesofadogcatratmalthouseandfellowcalledJackItisnotsimplythestringingtogetherofadogthatworriedacatacatthatkilledaratandsoforthAyoungchildthoughmightprocessitinthispiecemealwayasasuccessionofunrelatedevents
Recursion and evolutionary psychology
Inemphasizingrecursionasaunifyingconcepttheapproachtakeninthisbookcontrastswiththatadoptedbyso-calledevolutionarypsychologistswhohavearguedthatthemindhasmultiplefacetsThebasictenetsofevolutionarypsychologywerelaidoutinthe1992volumeThe Adapted MindeditedbyJeromeBarkowLedaCosmidesandJohnToobyandpopularizedbyStevenPinkerinhis influential 1997 book How the Mind Works20 Thus Pinkerwritesthatthehumanmindldquoisnotasingleorganbutasystemoforganswhichwecanthinkofaspsychologicalfacultiesormentalmodulesrdquo21 Inexaminingpresent-dayhumanbehaviortheevolu-tionarypsychologistrsquosagendaistodiscoverindependentprocessesasthebasicmodulesandrelatethemtoconditionsthatprevailedin the Pleistocene when humans existed primarily as hunter-gatherersAs Pinker puts it the aim is to carve the mind at itsjointssotospeakandldquoreverse-engineerrdquoitscomponentsormod-ulesbacktotheepochduringwhichthehumanmindwasformed
What Is RecuRsIon 1
In this view the mind is really a collection of miniminds eachbeaveringawayonitsownspecificproblemamongwhicharelan-guageandtheoryofmindThishasbeencalledtheSwiss-army-knifemodelofthemindwithabladeforeverypurpose22
Thedangerwith thisapproach is that itbecomes tooeasy topostulate modules and to tellldquojust sordquostories about how theyevolvedsothatthereisariskofreturningtothenow-abandonedinstinctpsychologyoftheearlytwentiethcentury23 Instinctpsy-chologyperishedunderthesheerweightofnumbersmdashtheauthorof one text counted 1594 instincts that had been attributed toanimals and humans24mdashand evolutionary psychology may alsodrowninaseaofmodulesifnotofmixedmetaphorsPinkersug-geststhatwelikepotatochipsbecausefattyfoodswerenutrition-allyvaluableduringthePleistocenebutscarceenoughthattherewasnodangerofobesitywelike landscapeswithtreesbecausetreesprovidesshadeandescapefromdangerouscarnivoresontheAfrica savanna flowers please us because they are markers foredible fruits nutsor tubers amid the greenery of the savannaandsoonldquoTherearemodulesrdquohewritesldquoforobjectsandforcesforanimatebeingsformindsandfornaturalkindslikeanimalsplantsandmineralsrdquo25
This isnot to say that theSwiss-army-knifemodel iswithoutmeritSomeofthepostulatedmodulesdoprovideinsightintothehumanconditionandarereasonablywellfoundedForexamplefoundationalworkinevolutionarypsychologybyLedaCosmidesprovidedgoodevidenceforaldquocheater-detectionmodulerdquomdashanin-stinctiveability todetect thosewhoflout social conventions fortheirowngain26 Arecentstudysuggests thathumanspossessaldquocategory-specificattentionsystemrdquothatisespeciallyadaptedforattendingtoanimals27 oneoftheauthorsJohnToobyisquotedas sayingldquoEvendullanimals likepigeons recruita surpris-ingamountofattentionmdashasdoturtlesresemblingrocksrdquo28 Thisbookisnotintendedtodenythattherearemanyspecificdisposi-tionsthatshapeourmentalandsociallivesrathermyaimistosuggest that therearedeeperaspectsofhumanthoughtthataregovernedbysimilarprinciplesandthatrecursionisoneofthoseprinciplesmdashandperhapsthemostimportantone
1 chapteR 1
Tobefairtoonotallevolutionarypsychologistshaveinsistedthatmodulesarecompletelyencapsulatedshutofffromanycom-munication with one anotherEven Steven Pinker for examplewrites ldquo[Modules] accomplish specialized functions thanks totheirspecializedstructuresbutdonrsquotnecessarilycomeinencap-sulated packagesrdquo29 Steven Mithen although scarcely a card-carryingevolutionarypsychologistarguedthatthehumanmindevolved itsdistinctive characterpreciselybecausepreviously en-capsulatedmodulesbegantoldquoleakrdquocreatingwhathecallsldquocogni-tivefluidityrdquo30 ItisasthoughthemodulesstoppedmindingtheirownbusinessesandbegantogossipMyapproachinthisbookisnotentirelyatoddswiththisviewinthatIargueacommonprin-ciplemightunderlieanumberofourdistinctiveabilities
Others are also beginning to question the Swiss-army-knifemodelofthehumanmindmorestarklyDavidPremackforex-ampleadoptsanapproachsimilartothatofferedinthisbookRe-viewingtheevidencefordiscontinuitybetweenhumansandotheranimalshewritesldquoAnimalcompetenciesaremainlyadaptationsrestrictedtoasinglegoalHumancompetenciesaredomain-generalandservenumerousgoalsrdquo31Thisineffectreversestheevolution-arypsychologyargumentmdashthemindhasbecomelessratherthanmoremodularThetidemaywellbeturning
In any event it is unlikely that recursion canbe considered amoduleAsweshallseerecursionseemstobeanorganizingprin-cipleinverydifferentspheresofhumanmentalactivityfromlan-guage tomemory tomind readingRecursive thinkingprobablydependsonothermentalattributesOneoftheseiswhathasbeentermedworkingmemorywhichholdsinformationinconscious-nessInordertoembedprocesseswithinprocessesitisnecessarytorememberwhereonehadgottointheearlierprocesswhenanembeddedprocesshasbeencompletedForinstanceinasentencelike My dog who eats bananas often gets sickone must holdthe early part of the sentence (My dog) and link it to the nextpart(often gets sick)DwightWReadhasarguedthatnonhumanprimatesevenourclosestrelativesthechimpanzeeshaveawork-ingmemorythatistoolimitedtoallowthiskindofembedding32
Recursionprobablyalsodependsonanexecutiveprocessthator-
What Is RecuRsIon 1
ganizeswhatistobeembeddedinwhatandthismaydependonthefrontallobesofthebrainTheabilitytoorganizeandcarryoutrecursiveoperationsmaythereforedependonseveralprocesses
AlthoughIdonotembracethemodularviewassumedbyevo-lutionarypsychologistsIamatonewiththeminproposingthata distinctively human mind evolved during the Pleistocene theepochthatstretchedfromaround26millionyearsagotosome12000 years agoHow that happened will be told in the laterchaptersofthisbook
plan of the Book
ThebookisdividedintofourpartsPart1dealswithlanguageAlthoughrecursionisnotlimitedto
languageitismostcommonlyinvokedtoexplainwhyhumanlan-guagediffersfromotherformsofanimalcommunicationanin-sightlargelyattributabletoNoamChomskyChapter2discussesthe nature of languagewithparticular emphasis on the role ofrecursionChapter3thenraisestheage-oldquestionofwhetherotheranimalshaveanythingresemblinghumanlanguageChapter4developstheideathatlanguageevolvedfrommanualgesturesmdashanideathatsuggestsgreaterevolutionarycontinuitybetweenhu-mansandotherprimatesthanthemorecommonassumptionthatlanguageemergedfromvocalcalls
Part2dealswithmentaltimetraveltheabilitytobringtomindeventsremovedfromthepresentinbothtimeandplaceChapter5startswithmemoryanddevelopstheideathatmemoryforspecificepisodesisuniquetohumansChapter6extendsthenotionofepi-sodicmemorytotheimaginingofpossiblefutureeventsleadingtotheconceptoftheselfasexistingthroughtimeThisleadstothenotiondiscussedinchapter7thatlanguageitselfevolvedtoenablepeopletosharetheirmemoriesandplansandsotocom-municateabouteventsthatarenotpresentintheimmediateenvi-ronmentThisleadsalsotofictionmdashthetellingofstoriesthatneednotbebasedonfactbutthatnonethelesshonethecapacitytodealwiththeepisodicexigenciesofhumansociallife
1 chapteR 1
Part3dealsanotherrecursiveaspectofhumanthoughtnamelytheoryofmindmdashortheabilitytounderstandwhatothersarethink-ingorfeelingChapter8introducesmind-readingnotasapsychicphenomenonbutasanaturalabilitytoinferthementalperspec-tivesofotherpeopleThisabilityisagaincriticaltosocialcohesionandcooperationChapter9explainshowtheoryofmindwasalsocriticaltotheemergenceoflanguage
Part4delvesmorespecificallyintothequestionofhowthere-cursivemindevolvedChapter10setsthisquestioninthecontextoftheclassicdebatebetweenCartesiandiscontinuityandDarwin-iancontinuityChapter11examinessomeofthestepsbywhichthe hominins33 after splitting from the line leading to modernchimpanzees and bonobos began to assume human-like attri-butesChapter12thenconsidersthefinalsteptoldquomodernrdquoHomo sapiensthesolesurvivinghomininspeciesmdashdominantmanipula-tiveMachiavellianandcapableofponderingourownnatureandstatusontheplanetThatperhapsistheultimatetriumphoftherecursivemind
Chapter13presentsthefinalsummaryandconclusions
chapteR 1
Figure1Thethinkerthinksofthinkingofthinking(authorrsquosdrawing)
fromwhichyoumayneverescapeinordertoreadtheotherstuffinthisbookThefollowingvariantsuggestsawayout
Recursion(rĭ-kucircrrsquo-zhən)nounIfyoustilldonrsquotgetitseerecursion
ThisbanksonthepossibilitythatifyoudogetitafteraroundortwoyoucanescapeandmoveonIfyoudonrsquotwellIrsquomsorry
ThepostmodernnovelistJohnBarthconcoctedwhatisprobablyboththeshortestandthelongeststoryeverwrittencalledFrameshyTaleItcanbereproducedasfollowsWritethesentenceONCE UPON A TIME THERE ononesideofastripofpaperandWAS A STORY THAT BEGAN ontheothersideThentwistoneendonceandattachittotheotherendtoformaMobiusstripAsyouworkyourwayroundthestripthestorygoesonforever
AsimilarexamplecomesfromananonymousparodyofthefirstlineofBulwer-LyttonrsquosinfamousnovelPaul Clifford
ItwasadarkandstormynightandwesaidtothecaptainldquoTellusa storyrdquoAnd this is the story thecaptain toldldquoItwasadarkandstormynightandwesaidtothecaptainlsquoTellusastoryrsquoAndthisisthestorythecaptaintoldlsquoItwasadarkrsquordquo
Anotheramusingexample isprovidedbyacompetitionrunbyThe Spectator magazinewhichaskedreaderstostatewhatthey
What Is RecuRsIon
wouldmostliketoreadonopeningthemorningpaperThewin-ningentryreadasfollows
OurSecondCompetition
TheFirstPrize inthesecondof thisyearrsquoscompetitionsgoes toMrArthurRobinsonwhosewittyentrywaseasilythebestofthosewereceivedHischoiceofwhathewouldliketoreadwhenopeningthepaperwasheadedldquoOurSecondCompetitionrdquoandwasas followsldquoTheFirstprizeinthesecondofthisyearrsquoscompetitionsgoestoMrArthurRobinsonwhosewittyentrywaseasilythebestofthosewereceivedHischoiceofwhathewouldliketoreadwhenopeningthepaperwasheadedlsquoOurSecondCompetitionrsquobutowingtopaperre-strictionswecannotprintallofitrdquo2
Taking a different tack John Barthrsquos story Autobiography A Selfshyrecorded Fiction is a recursive tale inwhich thenarrator isostensiblythestoryitselfwritingaboutitself3Itendsrecursivelyinitsownend
NonsenseIrsquollmutter to the endonewordafter anotherstring therascalsoutmadornotheardornotmylastwordswillbemylastwords
Tomyknowledgenostoryhasyetattemptedtowriteastoryofastorythatwritesaboutitself
AndthenthereistherecurringproblemoffleasaspennedbytheVictorianmathematicianAugustusdeMorgan
GreatfleashavelittlefleasupontheirbackstobitersquoemAndlittlefleashavesmallerfleasandsoad infinitumAndthegreatfleasthemselvesinturnhavegreaterfleastogoonWhiletheseagainhavegreaterstillandgreaterstillandsoon4
Thisnotionof insertingprogressivelysmallerentities into largeronesad infinitum canalsogiverisetointerestingvisualeffectsasintheexamplesshowninfigure2
chapteR 1
Apollonian gasket Sierpinski triangle
Figure2Twofiguresshowingrecursiveinsertionsofcircles(left)andtriangles(right)TheApolloniangasketderivesfromApolloniusaGreekgeometerfromthethirdcenturyBCwhostudiedtheproblemofhowtodrawacirclethatistangentialtothreecirclesStartingwiththreecirclesthataretangentialtooneanotheronecancontinuetheprocessofconstructingcirclestangentialtoalltripletsad infinitumTheresultingfigureservesasamathematicalmodelforfoam(seeMackenzie2009formoreinformation)(2DApolloniangasketwithfourinitialcirclescourtesyofGuillaumeJacquenot)
TheuseofrecursiontocreateinfinitesequencesisalsoexploitedbymathematicsOnesuchsequenceisthesetofnatural(iewhole)numberswhich IrsquollwriteasNThuswecangenerateallof thepositivenaturalnumbersbythedefinitions
1isinNIfnisinNthen(n+1)isinN
ThisseconddefinitionisrecursivebecauseNappearsinthecondi-tionthatneedstobesatisfiedforN
YoumayrememberpossiblyfromschooldayswhatfactorialsareAsaschoolboy I foundthemamusing inachildishkindofwaybecausetheyweresignaledwithexclamationmarksthusfac-torial3usuallywritten3is321andequals65Similarlywecancomputethefollowing
5=54321=1208=87654321=40320
What Is RecuRsIon
Clearlythiscangoonforeverbutwecancapturetheentiresetbyusingjusttwodefiningequations
0=1n=n(nminus1) [wherengt0]
ThissecondequationisrecursiveinthatafactorialisdefinedintermsofafactorialWeneedthefirstequationtokickthethingoff
ThenextexampleisforrabbitsandiscalledtheFibonaccise-riesdefinedbythefollowingthreeequations
fibonacci(0)=1fibonacci(1)=1fibonacci(n)=fibonacci(n minus1)+fibonacci(n minus2)[wherengt1]
Ifyouarefollowingmeyoushouldbeabletocomputetheserieswhichgoes11235813WhatthedefinitionsaysthenisthateachnumberintheseriesisthesumofthetwopreviousonesWhyrabbitsFibonacci(c1170ndash1250)wasanItalianmathemati-cianwhousedtheseriestopredictthegrowthofahypotheticalpopulationofrabbits6
ForafinalinformalexampleItakeyoutoKyotoJapanwhereIoncehappeneduponasignonagatethatwaswritteninKanjiscriptIaskedwhatitmeantandmyguidetoldmeIhopecorrectlythatitmeantPost no billsThereisaparadoxhereinthatthesignwas itselfabilltherebycontravening itsownpresencePerhapsthereneededtobeanothersignthatsaidPost no ldquoPost no billsrdquo billsButofcoursethisisitselfinviolationofitsownmessagesowemightenvisageanothersignthatreadsPost no lsquoPost ldquoPost no billsrdquo billsrsquo billsThereisnoendtothisprocesssoitmighthavebeenmoresensible tohaveallowedbillson thegate in thefirstplaceInpracticethoughlimitationsoftimespaceormemorywillpreventarecursivesequenceofstructurecontinuingforever
toward a Working Definition
Oneof thecharacteristicsofrecursionthenis that itcantakesits own output as the next input a loop that can be extended
chapteR 1
indefinitelytocreatesequencesorstructuresofunboundedlengthorcomplexityInpracticeofcoursewedonotgetcaughtupininfiniteloopsmdashlifeissimplytooshortforthatForthepurposesofthisbookthenweshallnotbeinterestedsomuchinthegenera-tionofinfinitesequencesasinadefinitionthatmightapplyuse-fullytohumanthoughtAdefinitionthatmeetsthisrequirementissuggestedbyStevenPinkerandRayJackendoffwhodefinere-cursionasldquoaprocedurethatcallsitselforaconstituentthatcontainsaconstituentofthesamekindrdquo7
Thesecondpartofthisdefinitionisimportantespeciallyinlan-guagebecauseitallowsthatrecursiveconstructionsneednotin-volvetheembeddingofthesame constituentsasintheexampleof the gate inKyotobutmay contain constituents of the samekindmdasha process sometimes known asldquoself-similar embeddingrdquoForexamplenounphrasescanbebuiltfromnounphrasesinre-cursivefashionTecumsehFitchgivestheexampleofsimplenounphrasessuchasthe dogthe catthe treethe lakeandonecanthencreatenewnounphrasesbyplacingthewordbeside betweenanypairthe dog beside the treethe cat beside the lake8OronemighthavetwosentencesJane loves John andJane flies airplanesandembedone in theother (withappropriatemodification)asJane who flies airplanes loves JohnThesecanbeextendedrecursivelytowhateverlevelofcomplexityisdesiredForexamplewecouldextendthenounphrasetothe dog beside the tree beside the lakeor thesentenceaboutJaneandJohntoJane who flies airplanes that exceed the sound barrier loves John who is prone to selfshydoubtMost languagesmakeuseof recursiveoperationsof thissortmdashalthoughweshallseeinthenextchapterthattheremaybeafewlanguagesthatdonrsquotoperateinthisway
AlthoughitiscommontoprovideillustrationsfromlanguagethemainthemeofthisbookisthatitisinthoughtratherthaninlanguagethatrecursionoriginatesAsPinkerandJackendoffputitldquoTheonlyreasonlanguageneedstoberecursiveisbecauseitsfunction is to express recursive thoughts If there were not anyrecursivethoughtsthemeansofexpressionwouldnotneedrecur-sioneitherrdquo9 Inrememberingepisodesfromthepastforinstanceweessentiallyinsertsequencesofpastconsciousnessintopresent
What Is RecuRsIon
consciousnessor inour interactionswithotherpeoplewemayinsertwhattheyarethinkingintoourownthinkingThesethemesareexploredinlaterchapters
Process and Structure
AssuggestedbyPinkerandJackendoffrsquosdualdefinitionrecursioncanbeunderstoodeitherasaprocess orasastructureThedistinc-tioncanbeimportantArecursiveprocessmayleadtoastructurethatneednotbeseenasitselfrecursiveForexamplesupposeweconstructasequenceofmusicalnoteswithanembeddingroutinebypairingpairsofnoteseachconsistingofa randomlychosennoteplayedonapianowitharandomlychosennoteplayedonaviolinThefirstpairisembeddedinanotherpairandthefour-noteoutputthenembeddedinanotherpairThisprocesscanbecontin-uedindefinitelytocreateasequenceofnotesAsillustratedinfig-ure3thoughthesequencecanbeinterpretednotasarecursivelyembeddedstructurebutasasequenceofpianonotesfollowedbyanequallylongsequenceofviolinnotesThisfailuretodistinguishrecursiveembeddingfromrecursivestructurehasledtosomecon-fusionespeciallyinclaimsaboutrecursioninnonhumanspecies10
Again in his most recent theory on the nature of languageknownastheMinimalistProgram11 NoamChomskyhasarguedthathumanthoughtisgeneratedbyaMergeoperationappliedre-cursivelyThatisunitsaremergedtoformlargerentitiesandthemergedentitiescanbethemselvesmergedtoformstilllargerenti-tiesandsoonThisoperationunderliestheembeddedstructureofhumanlanguagealthoughinChomskyrsquostheoryitappliesstrictlytowhathecallsIshylanguagewhichisthethoughtprocesspreced-ingEshylanguagetheexternal language that isactually spokenorsignedMergecanproducestringsofelementsbetheywordsorelementsofthoughtandalthoughitmaybeappliedrecursivelytoproducehierarchicalstructurethatstructuremaynotbeevidentinthefinaloutputForinstanceevensentencesmightberegardedsimplyaswordsallmergedinunstructuredsequenceasinritual-izedsongsorprayersEverydaylanguagetoomayincludemen-tally undifferentiated clicheacutes and slogansor sequences that are
chapteR 1
P P P P V V V V P P P P V V V V
Figure3ThesequenceofPsandVscanbecreatedeitherbyrecursivelynest-ingPVpairsinPVpairs(left)orbyarrangingasequenceofPsfollowedbyasequenceofanequalnumberofVs(right)Thesequencemightbegeneratedasintheleftpanelbutinterpretedasintherightpanel
highlyautomatedPoliticiansmaybeespeciallypronetothiskindoftalk
Asnotedaboverecursiveprocessesandstructurescaninprin-cipleextendwithoutlimitbutarelimitedinpracticeNeverthelessrecursiondoes give rise to theconcept of infinity itself perhapslimited to the human imaginationAfter all only humans haveacquired theability to count indefinitelyand tounderstand thenatureofinfiniteserieswhereasotherspeciescanatbestmerelyestimatequantityandareaccurateonlyup to somesmallfinitenumber12EveninlanguageweunderstandthatasentencecaninprinciplebeextendedindefinitelyeventhoughinpracticeitcannotbemdashalthoughthenovelistHenryJameshadadamngoodtrySuchunderstandingsareindeedpartofhumanmentalachievementanddependonahumancapacityforrecursivethoughtNeverthelesstheyarenottheprimaryconcernsofthisbook
The appealing aspect of recursion is precisely that it can in principle extend indefinitely to create thoughts (and sentences)ofwhatevercomplexityisrequiredTheideahasanelegantsim-plicitygivingrisetowhatChomskycalledldquodiscreteinfinityrdquo13orWilhelmHumboldt(1767ndash1835)famouslycalledldquotheinfiniteuseoffinitemeansrdquoAndalthoughrecursionislimitedinpracticewecanneverthelessachieveconsiderabledepthsofrecursivethoughtarguablyunsurpassedinanyotherspeciesInchessforexampleaplayermaybeabletothinkrecursivelythreeorfourstepsahead
What Is RecuRsIon
examiningpossiblemovesandcountermovesbutthenumberofpossibilitiessoonmultipliesbeyondthecapacityofthemindtoholdthem
Deeperlevelsofrecursionmaybepossiblewiththeaidofwrit-ingorsimplywithextendedtimeforrehearsalandcontemplationorextendedmemorycapacitythroughartificialmeansTheslowdevelopmentofacomplexmathematicalproofforexamplemayrequiresubtheoremswithinsubtheoremsPlaysornovelsmayin-volverecursiveloopsthatbuildslowlymdashinShakespearersquosTwelfth NightforexampleMariaforeseesthatSirTobywilleagerlyan-ticipate thatOliviawill judgeMalvolio absurdly impertinent tosuppose that she wishes him to regard himself as her preferredsuitor14Asinfictionsoinlifeweallliveinawebofcomplexre-cursiverelationshipsandplanningadinnerpartymayneedcare-fulattentiontowhothinkswhatofwhom
ThestructuresresultingfromrecursiveprocessesneednotrevealthenatureofthoseprocessesjustasaloafofbreadmaynotrevealtheprocessesofkneadingthatwentintothemakingofthebreadorthetasteofwinethepickingandtramplingofthegrapesOftenthoughthestructureofasentenceorstreamofthoughtmayrevealrecursiveembeddingmdashinterpretationofasentencemayrequiretheunderstandingofphrasesembeddedinphrasesregardlessofhowtheembeddingwasactuallyaccomplishedandaninternalunder-standingofastreamofthoughtmayrequirethesegmentationofepisodeswithinepisodes
What Recursion Is not
Recursionisnottheonlydeviceforcreatingsequencesorstruc-tures of potentially infinite length or size I now consider someexamplesthatdonotmeetthecriteriaforrecursion
Repetition
Simple repetition can lead to sequences of potentially infinitelengthbutdoesnot classifyas true recursionForexamplethesentence thatopenschapter9ofAAMilnersquosWinnie the Pooh
10 chapteR 1
goesIt rained and it rained and it rained Thiscouldgoonfor-evermdashoratleastuntilPigletisdrownedmdashbuttherepetitionsimplyconveys the information that it rained rathera lotcausingPig-letsomeennuiItisnotrecursivebecauseeachadditionofand it rained isnotdrivenbythepreviousoneitissimplyaddedatthediscretionofthewriter
InanyeventrepetitiondoesnotdistinguishhumanactivityfromthatofnonhumananimalsBirdsongforexampleisrelentlesslyrepetitivebuteachrepeatedthemedoesnotembellishorqualifythepreviousoneAtmosttherepetitionmightsignalurgencyorsimplysignalcontinuingpresenceasonemightrepeatedlyknockonadoorinthehopeofarousingsomeoneinsideRepetitionisubiquitousinhumanandanimallifeinactivitiesrangingfromtherepeatedjawmovementsineatingtothecuriouslyrepetitivena-tureofsexualactivityThespidernolessiscapableofrepetitionasinWaltWhitmanrsquosLeaves of Grass15
ANOISELESSpatientspiderImarkrsquodwhereonalittlepromontoryitstoodisolatedMarkrsquodhowtoexplorethevacantvastsurroundingItlaunchrsquodforthfilamentfilamentfilamentoutofitselfEverunreelingthemmdashevertirelesslyspeedingthem
Informationcanalsobeaggregatedinnonrecursivefashionaswhentheshort-storywriterSaki(HHMunro)wroteldquoHungerfatigueanddespairinghopelessnesshadnumbedhisbrainrdquo16Ag-gregationofdifferentphrases similarlycompoundsmeaningad-ditivelyaswhenthehistorianPeterHennessywrote
ThemodelofamodernPrimeMinisterwouldbeakindofgrotesquecompositefreakmdashsomeonewiththededicationtodutyofaPeelthephysicalenergyofaGladstonethedetachmentofaSalisburytheballsofaLloydGeorgetheword-powerofaChurchilltheadministrativegiftsof anAttleethe styleof aMacmillanthemanagerialismof aHeathandthesleeprequirementsofaThatcher17
ThesentenceitselfhasrecursiveelementsbuttheaggregationofphrasestodescribethefreakishcompositeisnotrecursiveinthateachdoesnotcallthenextInsteadtheyareeffectivelyelements
What Is RecuRsIon 11
inalistinsertedtoaddinformationNonhumanspeciesmaywellhaveasimilarabilitytoaccumulateinformationaswhenunder-standingapredatoraslargefierceandwithsharpteethandclaws
Iteration
AslightlymoresubtlevariantonrepetitionandaggregationisitshyerationwhereaprocessisrepeatedbutinthiscasethereisinputfromthepreviousapplicationoftheprocessInthisrespect it islikerecursionandindeedconsideredbymathematicianstobelongtotheclassofldquogeneralrecursivefunctionsrdquoForthemainpurposesofthisbookthoughitdoesnotqualifyastruerecursionbecauseeachoutputisdiscardedonceithasbeenenteredintothenextap-plicationThedictionarydefinitionofrecursionthatIgaveearlierinthischapterwasalsoreallyanexampleofiterationratherthanrecursionbecauseyoujustkeepgoingroundandroundtheloopwithoutanyaddedstructureTheiterationsthereforedonotleadtoaddedcomplexity18
Iterativeproceduresareusedincomputationalmathematicstoarriveat increasinglyaccuratesolutionstoaproblemThebasicideaistostartwithapreliminarysolutionmdashperhapsaguessmdashandthenuseaproceduretocomputeanewsolutionThissolutionisthenusedasthestartingpointforthenextcomputationandthenew solution is then the starting point for the next roundThecycle is repeateduntil the solutions stabilize to someacceptablecriterion19Feedbacksystemsoperateinmuchthesamewaytypi-callyasameansofmaintaininghomeostasisForexampleather-mostatmayinvolveasystemforraisingorloweringtemperatureandthegoalistoachievesomegiventemperatureTheactualtem-peratureisfedintothesystemwhichoperatestoraiseorlowerthe temperatureuntil thedesired is reachedThebody isawashwith feedback systems to maintain homeostasis of temperatureironenergybloodcompositionandsoonThemainregulatoristhehypothalamusinthelimbicsystemofthebrainSuchsystemsagaindonotdifferentiatehumansfromotheranimals
Sometimesthedistinctionbetweenrecursionanditerationmaybeamatterof interpretationIn the infinite loopcreatedby the
1 chapteR 1
parodyofPaul CliffordonemightsaythateachbeginningofthestoryisinitiatedbythepreviousonewhichisthenforgottenTheparodyisbestappreciatedthoughifthestoryisseenasanendlessever deepening whirlpoolwith each segment remaining as partofitIrsquomtoldthestoryworksbestifeachsegmentisspokenwithadifferentaccent
Considertoothislinefromawell-knownchildrenrsquosverse
ThisisthedogthatworriedthecatthatkilledtheratthatatethemaltthatlayinthehousethatJackbuilt
TounderstandthissentenceastrulyrecursiveonemustappreciatethatitdescribesastateofaffairsasacomplexwholeandreferstoparticularcasesofadogcatratmalthouseandfellowcalledJackItisnotsimplythestringingtogetherofadogthatworriedacatacatthatkilledaratandsoforthAyoungchildthoughmightprocessitinthispiecemealwayasasuccessionofunrelatedevents
Recursion and evolutionary psychology
Inemphasizingrecursionasaunifyingconcepttheapproachtakeninthisbookcontrastswiththatadoptedbyso-calledevolutionarypsychologistswhohavearguedthatthemindhasmultiplefacetsThebasictenetsofevolutionarypsychologywerelaidoutinthe1992volumeThe Adapted MindeditedbyJeromeBarkowLedaCosmidesandJohnToobyandpopularizedbyStevenPinkerinhis influential 1997 book How the Mind Works20 Thus Pinkerwritesthatthehumanmindldquoisnotasingleorganbutasystemoforganswhichwecanthinkofaspsychologicalfacultiesormentalmodulesrdquo21 Inexaminingpresent-dayhumanbehaviortheevolu-tionarypsychologistrsquosagendaistodiscoverindependentprocessesasthebasicmodulesandrelatethemtoconditionsthatprevailedin the Pleistocene when humans existed primarily as hunter-gatherersAs Pinker puts it the aim is to carve the mind at itsjointssotospeakandldquoreverse-engineerrdquoitscomponentsormod-ulesbacktotheepochduringwhichthehumanmindwasformed
What Is RecuRsIon 1
In this view the mind is really a collection of miniminds eachbeaveringawayonitsownspecificproblemamongwhicharelan-guageandtheoryofmindThishasbeencalledtheSwiss-army-knifemodelofthemindwithabladeforeverypurpose22
Thedangerwith thisapproach is that itbecomes tooeasy topostulate modules and to tellldquojust sordquostories about how theyevolvedsothatthereisariskofreturningtothenow-abandonedinstinctpsychologyoftheearlytwentiethcentury23 Instinctpsy-chologyperishedunderthesheerweightofnumbersmdashtheauthorof one text counted 1594 instincts that had been attributed toanimals and humans24mdashand evolutionary psychology may alsodrowninaseaofmodulesifnotofmixedmetaphorsPinkersug-geststhatwelikepotatochipsbecausefattyfoodswerenutrition-allyvaluableduringthePleistocenebutscarceenoughthattherewasnodangerofobesitywelike landscapeswithtreesbecausetreesprovidesshadeandescapefromdangerouscarnivoresontheAfrica savanna flowers please us because they are markers foredible fruits nutsor tubers amid the greenery of the savannaandsoonldquoTherearemodulesrdquohewritesldquoforobjectsandforcesforanimatebeingsformindsandfornaturalkindslikeanimalsplantsandmineralsrdquo25
This isnot to say that theSwiss-army-knifemodel iswithoutmeritSomeofthepostulatedmodulesdoprovideinsightintothehumanconditionandarereasonablywellfoundedForexamplefoundationalworkinevolutionarypsychologybyLedaCosmidesprovidedgoodevidenceforaldquocheater-detectionmodulerdquomdashanin-stinctiveability todetect thosewhoflout social conventions fortheirowngain26 Arecentstudysuggests thathumanspossessaldquocategory-specificattentionsystemrdquothatisespeciallyadaptedforattendingtoanimals27 oneoftheauthorsJohnToobyisquotedas sayingldquoEvendullanimals likepigeons recruita surpris-ingamountofattentionmdashasdoturtlesresemblingrocksrdquo28 Thisbookisnotintendedtodenythattherearemanyspecificdisposi-tionsthatshapeourmentalandsociallivesrathermyaimistosuggest that therearedeeperaspectsofhumanthoughtthataregovernedbysimilarprinciplesandthatrecursionisoneofthoseprinciplesmdashandperhapsthemostimportantone
1 chapteR 1
Tobefairtoonotallevolutionarypsychologistshaveinsistedthatmodulesarecompletelyencapsulatedshutofffromanycom-munication with one anotherEven Steven Pinker for examplewrites ldquo[Modules] accomplish specialized functions thanks totheirspecializedstructuresbutdonrsquotnecessarilycomeinencap-sulated packagesrdquo29 Steven Mithen although scarcely a card-carryingevolutionarypsychologistarguedthatthehumanmindevolved itsdistinctive characterpreciselybecausepreviously en-capsulatedmodulesbegantoldquoleakrdquocreatingwhathecallsldquocogni-tivefluidityrdquo30 ItisasthoughthemodulesstoppedmindingtheirownbusinessesandbegantogossipMyapproachinthisbookisnotentirelyatoddswiththisviewinthatIargueacommonprin-ciplemightunderlieanumberofourdistinctiveabilities
Others are also beginning to question the Swiss-army-knifemodelofthehumanmindmorestarklyDavidPremackforex-ampleadoptsanapproachsimilartothatofferedinthisbookRe-viewingtheevidencefordiscontinuitybetweenhumansandotheranimalshewritesldquoAnimalcompetenciesaremainlyadaptationsrestrictedtoasinglegoalHumancompetenciesaredomain-generalandservenumerousgoalsrdquo31Thisineffectreversestheevolution-arypsychologyargumentmdashthemindhasbecomelessratherthanmoremodularThetidemaywellbeturning
In any event it is unlikely that recursion canbe considered amoduleAsweshallseerecursionseemstobeanorganizingprin-cipleinverydifferentspheresofhumanmentalactivityfromlan-guage tomemory tomind readingRecursive thinkingprobablydependsonothermentalattributesOneoftheseiswhathasbeentermedworkingmemorywhichholdsinformationinconscious-nessInordertoembedprocesseswithinprocessesitisnecessarytorememberwhereonehadgottointheearlierprocesswhenanembeddedprocesshasbeencompletedForinstanceinasentencelike My dog who eats bananas often gets sickone must holdthe early part of the sentence (My dog) and link it to the nextpart(often gets sick)DwightWReadhasarguedthatnonhumanprimatesevenourclosestrelativesthechimpanzeeshaveawork-ingmemorythatistoolimitedtoallowthiskindofembedding32
Recursionprobablyalsodependsonanexecutiveprocessthator-
What Is RecuRsIon 1
ganizeswhatistobeembeddedinwhatandthismaydependonthefrontallobesofthebrainTheabilitytoorganizeandcarryoutrecursiveoperationsmaythereforedependonseveralprocesses
AlthoughIdonotembracethemodularviewassumedbyevo-lutionarypsychologistsIamatonewiththeminproposingthata distinctively human mind evolved during the Pleistocene theepochthatstretchedfromaround26millionyearsagotosome12000 years agoHow that happened will be told in the laterchaptersofthisbook
plan of the Book
ThebookisdividedintofourpartsPart1dealswithlanguageAlthoughrecursionisnotlimitedto
languageitismostcommonlyinvokedtoexplainwhyhumanlan-guagediffersfromotherformsofanimalcommunicationanin-sightlargelyattributabletoNoamChomskyChapter2discussesthe nature of languagewithparticular emphasis on the role ofrecursionChapter3thenraisestheage-oldquestionofwhetherotheranimalshaveanythingresemblinghumanlanguageChapter4developstheideathatlanguageevolvedfrommanualgesturesmdashanideathatsuggestsgreaterevolutionarycontinuitybetweenhu-mansandotherprimatesthanthemorecommonassumptionthatlanguageemergedfromvocalcalls
Part2dealswithmentaltimetraveltheabilitytobringtomindeventsremovedfromthepresentinbothtimeandplaceChapter5startswithmemoryanddevelopstheideathatmemoryforspecificepisodesisuniquetohumansChapter6extendsthenotionofepi-sodicmemorytotheimaginingofpossiblefutureeventsleadingtotheconceptoftheselfasexistingthroughtimeThisleadstothenotiondiscussedinchapter7thatlanguageitselfevolvedtoenablepeopletosharetheirmemoriesandplansandsotocom-municateabouteventsthatarenotpresentintheimmediateenvi-ronmentThisleadsalsotofictionmdashthetellingofstoriesthatneednotbebasedonfactbutthatnonethelesshonethecapacitytodealwiththeepisodicexigenciesofhumansociallife
1 chapteR 1
Part3dealsanotherrecursiveaspectofhumanthoughtnamelytheoryofmindmdashortheabilitytounderstandwhatothersarethink-ingorfeelingChapter8introducesmind-readingnotasapsychicphenomenonbutasanaturalabilitytoinferthementalperspec-tivesofotherpeopleThisabilityisagaincriticaltosocialcohesionandcooperationChapter9explainshowtheoryofmindwasalsocriticaltotheemergenceoflanguage
Part4delvesmorespecificallyintothequestionofhowthere-cursivemindevolvedChapter10setsthisquestioninthecontextoftheclassicdebatebetweenCartesiandiscontinuityandDarwin-iancontinuityChapter11examinessomeofthestepsbywhichthe hominins33 after splitting from the line leading to modernchimpanzees and bonobos began to assume human-like attri-butesChapter12thenconsidersthefinalsteptoldquomodernrdquoHomo sapiensthesolesurvivinghomininspeciesmdashdominantmanipula-tiveMachiavellianandcapableofponderingourownnatureandstatusontheplanetThatperhapsistheultimatetriumphoftherecursivemind
Chapter13presentsthefinalsummaryandconclusions
What Is RecuRsIon
wouldmostliketoreadonopeningthemorningpaperThewin-ningentryreadasfollows
OurSecondCompetition
TheFirstPrize inthesecondof thisyearrsquoscompetitionsgoes toMrArthurRobinsonwhosewittyentrywaseasilythebestofthosewereceivedHischoiceofwhathewouldliketoreadwhenopeningthepaperwasheadedldquoOurSecondCompetitionrdquoandwasas followsldquoTheFirstprizeinthesecondofthisyearrsquoscompetitionsgoestoMrArthurRobinsonwhosewittyentrywaseasilythebestofthosewereceivedHischoiceofwhathewouldliketoreadwhenopeningthepaperwasheadedlsquoOurSecondCompetitionrsquobutowingtopaperre-strictionswecannotprintallofitrdquo2
Taking a different tack John Barthrsquos story Autobiography A Selfshyrecorded Fiction is a recursive tale inwhich thenarrator isostensiblythestoryitselfwritingaboutitself3Itendsrecursivelyinitsownend
NonsenseIrsquollmutter to the endonewordafter anotherstring therascalsoutmadornotheardornotmylastwordswillbemylastwords
Tomyknowledgenostoryhasyetattemptedtowriteastoryofastorythatwritesaboutitself
AndthenthereistherecurringproblemoffleasaspennedbytheVictorianmathematicianAugustusdeMorgan
GreatfleashavelittlefleasupontheirbackstobitersquoemAndlittlefleashavesmallerfleasandsoad infinitumAndthegreatfleasthemselvesinturnhavegreaterfleastogoonWhiletheseagainhavegreaterstillandgreaterstillandsoon4
Thisnotionof insertingprogressivelysmallerentities into largeronesad infinitum canalsogiverisetointerestingvisualeffectsasintheexamplesshowninfigure2
chapteR 1
Apollonian gasket Sierpinski triangle
Figure2Twofiguresshowingrecursiveinsertionsofcircles(left)andtriangles(right)TheApolloniangasketderivesfromApolloniusaGreekgeometerfromthethirdcenturyBCwhostudiedtheproblemofhowtodrawacirclethatistangentialtothreecirclesStartingwiththreecirclesthataretangentialtooneanotheronecancontinuetheprocessofconstructingcirclestangentialtoalltripletsad infinitumTheresultingfigureservesasamathematicalmodelforfoam(seeMackenzie2009formoreinformation)(2DApolloniangasketwithfourinitialcirclescourtesyofGuillaumeJacquenot)
TheuseofrecursiontocreateinfinitesequencesisalsoexploitedbymathematicsOnesuchsequenceisthesetofnatural(iewhole)numberswhich IrsquollwriteasNThuswecangenerateallof thepositivenaturalnumbersbythedefinitions
1isinNIfnisinNthen(n+1)isinN
ThisseconddefinitionisrecursivebecauseNappearsinthecondi-tionthatneedstobesatisfiedforN
YoumayrememberpossiblyfromschooldayswhatfactorialsareAsaschoolboy I foundthemamusing inachildishkindofwaybecausetheyweresignaledwithexclamationmarksthusfac-torial3usuallywritten3is321andequals65Similarlywecancomputethefollowing
5=54321=1208=87654321=40320
What Is RecuRsIon
Clearlythiscangoonforeverbutwecancapturetheentiresetbyusingjusttwodefiningequations
0=1n=n(nminus1) [wherengt0]
ThissecondequationisrecursiveinthatafactorialisdefinedintermsofafactorialWeneedthefirstequationtokickthethingoff
ThenextexampleisforrabbitsandiscalledtheFibonaccise-riesdefinedbythefollowingthreeequations
fibonacci(0)=1fibonacci(1)=1fibonacci(n)=fibonacci(n minus1)+fibonacci(n minus2)[wherengt1]
Ifyouarefollowingmeyoushouldbeabletocomputetheserieswhichgoes11235813WhatthedefinitionsaysthenisthateachnumberintheseriesisthesumofthetwopreviousonesWhyrabbitsFibonacci(c1170ndash1250)wasanItalianmathemati-cianwhousedtheseriestopredictthegrowthofahypotheticalpopulationofrabbits6
ForafinalinformalexampleItakeyoutoKyotoJapanwhereIoncehappeneduponasignonagatethatwaswritteninKanjiscriptIaskedwhatitmeantandmyguidetoldmeIhopecorrectlythatitmeantPost no billsThereisaparadoxhereinthatthesignwas itselfabilltherebycontravening itsownpresencePerhapsthereneededtobeanothersignthatsaidPost no ldquoPost no billsrdquo billsButofcoursethisisitselfinviolationofitsownmessagesowemightenvisageanothersignthatreadsPost no lsquoPost ldquoPost no billsrdquo billsrsquo billsThereisnoendtothisprocesssoitmighthavebeenmoresensible tohaveallowedbillson thegate in thefirstplaceInpracticethoughlimitationsoftimespaceormemorywillpreventarecursivesequenceofstructurecontinuingforever
toward a Working Definition
Oneof thecharacteristicsofrecursionthenis that itcantakesits own output as the next input a loop that can be extended
chapteR 1
indefinitelytocreatesequencesorstructuresofunboundedlengthorcomplexityInpracticeofcoursewedonotgetcaughtupininfiniteloopsmdashlifeissimplytooshortforthatForthepurposesofthisbookthenweshallnotbeinterestedsomuchinthegenera-tionofinfinitesequencesasinadefinitionthatmightapplyuse-fullytohumanthoughtAdefinitionthatmeetsthisrequirementissuggestedbyStevenPinkerandRayJackendoffwhodefinere-cursionasldquoaprocedurethatcallsitselforaconstituentthatcontainsaconstituentofthesamekindrdquo7
Thesecondpartofthisdefinitionisimportantespeciallyinlan-guagebecauseitallowsthatrecursiveconstructionsneednotin-volvetheembeddingofthesame constituentsasintheexampleof the gate inKyotobutmay contain constituents of the samekindmdasha process sometimes known asldquoself-similar embeddingrdquoForexamplenounphrasescanbebuiltfromnounphrasesinre-cursivefashionTecumsehFitchgivestheexampleofsimplenounphrasessuchasthe dogthe catthe treethe lakeandonecanthencreatenewnounphrasesbyplacingthewordbeside betweenanypairthe dog beside the treethe cat beside the lake8OronemighthavetwosentencesJane loves John andJane flies airplanesandembedone in theother (withappropriatemodification)asJane who flies airplanes loves JohnThesecanbeextendedrecursivelytowhateverlevelofcomplexityisdesiredForexamplewecouldextendthenounphrasetothe dog beside the tree beside the lakeor thesentenceaboutJaneandJohntoJane who flies airplanes that exceed the sound barrier loves John who is prone to selfshydoubtMost languagesmakeuseof recursiveoperationsof thissortmdashalthoughweshallseeinthenextchapterthattheremaybeafewlanguagesthatdonrsquotoperateinthisway
AlthoughitiscommontoprovideillustrationsfromlanguagethemainthemeofthisbookisthatitisinthoughtratherthaninlanguagethatrecursionoriginatesAsPinkerandJackendoffputitldquoTheonlyreasonlanguageneedstoberecursiveisbecauseitsfunction is to express recursive thoughts If there were not anyrecursivethoughtsthemeansofexpressionwouldnotneedrecur-sioneitherrdquo9 Inrememberingepisodesfromthepastforinstanceweessentiallyinsertsequencesofpastconsciousnessintopresent
What Is RecuRsIon
consciousnessor inour interactionswithotherpeoplewemayinsertwhattheyarethinkingintoourownthinkingThesethemesareexploredinlaterchapters
Process and Structure
AssuggestedbyPinkerandJackendoffrsquosdualdefinitionrecursioncanbeunderstoodeitherasaprocess orasastructureThedistinc-tioncanbeimportantArecursiveprocessmayleadtoastructurethatneednotbeseenasitselfrecursiveForexamplesupposeweconstructasequenceofmusicalnoteswithanembeddingroutinebypairingpairsofnoteseachconsistingofa randomlychosennoteplayedonapianowitharandomlychosennoteplayedonaviolinThefirstpairisembeddedinanotherpairandthefour-noteoutputthenembeddedinanotherpairThisprocesscanbecontin-uedindefinitelytocreateasequenceofnotesAsillustratedinfig-ure3thoughthesequencecanbeinterpretednotasarecursivelyembeddedstructurebutasasequenceofpianonotesfollowedbyanequallylongsequenceofviolinnotesThisfailuretodistinguishrecursiveembeddingfromrecursivestructurehasledtosomecon-fusionespeciallyinclaimsaboutrecursioninnonhumanspecies10
Again in his most recent theory on the nature of languageknownastheMinimalistProgram11 NoamChomskyhasarguedthathumanthoughtisgeneratedbyaMergeoperationappliedre-cursivelyThatisunitsaremergedtoformlargerentitiesandthemergedentitiescanbethemselvesmergedtoformstilllargerenti-tiesandsoonThisoperationunderliestheembeddedstructureofhumanlanguagealthoughinChomskyrsquostheoryitappliesstrictlytowhathecallsIshylanguagewhichisthethoughtprocesspreced-ingEshylanguagetheexternal language that isactually spokenorsignedMergecanproducestringsofelementsbetheywordsorelementsofthoughtandalthoughitmaybeappliedrecursivelytoproducehierarchicalstructurethatstructuremaynotbeevidentinthefinaloutputForinstanceevensentencesmightberegardedsimplyaswordsallmergedinunstructuredsequenceasinritual-izedsongsorprayersEverydaylanguagetoomayincludemen-tally undifferentiated clicheacutes and slogansor sequences that are
chapteR 1
P P P P V V V V P P P P V V V V
Figure3ThesequenceofPsandVscanbecreatedeitherbyrecursivelynest-ingPVpairsinPVpairs(left)orbyarrangingasequenceofPsfollowedbyasequenceofanequalnumberofVs(right)Thesequencemightbegeneratedasintheleftpanelbutinterpretedasintherightpanel
highlyautomatedPoliticiansmaybeespeciallypronetothiskindoftalk
Asnotedaboverecursiveprocessesandstructurescaninprin-cipleextendwithoutlimitbutarelimitedinpracticeNeverthelessrecursiondoes give rise to theconcept of infinity itself perhapslimited to the human imaginationAfter all only humans haveacquired theability to count indefinitelyand tounderstand thenatureofinfiniteserieswhereasotherspeciescanatbestmerelyestimatequantityandareaccurateonlyup to somesmallfinitenumber12EveninlanguageweunderstandthatasentencecaninprinciplebeextendedindefinitelyeventhoughinpracticeitcannotbemdashalthoughthenovelistHenryJameshadadamngoodtrySuchunderstandingsareindeedpartofhumanmentalachievementanddependonahumancapacityforrecursivethoughtNeverthelesstheyarenottheprimaryconcernsofthisbook
The appealing aspect of recursion is precisely that it can in principle extend indefinitely to create thoughts (and sentences)ofwhatevercomplexityisrequiredTheideahasanelegantsim-plicitygivingrisetowhatChomskycalledldquodiscreteinfinityrdquo13orWilhelmHumboldt(1767ndash1835)famouslycalledldquotheinfiniteuseoffinitemeansrdquoAndalthoughrecursionislimitedinpracticewecanneverthelessachieveconsiderabledepthsofrecursivethoughtarguablyunsurpassedinanyotherspeciesInchessforexampleaplayermaybeabletothinkrecursivelythreeorfourstepsahead
What Is RecuRsIon
examiningpossiblemovesandcountermovesbutthenumberofpossibilitiessoonmultipliesbeyondthecapacityofthemindtoholdthem
Deeperlevelsofrecursionmaybepossiblewiththeaidofwrit-ingorsimplywithextendedtimeforrehearsalandcontemplationorextendedmemorycapacitythroughartificialmeansTheslowdevelopmentofacomplexmathematicalproofforexamplemayrequiresubtheoremswithinsubtheoremsPlaysornovelsmayin-volverecursiveloopsthatbuildslowlymdashinShakespearersquosTwelfth NightforexampleMariaforeseesthatSirTobywilleagerlyan-ticipate thatOliviawill judgeMalvolio absurdly impertinent tosuppose that she wishes him to regard himself as her preferredsuitor14Asinfictionsoinlifeweallliveinawebofcomplexre-cursiverelationshipsandplanningadinnerpartymayneedcare-fulattentiontowhothinkswhatofwhom
ThestructuresresultingfromrecursiveprocessesneednotrevealthenatureofthoseprocessesjustasaloafofbreadmaynotrevealtheprocessesofkneadingthatwentintothemakingofthebreadorthetasteofwinethepickingandtramplingofthegrapesOftenthoughthestructureofasentenceorstreamofthoughtmayrevealrecursiveembeddingmdashinterpretationofasentencemayrequiretheunderstandingofphrasesembeddedinphrasesregardlessofhowtheembeddingwasactuallyaccomplishedandaninternalunder-standingofastreamofthoughtmayrequirethesegmentationofepisodeswithinepisodes
What Recursion Is not
Recursionisnottheonlydeviceforcreatingsequencesorstruc-tures of potentially infinite length or size I now consider someexamplesthatdonotmeetthecriteriaforrecursion
Repetition
Simple repetition can lead to sequences of potentially infinitelengthbutdoesnot classifyas true recursionForexamplethesentence thatopenschapter9ofAAMilnersquosWinnie the Pooh
10 chapteR 1
goesIt rained and it rained and it rained Thiscouldgoonfor-evermdashoratleastuntilPigletisdrownedmdashbuttherepetitionsimplyconveys the information that it rained rathera lotcausingPig-letsomeennuiItisnotrecursivebecauseeachadditionofand it rained isnotdrivenbythepreviousoneitissimplyaddedatthediscretionofthewriter
InanyeventrepetitiondoesnotdistinguishhumanactivityfromthatofnonhumananimalsBirdsongforexampleisrelentlesslyrepetitivebuteachrepeatedthemedoesnotembellishorqualifythepreviousoneAtmosttherepetitionmightsignalurgencyorsimplysignalcontinuingpresenceasonemightrepeatedlyknockonadoorinthehopeofarousingsomeoneinsideRepetitionisubiquitousinhumanandanimallifeinactivitiesrangingfromtherepeatedjawmovementsineatingtothecuriouslyrepetitivena-tureofsexualactivityThespidernolessiscapableofrepetitionasinWaltWhitmanrsquosLeaves of Grass15
ANOISELESSpatientspiderImarkrsquodwhereonalittlepromontoryitstoodisolatedMarkrsquodhowtoexplorethevacantvastsurroundingItlaunchrsquodforthfilamentfilamentfilamentoutofitselfEverunreelingthemmdashevertirelesslyspeedingthem
Informationcanalsobeaggregatedinnonrecursivefashionaswhentheshort-storywriterSaki(HHMunro)wroteldquoHungerfatigueanddespairinghopelessnesshadnumbedhisbrainrdquo16Ag-gregationofdifferentphrases similarlycompoundsmeaningad-ditivelyaswhenthehistorianPeterHennessywrote
ThemodelofamodernPrimeMinisterwouldbeakindofgrotesquecompositefreakmdashsomeonewiththededicationtodutyofaPeelthephysicalenergyofaGladstonethedetachmentofaSalisburytheballsofaLloydGeorgetheword-powerofaChurchilltheadministrativegiftsof anAttleethe styleof aMacmillanthemanagerialismof aHeathandthesleeprequirementsofaThatcher17
ThesentenceitselfhasrecursiveelementsbuttheaggregationofphrasestodescribethefreakishcompositeisnotrecursiveinthateachdoesnotcallthenextInsteadtheyareeffectivelyelements
What Is RecuRsIon 11
inalistinsertedtoaddinformationNonhumanspeciesmaywellhaveasimilarabilitytoaccumulateinformationaswhenunder-standingapredatoraslargefierceandwithsharpteethandclaws
Iteration
AslightlymoresubtlevariantonrepetitionandaggregationisitshyerationwhereaprocessisrepeatedbutinthiscasethereisinputfromthepreviousapplicationoftheprocessInthisrespect it islikerecursionandindeedconsideredbymathematicianstobelongtotheclassofldquogeneralrecursivefunctionsrdquoForthemainpurposesofthisbookthoughitdoesnotqualifyastruerecursionbecauseeachoutputisdiscardedonceithasbeenenteredintothenextap-plicationThedictionarydefinitionofrecursionthatIgaveearlierinthischapterwasalsoreallyanexampleofiterationratherthanrecursionbecauseyoujustkeepgoingroundandroundtheloopwithoutanyaddedstructureTheiterationsthereforedonotleadtoaddedcomplexity18
Iterativeproceduresareusedincomputationalmathematicstoarriveat increasinglyaccuratesolutionstoaproblemThebasicideaistostartwithapreliminarysolutionmdashperhapsaguessmdashandthenuseaproceduretocomputeanewsolutionThissolutionisthenusedasthestartingpointforthenextcomputationandthenew solution is then the starting point for the next roundThecycle is repeateduntil the solutions stabilize to someacceptablecriterion19Feedbacksystemsoperateinmuchthesamewaytypi-callyasameansofmaintaininghomeostasisForexampleather-mostatmayinvolveasystemforraisingorloweringtemperatureandthegoalistoachievesomegiventemperatureTheactualtem-peratureisfedintothesystemwhichoperatestoraiseorlowerthe temperatureuntil thedesired is reachedThebody isawashwith feedback systems to maintain homeostasis of temperatureironenergybloodcompositionandsoonThemainregulatoristhehypothalamusinthelimbicsystemofthebrainSuchsystemsagaindonotdifferentiatehumansfromotheranimals
Sometimesthedistinctionbetweenrecursionanditerationmaybeamatterof interpretationIn the infinite loopcreatedby the
1 chapteR 1
parodyofPaul CliffordonemightsaythateachbeginningofthestoryisinitiatedbythepreviousonewhichisthenforgottenTheparodyisbestappreciatedthoughifthestoryisseenasanendlessever deepening whirlpoolwith each segment remaining as partofitIrsquomtoldthestoryworksbestifeachsegmentisspokenwithadifferentaccent
Considertoothislinefromawell-knownchildrenrsquosverse
ThisisthedogthatworriedthecatthatkilledtheratthatatethemaltthatlayinthehousethatJackbuilt
TounderstandthissentenceastrulyrecursiveonemustappreciatethatitdescribesastateofaffairsasacomplexwholeandreferstoparticularcasesofadogcatratmalthouseandfellowcalledJackItisnotsimplythestringingtogetherofadogthatworriedacatacatthatkilledaratandsoforthAyoungchildthoughmightprocessitinthispiecemealwayasasuccessionofunrelatedevents
Recursion and evolutionary psychology
Inemphasizingrecursionasaunifyingconcepttheapproachtakeninthisbookcontrastswiththatadoptedbyso-calledevolutionarypsychologistswhohavearguedthatthemindhasmultiplefacetsThebasictenetsofevolutionarypsychologywerelaidoutinthe1992volumeThe Adapted MindeditedbyJeromeBarkowLedaCosmidesandJohnToobyandpopularizedbyStevenPinkerinhis influential 1997 book How the Mind Works20 Thus Pinkerwritesthatthehumanmindldquoisnotasingleorganbutasystemoforganswhichwecanthinkofaspsychologicalfacultiesormentalmodulesrdquo21 Inexaminingpresent-dayhumanbehaviortheevolu-tionarypsychologistrsquosagendaistodiscoverindependentprocessesasthebasicmodulesandrelatethemtoconditionsthatprevailedin the Pleistocene when humans existed primarily as hunter-gatherersAs Pinker puts it the aim is to carve the mind at itsjointssotospeakandldquoreverse-engineerrdquoitscomponentsormod-ulesbacktotheepochduringwhichthehumanmindwasformed
What Is RecuRsIon 1
In this view the mind is really a collection of miniminds eachbeaveringawayonitsownspecificproblemamongwhicharelan-guageandtheoryofmindThishasbeencalledtheSwiss-army-knifemodelofthemindwithabladeforeverypurpose22
Thedangerwith thisapproach is that itbecomes tooeasy topostulate modules and to tellldquojust sordquostories about how theyevolvedsothatthereisariskofreturningtothenow-abandonedinstinctpsychologyoftheearlytwentiethcentury23 Instinctpsy-chologyperishedunderthesheerweightofnumbersmdashtheauthorof one text counted 1594 instincts that had been attributed toanimals and humans24mdashand evolutionary psychology may alsodrowninaseaofmodulesifnotofmixedmetaphorsPinkersug-geststhatwelikepotatochipsbecausefattyfoodswerenutrition-allyvaluableduringthePleistocenebutscarceenoughthattherewasnodangerofobesitywelike landscapeswithtreesbecausetreesprovidesshadeandescapefromdangerouscarnivoresontheAfrica savanna flowers please us because they are markers foredible fruits nutsor tubers amid the greenery of the savannaandsoonldquoTherearemodulesrdquohewritesldquoforobjectsandforcesforanimatebeingsformindsandfornaturalkindslikeanimalsplantsandmineralsrdquo25
This isnot to say that theSwiss-army-knifemodel iswithoutmeritSomeofthepostulatedmodulesdoprovideinsightintothehumanconditionandarereasonablywellfoundedForexamplefoundationalworkinevolutionarypsychologybyLedaCosmidesprovidedgoodevidenceforaldquocheater-detectionmodulerdquomdashanin-stinctiveability todetect thosewhoflout social conventions fortheirowngain26 Arecentstudysuggests thathumanspossessaldquocategory-specificattentionsystemrdquothatisespeciallyadaptedforattendingtoanimals27 oneoftheauthorsJohnToobyisquotedas sayingldquoEvendullanimals likepigeons recruita surpris-ingamountofattentionmdashasdoturtlesresemblingrocksrdquo28 Thisbookisnotintendedtodenythattherearemanyspecificdisposi-tionsthatshapeourmentalandsociallivesrathermyaimistosuggest that therearedeeperaspectsofhumanthoughtthataregovernedbysimilarprinciplesandthatrecursionisoneofthoseprinciplesmdashandperhapsthemostimportantone
1 chapteR 1
Tobefairtoonotallevolutionarypsychologistshaveinsistedthatmodulesarecompletelyencapsulatedshutofffromanycom-munication with one anotherEven Steven Pinker for examplewrites ldquo[Modules] accomplish specialized functions thanks totheirspecializedstructuresbutdonrsquotnecessarilycomeinencap-sulated packagesrdquo29 Steven Mithen although scarcely a card-carryingevolutionarypsychologistarguedthatthehumanmindevolved itsdistinctive characterpreciselybecausepreviously en-capsulatedmodulesbegantoldquoleakrdquocreatingwhathecallsldquocogni-tivefluidityrdquo30 ItisasthoughthemodulesstoppedmindingtheirownbusinessesandbegantogossipMyapproachinthisbookisnotentirelyatoddswiththisviewinthatIargueacommonprin-ciplemightunderlieanumberofourdistinctiveabilities
Others are also beginning to question the Swiss-army-knifemodelofthehumanmindmorestarklyDavidPremackforex-ampleadoptsanapproachsimilartothatofferedinthisbookRe-viewingtheevidencefordiscontinuitybetweenhumansandotheranimalshewritesldquoAnimalcompetenciesaremainlyadaptationsrestrictedtoasinglegoalHumancompetenciesaredomain-generalandservenumerousgoalsrdquo31Thisineffectreversestheevolution-arypsychologyargumentmdashthemindhasbecomelessratherthanmoremodularThetidemaywellbeturning
In any event it is unlikely that recursion canbe considered amoduleAsweshallseerecursionseemstobeanorganizingprin-cipleinverydifferentspheresofhumanmentalactivityfromlan-guage tomemory tomind readingRecursive thinkingprobablydependsonothermentalattributesOneoftheseiswhathasbeentermedworkingmemorywhichholdsinformationinconscious-nessInordertoembedprocesseswithinprocessesitisnecessarytorememberwhereonehadgottointheearlierprocesswhenanembeddedprocesshasbeencompletedForinstanceinasentencelike My dog who eats bananas often gets sickone must holdthe early part of the sentence (My dog) and link it to the nextpart(often gets sick)DwightWReadhasarguedthatnonhumanprimatesevenourclosestrelativesthechimpanzeeshaveawork-ingmemorythatistoolimitedtoallowthiskindofembedding32
Recursionprobablyalsodependsonanexecutiveprocessthator-
What Is RecuRsIon 1
ganizeswhatistobeembeddedinwhatandthismaydependonthefrontallobesofthebrainTheabilitytoorganizeandcarryoutrecursiveoperationsmaythereforedependonseveralprocesses
AlthoughIdonotembracethemodularviewassumedbyevo-lutionarypsychologistsIamatonewiththeminproposingthata distinctively human mind evolved during the Pleistocene theepochthatstretchedfromaround26millionyearsagotosome12000 years agoHow that happened will be told in the laterchaptersofthisbook
plan of the Book
ThebookisdividedintofourpartsPart1dealswithlanguageAlthoughrecursionisnotlimitedto
languageitismostcommonlyinvokedtoexplainwhyhumanlan-guagediffersfromotherformsofanimalcommunicationanin-sightlargelyattributabletoNoamChomskyChapter2discussesthe nature of languagewithparticular emphasis on the role ofrecursionChapter3thenraisestheage-oldquestionofwhetherotheranimalshaveanythingresemblinghumanlanguageChapter4developstheideathatlanguageevolvedfrommanualgesturesmdashanideathatsuggestsgreaterevolutionarycontinuitybetweenhu-mansandotherprimatesthanthemorecommonassumptionthatlanguageemergedfromvocalcalls
Part2dealswithmentaltimetraveltheabilitytobringtomindeventsremovedfromthepresentinbothtimeandplaceChapter5startswithmemoryanddevelopstheideathatmemoryforspecificepisodesisuniquetohumansChapter6extendsthenotionofepi-sodicmemorytotheimaginingofpossiblefutureeventsleadingtotheconceptoftheselfasexistingthroughtimeThisleadstothenotiondiscussedinchapter7thatlanguageitselfevolvedtoenablepeopletosharetheirmemoriesandplansandsotocom-municateabouteventsthatarenotpresentintheimmediateenvi-ronmentThisleadsalsotofictionmdashthetellingofstoriesthatneednotbebasedonfactbutthatnonethelesshonethecapacitytodealwiththeepisodicexigenciesofhumansociallife
1 chapteR 1
Part3dealsanotherrecursiveaspectofhumanthoughtnamelytheoryofmindmdashortheabilitytounderstandwhatothersarethink-ingorfeelingChapter8introducesmind-readingnotasapsychicphenomenonbutasanaturalabilitytoinferthementalperspec-tivesofotherpeopleThisabilityisagaincriticaltosocialcohesionandcooperationChapter9explainshowtheoryofmindwasalsocriticaltotheemergenceoflanguage
Part4delvesmorespecificallyintothequestionofhowthere-cursivemindevolvedChapter10setsthisquestioninthecontextoftheclassicdebatebetweenCartesiandiscontinuityandDarwin-iancontinuityChapter11examinessomeofthestepsbywhichthe hominins33 after splitting from the line leading to modernchimpanzees and bonobos began to assume human-like attri-butesChapter12thenconsidersthefinalsteptoldquomodernrdquoHomo sapiensthesolesurvivinghomininspeciesmdashdominantmanipula-tiveMachiavellianandcapableofponderingourownnatureandstatusontheplanetThatperhapsistheultimatetriumphoftherecursivemind
Chapter13presentsthefinalsummaryandconclusions
chapteR 1
Apollonian gasket Sierpinski triangle
Figure2Twofiguresshowingrecursiveinsertionsofcircles(left)andtriangles(right)TheApolloniangasketderivesfromApolloniusaGreekgeometerfromthethirdcenturyBCwhostudiedtheproblemofhowtodrawacirclethatistangentialtothreecirclesStartingwiththreecirclesthataretangentialtooneanotheronecancontinuetheprocessofconstructingcirclestangentialtoalltripletsad infinitumTheresultingfigureservesasamathematicalmodelforfoam(seeMackenzie2009formoreinformation)(2DApolloniangasketwithfourinitialcirclescourtesyofGuillaumeJacquenot)
TheuseofrecursiontocreateinfinitesequencesisalsoexploitedbymathematicsOnesuchsequenceisthesetofnatural(iewhole)numberswhich IrsquollwriteasNThuswecangenerateallof thepositivenaturalnumbersbythedefinitions
1isinNIfnisinNthen(n+1)isinN
ThisseconddefinitionisrecursivebecauseNappearsinthecondi-tionthatneedstobesatisfiedforN
YoumayrememberpossiblyfromschooldayswhatfactorialsareAsaschoolboy I foundthemamusing inachildishkindofwaybecausetheyweresignaledwithexclamationmarksthusfac-torial3usuallywritten3is321andequals65Similarlywecancomputethefollowing
5=54321=1208=87654321=40320
What Is RecuRsIon
Clearlythiscangoonforeverbutwecancapturetheentiresetbyusingjusttwodefiningequations
0=1n=n(nminus1) [wherengt0]
ThissecondequationisrecursiveinthatafactorialisdefinedintermsofafactorialWeneedthefirstequationtokickthethingoff
ThenextexampleisforrabbitsandiscalledtheFibonaccise-riesdefinedbythefollowingthreeequations
fibonacci(0)=1fibonacci(1)=1fibonacci(n)=fibonacci(n minus1)+fibonacci(n minus2)[wherengt1]
Ifyouarefollowingmeyoushouldbeabletocomputetheserieswhichgoes11235813WhatthedefinitionsaysthenisthateachnumberintheseriesisthesumofthetwopreviousonesWhyrabbitsFibonacci(c1170ndash1250)wasanItalianmathemati-cianwhousedtheseriestopredictthegrowthofahypotheticalpopulationofrabbits6
ForafinalinformalexampleItakeyoutoKyotoJapanwhereIoncehappeneduponasignonagatethatwaswritteninKanjiscriptIaskedwhatitmeantandmyguidetoldmeIhopecorrectlythatitmeantPost no billsThereisaparadoxhereinthatthesignwas itselfabilltherebycontravening itsownpresencePerhapsthereneededtobeanothersignthatsaidPost no ldquoPost no billsrdquo billsButofcoursethisisitselfinviolationofitsownmessagesowemightenvisageanothersignthatreadsPost no lsquoPost ldquoPost no billsrdquo billsrsquo billsThereisnoendtothisprocesssoitmighthavebeenmoresensible tohaveallowedbillson thegate in thefirstplaceInpracticethoughlimitationsoftimespaceormemorywillpreventarecursivesequenceofstructurecontinuingforever
toward a Working Definition
Oneof thecharacteristicsofrecursionthenis that itcantakesits own output as the next input a loop that can be extended
chapteR 1
indefinitelytocreatesequencesorstructuresofunboundedlengthorcomplexityInpracticeofcoursewedonotgetcaughtupininfiniteloopsmdashlifeissimplytooshortforthatForthepurposesofthisbookthenweshallnotbeinterestedsomuchinthegenera-tionofinfinitesequencesasinadefinitionthatmightapplyuse-fullytohumanthoughtAdefinitionthatmeetsthisrequirementissuggestedbyStevenPinkerandRayJackendoffwhodefinere-cursionasldquoaprocedurethatcallsitselforaconstituentthatcontainsaconstituentofthesamekindrdquo7
Thesecondpartofthisdefinitionisimportantespeciallyinlan-guagebecauseitallowsthatrecursiveconstructionsneednotin-volvetheembeddingofthesame constituentsasintheexampleof the gate inKyotobutmay contain constituents of the samekindmdasha process sometimes known asldquoself-similar embeddingrdquoForexamplenounphrasescanbebuiltfromnounphrasesinre-cursivefashionTecumsehFitchgivestheexampleofsimplenounphrasessuchasthe dogthe catthe treethe lakeandonecanthencreatenewnounphrasesbyplacingthewordbeside betweenanypairthe dog beside the treethe cat beside the lake8OronemighthavetwosentencesJane loves John andJane flies airplanesandembedone in theother (withappropriatemodification)asJane who flies airplanes loves JohnThesecanbeextendedrecursivelytowhateverlevelofcomplexityisdesiredForexamplewecouldextendthenounphrasetothe dog beside the tree beside the lakeor thesentenceaboutJaneandJohntoJane who flies airplanes that exceed the sound barrier loves John who is prone to selfshydoubtMost languagesmakeuseof recursiveoperationsof thissortmdashalthoughweshallseeinthenextchapterthattheremaybeafewlanguagesthatdonrsquotoperateinthisway
AlthoughitiscommontoprovideillustrationsfromlanguagethemainthemeofthisbookisthatitisinthoughtratherthaninlanguagethatrecursionoriginatesAsPinkerandJackendoffputitldquoTheonlyreasonlanguageneedstoberecursiveisbecauseitsfunction is to express recursive thoughts If there were not anyrecursivethoughtsthemeansofexpressionwouldnotneedrecur-sioneitherrdquo9 Inrememberingepisodesfromthepastforinstanceweessentiallyinsertsequencesofpastconsciousnessintopresent
What Is RecuRsIon
consciousnessor inour interactionswithotherpeoplewemayinsertwhattheyarethinkingintoourownthinkingThesethemesareexploredinlaterchapters
Process and Structure
AssuggestedbyPinkerandJackendoffrsquosdualdefinitionrecursioncanbeunderstoodeitherasaprocess orasastructureThedistinc-tioncanbeimportantArecursiveprocessmayleadtoastructurethatneednotbeseenasitselfrecursiveForexamplesupposeweconstructasequenceofmusicalnoteswithanembeddingroutinebypairingpairsofnoteseachconsistingofa randomlychosennoteplayedonapianowitharandomlychosennoteplayedonaviolinThefirstpairisembeddedinanotherpairandthefour-noteoutputthenembeddedinanotherpairThisprocesscanbecontin-uedindefinitelytocreateasequenceofnotesAsillustratedinfig-ure3thoughthesequencecanbeinterpretednotasarecursivelyembeddedstructurebutasasequenceofpianonotesfollowedbyanequallylongsequenceofviolinnotesThisfailuretodistinguishrecursiveembeddingfromrecursivestructurehasledtosomecon-fusionespeciallyinclaimsaboutrecursioninnonhumanspecies10
Again in his most recent theory on the nature of languageknownastheMinimalistProgram11 NoamChomskyhasarguedthathumanthoughtisgeneratedbyaMergeoperationappliedre-cursivelyThatisunitsaremergedtoformlargerentitiesandthemergedentitiescanbethemselvesmergedtoformstilllargerenti-tiesandsoonThisoperationunderliestheembeddedstructureofhumanlanguagealthoughinChomskyrsquostheoryitappliesstrictlytowhathecallsIshylanguagewhichisthethoughtprocesspreced-ingEshylanguagetheexternal language that isactually spokenorsignedMergecanproducestringsofelementsbetheywordsorelementsofthoughtandalthoughitmaybeappliedrecursivelytoproducehierarchicalstructurethatstructuremaynotbeevidentinthefinaloutputForinstanceevensentencesmightberegardedsimplyaswordsallmergedinunstructuredsequenceasinritual-izedsongsorprayersEverydaylanguagetoomayincludemen-tally undifferentiated clicheacutes and slogansor sequences that are
chapteR 1
P P P P V V V V P P P P V V V V
Figure3ThesequenceofPsandVscanbecreatedeitherbyrecursivelynest-ingPVpairsinPVpairs(left)orbyarrangingasequenceofPsfollowedbyasequenceofanequalnumberofVs(right)Thesequencemightbegeneratedasintheleftpanelbutinterpretedasintherightpanel
highlyautomatedPoliticiansmaybeespeciallypronetothiskindoftalk
Asnotedaboverecursiveprocessesandstructurescaninprin-cipleextendwithoutlimitbutarelimitedinpracticeNeverthelessrecursiondoes give rise to theconcept of infinity itself perhapslimited to the human imaginationAfter all only humans haveacquired theability to count indefinitelyand tounderstand thenatureofinfiniteserieswhereasotherspeciescanatbestmerelyestimatequantityandareaccurateonlyup to somesmallfinitenumber12EveninlanguageweunderstandthatasentencecaninprinciplebeextendedindefinitelyeventhoughinpracticeitcannotbemdashalthoughthenovelistHenryJameshadadamngoodtrySuchunderstandingsareindeedpartofhumanmentalachievementanddependonahumancapacityforrecursivethoughtNeverthelesstheyarenottheprimaryconcernsofthisbook
The appealing aspect of recursion is precisely that it can in principle extend indefinitely to create thoughts (and sentences)ofwhatevercomplexityisrequiredTheideahasanelegantsim-plicitygivingrisetowhatChomskycalledldquodiscreteinfinityrdquo13orWilhelmHumboldt(1767ndash1835)famouslycalledldquotheinfiniteuseoffinitemeansrdquoAndalthoughrecursionislimitedinpracticewecanneverthelessachieveconsiderabledepthsofrecursivethoughtarguablyunsurpassedinanyotherspeciesInchessforexampleaplayermaybeabletothinkrecursivelythreeorfourstepsahead
What Is RecuRsIon
examiningpossiblemovesandcountermovesbutthenumberofpossibilitiessoonmultipliesbeyondthecapacityofthemindtoholdthem
Deeperlevelsofrecursionmaybepossiblewiththeaidofwrit-ingorsimplywithextendedtimeforrehearsalandcontemplationorextendedmemorycapacitythroughartificialmeansTheslowdevelopmentofacomplexmathematicalproofforexamplemayrequiresubtheoremswithinsubtheoremsPlaysornovelsmayin-volverecursiveloopsthatbuildslowlymdashinShakespearersquosTwelfth NightforexampleMariaforeseesthatSirTobywilleagerlyan-ticipate thatOliviawill judgeMalvolio absurdly impertinent tosuppose that she wishes him to regard himself as her preferredsuitor14Asinfictionsoinlifeweallliveinawebofcomplexre-cursiverelationshipsandplanningadinnerpartymayneedcare-fulattentiontowhothinkswhatofwhom
ThestructuresresultingfromrecursiveprocessesneednotrevealthenatureofthoseprocessesjustasaloafofbreadmaynotrevealtheprocessesofkneadingthatwentintothemakingofthebreadorthetasteofwinethepickingandtramplingofthegrapesOftenthoughthestructureofasentenceorstreamofthoughtmayrevealrecursiveembeddingmdashinterpretationofasentencemayrequiretheunderstandingofphrasesembeddedinphrasesregardlessofhowtheembeddingwasactuallyaccomplishedandaninternalunder-standingofastreamofthoughtmayrequirethesegmentationofepisodeswithinepisodes
What Recursion Is not
Recursionisnottheonlydeviceforcreatingsequencesorstruc-tures of potentially infinite length or size I now consider someexamplesthatdonotmeetthecriteriaforrecursion
Repetition
Simple repetition can lead to sequences of potentially infinitelengthbutdoesnot classifyas true recursionForexamplethesentence thatopenschapter9ofAAMilnersquosWinnie the Pooh
10 chapteR 1
goesIt rained and it rained and it rained Thiscouldgoonfor-evermdashoratleastuntilPigletisdrownedmdashbuttherepetitionsimplyconveys the information that it rained rathera lotcausingPig-letsomeennuiItisnotrecursivebecauseeachadditionofand it rained isnotdrivenbythepreviousoneitissimplyaddedatthediscretionofthewriter
InanyeventrepetitiondoesnotdistinguishhumanactivityfromthatofnonhumananimalsBirdsongforexampleisrelentlesslyrepetitivebuteachrepeatedthemedoesnotembellishorqualifythepreviousoneAtmosttherepetitionmightsignalurgencyorsimplysignalcontinuingpresenceasonemightrepeatedlyknockonadoorinthehopeofarousingsomeoneinsideRepetitionisubiquitousinhumanandanimallifeinactivitiesrangingfromtherepeatedjawmovementsineatingtothecuriouslyrepetitivena-tureofsexualactivityThespidernolessiscapableofrepetitionasinWaltWhitmanrsquosLeaves of Grass15
ANOISELESSpatientspiderImarkrsquodwhereonalittlepromontoryitstoodisolatedMarkrsquodhowtoexplorethevacantvastsurroundingItlaunchrsquodforthfilamentfilamentfilamentoutofitselfEverunreelingthemmdashevertirelesslyspeedingthem
Informationcanalsobeaggregatedinnonrecursivefashionaswhentheshort-storywriterSaki(HHMunro)wroteldquoHungerfatigueanddespairinghopelessnesshadnumbedhisbrainrdquo16Ag-gregationofdifferentphrases similarlycompoundsmeaningad-ditivelyaswhenthehistorianPeterHennessywrote
ThemodelofamodernPrimeMinisterwouldbeakindofgrotesquecompositefreakmdashsomeonewiththededicationtodutyofaPeelthephysicalenergyofaGladstonethedetachmentofaSalisburytheballsofaLloydGeorgetheword-powerofaChurchilltheadministrativegiftsof anAttleethe styleof aMacmillanthemanagerialismof aHeathandthesleeprequirementsofaThatcher17
ThesentenceitselfhasrecursiveelementsbuttheaggregationofphrasestodescribethefreakishcompositeisnotrecursiveinthateachdoesnotcallthenextInsteadtheyareeffectivelyelements
What Is RecuRsIon 11
inalistinsertedtoaddinformationNonhumanspeciesmaywellhaveasimilarabilitytoaccumulateinformationaswhenunder-standingapredatoraslargefierceandwithsharpteethandclaws
Iteration
AslightlymoresubtlevariantonrepetitionandaggregationisitshyerationwhereaprocessisrepeatedbutinthiscasethereisinputfromthepreviousapplicationoftheprocessInthisrespect it islikerecursionandindeedconsideredbymathematicianstobelongtotheclassofldquogeneralrecursivefunctionsrdquoForthemainpurposesofthisbookthoughitdoesnotqualifyastruerecursionbecauseeachoutputisdiscardedonceithasbeenenteredintothenextap-plicationThedictionarydefinitionofrecursionthatIgaveearlierinthischapterwasalsoreallyanexampleofiterationratherthanrecursionbecauseyoujustkeepgoingroundandroundtheloopwithoutanyaddedstructureTheiterationsthereforedonotleadtoaddedcomplexity18
Iterativeproceduresareusedincomputationalmathematicstoarriveat increasinglyaccuratesolutionstoaproblemThebasicideaistostartwithapreliminarysolutionmdashperhapsaguessmdashandthenuseaproceduretocomputeanewsolutionThissolutionisthenusedasthestartingpointforthenextcomputationandthenew solution is then the starting point for the next roundThecycle is repeateduntil the solutions stabilize to someacceptablecriterion19Feedbacksystemsoperateinmuchthesamewaytypi-callyasameansofmaintaininghomeostasisForexampleather-mostatmayinvolveasystemforraisingorloweringtemperatureandthegoalistoachievesomegiventemperatureTheactualtem-peratureisfedintothesystemwhichoperatestoraiseorlowerthe temperatureuntil thedesired is reachedThebody isawashwith feedback systems to maintain homeostasis of temperatureironenergybloodcompositionandsoonThemainregulatoristhehypothalamusinthelimbicsystemofthebrainSuchsystemsagaindonotdifferentiatehumansfromotheranimals
Sometimesthedistinctionbetweenrecursionanditerationmaybeamatterof interpretationIn the infinite loopcreatedby the
1 chapteR 1
parodyofPaul CliffordonemightsaythateachbeginningofthestoryisinitiatedbythepreviousonewhichisthenforgottenTheparodyisbestappreciatedthoughifthestoryisseenasanendlessever deepening whirlpoolwith each segment remaining as partofitIrsquomtoldthestoryworksbestifeachsegmentisspokenwithadifferentaccent
Considertoothislinefromawell-knownchildrenrsquosverse
ThisisthedogthatworriedthecatthatkilledtheratthatatethemaltthatlayinthehousethatJackbuilt
TounderstandthissentenceastrulyrecursiveonemustappreciatethatitdescribesastateofaffairsasacomplexwholeandreferstoparticularcasesofadogcatratmalthouseandfellowcalledJackItisnotsimplythestringingtogetherofadogthatworriedacatacatthatkilledaratandsoforthAyoungchildthoughmightprocessitinthispiecemealwayasasuccessionofunrelatedevents
Recursion and evolutionary psychology
Inemphasizingrecursionasaunifyingconcepttheapproachtakeninthisbookcontrastswiththatadoptedbyso-calledevolutionarypsychologistswhohavearguedthatthemindhasmultiplefacetsThebasictenetsofevolutionarypsychologywerelaidoutinthe1992volumeThe Adapted MindeditedbyJeromeBarkowLedaCosmidesandJohnToobyandpopularizedbyStevenPinkerinhis influential 1997 book How the Mind Works20 Thus Pinkerwritesthatthehumanmindldquoisnotasingleorganbutasystemoforganswhichwecanthinkofaspsychologicalfacultiesormentalmodulesrdquo21 Inexaminingpresent-dayhumanbehaviortheevolu-tionarypsychologistrsquosagendaistodiscoverindependentprocessesasthebasicmodulesandrelatethemtoconditionsthatprevailedin the Pleistocene when humans existed primarily as hunter-gatherersAs Pinker puts it the aim is to carve the mind at itsjointssotospeakandldquoreverse-engineerrdquoitscomponentsormod-ulesbacktotheepochduringwhichthehumanmindwasformed
What Is RecuRsIon 1
In this view the mind is really a collection of miniminds eachbeaveringawayonitsownspecificproblemamongwhicharelan-guageandtheoryofmindThishasbeencalledtheSwiss-army-knifemodelofthemindwithabladeforeverypurpose22
Thedangerwith thisapproach is that itbecomes tooeasy topostulate modules and to tellldquojust sordquostories about how theyevolvedsothatthereisariskofreturningtothenow-abandonedinstinctpsychologyoftheearlytwentiethcentury23 Instinctpsy-chologyperishedunderthesheerweightofnumbersmdashtheauthorof one text counted 1594 instincts that had been attributed toanimals and humans24mdashand evolutionary psychology may alsodrowninaseaofmodulesifnotofmixedmetaphorsPinkersug-geststhatwelikepotatochipsbecausefattyfoodswerenutrition-allyvaluableduringthePleistocenebutscarceenoughthattherewasnodangerofobesitywelike landscapeswithtreesbecausetreesprovidesshadeandescapefromdangerouscarnivoresontheAfrica savanna flowers please us because they are markers foredible fruits nutsor tubers amid the greenery of the savannaandsoonldquoTherearemodulesrdquohewritesldquoforobjectsandforcesforanimatebeingsformindsandfornaturalkindslikeanimalsplantsandmineralsrdquo25
This isnot to say that theSwiss-army-knifemodel iswithoutmeritSomeofthepostulatedmodulesdoprovideinsightintothehumanconditionandarereasonablywellfoundedForexamplefoundationalworkinevolutionarypsychologybyLedaCosmidesprovidedgoodevidenceforaldquocheater-detectionmodulerdquomdashanin-stinctiveability todetect thosewhoflout social conventions fortheirowngain26 Arecentstudysuggests thathumanspossessaldquocategory-specificattentionsystemrdquothatisespeciallyadaptedforattendingtoanimals27 oneoftheauthorsJohnToobyisquotedas sayingldquoEvendullanimals likepigeons recruita surpris-ingamountofattentionmdashasdoturtlesresemblingrocksrdquo28 Thisbookisnotintendedtodenythattherearemanyspecificdisposi-tionsthatshapeourmentalandsociallivesrathermyaimistosuggest that therearedeeperaspectsofhumanthoughtthataregovernedbysimilarprinciplesandthatrecursionisoneofthoseprinciplesmdashandperhapsthemostimportantone
1 chapteR 1
Tobefairtoonotallevolutionarypsychologistshaveinsistedthatmodulesarecompletelyencapsulatedshutofffromanycom-munication with one anotherEven Steven Pinker for examplewrites ldquo[Modules] accomplish specialized functions thanks totheirspecializedstructuresbutdonrsquotnecessarilycomeinencap-sulated packagesrdquo29 Steven Mithen although scarcely a card-carryingevolutionarypsychologistarguedthatthehumanmindevolved itsdistinctive characterpreciselybecausepreviously en-capsulatedmodulesbegantoldquoleakrdquocreatingwhathecallsldquocogni-tivefluidityrdquo30 ItisasthoughthemodulesstoppedmindingtheirownbusinessesandbegantogossipMyapproachinthisbookisnotentirelyatoddswiththisviewinthatIargueacommonprin-ciplemightunderlieanumberofourdistinctiveabilities
Others are also beginning to question the Swiss-army-knifemodelofthehumanmindmorestarklyDavidPremackforex-ampleadoptsanapproachsimilartothatofferedinthisbookRe-viewingtheevidencefordiscontinuitybetweenhumansandotheranimalshewritesldquoAnimalcompetenciesaremainlyadaptationsrestrictedtoasinglegoalHumancompetenciesaredomain-generalandservenumerousgoalsrdquo31Thisineffectreversestheevolution-arypsychologyargumentmdashthemindhasbecomelessratherthanmoremodularThetidemaywellbeturning
In any event it is unlikely that recursion canbe considered amoduleAsweshallseerecursionseemstobeanorganizingprin-cipleinverydifferentspheresofhumanmentalactivityfromlan-guage tomemory tomind readingRecursive thinkingprobablydependsonothermentalattributesOneoftheseiswhathasbeentermedworkingmemorywhichholdsinformationinconscious-nessInordertoembedprocesseswithinprocessesitisnecessarytorememberwhereonehadgottointheearlierprocesswhenanembeddedprocesshasbeencompletedForinstanceinasentencelike My dog who eats bananas often gets sickone must holdthe early part of the sentence (My dog) and link it to the nextpart(often gets sick)DwightWReadhasarguedthatnonhumanprimatesevenourclosestrelativesthechimpanzeeshaveawork-ingmemorythatistoolimitedtoallowthiskindofembedding32
Recursionprobablyalsodependsonanexecutiveprocessthator-
What Is RecuRsIon 1
ganizeswhatistobeembeddedinwhatandthismaydependonthefrontallobesofthebrainTheabilitytoorganizeandcarryoutrecursiveoperationsmaythereforedependonseveralprocesses
AlthoughIdonotembracethemodularviewassumedbyevo-lutionarypsychologistsIamatonewiththeminproposingthata distinctively human mind evolved during the Pleistocene theepochthatstretchedfromaround26millionyearsagotosome12000 years agoHow that happened will be told in the laterchaptersofthisbook
plan of the Book
ThebookisdividedintofourpartsPart1dealswithlanguageAlthoughrecursionisnotlimitedto
languageitismostcommonlyinvokedtoexplainwhyhumanlan-guagediffersfromotherformsofanimalcommunicationanin-sightlargelyattributabletoNoamChomskyChapter2discussesthe nature of languagewithparticular emphasis on the role ofrecursionChapter3thenraisestheage-oldquestionofwhetherotheranimalshaveanythingresemblinghumanlanguageChapter4developstheideathatlanguageevolvedfrommanualgesturesmdashanideathatsuggestsgreaterevolutionarycontinuitybetweenhu-mansandotherprimatesthanthemorecommonassumptionthatlanguageemergedfromvocalcalls
Part2dealswithmentaltimetraveltheabilitytobringtomindeventsremovedfromthepresentinbothtimeandplaceChapter5startswithmemoryanddevelopstheideathatmemoryforspecificepisodesisuniquetohumansChapter6extendsthenotionofepi-sodicmemorytotheimaginingofpossiblefutureeventsleadingtotheconceptoftheselfasexistingthroughtimeThisleadstothenotiondiscussedinchapter7thatlanguageitselfevolvedtoenablepeopletosharetheirmemoriesandplansandsotocom-municateabouteventsthatarenotpresentintheimmediateenvi-ronmentThisleadsalsotofictionmdashthetellingofstoriesthatneednotbebasedonfactbutthatnonethelesshonethecapacitytodealwiththeepisodicexigenciesofhumansociallife
1 chapteR 1
Part3dealsanotherrecursiveaspectofhumanthoughtnamelytheoryofmindmdashortheabilitytounderstandwhatothersarethink-ingorfeelingChapter8introducesmind-readingnotasapsychicphenomenonbutasanaturalabilitytoinferthementalperspec-tivesofotherpeopleThisabilityisagaincriticaltosocialcohesionandcooperationChapter9explainshowtheoryofmindwasalsocriticaltotheemergenceoflanguage
Part4delvesmorespecificallyintothequestionofhowthere-cursivemindevolvedChapter10setsthisquestioninthecontextoftheclassicdebatebetweenCartesiandiscontinuityandDarwin-iancontinuityChapter11examinessomeofthestepsbywhichthe hominins33 after splitting from the line leading to modernchimpanzees and bonobos began to assume human-like attri-butesChapter12thenconsidersthefinalsteptoldquomodernrdquoHomo sapiensthesolesurvivinghomininspeciesmdashdominantmanipula-tiveMachiavellianandcapableofponderingourownnatureandstatusontheplanetThatperhapsistheultimatetriumphoftherecursivemind
Chapter13presentsthefinalsummaryandconclusions
What Is RecuRsIon
Clearlythiscangoonforeverbutwecancapturetheentiresetbyusingjusttwodefiningequations
0=1n=n(nminus1) [wherengt0]
ThissecondequationisrecursiveinthatafactorialisdefinedintermsofafactorialWeneedthefirstequationtokickthethingoff
ThenextexampleisforrabbitsandiscalledtheFibonaccise-riesdefinedbythefollowingthreeequations
fibonacci(0)=1fibonacci(1)=1fibonacci(n)=fibonacci(n minus1)+fibonacci(n minus2)[wherengt1]
Ifyouarefollowingmeyoushouldbeabletocomputetheserieswhichgoes11235813WhatthedefinitionsaysthenisthateachnumberintheseriesisthesumofthetwopreviousonesWhyrabbitsFibonacci(c1170ndash1250)wasanItalianmathemati-cianwhousedtheseriestopredictthegrowthofahypotheticalpopulationofrabbits6
ForafinalinformalexampleItakeyoutoKyotoJapanwhereIoncehappeneduponasignonagatethatwaswritteninKanjiscriptIaskedwhatitmeantandmyguidetoldmeIhopecorrectlythatitmeantPost no billsThereisaparadoxhereinthatthesignwas itselfabilltherebycontravening itsownpresencePerhapsthereneededtobeanothersignthatsaidPost no ldquoPost no billsrdquo billsButofcoursethisisitselfinviolationofitsownmessagesowemightenvisageanothersignthatreadsPost no lsquoPost ldquoPost no billsrdquo billsrsquo billsThereisnoendtothisprocesssoitmighthavebeenmoresensible tohaveallowedbillson thegate in thefirstplaceInpracticethoughlimitationsoftimespaceormemorywillpreventarecursivesequenceofstructurecontinuingforever
toward a Working Definition
Oneof thecharacteristicsofrecursionthenis that itcantakesits own output as the next input a loop that can be extended
chapteR 1
indefinitelytocreatesequencesorstructuresofunboundedlengthorcomplexityInpracticeofcoursewedonotgetcaughtupininfiniteloopsmdashlifeissimplytooshortforthatForthepurposesofthisbookthenweshallnotbeinterestedsomuchinthegenera-tionofinfinitesequencesasinadefinitionthatmightapplyuse-fullytohumanthoughtAdefinitionthatmeetsthisrequirementissuggestedbyStevenPinkerandRayJackendoffwhodefinere-cursionasldquoaprocedurethatcallsitselforaconstituentthatcontainsaconstituentofthesamekindrdquo7
Thesecondpartofthisdefinitionisimportantespeciallyinlan-guagebecauseitallowsthatrecursiveconstructionsneednotin-volvetheembeddingofthesame constituentsasintheexampleof the gate inKyotobutmay contain constituents of the samekindmdasha process sometimes known asldquoself-similar embeddingrdquoForexamplenounphrasescanbebuiltfromnounphrasesinre-cursivefashionTecumsehFitchgivestheexampleofsimplenounphrasessuchasthe dogthe catthe treethe lakeandonecanthencreatenewnounphrasesbyplacingthewordbeside betweenanypairthe dog beside the treethe cat beside the lake8OronemighthavetwosentencesJane loves John andJane flies airplanesandembedone in theother (withappropriatemodification)asJane who flies airplanes loves JohnThesecanbeextendedrecursivelytowhateverlevelofcomplexityisdesiredForexamplewecouldextendthenounphrasetothe dog beside the tree beside the lakeor thesentenceaboutJaneandJohntoJane who flies airplanes that exceed the sound barrier loves John who is prone to selfshydoubtMost languagesmakeuseof recursiveoperationsof thissortmdashalthoughweshallseeinthenextchapterthattheremaybeafewlanguagesthatdonrsquotoperateinthisway
AlthoughitiscommontoprovideillustrationsfromlanguagethemainthemeofthisbookisthatitisinthoughtratherthaninlanguagethatrecursionoriginatesAsPinkerandJackendoffputitldquoTheonlyreasonlanguageneedstoberecursiveisbecauseitsfunction is to express recursive thoughts If there were not anyrecursivethoughtsthemeansofexpressionwouldnotneedrecur-sioneitherrdquo9 Inrememberingepisodesfromthepastforinstanceweessentiallyinsertsequencesofpastconsciousnessintopresent
What Is RecuRsIon
consciousnessor inour interactionswithotherpeoplewemayinsertwhattheyarethinkingintoourownthinkingThesethemesareexploredinlaterchapters
Process and Structure
AssuggestedbyPinkerandJackendoffrsquosdualdefinitionrecursioncanbeunderstoodeitherasaprocess orasastructureThedistinc-tioncanbeimportantArecursiveprocessmayleadtoastructurethatneednotbeseenasitselfrecursiveForexamplesupposeweconstructasequenceofmusicalnoteswithanembeddingroutinebypairingpairsofnoteseachconsistingofa randomlychosennoteplayedonapianowitharandomlychosennoteplayedonaviolinThefirstpairisembeddedinanotherpairandthefour-noteoutputthenembeddedinanotherpairThisprocesscanbecontin-uedindefinitelytocreateasequenceofnotesAsillustratedinfig-ure3thoughthesequencecanbeinterpretednotasarecursivelyembeddedstructurebutasasequenceofpianonotesfollowedbyanequallylongsequenceofviolinnotesThisfailuretodistinguishrecursiveembeddingfromrecursivestructurehasledtosomecon-fusionespeciallyinclaimsaboutrecursioninnonhumanspecies10
Again in his most recent theory on the nature of languageknownastheMinimalistProgram11 NoamChomskyhasarguedthathumanthoughtisgeneratedbyaMergeoperationappliedre-cursivelyThatisunitsaremergedtoformlargerentitiesandthemergedentitiescanbethemselvesmergedtoformstilllargerenti-tiesandsoonThisoperationunderliestheembeddedstructureofhumanlanguagealthoughinChomskyrsquostheoryitappliesstrictlytowhathecallsIshylanguagewhichisthethoughtprocesspreced-ingEshylanguagetheexternal language that isactually spokenorsignedMergecanproducestringsofelementsbetheywordsorelementsofthoughtandalthoughitmaybeappliedrecursivelytoproducehierarchicalstructurethatstructuremaynotbeevidentinthefinaloutputForinstanceevensentencesmightberegardedsimplyaswordsallmergedinunstructuredsequenceasinritual-izedsongsorprayersEverydaylanguagetoomayincludemen-tally undifferentiated clicheacutes and slogansor sequences that are
chapteR 1
P P P P V V V V P P P P V V V V
Figure3ThesequenceofPsandVscanbecreatedeitherbyrecursivelynest-ingPVpairsinPVpairs(left)orbyarrangingasequenceofPsfollowedbyasequenceofanequalnumberofVs(right)Thesequencemightbegeneratedasintheleftpanelbutinterpretedasintherightpanel
highlyautomatedPoliticiansmaybeespeciallypronetothiskindoftalk
Asnotedaboverecursiveprocessesandstructurescaninprin-cipleextendwithoutlimitbutarelimitedinpracticeNeverthelessrecursiondoes give rise to theconcept of infinity itself perhapslimited to the human imaginationAfter all only humans haveacquired theability to count indefinitelyand tounderstand thenatureofinfiniteserieswhereasotherspeciescanatbestmerelyestimatequantityandareaccurateonlyup to somesmallfinitenumber12EveninlanguageweunderstandthatasentencecaninprinciplebeextendedindefinitelyeventhoughinpracticeitcannotbemdashalthoughthenovelistHenryJameshadadamngoodtrySuchunderstandingsareindeedpartofhumanmentalachievementanddependonahumancapacityforrecursivethoughtNeverthelesstheyarenottheprimaryconcernsofthisbook
The appealing aspect of recursion is precisely that it can in principle extend indefinitely to create thoughts (and sentences)ofwhatevercomplexityisrequiredTheideahasanelegantsim-plicitygivingrisetowhatChomskycalledldquodiscreteinfinityrdquo13orWilhelmHumboldt(1767ndash1835)famouslycalledldquotheinfiniteuseoffinitemeansrdquoAndalthoughrecursionislimitedinpracticewecanneverthelessachieveconsiderabledepthsofrecursivethoughtarguablyunsurpassedinanyotherspeciesInchessforexampleaplayermaybeabletothinkrecursivelythreeorfourstepsahead
What Is RecuRsIon
examiningpossiblemovesandcountermovesbutthenumberofpossibilitiessoonmultipliesbeyondthecapacityofthemindtoholdthem
Deeperlevelsofrecursionmaybepossiblewiththeaidofwrit-ingorsimplywithextendedtimeforrehearsalandcontemplationorextendedmemorycapacitythroughartificialmeansTheslowdevelopmentofacomplexmathematicalproofforexamplemayrequiresubtheoremswithinsubtheoremsPlaysornovelsmayin-volverecursiveloopsthatbuildslowlymdashinShakespearersquosTwelfth NightforexampleMariaforeseesthatSirTobywilleagerlyan-ticipate thatOliviawill judgeMalvolio absurdly impertinent tosuppose that she wishes him to regard himself as her preferredsuitor14Asinfictionsoinlifeweallliveinawebofcomplexre-cursiverelationshipsandplanningadinnerpartymayneedcare-fulattentiontowhothinkswhatofwhom
ThestructuresresultingfromrecursiveprocessesneednotrevealthenatureofthoseprocessesjustasaloafofbreadmaynotrevealtheprocessesofkneadingthatwentintothemakingofthebreadorthetasteofwinethepickingandtramplingofthegrapesOftenthoughthestructureofasentenceorstreamofthoughtmayrevealrecursiveembeddingmdashinterpretationofasentencemayrequiretheunderstandingofphrasesembeddedinphrasesregardlessofhowtheembeddingwasactuallyaccomplishedandaninternalunder-standingofastreamofthoughtmayrequirethesegmentationofepisodeswithinepisodes
What Recursion Is not
Recursionisnottheonlydeviceforcreatingsequencesorstruc-tures of potentially infinite length or size I now consider someexamplesthatdonotmeetthecriteriaforrecursion
Repetition
Simple repetition can lead to sequences of potentially infinitelengthbutdoesnot classifyas true recursionForexamplethesentence thatopenschapter9ofAAMilnersquosWinnie the Pooh
10 chapteR 1
goesIt rained and it rained and it rained Thiscouldgoonfor-evermdashoratleastuntilPigletisdrownedmdashbuttherepetitionsimplyconveys the information that it rained rathera lotcausingPig-letsomeennuiItisnotrecursivebecauseeachadditionofand it rained isnotdrivenbythepreviousoneitissimplyaddedatthediscretionofthewriter
InanyeventrepetitiondoesnotdistinguishhumanactivityfromthatofnonhumananimalsBirdsongforexampleisrelentlesslyrepetitivebuteachrepeatedthemedoesnotembellishorqualifythepreviousoneAtmosttherepetitionmightsignalurgencyorsimplysignalcontinuingpresenceasonemightrepeatedlyknockonadoorinthehopeofarousingsomeoneinsideRepetitionisubiquitousinhumanandanimallifeinactivitiesrangingfromtherepeatedjawmovementsineatingtothecuriouslyrepetitivena-tureofsexualactivityThespidernolessiscapableofrepetitionasinWaltWhitmanrsquosLeaves of Grass15
ANOISELESSpatientspiderImarkrsquodwhereonalittlepromontoryitstoodisolatedMarkrsquodhowtoexplorethevacantvastsurroundingItlaunchrsquodforthfilamentfilamentfilamentoutofitselfEverunreelingthemmdashevertirelesslyspeedingthem
Informationcanalsobeaggregatedinnonrecursivefashionaswhentheshort-storywriterSaki(HHMunro)wroteldquoHungerfatigueanddespairinghopelessnesshadnumbedhisbrainrdquo16Ag-gregationofdifferentphrases similarlycompoundsmeaningad-ditivelyaswhenthehistorianPeterHennessywrote
ThemodelofamodernPrimeMinisterwouldbeakindofgrotesquecompositefreakmdashsomeonewiththededicationtodutyofaPeelthephysicalenergyofaGladstonethedetachmentofaSalisburytheballsofaLloydGeorgetheword-powerofaChurchilltheadministrativegiftsof anAttleethe styleof aMacmillanthemanagerialismof aHeathandthesleeprequirementsofaThatcher17
ThesentenceitselfhasrecursiveelementsbuttheaggregationofphrasestodescribethefreakishcompositeisnotrecursiveinthateachdoesnotcallthenextInsteadtheyareeffectivelyelements
What Is RecuRsIon 11
inalistinsertedtoaddinformationNonhumanspeciesmaywellhaveasimilarabilitytoaccumulateinformationaswhenunder-standingapredatoraslargefierceandwithsharpteethandclaws
Iteration
AslightlymoresubtlevariantonrepetitionandaggregationisitshyerationwhereaprocessisrepeatedbutinthiscasethereisinputfromthepreviousapplicationoftheprocessInthisrespect it islikerecursionandindeedconsideredbymathematicianstobelongtotheclassofldquogeneralrecursivefunctionsrdquoForthemainpurposesofthisbookthoughitdoesnotqualifyastruerecursionbecauseeachoutputisdiscardedonceithasbeenenteredintothenextap-plicationThedictionarydefinitionofrecursionthatIgaveearlierinthischapterwasalsoreallyanexampleofiterationratherthanrecursionbecauseyoujustkeepgoingroundandroundtheloopwithoutanyaddedstructureTheiterationsthereforedonotleadtoaddedcomplexity18
Iterativeproceduresareusedincomputationalmathematicstoarriveat increasinglyaccuratesolutionstoaproblemThebasicideaistostartwithapreliminarysolutionmdashperhapsaguessmdashandthenuseaproceduretocomputeanewsolutionThissolutionisthenusedasthestartingpointforthenextcomputationandthenew solution is then the starting point for the next roundThecycle is repeateduntil the solutions stabilize to someacceptablecriterion19Feedbacksystemsoperateinmuchthesamewaytypi-callyasameansofmaintaininghomeostasisForexampleather-mostatmayinvolveasystemforraisingorloweringtemperatureandthegoalistoachievesomegiventemperatureTheactualtem-peratureisfedintothesystemwhichoperatestoraiseorlowerthe temperatureuntil thedesired is reachedThebody isawashwith feedback systems to maintain homeostasis of temperatureironenergybloodcompositionandsoonThemainregulatoristhehypothalamusinthelimbicsystemofthebrainSuchsystemsagaindonotdifferentiatehumansfromotheranimals
Sometimesthedistinctionbetweenrecursionanditerationmaybeamatterof interpretationIn the infinite loopcreatedby the
1 chapteR 1
parodyofPaul CliffordonemightsaythateachbeginningofthestoryisinitiatedbythepreviousonewhichisthenforgottenTheparodyisbestappreciatedthoughifthestoryisseenasanendlessever deepening whirlpoolwith each segment remaining as partofitIrsquomtoldthestoryworksbestifeachsegmentisspokenwithadifferentaccent
Considertoothislinefromawell-knownchildrenrsquosverse
ThisisthedogthatworriedthecatthatkilledtheratthatatethemaltthatlayinthehousethatJackbuilt
TounderstandthissentenceastrulyrecursiveonemustappreciatethatitdescribesastateofaffairsasacomplexwholeandreferstoparticularcasesofadogcatratmalthouseandfellowcalledJackItisnotsimplythestringingtogetherofadogthatworriedacatacatthatkilledaratandsoforthAyoungchildthoughmightprocessitinthispiecemealwayasasuccessionofunrelatedevents
Recursion and evolutionary psychology
Inemphasizingrecursionasaunifyingconcepttheapproachtakeninthisbookcontrastswiththatadoptedbyso-calledevolutionarypsychologistswhohavearguedthatthemindhasmultiplefacetsThebasictenetsofevolutionarypsychologywerelaidoutinthe1992volumeThe Adapted MindeditedbyJeromeBarkowLedaCosmidesandJohnToobyandpopularizedbyStevenPinkerinhis influential 1997 book How the Mind Works20 Thus Pinkerwritesthatthehumanmindldquoisnotasingleorganbutasystemoforganswhichwecanthinkofaspsychologicalfacultiesormentalmodulesrdquo21 Inexaminingpresent-dayhumanbehaviortheevolu-tionarypsychologistrsquosagendaistodiscoverindependentprocessesasthebasicmodulesandrelatethemtoconditionsthatprevailedin the Pleistocene when humans existed primarily as hunter-gatherersAs Pinker puts it the aim is to carve the mind at itsjointssotospeakandldquoreverse-engineerrdquoitscomponentsormod-ulesbacktotheepochduringwhichthehumanmindwasformed
What Is RecuRsIon 1
In this view the mind is really a collection of miniminds eachbeaveringawayonitsownspecificproblemamongwhicharelan-guageandtheoryofmindThishasbeencalledtheSwiss-army-knifemodelofthemindwithabladeforeverypurpose22
Thedangerwith thisapproach is that itbecomes tooeasy topostulate modules and to tellldquojust sordquostories about how theyevolvedsothatthereisariskofreturningtothenow-abandonedinstinctpsychologyoftheearlytwentiethcentury23 Instinctpsy-chologyperishedunderthesheerweightofnumbersmdashtheauthorof one text counted 1594 instincts that had been attributed toanimals and humans24mdashand evolutionary psychology may alsodrowninaseaofmodulesifnotofmixedmetaphorsPinkersug-geststhatwelikepotatochipsbecausefattyfoodswerenutrition-allyvaluableduringthePleistocenebutscarceenoughthattherewasnodangerofobesitywelike landscapeswithtreesbecausetreesprovidesshadeandescapefromdangerouscarnivoresontheAfrica savanna flowers please us because they are markers foredible fruits nutsor tubers amid the greenery of the savannaandsoonldquoTherearemodulesrdquohewritesldquoforobjectsandforcesforanimatebeingsformindsandfornaturalkindslikeanimalsplantsandmineralsrdquo25
This isnot to say that theSwiss-army-knifemodel iswithoutmeritSomeofthepostulatedmodulesdoprovideinsightintothehumanconditionandarereasonablywellfoundedForexamplefoundationalworkinevolutionarypsychologybyLedaCosmidesprovidedgoodevidenceforaldquocheater-detectionmodulerdquomdashanin-stinctiveability todetect thosewhoflout social conventions fortheirowngain26 Arecentstudysuggests thathumanspossessaldquocategory-specificattentionsystemrdquothatisespeciallyadaptedforattendingtoanimals27 oneoftheauthorsJohnToobyisquotedas sayingldquoEvendullanimals likepigeons recruita surpris-ingamountofattentionmdashasdoturtlesresemblingrocksrdquo28 Thisbookisnotintendedtodenythattherearemanyspecificdisposi-tionsthatshapeourmentalandsociallivesrathermyaimistosuggest that therearedeeperaspectsofhumanthoughtthataregovernedbysimilarprinciplesandthatrecursionisoneofthoseprinciplesmdashandperhapsthemostimportantone
1 chapteR 1
Tobefairtoonotallevolutionarypsychologistshaveinsistedthatmodulesarecompletelyencapsulatedshutofffromanycom-munication with one anotherEven Steven Pinker for examplewrites ldquo[Modules] accomplish specialized functions thanks totheirspecializedstructuresbutdonrsquotnecessarilycomeinencap-sulated packagesrdquo29 Steven Mithen although scarcely a card-carryingevolutionarypsychologistarguedthatthehumanmindevolved itsdistinctive characterpreciselybecausepreviously en-capsulatedmodulesbegantoldquoleakrdquocreatingwhathecallsldquocogni-tivefluidityrdquo30 ItisasthoughthemodulesstoppedmindingtheirownbusinessesandbegantogossipMyapproachinthisbookisnotentirelyatoddswiththisviewinthatIargueacommonprin-ciplemightunderlieanumberofourdistinctiveabilities
Others are also beginning to question the Swiss-army-knifemodelofthehumanmindmorestarklyDavidPremackforex-ampleadoptsanapproachsimilartothatofferedinthisbookRe-viewingtheevidencefordiscontinuitybetweenhumansandotheranimalshewritesldquoAnimalcompetenciesaremainlyadaptationsrestrictedtoasinglegoalHumancompetenciesaredomain-generalandservenumerousgoalsrdquo31Thisineffectreversestheevolution-arypsychologyargumentmdashthemindhasbecomelessratherthanmoremodularThetidemaywellbeturning
In any event it is unlikely that recursion canbe considered amoduleAsweshallseerecursionseemstobeanorganizingprin-cipleinverydifferentspheresofhumanmentalactivityfromlan-guage tomemory tomind readingRecursive thinkingprobablydependsonothermentalattributesOneoftheseiswhathasbeentermedworkingmemorywhichholdsinformationinconscious-nessInordertoembedprocesseswithinprocessesitisnecessarytorememberwhereonehadgottointheearlierprocesswhenanembeddedprocesshasbeencompletedForinstanceinasentencelike My dog who eats bananas often gets sickone must holdthe early part of the sentence (My dog) and link it to the nextpart(often gets sick)DwightWReadhasarguedthatnonhumanprimatesevenourclosestrelativesthechimpanzeeshaveawork-ingmemorythatistoolimitedtoallowthiskindofembedding32
Recursionprobablyalsodependsonanexecutiveprocessthator-
What Is RecuRsIon 1
ganizeswhatistobeembeddedinwhatandthismaydependonthefrontallobesofthebrainTheabilitytoorganizeandcarryoutrecursiveoperationsmaythereforedependonseveralprocesses
AlthoughIdonotembracethemodularviewassumedbyevo-lutionarypsychologistsIamatonewiththeminproposingthata distinctively human mind evolved during the Pleistocene theepochthatstretchedfromaround26millionyearsagotosome12000 years agoHow that happened will be told in the laterchaptersofthisbook
plan of the Book
ThebookisdividedintofourpartsPart1dealswithlanguageAlthoughrecursionisnotlimitedto
languageitismostcommonlyinvokedtoexplainwhyhumanlan-guagediffersfromotherformsofanimalcommunicationanin-sightlargelyattributabletoNoamChomskyChapter2discussesthe nature of languagewithparticular emphasis on the role ofrecursionChapter3thenraisestheage-oldquestionofwhetherotheranimalshaveanythingresemblinghumanlanguageChapter4developstheideathatlanguageevolvedfrommanualgesturesmdashanideathatsuggestsgreaterevolutionarycontinuitybetweenhu-mansandotherprimatesthanthemorecommonassumptionthatlanguageemergedfromvocalcalls
Part2dealswithmentaltimetraveltheabilitytobringtomindeventsremovedfromthepresentinbothtimeandplaceChapter5startswithmemoryanddevelopstheideathatmemoryforspecificepisodesisuniquetohumansChapter6extendsthenotionofepi-sodicmemorytotheimaginingofpossiblefutureeventsleadingtotheconceptoftheselfasexistingthroughtimeThisleadstothenotiondiscussedinchapter7thatlanguageitselfevolvedtoenablepeopletosharetheirmemoriesandplansandsotocom-municateabouteventsthatarenotpresentintheimmediateenvi-ronmentThisleadsalsotofictionmdashthetellingofstoriesthatneednotbebasedonfactbutthatnonethelesshonethecapacitytodealwiththeepisodicexigenciesofhumansociallife
1 chapteR 1
Part3dealsanotherrecursiveaspectofhumanthoughtnamelytheoryofmindmdashortheabilitytounderstandwhatothersarethink-ingorfeelingChapter8introducesmind-readingnotasapsychicphenomenonbutasanaturalabilitytoinferthementalperspec-tivesofotherpeopleThisabilityisagaincriticaltosocialcohesionandcooperationChapter9explainshowtheoryofmindwasalsocriticaltotheemergenceoflanguage
Part4delvesmorespecificallyintothequestionofhowthere-cursivemindevolvedChapter10setsthisquestioninthecontextoftheclassicdebatebetweenCartesiandiscontinuityandDarwin-iancontinuityChapter11examinessomeofthestepsbywhichthe hominins33 after splitting from the line leading to modernchimpanzees and bonobos began to assume human-like attri-butesChapter12thenconsidersthefinalsteptoldquomodernrdquoHomo sapiensthesolesurvivinghomininspeciesmdashdominantmanipula-tiveMachiavellianandcapableofponderingourownnatureandstatusontheplanetThatperhapsistheultimatetriumphoftherecursivemind
Chapter13presentsthefinalsummaryandconclusions
chapteR 1
indefinitelytocreatesequencesorstructuresofunboundedlengthorcomplexityInpracticeofcoursewedonotgetcaughtupininfiniteloopsmdashlifeissimplytooshortforthatForthepurposesofthisbookthenweshallnotbeinterestedsomuchinthegenera-tionofinfinitesequencesasinadefinitionthatmightapplyuse-fullytohumanthoughtAdefinitionthatmeetsthisrequirementissuggestedbyStevenPinkerandRayJackendoffwhodefinere-cursionasldquoaprocedurethatcallsitselforaconstituentthatcontainsaconstituentofthesamekindrdquo7
Thesecondpartofthisdefinitionisimportantespeciallyinlan-guagebecauseitallowsthatrecursiveconstructionsneednotin-volvetheembeddingofthesame constituentsasintheexampleof the gate inKyotobutmay contain constituents of the samekindmdasha process sometimes known asldquoself-similar embeddingrdquoForexamplenounphrasescanbebuiltfromnounphrasesinre-cursivefashionTecumsehFitchgivestheexampleofsimplenounphrasessuchasthe dogthe catthe treethe lakeandonecanthencreatenewnounphrasesbyplacingthewordbeside betweenanypairthe dog beside the treethe cat beside the lake8OronemighthavetwosentencesJane loves John andJane flies airplanesandembedone in theother (withappropriatemodification)asJane who flies airplanes loves JohnThesecanbeextendedrecursivelytowhateverlevelofcomplexityisdesiredForexamplewecouldextendthenounphrasetothe dog beside the tree beside the lakeor thesentenceaboutJaneandJohntoJane who flies airplanes that exceed the sound barrier loves John who is prone to selfshydoubtMost languagesmakeuseof recursiveoperationsof thissortmdashalthoughweshallseeinthenextchapterthattheremaybeafewlanguagesthatdonrsquotoperateinthisway
AlthoughitiscommontoprovideillustrationsfromlanguagethemainthemeofthisbookisthatitisinthoughtratherthaninlanguagethatrecursionoriginatesAsPinkerandJackendoffputitldquoTheonlyreasonlanguageneedstoberecursiveisbecauseitsfunction is to express recursive thoughts If there were not anyrecursivethoughtsthemeansofexpressionwouldnotneedrecur-sioneitherrdquo9 Inrememberingepisodesfromthepastforinstanceweessentiallyinsertsequencesofpastconsciousnessintopresent
What Is RecuRsIon
consciousnessor inour interactionswithotherpeoplewemayinsertwhattheyarethinkingintoourownthinkingThesethemesareexploredinlaterchapters
Process and Structure
AssuggestedbyPinkerandJackendoffrsquosdualdefinitionrecursioncanbeunderstoodeitherasaprocess orasastructureThedistinc-tioncanbeimportantArecursiveprocessmayleadtoastructurethatneednotbeseenasitselfrecursiveForexamplesupposeweconstructasequenceofmusicalnoteswithanembeddingroutinebypairingpairsofnoteseachconsistingofa randomlychosennoteplayedonapianowitharandomlychosennoteplayedonaviolinThefirstpairisembeddedinanotherpairandthefour-noteoutputthenembeddedinanotherpairThisprocesscanbecontin-uedindefinitelytocreateasequenceofnotesAsillustratedinfig-ure3thoughthesequencecanbeinterpretednotasarecursivelyembeddedstructurebutasasequenceofpianonotesfollowedbyanequallylongsequenceofviolinnotesThisfailuretodistinguishrecursiveembeddingfromrecursivestructurehasledtosomecon-fusionespeciallyinclaimsaboutrecursioninnonhumanspecies10
Again in his most recent theory on the nature of languageknownastheMinimalistProgram11 NoamChomskyhasarguedthathumanthoughtisgeneratedbyaMergeoperationappliedre-cursivelyThatisunitsaremergedtoformlargerentitiesandthemergedentitiescanbethemselvesmergedtoformstilllargerenti-tiesandsoonThisoperationunderliestheembeddedstructureofhumanlanguagealthoughinChomskyrsquostheoryitappliesstrictlytowhathecallsIshylanguagewhichisthethoughtprocesspreced-ingEshylanguagetheexternal language that isactually spokenorsignedMergecanproducestringsofelementsbetheywordsorelementsofthoughtandalthoughitmaybeappliedrecursivelytoproducehierarchicalstructurethatstructuremaynotbeevidentinthefinaloutputForinstanceevensentencesmightberegardedsimplyaswordsallmergedinunstructuredsequenceasinritual-izedsongsorprayersEverydaylanguagetoomayincludemen-tally undifferentiated clicheacutes and slogansor sequences that are
chapteR 1
P P P P V V V V P P P P V V V V
Figure3ThesequenceofPsandVscanbecreatedeitherbyrecursivelynest-ingPVpairsinPVpairs(left)orbyarrangingasequenceofPsfollowedbyasequenceofanequalnumberofVs(right)Thesequencemightbegeneratedasintheleftpanelbutinterpretedasintherightpanel
highlyautomatedPoliticiansmaybeespeciallypronetothiskindoftalk
Asnotedaboverecursiveprocessesandstructurescaninprin-cipleextendwithoutlimitbutarelimitedinpracticeNeverthelessrecursiondoes give rise to theconcept of infinity itself perhapslimited to the human imaginationAfter all only humans haveacquired theability to count indefinitelyand tounderstand thenatureofinfiniteserieswhereasotherspeciescanatbestmerelyestimatequantityandareaccurateonlyup to somesmallfinitenumber12EveninlanguageweunderstandthatasentencecaninprinciplebeextendedindefinitelyeventhoughinpracticeitcannotbemdashalthoughthenovelistHenryJameshadadamngoodtrySuchunderstandingsareindeedpartofhumanmentalachievementanddependonahumancapacityforrecursivethoughtNeverthelesstheyarenottheprimaryconcernsofthisbook
The appealing aspect of recursion is precisely that it can in principle extend indefinitely to create thoughts (and sentences)ofwhatevercomplexityisrequiredTheideahasanelegantsim-plicitygivingrisetowhatChomskycalledldquodiscreteinfinityrdquo13orWilhelmHumboldt(1767ndash1835)famouslycalledldquotheinfiniteuseoffinitemeansrdquoAndalthoughrecursionislimitedinpracticewecanneverthelessachieveconsiderabledepthsofrecursivethoughtarguablyunsurpassedinanyotherspeciesInchessforexampleaplayermaybeabletothinkrecursivelythreeorfourstepsahead
What Is RecuRsIon
examiningpossiblemovesandcountermovesbutthenumberofpossibilitiessoonmultipliesbeyondthecapacityofthemindtoholdthem
Deeperlevelsofrecursionmaybepossiblewiththeaidofwrit-ingorsimplywithextendedtimeforrehearsalandcontemplationorextendedmemorycapacitythroughartificialmeansTheslowdevelopmentofacomplexmathematicalproofforexamplemayrequiresubtheoremswithinsubtheoremsPlaysornovelsmayin-volverecursiveloopsthatbuildslowlymdashinShakespearersquosTwelfth NightforexampleMariaforeseesthatSirTobywilleagerlyan-ticipate thatOliviawill judgeMalvolio absurdly impertinent tosuppose that she wishes him to regard himself as her preferredsuitor14Asinfictionsoinlifeweallliveinawebofcomplexre-cursiverelationshipsandplanningadinnerpartymayneedcare-fulattentiontowhothinkswhatofwhom
ThestructuresresultingfromrecursiveprocessesneednotrevealthenatureofthoseprocessesjustasaloafofbreadmaynotrevealtheprocessesofkneadingthatwentintothemakingofthebreadorthetasteofwinethepickingandtramplingofthegrapesOftenthoughthestructureofasentenceorstreamofthoughtmayrevealrecursiveembeddingmdashinterpretationofasentencemayrequiretheunderstandingofphrasesembeddedinphrasesregardlessofhowtheembeddingwasactuallyaccomplishedandaninternalunder-standingofastreamofthoughtmayrequirethesegmentationofepisodeswithinepisodes
What Recursion Is not
Recursionisnottheonlydeviceforcreatingsequencesorstruc-tures of potentially infinite length or size I now consider someexamplesthatdonotmeetthecriteriaforrecursion
Repetition
Simple repetition can lead to sequences of potentially infinitelengthbutdoesnot classifyas true recursionForexamplethesentence thatopenschapter9ofAAMilnersquosWinnie the Pooh
10 chapteR 1
goesIt rained and it rained and it rained Thiscouldgoonfor-evermdashoratleastuntilPigletisdrownedmdashbuttherepetitionsimplyconveys the information that it rained rathera lotcausingPig-letsomeennuiItisnotrecursivebecauseeachadditionofand it rained isnotdrivenbythepreviousoneitissimplyaddedatthediscretionofthewriter
InanyeventrepetitiondoesnotdistinguishhumanactivityfromthatofnonhumananimalsBirdsongforexampleisrelentlesslyrepetitivebuteachrepeatedthemedoesnotembellishorqualifythepreviousoneAtmosttherepetitionmightsignalurgencyorsimplysignalcontinuingpresenceasonemightrepeatedlyknockonadoorinthehopeofarousingsomeoneinsideRepetitionisubiquitousinhumanandanimallifeinactivitiesrangingfromtherepeatedjawmovementsineatingtothecuriouslyrepetitivena-tureofsexualactivityThespidernolessiscapableofrepetitionasinWaltWhitmanrsquosLeaves of Grass15
ANOISELESSpatientspiderImarkrsquodwhereonalittlepromontoryitstoodisolatedMarkrsquodhowtoexplorethevacantvastsurroundingItlaunchrsquodforthfilamentfilamentfilamentoutofitselfEverunreelingthemmdashevertirelesslyspeedingthem
Informationcanalsobeaggregatedinnonrecursivefashionaswhentheshort-storywriterSaki(HHMunro)wroteldquoHungerfatigueanddespairinghopelessnesshadnumbedhisbrainrdquo16Ag-gregationofdifferentphrases similarlycompoundsmeaningad-ditivelyaswhenthehistorianPeterHennessywrote
ThemodelofamodernPrimeMinisterwouldbeakindofgrotesquecompositefreakmdashsomeonewiththededicationtodutyofaPeelthephysicalenergyofaGladstonethedetachmentofaSalisburytheballsofaLloydGeorgetheword-powerofaChurchilltheadministrativegiftsof anAttleethe styleof aMacmillanthemanagerialismof aHeathandthesleeprequirementsofaThatcher17
ThesentenceitselfhasrecursiveelementsbuttheaggregationofphrasestodescribethefreakishcompositeisnotrecursiveinthateachdoesnotcallthenextInsteadtheyareeffectivelyelements
What Is RecuRsIon 11
inalistinsertedtoaddinformationNonhumanspeciesmaywellhaveasimilarabilitytoaccumulateinformationaswhenunder-standingapredatoraslargefierceandwithsharpteethandclaws
Iteration
AslightlymoresubtlevariantonrepetitionandaggregationisitshyerationwhereaprocessisrepeatedbutinthiscasethereisinputfromthepreviousapplicationoftheprocessInthisrespect it islikerecursionandindeedconsideredbymathematicianstobelongtotheclassofldquogeneralrecursivefunctionsrdquoForthemainpurposesofthisbookthoughitdoesnotqualifyastruerecursionbecauseeachoutputisdiscardedonceithasbeenenteredintothenextap-plicationThedictionarydefinitionofrecursionthatIgaveearlierinthischapterwasalsoreallyanexampleofiterationratherthanrecursionbecauseyoujustkeepgoingroundandroundtheloopwithoutanyaddedstructureTheiterationsthereforedonotleadtoaddedcomplexity18
Iterativeproceduresareusedincomputationalmathematicstoarriveat increasinglyaccuratesolutionstoaproblemThebasicideaistostartwithapreliminarysolutionmdashperhapsaguessmdashandthenuseaproceduretocomputeanewsolutionThissolutionisthenusedasthestartingpointforthenextcomputationandthenew solution is then the starting point for the next roundThecycle is repeateduntil the solutions stabilize to someacceptablecriterion19Feedbacksystemsoperateinmuchthesamewaytypi-callyasameansofmaintaininghomeostasisForexampleather-mostatmayinvolveasystemforraisingorloweringtemperatureandthegoalistoachievesomegiventemperatureTheactualtem-peratureisfedintothesystemwhichoperatestoraiseorlowerthe temperatureuntil thedesired is reachedThebody isawashwith feedback systems to maintain homeostasis of temperatureironenergybloodcompositionandsoonThemainregulatoristhehypothalamusinthelimbicsystemofthebrainSuchsystemsagaindonotdifferentiatehumansfromotheranimals
Sometimesthedistinctionbetweenrecursionanditerationmaybeamatterof interpretationIn the infinite loopcreatedby the
1 chapteR 1
parodyofPaul CliffordonemightsaythateachbeginningofthestoryisinitiatedbythepreviousonewhichisthenforgottenTheparodyisbestappreciatedthoughifthestoryisseenasanendlessever deepening whirlpoolwith each segment remaining as partofitIrsquomtoldthestoryworksbestifeachsegmentisspokenwithadifferentaccent
Considertoothislinefromawell-knownchildrenrsquosverse
ThisisthedogthatworriedthecatthatkilledtheratthatatethemaltthatlayinthehousethatJackbuilt
TounderstandthissentenceastrulyrecursiveonemustappreciatethatitdescribesastateofaffairsasacomplexwholeandreferstoparticularcasesofadogcatratmalthouseandfellowcalledJackItisnotsimplythestringingtogetherofadogthatworriedacatacatthatkilledaratandsoforthAyoungchildthoughmightprocessitinthispiecemealwayasasuccessionofunrelatedevents
Recursion and evolutionary psychology
Inemphasizingrecursionasaunifyingconcepttheapproachtakeninthisbookcontrastswiththatadoptedbyso-calledevolutionarypsychologistswhohavearguedthatthemindhasmultiplefacetsThebasictenetsofevolutionarypsychologywerelaidoutinthe1992volumeThe Adapted MindeditedbyJeromeBarkowLedaCosmidesandJohnToobyandpopularizedbyStevenPinkerinhis influential 1997 book How the Mind Works20 Thus Pinkerwritesthatthehumanmindldquoisnotasingleorganbutasystemoforganswhichwecanthinkofaspsychologicalfacultiesormentalmodulesrdquo21 Inexaminingpresent-dayhumanbehaviortheevolu-tionarypsychologistrsquosagendaistodiscoverindependentprocessesasthebasicmodulesandrelatethemtoconditionsthatprevailedin the Pleistocene when humans existed primarily as hunter-gatherersAs Pinker puts it the aim is to carve the mind at itsjointssotospeakandldquoreverse-engineerrdquoitscomponentsormod-ulesbacktotheepochduringwhichthehumanmindwasformed
What Is RecuRsIon 1
In this view the mind is really a collection of miniminds eachbeaveringawayonitsownspecificproblemamongwhicharelan-guageandtheoryofmindThishasbeencalledtheSwiss-army-knifemodelofthemindwithabladeforeverypurpose22
Thedangerwith thisapproach is that itbecomes tooeasy topostulate modules and to tellldquojust sordquostories about how theyevolvedsothatthereisariskofreturningtothenow-abandonedinstinctpsychologyoftheearlytwentiethcentury23 Instinctpsy-chologyperishedunderthesheerweightofnumbersmdashtheauthorof one text counted 1594 instincts that had been attributed toanimals and humans24mdashand evolutionary psychology may alsodrowninaseaofmodulesifnotofmixedmetaphorsPinkersug-geststhatwelikepotatochipsbecausefattyfoodswerenutrition-allyvaluableduringthePleistocenebutscarceenoughthattherewasnodangerofobesitywelike landscapeswithtreesbecausetreesprovidesshadeandescapefromdangerouscarnivoresontheAfrica savanna flowers please us because they are markers foredible fruits nutsor tubers amid the greenery of the savannaandsoonldquoTherearemodulesrdquohewritesldquoforobjectsandforcesforanimatebeingsformindsandfornaturalkindslikeanimalsplantsandmineralsrdquo25
This isnot to say that theSwiss-army-knifemodel iswithoutmeritSomeofthepostulatedmodulesdoprovideinsightintothehumanconditionandarereasonablywellfoundedForexamplefoundationalworkinevolutionarypsychologybyLedaCosmidesprovidedgoodevidenceforaldquocheater-detectionmodulerdquomdashanin-stinctiveability todetect thosewhoflout social conventions fortheirowngain26 Arecentstudysuggests thathumanspossessaldquocategory-specificattentionsystemrdquothatisespeciallyadaptedforattendingtoanimals27 oneoftheauthorsJohnToobyisquotedas sayingldquoEvendullanimals likepigeons recruita surpris-ingamountofattentionmdashasdoturtlesresemblingrocksrdquo28 Thisbookisnotintendedtodenythattherearemanyspecificdisposi-tionsthatshapeourmentalandsociallivesrathermyaimistosuggest that therearedeeperaspectsofhumanthoughtthataregovernedbysimilarprinciplesandthatrecursionisoneofthoseprinciplesmdashandperhapsthemostimportantone
1 chapteR 1
Tobefairtoonotallevolutionarypsychologistshaveinsistedthatmodulesarecompletelyencapsulatedshutofffromanycom-munication with one anotherEven Steven Pinker for examplewrites ldquo[Modules] accomplish specialized functions thanks totheirspecializedstructuresbutdonrsquotnecessarilycomeinencap-sulated packagesrdquo29 Steven Mithen although scarcely a card-carryingevolutionarypsychologistarguedthatthehumanmindevolved itsdistinctive characterpreciselybecausepreviously en-capsulatedmodulesbegantoldquoleakrdquocreatingwhathecallsldquocogni-tivefluidityrdquo30 ItisasthoughthemodulesstoppedmindingtheirownbusinessesandbegantogossipMyapproachinthisbookisnotentirelyatoddswiththisviewinthatIargueacommonprin-ciplemightunderlieanumberofourdistinctiveabilities
Others are also beginning to question the Swiss-army-knifemodelofthehumanmindmorestarklyDavidPremackforex-ampleadoptsanapproachsimilartothatofferedinthisbookRe-viewingtheevidencefordiscontinuitybetweenhumansandotheranimalshewritesldquoAnimalcompetenciesaremainlyadaptationsrestrictedtoasinglegoalHumancompetenciesaredomain-generalandservenumerousgoalsrdquo31Thisineffectreversestheevolution-arypsychologyargumentmdashthemindhasbecomelessratherthanmoremodularThetidemaywellbeturning
In any event it is unlikely that recursion canbe considered amoduleAsweshallseerecursionseemstobeanorganizingprin-cipleinverydifferentspheresofhumanmentalactivityfromlan-guage tomemory tomind readingRecursive thinkingprobablydependsonothermentalattributesOneoftheseiswhathasbeentermedworkingmemorywhichholdsinformationinconscious-nessInordertoembedprocesseswithinprocessesitisnecessarytorememberwhereonehadgottointheearlierprocesswhenanembeddedprocesshasbeencompletedForinstanceinasentencelike My dog who eats bananas often gets sickone must holdthe early part of the sentence (My dog) and link it to the nextpart(often gets sick)DwightWReadhasarguedthatnonhumanprimatesevenourclosestrelativesthechimpanzeeshaveawork-ingmemorythatistoolimitedtoallowthiskindofembedding32
Recursionprobablyalsodependsonanexecutiveprocessthator-
What Is RecuRsIon 1
ganizeswhatistobeembeddedinwhatandthismaydependonthefrontallobesofthebrainTheabilitytoorganizeandcarryoutrecursiveoperationsmaythereforedependonseveralprocesses
AlthoughIdonotembracethemodularviewassumedbyevo-lutionarypsychologistsIamatonewiththeminproposingthata distinctively human mind evolved during the Pleistocene theepochthatstretchedfromaround26millionyearsagotosome12000 years agoHow that happened will be told in the laterchaptersofthisbook
plan of the Book
ThebookisdividedintofourpartsPart1dealswithlanguageAlthoughrecursionisnotlimitedto
languageitismostcommonlyinvokedtoexplainwhyhumanlan-guagediffersfromotherformsofanimalcommunicationanin-sightlargelyattributabletoNoamChomskyChapter2discussesthe nature of languagewithparticular emphasis on the role ofrecursionChapter3thenraisestheage-oldquestionofwhetherotheranimalshaveanythingresemblinghumanlanguageChapter4developstheideathatlanguageevolvedfrommanualgesturesmdashanideathatsuggestsgreaterevolutionarycontinuitybetweenhu-mansandotherprimatesthanthemorecommonassumptionthatlanguageemergedfromvocalcalls
Part2dealswithmentaltimetraveltheabilitytobringtomindeventsremovedfromthepresentinbothtimeandplaceChapter5startswithmemoryanddevelopstheideathatmemoryforspecificepisodesisuniquetohumansChapter6extendsthenotionofepi-sodicmemorytotheimaginingofpossiblefutureeventsleadingtotheconceptoftheselfasexistingthroughtimeThisleadstothenotiondiscussedinchapter7thatlanguageitselfevolvedtoenablepeopletosharetheirmemoriesandplansandsotocom-municateabouteventsthatarenotpresentintheimmediateenvi-ronmentThisleadsalsotofictionmdashthetellingofstoriesthatneednotbebasedonfactbutthatnonethelesshonethecapacitytodealwiththeepisodicexigenciesofhumansociallife
1 chapteR 1
Part3dealsanotherrecursiveaspectofhumanthoughtnamelytheoryofmindmdashortheabilitytounderstandwhatothersarethink-ingorfeelingChapter8introducesmind-readingnotasapsychicphenomenonbutasanaturalabilitytoinferthementalperspec-tivesofotherpeopleThisabilityisagaincriticaltosocialcohesionandcooperationChapter9explainshowtheoryofmindwasalsocriticaltotheemergenceoflanguage
Part4delvesmorespecificallyintothequestionofhowthere-cursivemindevolvedChapter10setsthisquestioninthecontextoftheclassicdebatebetweenCartesiandiscontinuityandDarwin-iancontinuityChapter11examinessomeofthestepsbywhichthe hominins33 after splitting from the line leading to modernchimpanzees and bonobos began to assume human-like attri-butesChapter12thenconsidersthefinalsteptoldquomodernrdquoHomo sapiensthesolesurvivinghomininspeciesmdashdominantmanipula-tiveMachiavellianandcapableofponderingourownnatureandstatusontheplanetThatperhapsistheultimatetriumphoftherecursivemind
Chapter13presentsthefinalsummaryandconclusions
What Is RecuRsIon
consciousnessor inour interactionswithotherpeoplewemayinsertwhattheyarethinkingintoourownthinkingThesethemesareexploredinlaterchapters
Process and Structure
AssuggestedbyPinkerandJackendoffrsquosdualdefinitionrecursioncanbeunderstoodeitherasaprocess orasastructureThedistinc-tioncanbeimportantArecursiveprocessmayleadtoastructurethatneednotbeseenasitselfrecursiveForexamplesupposeweconstructasequenceofmusicalnoteswithanembeddingroutinebypairingpairsofnoteseachconsistingofa randomlychosennoteplayedonapianowitharandomlychosennoteplayedonaviolinThefirstpairisembeddedinanotherpairandthefour-noteoutputthenembeddedinanotherpairThisprocesscanbecontin-uedindefinitelytocreateasequenceofnotesAsillustratedinfig-ure3thoughthesequencecanbeinterpretednotasarecursivelyembeddedstructurebutasasequenceofpianonotesfollowedbyanequallylongsequenceofviolinnotesThisfailuretodistinguishrecursiveembeddingfromrecursivestructurehasledtosomecon-fusionespeciallyinclaimsaboutrecursioninnonhumanspecies10
Again in his most recent theory on the nature of languageknownastheMinimalistProgram11 NoamChomskyhasarguedthathumanthoughtisgeneratedbyaMergeoperationappliedre-cursivelyThatisunitsaremergedtoformlargerentitiesandthemergedentitiescanbethemselvesmergedtoformstilllargerenti-tiesandsoonThisoperationunderliestheembeddedstructureofhumanlanguagealthoughinChomskyrsquostheoryitappliesstrictlytowhathecallsIshylanguagewhichisthethoughtprocesspreced-ingEshylanguagetheexternal language that isactually spokenorsignedMergecanproducestringsofelementsbetheywordsorelementsofthoughtandalthoughitmaybeappliedrecursivelytoproducehierarchicalstructurethatstructuremaynotbeevidentinthefinaloutputForinstanceevensentencesmightberegardedsimplyaswordsallmergedinunstructuredsequenceasinritual-izedsongsorprayersEverydaylanguagetoomayincludemen-tally undifferentiated clicheacutes and slogansor sequences that are
chapteR 1
P P P P V V V V P P P P V V V V
Figure3ThesequenceofPsandVscanbecreatedeitherbyrecursivelynest-ingPVpairsinPVpairs(left)orbyarrangingasequenceofPsfollowedbyasequenceofanequalnumberofVs(right)Thesequencemightbegeneratedasintheleftpanelbutinterpretedasintherightpanel
highlyautomatedPoliticiansmaybeespeciallypronetothiskindoftalk
Asnotedaboverecursiveprocessesandstructurescaninprin-cipleextendwithoutlimitbutarelimitedinpracticeNeverthelessrecursiondoes give rise to theconcept of infinity itself perhapslimited to the human imaginationAfter all only humans haveacquired theability to count indefinitelyand tounderstand thenatureofinfiniteserieswhereasotherspeciescanatbestmerelyestimatequantityandareaccurateonlyup to somesmallfinitenumber12EveninlanguageweunderstandthatasentencecaninprinciplebeextendedindefinitelyeventhoughinpracticeitcannotbemdashalthoughthenovelistHenryJameshadadamngoodtrySuchunderstandingsareindeedpartofhumanmentalachievementanddependonahumancapacityforrecursivethoughtNeverthelesstheyarenottheprimaryconcernsofthisbook
The appealing aspect of recursion is precisely that it can in principle extend indefinitely to create thoughts (and sentences)ofwhatevercomplexityisrequiredTheideahasanelegantsim-plicitygivingrisetowhatChomskycalledldquodiscreteinfinityrdquo13orWilhelmHumboldt(1767ndash1835)famouslycalledldquotheinfiniteuseoffinitemeansrdquoAndalthoughrecursionislimitedinpracticewecanneverthelessachieveconsiderabledepthsofrecursivethoughtarguablyunsurpassedinanyotherspeciesInchessforexampleaplayermaybeabletothinkrecursivelythreeorfourstepsahead
What Is RecuRsIon
examiningpossiblemovesandcountermovesbutthenumberofpossibilitiessoonmultipliesbeyondthecapacityofthemindtoholdthem
Deeperlevelsofrecursionmaybepossiblewiththeaidofwrit-ingorsimplywithextendedtimeforrehearsalandcontemplationorextendedmemorycapacitythroughartificialmeansTheslowdevelopmentofacomplexmathematicalproofforexamplemayrequiresubtheoremswithinsubtheoremsPlaysornovelsmayin-volverecursiveloopsthatbuildslowlymdashinShakespearersquosTwelfth NightforexampleMariaforeseesthatSirTobywilleagerlyan-ticipate thatOliviawill judgeMalvolio absurdly impertinent tosuppose that she wishes him to regard himself as her preferredsuitor14Asinfictionsoinlifeweallliveinawebofcomplexre-cursiverelationshipsandplanningadinnerpartymayneedcare-fulattentiontowhothinkswhatofwhom
ThestructuresresultingfromrecursiveprocessesneednotrevealthenatureofthoseprocessesjustasaloafofbreadmaynotrevealtheprocessesofkneadingthatwentintothemakingofthebreadorthetasteofwinethepickingandtramplingofthegrapesOftenthoughthestructureofasentenceorstreamofthoughtmayrevealrecursiveembeddingmdashinterpretationofasentencemayrequiretheunderstandingofphrasesembeddedinphrasesregardlessofhowtheembeddingwasactuallyaccomplishedandaninternalunder-standingofastreamofthoughtmayrequirethesegmentationofepisodeswithinepisodes
What Recursion Is not
Recursionisnottheonlydeviceforcreatingsequencesorstruc-tures of potentially infinite length or size I now consider someexamplesthatdonotmeetthecriteriaforrecursion
Repetition
Simple repetition can lead to sequences of potentially infinitelengthbutdoesnot classifyas true recursionForexamplethesentence thatopenschapter9ofAAMilnersquosWinnie the Pooh
10 chapteR 1
goesIt rained and it rained and it rained Thiscouldgoonfor-evermdashoratleastuntilPigletisdrownedmdashbuttherepetitionsimplyconveys the information that it rained rathera lotcausingPig-letsomeennuiItisnotrecursivebecauseeachadditionofand it rained isnotdrivenbythepreviousoneitissimplyaddedatthediscretionofthewriter
InanyeventrepetitiondoesnotdistinguishhumanactivityfromthatofnonhumananimalsBirdsongforexampleisrelentlesslyrepetitivebuteachrepeatedthemedoesnotembellishorqualifythepreviousoneAtmosttherepetitionmightsignalurgencyorsimplysignalcontinuingpresenceasonemightrepeatedlyknockonadoorinthehopeofarousingsomeoneinsideRepetitionisubiquitousinhumanandanimallifeinactivitiesrangingfromtherepeatedjawmovementsineatingtothecuriouslyrepetitivena-tureofsexualactivityThespidernolessiscapableofrepetitionasinWaltWhitmanrsquosLeaves of Grass15
ANOISELESSpatientspiderImarkrsquodwhereonalittlepromontoryitstoodisolatedMarkrsquodhowtoexplorethevacantvastsurroundingItlaunchrsquodforthfilamentfilamentfilamentoutofitselfEverunreelingthemmdashevertirelesslyspeedingthem
Informationcanalsobeaggregatedinnonrecursivefashionaswhentheshort-storywriterSaki(HHMunro)wroteldquoHungerfatigueanddespairinghopelessnesshadnumbedhisbrainrdquo16Ag-gregationofdifferentphrases similarlycompoundsmeaningad-ditivelyaswhenthehistorianPeterHennessywrote
ThemodelofamodernPrimeMinisterwouldbeakindofgrotesquecompositefreakmdashsomeonewiththededicationtodutyofaPeelthephysicalenergyofaGladstonethedetachmentofaSalisburytheballsofaLloydGeorgetheword-powerofaChurchilltheadministrativegiftsof anAttleethe styleof aMacmillanthemanagerialismof aHeathandthesleeprequirementsofaThatcher17
ThesentenceitselfhasrecursiveelementsbuttheaggregationofphrasestodescribethefreakishcompositeisnotrecursiveinthateachdoesnotcallthenextInsteadtheyareeffectivelyelements
What Is RecuRsIon 11
inalistinsertedtoaddinformationNonhumanspeciesmaywellhaveasimilarabilitytoaccumulateinformationaswhenunder-standingapredatoraslargefierceandwithsharpteethandclaws
Iteration
AslightlymoresubtlevariantonrepetitionandaggregationisitshyerationwhereaprocessisrepeatedbutinthiscasethereisinputfromthepreviousapplicationoftheprocessInthisrespect it islikerecursionandindeedconsideredbymathematicianstobelongtotheclassofldquogeneralrecursivefunctionsrdquoForthemainpurposesofthisbookthoughitdoesnotqualifyastruerecursionbecauseeachoutputisdiscardedonceithasbeenenteredintothenextap-plicationThedictionarydefinitionofrecursionthatIgaveearlierinthischapterwasalsoreallyanexampleofiterationratherthanrecursionbecauseyoujustkeepgoingroundandroundtheloopwithoutanyaddedstructureTheiterationsthereforedonotleadtoaddedcomplexity18
Iterativeproceduresareusedincomputationalmathematicstoarriveat increasinglyaccuratesolutionstoaproblemThebasicideaistostartwithapreliminarysolutionmdashperhapsaguessmdashandthenuseaproceduretocomputeanewsolutionThissolutionisthenusedasthestartingpointforthenextcomputationandthenew solution is then the starting point for the next roundThecycle is repeateduntil the solutions stabilize to someacceptablecriterion19Feedbacksystemsoperateinmuchthesamewaytypi-callyasameansofmaintaininghomeostasisForexampleather-mostatmayinvolveasystemforraisingorloweringtemperatureandthegoalistoachievesomegiventemperatureTheactualtem-peratureisfedintothesystemwhichoperatestoraiseorlowerthe temperatureuntil thedesired is reachedThebody isawashwith feedback systems to maintain homeostasis of temperatureironenergybloodcompositionandsoonThemainregulatoristhehypothalamusinthelimbicsystemofthebrainSuchsystemsagaindonotdifferentiatehumansfromotheranimals
Sometimesthedistinctionbetweenrecursionanditerationmaybeamatterof interpretationIn the infinite loopcreatedby the
1 chapteR 1
parodyofPaul CliffordonemightsaythateachbeginningofthestoryisinitiatedbythepreviousonewhichisthenforgottenTheparodyisbestappreciatedthoughifthestoryisseenasanendlessever deepening whirlpoolwith each segment remaining as partofitIrsquomtoldthestoryworksbestifeachsegmentisspokenwithadifferentaccent
Considertoothislinefromawell-knownchildrenrsquosverse
ThisisthedogthatworriedthecatthatkilledtheratthatatethemaltthatlayinthehousethatJackbuilt
TounderstandthissentenceastrulyrecursiveonemustappreciatethatitdescribesastateofaffairsasacomplexwholeandreferstoparticularcasesofadogcatratmalthouseandfellowcalledJackItisnotsimplythestringingtogetherofadogthatworriedacatacatthatkilledaratandsoforthAyoungchildthoughmightprocessitinthispiecemealwayasasuccessionofunrelatedevents
Recursion and evolutionary psychology
Inemphasizingrecursionasaunifyingconcepttheapproachtakeninthisbookcontrastswiththatadoptedbyso-calledevolutionarypsychologistswhohavearguedthatthemindhasmultiplefacetsThebasictenetsofevolutionarypsychologywerelaidoutinthe1992volumeThe Adapted MindeditedbyJeromeBarkowLedaCosmidesandJohnToobyandpopularizedbyStevenPinkerinhis influential 1997 book How the Mind Works20 Thus Pinkerwritesthatthehumanmindldquoisnotasingleorganbutasystemoforganswhichwecanthinkofaspsychologicalfacultiesormentalmodulesrdquo21 Inexaminingpresent-dayhumanbehaviortheevolu-tionarypsychologistrsquosagendaistodiscoverindependentprocessesasthebasicmodulesandrelatethemtoconditionsthatprevailedin the Pleistocene when humans existed primarily as hunter-gatherersAs Pinker puts it the aim is to carve the mind at itsjointssotospeakandldquoreverse-engineerrdquoitscomponentsormod-ulesbacktotheepochduringwhichthehumanmindwasformed
What Is RecuRsIon 1
In this view the mind is really a collection of miniminds eachbeaveringawayonitsownspecificproblemamongwhicharelan-guageandtheoryofmindThishasbeencalledtheSwiss-army-knifemodelofthemindwithabladeforeverypurpose22
Thedangerwith thisapproach is that itbecomes tooeasy topostulate modules and to tellldquojust sordquostories about how theyevolvedsothatthereisariskofreturningtothenow-abandonedinstinctpsychologyoftheearlytwentiethcentury23 Instinctpsy-chologyperishedunderthesheerweightofnumbersmdashtheauthorof one text counted 1594 instincts that had been attributed toanimals and humans24mdashand evolutionary psychology may alsodrowninaseaofmodulesifnotofmixedmetaphorsPinkersug-geststhatwelikepotatochipsbecausefattyfoodswerenutrition-allyvaluableduringthePleistocenebutscarceenoughthattherewasnodangerofobesitywelike landscapeswithtreesbecausetreesprovidesshadeandescapefromdangerouscarnivoresontheAfrica savanna flowers please us because they are markers foredible fruits nutsor tubers amid the greenery of the savannaandsoonldquoTherearemodulesrdquohewritesldquoforobjectsandforcesforanimatebeingsformindsandfornaturalkindslikeanimalsplantsandmineralsrdquo25
This isnot to say that theSwiss-army-knifemodel iswithoutmeritSomeofthepostulatedmodulesdoprovideinsightintothehumanconditionandarereasonablywellfoundedForexamplefoundationalworkinevolutionarypsychologybyLedaCosmidesprovidedgoodevidenceforaldquocheater-detectionmodulerdquomdashanin-stinctiveability todetect thosewhoflout social conventions fortheirowngain26 Arecentstudysuggests thathumanspossessaldquocategory-specificattentionsystemrdquothatisespeciallyadaptedforattendingtoanimals27 oneoftheauthorsJohnToobyisquotedas sayingldquoEvendullanimals likepigeons recruita surpris-ingamountofattentionmdashasdoturtlesresemblingrocksrdquo28 Thisbookisnotintendedtodenythattherearemanyspecificdisposi-tionsthatshapeourmentalandsociallivesrathermyaimistosuggest that therearedeeperaspectsofhumanthoughtthataregovernedbysimilarprinciplesandthatrecursionisoneofthoseprinciplesmdashandperhapsthemostimportantone
1 chapteR 1
Tobefairtoonotallevolutionarypsychologistshaveinsistedthatmodulesarecompletelyencapsulatedshutofffromanycom-munication with one anotherEven Steven Pinker for examplewrites ldquo[Modules] accomplish specialized functions thanks totheirspecializedstructuresbutdonrsquotnecessarilycomeinencap-sulated packagesrdquo29 Steven Mithen although scarcely a card-carryingevolutionarypsychologistarguedthatthehumanmindevolved itsdistinctive characterpreciselybecausepreviously en-capsulatedmodulesbegantoldquoleakrdquocreatingwhathecallsldquocogni-tivefluidityrdquo30 ItisasthoughthemodulesstoppedmindingtheirownbusinessesandbegantogossipMyapproachinthisbookisnotentirelyatoddswiththisviewinthatIargueacommonprin-ciplemightunderlieanumberofourdistinctiveabilities
Others are also beginning to question the Swiss-army-knifemodelofthehumanmindmorestarklyDavidPremackforex-ampleadoptsanapproachsimilartothatofferedinthisbookRe-viewingtheevidencefordiscontinuitybetweenhumansandotheranimalshewritesldquoAnimalcompetenciesaremainlyadaptationsrestrictedtoasinglegoalHumancompetenciesaredomain-generalandservenumerousgoalsrdquo31Thisineffectreversestheevolution-arypsychologyargumentmdashthemindhasbecomelessratherthanmoremodularThetidemaywellbeturning
In any event it is unlikely that recursion canbe considered amoduleAsweshallseerecursionseemstobeanorganizingprin-cipleinverydifferentspheresofhumanmentalactivityfromlan-guage tomemory tomind readingRecursive thinkingprobablydependsonothermentalattributesOneoftheseiswhathasbeentermedworkingmemorywhichholdsinformationinconscious-nessInordertoembedprocesseswithinprocessesitisnecessarytorememberwhereonehadgottointheearlierprocesswhenanembeddedprocesshasbeencompletedForinstanceinasentencelike My dog who eats bananas often gets sickone must holdthe early part of the sentence (My dog) and link it to the nextpart(often gets sick)DwightWReadhasarguedthatnonhumanprimatesevenourclosestrelativesthechimpanzeeshaveawork-ingmemorythatistoolimitedtoallowthiskindofembedding32
Recursionprobablyalsodependsonanexecutiveprocessthator-
What Is RecuRsIon 1
ganizeswhatistobeembeddedinwhatandthismaydependonthefrontallobesofthebrainTheabilitytoorganizeandcarryoutrecursiveoperationsmaythereforedependonseveralprocesses
AlthoughIdonotembracethemodularviewassumedbyevo-lutionarypsychologistsIamatonewiththeminproposingthata distinctively human mind evolved during the Pleistocene theepochthatstretchedfromaround26millionyearsagotosome12000 years agoHow that happened will be told in the laterchaptersofthisbook
plan of the Book
ThebookisdividedintofourpartsPart1dealswithlanguageAlthoughrecursionisnotlimitedto
languageitismostcommonlyinvokedtoexplainwhyhumanlan-guagediffersfromotherformsofanimalcommunicationanin-sightlargelyattributabletoNoamChomskyChapter2discussesthe nature of languagewithparticular emphasis on the role ofrecursionChapter3thenraisestheage-oldquestionofwhetherotheranimalshaveanythingresemblinghumanlanguageChapter4developstheideathatlanguageevolvedfrommanualgesturesmdashanideathatsuggestsgreaterevolutionarycontinuitybetweenhu-mansandotherprimatesthanthemorecommonassumptionthatlanguageemergedfromvocalcalls
Part2dealswithmentaltimetraveltheabilitytobringtomindeventsremovedfromthepresentinbothtimeandplaceChapter5startswithmemoryanddevelopstheideathatmemoryforspecificepisodesisuniquetohumansChapter6extendsthenotionofepi-sodicmemorytotheimaginingofpossiblefutureeventsleadingtotheconceptoftheselfasexistingthroughtimeThisleadstothenotiondiscussedinchapter7thatlanguageitselfevolvedtoenablepeopletosharetheirmemoriesandplansandsotocom-municateabouteventsthatarenotpresentintheimmediateenvi-ronmentThisleadsalsotofictionmdashthetellingofstoriesthatneednotbebasedonfactbutthatnonethelesshonethecapacitytodealwiththeepisodicexigenciesofhumansociallife
1 chapteR 1
Part3dealsanotherrecursiveaspectofhumanthoughtnamelytheoryofmindmdashortheabilitytounderstandwhatothersarethink-ingorfeelingChapter8introducesmind-readingnotasapsychicphenomenonbutasanaturalabilitytoinferthementalperspec-tivesofotherpeopleThisabilityisagaincriticaltosocialcohesionandcooperationChapter9explainshowtheoryofmindwasalsocriticaltotheemergenceoflanguage
Part4delvesmorespecificallyintothequestionofhowthere-cursivemindevolvedChapter10setsthisquestioninthecontextoftheclassicdebatebetweenCartesiandiscontinuityandDarwin-iancontinuityChapter11examinessomeofthestepsbywhichthe hominins33 after splitting from the line leading to modernchimpanzees and bonobos began to assume human-like attri-butesChapter12thenconsidersthefinalsteptoldquomodernrdquoHomo sapiensthesolesurvivinghomininspeciesmdashdominantmanipula-tiveMachiavellianandcapableofponderingourownnatureandstatusontheplanetThatperhapsistheultimatetriumphoftherecursivemind
Chapter13presentsthefinalsummaryandconclusions
chapteR 1
P P P P V V V V P P P P V V V V
Figure3ThesequenceofPsandVscanbecreatedeitherbyrecursivelynest-ingPVpairsinPVpairs(left)orbyarrangingasequenceofPsfollowedbyasequenceofanequalnumberofVs(right)Thesequencemightbegeneratedasintheleftpanelbutinterpretedasintherightpanel
highlyautomatedPoliticiansmaybeespeciallypronetothiskindoftalk
Asnotedaboverecursiveprocessesandstructurescaninprin-cipleextendwithoutlimitbutarelimitedinpracticeNeverthelessrecursiondoes give rise to theconcept of infinity itself perhapslimited to the human imaginationAfter all only humans haveacquired theability to count indefinitelyand tounderstand thenatureofinfiniteserieswhereasotherspeciescanatbestmerelyestimatequantityandareaccurateonlyup to somesmallfinitenumber12EveninlanguageweunderstandthatasentencecaninprinciplebeextendedindefinitelyeventhoughinpracticeitcannotbemdashalthoughthenovelistHenryJameshadadamngoodtrySuchunderstandingsareindeedpartofhumanmentalachievementanddependonahumancapacityforrecursivethoughtNeverthelesstheyarenottheprimaryconcernsofthisbook
The appealing aspect of recursion is precisely that it can in principle extend indefinitely to create thoughts (and sentences)ofwhatevercomplexityisrequiredTheideahasanelegantsim-plicitygivingrisetowhatChomskycalledldquodiscreteinfinityrdquo13orWilhelmHumboldt(1767ndash1835)famouslycalledldquotheinfiniteuseoffinitemeansrdquoAndalthoughrecursionislimitedinpracticewecanneverthelessachieveconsiderabledepthsofrecursivethoughtarguablyunsurpassedinanyotherspeciesInchessforexampleaplayermaybeabletothinkrecursivelythreeorfourstepsahead
What Is RecuRsIon
examiningpossiblemovesandcountermovesbutthenumberofpossibilitiessoonmultipliesbeyondthecapacityofthemindtoholdthem
Deeperlevelsofrecursionmaybepossiblewiththeaidofwrit-ingorsimplywithextendedtimeforrehearsalandcontemplationorextendedmemorycapacitythroughartificialmeansTheslowdevelopmentofacomplexmathematicalproofforexamplemayrequiresubtheoremswithinsubtheoremsPlaysornovelsmayin-volverecursiveloopsthatbuildslowlymdashinShakespearersquosTwelfth NightforexampleMariaforeseesthatSirTobywilleagerlyan-ticipate thatOliviawill judgeMalvolio absurdly impertinent tosuppose that she wishes him to regard himself as her preferredsuitor14Asinfictionsoinlifeweallliveinawebofcomplexre-cursiverelationshipsandplanningadinnerpartymayneedcare-fulattentiontowhothinkswhatofwhom
ThestructuresresultingfromrecursiveprocessesneednotrevealthenatureofthoseprocessesjustasaloafofbreadmaynotrevealtheprocessesofkneadingthatwentintothemakingofthebreadorthetasteofwinethepickingandtramplingofthegrapesOftenthoughthestructureofasentenceorstreamofthoughtmayrevealrecursiveembeddingmdashinterpretationofasentencemayrequiretheunderstandingofphrasesembeddedinphrasesregardlessofhowtheembeddingwasactuallyaccomplishedandaninternalunder-standingofastreamofthoughtmayrequirethesegmentationofepisodeswithinepisodes
What Recursion Is not
Recursionisnottheonlydeviceforcreatingsequencesorstruc-tures of potentially infinite length or size I now consider someexamplesthatdonotmeetthecriteriaforrecursion
Repetition
Simple repetition can lead to sequences of potentially infinitelengthbutdoesnot classifyas true recursionForexamplethesentence thatopenschapter9ofAAMilnersquosWinnie the Pooh
10 chapteR 1
goesIt rained and it rained and it rained Thiscouldgoonfor-evermdashoratleastuntilPigletisdrownedmdashbuttherepetitionsimplyconveys the information that it rained rathera lotcausingPig-letsomeennuiItisnotrecursivebecauseeachadditionofand it rained isnotdrivenbythepreviousoneitissimplyaddedatthediscretionofthewriter
InanyeventrepetitiondoesnotdistinguishhumanactivityfromthatofnonhumananimalsBirdsongforexampleisrelentlesslyrepetitivebuteachrepeatedthemedoesnotembellishorqualifythepreviousoneAtmosttherepetitionmightsignalurgencyorsimplysignalcontinuingpresenceasonemightrepeatedlyknockonadoorinthehopeofarousingsomeoneinsideRepetitionisubiquitousinhumanandanimallifeinactivitiesrangingfromtherepeatedjawmovementsineatingtothecuriouslyrepetitivena-tureofsexualactivityThespidernolessiscapableofrepetitionasinWaltWhitmanrsquosLeaves of Grass15
ANOISELESSpatientspiderImarkrsquodwhereonalittlepromontoryitstoodisolatedMarkrsquodhowtoexplorethevacantvastsurroundingItlaunchrsquodforthfilamentfilamentfilamentoutofitselfEverunreelingthemmdashevertirelesslyspeedingthem
Informationcanalsobeaggregatedinnonrecursivefashionaswhentheshort-storywriterSaki(HHMunro)wroteldquoHungerfatigueanddespairinghopelessnesshadnumbedhisbrainrdquo16Ag-gregationofdifferentphrases similarlycompoundsmeaningad-ditivelyaswhenthehistorianPeterHennessywrote
ThemodelofamodernPrimeMinisterwouldbeakindofgrotesquecompositefreakmdashsomeonewiththededicationtodutyofaPeelthephysicalenergyofaGladstonethedetachmentofaSalisburytheballsofaLloydGeorgetheword-powerofaChurchilltheadministrativegiftsof anAttleethe styleof aMacmillanthemanagerialismof aHeathandthesleeprequirementsofaThatcher17
ThesentenceitselfhasrecursiveelementsbuttheaggregationofphrasestodescribethefreakishcompositeisnotrecursiveinthateachdoesnotcallthenextInsteadtheyareeffectivelyelements
What Is RecuRsIon 11
inalistinsertedtoaddinformationNonhumanspeciesmaywellhaveasimilarabilitytoaccumulateinformationaswhenunder-standingapredatoraslargefierceandwithsharpteethandclaws
Iteration
AslightlymoresubtlevariantonrepetitionandaggregationisitshyerationwhereaprocessisrepeatedbutinthiscasethereisinputfromthepreviousapplicationoftheprocessInthisrespect it islikerecursionandindeedconsideredbymathematicianstobelongtotheclassofldquogeneralrecursivefunctionsrdquoForthemainpurposesofthisbookthoughitdoesnotqualifyastruerecursionbecauseeachoutputisdiscardedonceithasbeenenteredintothenextap-plicationThedictionarydefinitionofrecursionthatIgaveearlierinthischapterwasalsoreallyanexampleofiterationratherthanrecursionbecauseyoujustkeepgoingroundandroundtheloopwithoutanyaddedstructureTheiterationsthereforedonotleadtoaddedcomplexity18
Iterativeproceduresareusedincomputationalmathematicstoarriveat increasinglyaccuratesolutionstoaproblemThebasicideaistostartwithapreliminarysolutionmdashperhapsaguessmdashandthenuseaproceduretocomputeanewsolutionThissolutionisthenusedasthestartingpointforthenextcomputationandthenew solution is then the starting point for the next roundThecycle is repeateduntil the solutions stabilize to someacceptablecriterion19Feedbacksystemsoperateinmuchthesamewaytypi-callyasameansofmaintaininghomeostasisForexampleather-mostatmayinvolveasystemforraisingorloweringtemperatureandthegoalistoachievesomegiventemperatureTheactualtem-peratureisfedintothesystemwhichoperatestoraiseorlowerthe temperatureuntil thedesired is reachedThebody isawashwith feedback systems to maintain homeostasis of temperatureironenergybloodcompositionandsoonThemainregulatoristhehypothalamusinthelimbicsystemofthebrainSuchsystemsagaindonotdifferentiatehumansfromotheranimals
Sometimesthedistinctionbetweenrecursionanditerationmaybeamatterof interpretationIn the infinite loopcreatedby the
1 chapteR 1
parodyofPaul CliffordonemightsaythateachbeginningofthestoryisinitiatedbythepreviousonewhichisthenforgottenTheparodyisbestappreciatedthoughifthestoryisseenasanendlessever deepening whirlpoolwith each segment remaining as partofitIrsquomtoldthestoryworksbestifeachsegmentisspokenwithadifferentaccent
Considertoothislinefromawell-knownchildrenrsquosverse
ThisisthedogthatworriedthecatthatkilledtheratthatatethemaltthatlayinthehousethatJackbuilt
TounderstandthissentenceastrulyrecursiveonemustappreciatethatitdescribesastateofaffairsasacomplexwholeandreferstoparticularcasesofadogcatratmalthouseandfellowcalledJackItisnotsimplythestringingtogetherofadogthatworriedacatacatthatkilledaratandsoforthAyoungchildthoughmightprocessitinthispiecemealwayasasuccessionofunrelatedevents
Recursion and evolutionary psychology
Inemphasizingrecursionasaunifyingconcepttheapproachtakeninthisbookcontrastswiththatadoptedbyso-calledevolutionarypsychologistswhohavearguedthatthemindhasmultiplefacetsThebasictenetsofevolutionarypsychologywerelaidoutinthe1992volumeThe Adapted MindeditedbyJeromeBarkowLedaCosmidesandJohnToobyandpopularizedbyStevenPinkerinhis influential 1997 book How the Mind Works20 Thus Pinkerwritesthatthehumanmindldquoisnotasingleorganbutasystemoforganswhichwecanthinkofaspsychologicalfacultiesormentalmodulesrdquo21 Inexaminingpresent-dayhumanbehaviortheevolu-tionarypsychologistrsquosagendaistodiscoverindependentprocessesasthebasicmodulesandrelatethemtoconditionsthatprevailedin the Pleistocene when humans existed primarily as hunter-gatherersAs Pinker puts it the aim is to carve the mind at itsjointssotospeakandldquoreverse-engineerrdquoitscomponentsormod-ulesbacktotheepochduringwhichthehumanmindwasformed
What Is RecuRsIon 1
In this view the mind is really a collection of miniminds eachbeaveringawayonitsownspecificproblemamongwhicharelan-guageandtheoryofmindThishasbeencalledtheSwiss-army-knifemodelofthemindwithabladeforeverypurpose22
Thedangerwith thisapproach is that itbecomes tooeasy topostulate modules and to tellldquojust sordquostories about how theyevolvedsothatthereisariskofreturningtothenow-abandonedinstinctpsychologyoftheearlytwentiethcentury23 Instinctpsy-chologyperishedunderthesheerweightofnumbersmdashtheauthorof one text counted 1594 instincts that had been attributed toanimals and humans24mdashand evolutionary psychology may alsodrowninaseaofmodulesifnotofmixedmetaphorsPinkersug-geststhatwelikepotatochipsbecausefattyfoodswerenutrition-allyvaluableduringthePleistocenebutscarceenoughthattherewasnodangerofobesitywelike landscapeswithtreesbecausetreesprovidesshadeandescapefromdangerouscarnivoresontheAfrica savanna flowers please us because they are markers foredible fruits nutsor tubers amid the greenery of the savannaandsoonldquoTherearemodulesrdquohewritesldquoforobjectsandforcesforanimatebeingsformindsandfornaturalkindslikeanimalsplantsandmineralsrdquo25
This isnot to say that theSwiss-army-knifemodel iswithoutmeritSomeofthepostulatedmodulesdoprovideinsightintothehumanconditionandarereasonablywellfoundedForexamplefoundationalworkinevolutionarypsychologybyLedaCosmidesprovidedgoodevidenceforaldquocheater-detectionmodulerdquomdashanin-stinctiveability todetect thosewhoflout social conventions fortheirowngain26 Arecentstudysuggests thathumanspossessaldquocategory-specificattentionsystemrdquothatisespeciallyadaptedforattendingtoanimals27 oneoftheauthorsJohnToobyisquotedas sayingldquoEvendullanimals likepigeons recruita surpris-ingamountofattentionmdashasdoturtlesresemblingrocksrdquo28 Thisbookisnotintendedtodenythattherearemanyspecificdisposi-tionsthatshapeourmentalandsociallivesrathermyaimistosuggest that therearedeeperaspectsofhumanthoughtthataregovernedbysimilarprinciplesandthatrecursionisoneofthoseprinciplesmdashandperhapsthemostimportantone
1 chapteR 1
Tobefairtoonotallevolutionarypsychologistshaveinsistedthatmodulesarecompletelyencapsulatedshutofffromanycom-munication with one anotherEven Steven Pinker for examplewrites ldquo[Modules] accomplish specialized functions thanks totheirspecializedstructuresbutdonrsquotnecessarilycomeinencap-sulated packagesrdquo29 Steven Mithen although scarcely a card-carryingevolutionarypsychologistarguedthatthehumanmindevolved itsdistinctive characterpreciselybecausepreviously en-capsulatedmodulesbegantoldquoleakrdquocreatingwhathecallsldquocogni-tivefluidityrdquo30 ItisasthoughthemodulesstoppedmindingtheirownbusinessesandbegantogossipMyapproachinthisbookisnotentirelyatoddswiththisviewinthatIargueacommonprin-ciplemightunderlieanumberofourdistinctiveabilities
Others are also beginning to question the Swiss-army-knifemodelofthehumanmindmorestarklyDavidPremackforex-ampleadoptsanapproachsimilartothatofferedinthisbookRe-viewingtheevidencefordiscontinuitybetweenhumansandotheranimalshewritesldquoAnimalcompetenciesaremainlyadaptationsrestrictedtoasinglegoalHumancompetenciesaredomain-generalandservenumerousgoalsrdquo31Thisineffectreversestheevolution-arypsychologyargumentmdashthemindhasbecomelessratherthanmoremodularThetidemaywellbeturning
In any event it is unlikely that recursion canbe considered amoduleAsweshallseerecursionseemstobeanorganizingprin-cipleinverydifferentspheresofhumanmentalactivityfromlan-guage tomemory tomind readingRecursive thinkingprobablydependsonothermentalattributesOneoftheseiswhathasbeentermedworkingmemorywhichholdsinformationinconscious-nessInordertoembedprocesseswithinprocessesitisnecessarytorememberwhereonehadgottointheearlierprocesswhenanembeddedprocesshasbeencompletedForinstanceinasentencelike My dog who eats bananas often gets sickone must holdthe early part of the sentence (My dog) and link it to the nextpart(often gets sick)DwightWReadhasarguedthatnonhumanprimatesevenourclosestrelativesthechimpanzeeshaveawork-ingmemorythatistoolimitedtoallowthiskindofembedding32
Recursionprobablyalsodependsonanexecutiveprocessthator-
What Is RecuRsIon 1
ganizeswhatistobeembeddedinwhatandthismaydependonthefrontallobesofthebrainTheabilitytoorganizeandcarryoutrecursiveoperationsmaythereforedependonseveralprocesses
AlthoughIdonotembracethemodularviewassumedbyevo-lutionarypsychologistsIamatonewiththeminproposingthata distinctively human mind evolved during the Pleistocene theepochthatstretchedfromaround26millionyearsagotosome12000 years agoHow that happened will be told in the laterchaptersofthisbook
plan of the Book
ThebookisdividedintofourpartsPart1dealswithlanguageAlthoughrecursionisnotlimitedto
languageitismostcommonlyinvokedtoexplainwhyhumanlan-guagediffersfromotherformsofanimalcommunicationanin-sightlargelyattributabletoNoamChomskyChapter2discussesthe nature of languagewithparticular emphasis on the role ofrecursionChapter3thenraisestheage-oldquestionofwhetherotheranimalshaveanythingresemblinghumanlanguageChapter4developstheideathatlanguageevolvedfrommanualgesturesmdashanideathatsuggestsgreaterevolutionarycontinuitybetweenhu-mansandotherprimatesthanthemorecommonassumptionthatlanguageemergedfromvocalcalls
Part2dealswithmentaltimetraveltheabilitytobringtomindeventsremovedfromthepresentinbothtimeandplaceChapter5startswithmemoryanddevelopstheideathatmemoryforspecificepisodesisuniquetohumansChapter6extendsthenotionofepi-sodicmemorytotheimaginingofpossiblefutureeventsleadingtotheconceptoftheselfasexistingthroughtimeThisleadstothenotiondiscussedinchapter7thatlanguageitselfevolvedtoenablepeopletosharetheirmemoriesandplansandsotocom-municateabouteventsthatarenotpresentintheimmediateenvi-ronmentThisleadsalsotofictionmdashthetellingofstoriesthatneednotbebasedonfactbutthatnonethelesshonethecapacitytodealwiththeepisodicexigenciesofhumansociallife
1 chapteR 1
Part3dealsanotherrecursiveaspectofhumanthoughtnamelytheoryofmindmdashortheabilitytounderstandwhatothersarethink-ingorfeelingChapter8introducesmind-readingnotasapsychicphenomenonbutasanaturalabilitytoinferthementalperspec-tivesofotherpeopleThisabilityisagaincriticaltosocialcohesionandcooperationChapter9explainshowtheoryofmindwasalsocriticaltotheemergenceoflanguage
Part4delvesmorespecificallyintothequestionofhowthere-cursivemindevolvedChapter10setsthisquestioninthecontextoftheclassicdebatebetweenCartesiandiscontinuityandDarwin-iancontinuityChapter11examinessomeofthestepsbywhichthe hominins33 after splitting from the line leading to modernchimpanzees and bonobos began to assume human-like attri-butesChapter12thenconsidersthefinalsteptoldquomodernrdquoHomo sapiensthesolesurvivinghomininspeciesmdashdominantmanipula-tiveMachiavellianandcapableofponderingourownnatureandstatusontheplanetThatperhapsistheultimatetriumphoftherecursivemind
Chapter13presentsthefinalsummaryandconclusions
What Is RecuRsIon
examiningpossiblemovesandcountermovesbutthenumberofpossibilitiessoonmultipliesbeyondthecapacityofthemindtoholdthem
Deeperlevelsofrecursionmaybepossiblewiththeaidofwrit-ingorsimplywithextendedtimeforrehearsalandcontemplationorextendedmemorycapacitythroughartificialmeansTheslowdevelopmentofacomplexmathematicalproofforexamplemayrequiresubtheoremswithinsubtheoremsPlaysornovelsmayin-volverecursiveloopsthatbuildslowlymdashinShakespearersquosTwelfth NightforexampleMariaforeseesthatSirTobywilleagerlyan-ticipate thatOliviawill judgeMalvolio absurdly impertinent tosuppose that she wishes him to regard himself as her preferredsuitor14Asinfictionsoinlifeweallliveinawebofcomplexre-cursiverelationshipsandplanningadinnerpartymayneedcare-fulattentiontowhothinkswhatofwhom
ThestructuresresultingfromrecursiveprocessesneednotrevealthenatureofthoseprocessesjustasaloafofbreadmaynotrevealtheprocessesofkneadingthatwentintothemakingofthebreadorthetasteofwinethepickingandtramplingofthegrapesOftenthoughthestructureofasentenceorstreamofthoughtmayrevealrecursiveembeddingmdashinterpretationofasentencemayrequiretheunderstandingofphrasesembeddedinphrasesregardlessofhowtheembeddingwasactuallyaccomplishedandaninternalunder-standingofastreamofthoughtmayrequirethesegmentationofepisodeswithinepisodes
What Recursion Is not
Recursionisnottheonlydeviceforcreatingsequencesorstruc-tures of potentially infinite length or size I now consider someexamplesthatdonotmeetthecriteriaforrecursion
Repetition
Simple repetition can lead to sequences of potentially infinitelengthbutdoesnot classifyas true recursionForexamplethesentence thatopenschapter9ofAAMilnersquosWinnie the Pooh
10 chapteR 1
goesIt rained and it rained and it rained Thiscouldgoonfor-evermdashoratleastuntilPigletisdrownedmdashbuttherepetitionsimplyconveys the information that it rained rathera lotcausingPig-letsomeennuiItisnotrecursivebecauseeachadditionofand it rained isnotdrivenbythepreviousoneitissimplyaddedatthediscretionofthewriter
InanyeventrepetitiondoesnotdistinguishhumanactivityfromthatofnonhumananimalsBirdsongforexampleisrelentlesslyrepetitivebuteachrepeatedthemedoesnotembellishorqualifythepreviousoneAtmosttherepetitionmightsignalurgencyorsimplysignalcontinuingpresenceasonemightrepeatedlyknockonadoorinthehopeofarousingsomeoneinsideRepetitionisubiquitousinhumanandanimallifeinactivitiesrangingfromtherepeatedjawmovementsineatingtothecuriouslyrepetitivena-tureofsexualactivityThespidernolessiscapableofrepetitionasinWaltWhitmanrsquosLeaves of Grass15
ANOISELESSpatientspiderImarkrsquodwhereonalittlepromontoryitstoodisolatedMarkrsquodhowtoexplorethevacantvastsurroundingItlaunchrsquodforthfilamentfilamentfilamentoutofitselfEverunreelingthemmdashevertirelesslyspeedingthem
Informationcanalsobeaggregatedinnonrecursivefashionaswhentheshort-storywriterSaki(HHMunro)wroteldquoHungerfatigueanddespairinghopelessnesshadnumbedhisbrainrdquo16Ag-gregationofdifferentphrases similarlycompoundsmeaningad-ditivelyaswhenthehistorianPeterHennessywrote
ThemodelofamodernPrimeMinisterwouldbeakindofgrotesquecompositefreakmdashsomeonewiththededicationtodutyofaPeelthephysicalenergyofaGladstonethedetachmentofaSalisburytheballsofaLloydGeorgetheword-powerofaChurchilltheadministrativegiftsof anAttleethe styleof aMacmillanthemanagerialismof aHeathandthesleeprequirementsofaThatcher17
ThesentenceitselfhasrecursiveelementsbuttheaggregationofphrasestodescribethefreakishcompositeisnotrecursiveinthateachdoesnotcallthenextInsteadtheyareeffectivelyelements
What Is RecuRsIon 11
inalistinsertedtoaddinformationNonhumanspeciesmaywellhaveasimilarabilitytoaccumulateinformationaswhenunder-standingapredatoraslargefierceandwithsharpteethandclaws
Iteration
AslightlymoresubtlevariantonrepetitionandaggregationisitshyerationwhereaprocessisrepeatedbutinthiscasethereisinputfromthepreviousapplicationoftheprocessInthisrespect it islikerecursionandindeedconsideredbymathematicianstobelongtotheclassofldquogeneralrecursivefunctionsrdquoForthemainpurposesofthisbookthoughitdoesnotqualifyastruerecursionbecauseeachoutputisdiscardedonceithasbeenenteredintothenextap-plicationThedictionarydefinitionofrecursionthatIgaveearlierinthischapterwasalsoreallyanexampleofiterationratherthanrecursionbecauseyoujustkeepgoingroundandroundtheloopwithoutanyaddedstructureTheiterationsthereforedonotleadtoaddedcomplexity18
Iterativeproceduresareusedincomputationalmathematicstoarriveat increasinglyaccuratesolutionstoaproblemThebasicideaistostartwithapreliminarysolutionmdashperhapsaguessmdashandthenuseaproceduretocomputeanewsolutionThissolutionisthenusedasthestartingpointforthenextcomputationandthenew solution is then the starting point for the next roundThecycle is repeateduntil the solutions stabilize to someacceptablecriterion19Feedbacksystemsoperateinmuchthesamewaytypi-callyasameansofmaintaininghomeostasisForexampleather-mostatmayinvolveasystemforraisingorloweringtemperatureandthegoalistoachievesomegiventemperatureTheactualtem-peratureisfedintothesystemwhichoperatestoraiseorlowerthe temperatureuntil thedesired is reachedThebody isawashwith feedback systems to maintain homeostasis of temperatureironenergybloodcompositionandsoonThemainregulatoristhehypothalamusinthelimbicsystemofthebrainSuchsystemsagaindonotdifferentiatehumansfromotheranimals
Sometimesthedistinctionbetweenrecursionanditerationmaybeamatterof interpretationIn the infinite loopcreatedby the
1 chapteR 1
parodyofPaul CliffordonemightsaythateachbeginningofthestoryisinitiatedbythepreviousonewhichisthenforgottenTheparodyisbestappreciatedthoughifthestoryisseenasanendlessever deepening whirlpoolwith each segment remaining as partofitIrsquomtoldthestoryworksbestifeachsegmentisspokenwithadifferentaccent
Considertoothislinefromawell-knownchildrenrsquosverse
ThisisthedogthatworriedthecatthatkilledtheratthatatethemaltthatlayinthehousethatJackbuilt
TounderstandthissentenceastrulyrecursiveonemustappreciatethatitdescribesastateofaffairsasacomplexwholeandreferstoparticularcasesofadogcatratmalthouseandfellowcalledJackItisnotsimplythestringingtogetherofadogthatworriedacatacatthatkilledaratandsoforthAyoungchildthoughmightprocessitinthispiecemealwayasasuccessionofunrelatedevents
Recursion and evolutionary psychology
Inemphasizingrecursionasaunifyingconcepttheapproachtakeninthisbookcontrastswiththatadoptedbyso-calledevolutionarypsychologistswhohavearguedthatthemindhasmultiplefacetsThebasictenetsofevolutionarypsychologywerelaidoutinthe1992volumeThe Adapted MindeditedbyJeromeBarkowLedaCosmidesandJohnToobyandpopularizedbyStevenPinkerinhis influential 1997 book How the Mind Works20 Thus Pinkerwritesthatthehumanmindldquoisnotasingleorganbutasystemoforganswhichwecanthinkofaspsychologicalfacultiesormentalmodulesrdquo21 Inexaminingpresent-dayhumanbehaviortheevolu-tionarypsychologistrsquosagendaistodiscoverindependentprocessesasthebasicmodulesandrelatethemtoconditionsthatprevailedin the Pleistocene when humans existed primarily as hunter-gatherersAs Pinker puts it the aim is to carve the mind at itsjointssotospeakandldquoreverse-engineerrdquoitscomponentsormod-ulesbacktotheepochduringwhichthehumanmindwasformed
What Is RecuRsIon 1
In this view the mind is really a collection of miniminds eachbeaveringawayonitsownspecificproblemamongwhicharelan-guageandtheoryofmindThishasbeencalledtheSwiss-army-knifemodelofthemindwithabladeforeverypurpose22
Thedangerwith thisapproach is that itbecomes tooeasy topostulate modules and to tellldquojust sordquostories about how theyevolvedsothatthereisariskofreturningtothenow-abandonedinstinctpsychologyoftheearlytwentiethcentury23 Instinctpsy-chologyperishedunderthesheerweightofnumbersmdashtheauthorof one text counted 1594 instincts that had been attributed toanimals and humans24mdashand evolutionary psychology may alsodrowninaseaofmodulesifnotofmixedmetaphorsPinkersug-geststhatwelikepotatochipsbecausefattyfoodswerenutrition-allyvaluableduringthePleistocenebutscarceenoughthattherewasnodangerofobesitywelike landscapeswithtreesbecausetreesprovidesshadeandescapefromdangerouscarnivoresontheAfrica savanna flowers please us because they are markers foredible fruits nutsor tubers amid the greenery of the savannaandsoonldquoTherearemodulesrdquohewritesldquoforobjectsandforcesforanimatebeingsformindsandfornaturalkindslikeanimalsplantsandmineralsrdquo25
This isnot to say that theSwiss-army-knifemodel iswithoutmeritSomeofthepostulatedmodulesdoprovideinsightintothehumanconditionandarereasonablywellfoundedForexamplefoundationalworkinevolutionarypsychologybyLedaCosmidesprovidedgoodevidenceforaldquocheater-detectionmodulerdquomdashanin-stinctiveability todetect thosewhoflout social conventions fortheirowngain26 Arecentstudysuggests thathumanspossessaldquocategory-specificattentionsystemrdquothatisespeciallyadaptedforattendingtoanimals27 oneoftheauthorsJohnToobyisquotedas sayingldquoEvendullanimals likepigeons recruita surpris-ingamountofattentionmdashasdoturtlesresemblingrocksrdquo28 Thisbookisnotintendedtodenythattherearemanyspecificdisposi-tionsthatshapeourmentalandsociallivesrathermyaimistosuggest that therearedeeperaspectsofhumanthoughtthataregovernedbysimilarprinciplesandthatrecursionisoneofthoseprinciplesmdashandperhapsthemostimportantone
1 chapteR 1
Tobefairtoonotallevolutionarypsychologistshaveinsistedthatmodulesarecompletelyencapsulatedshutofffromanycom-munication with one anotherEven Steven Pinker for examplewrites ldquo[Modules] accomplish specialized functions thanks totheirspecializedstructuresbutdonrsquotnecessarilycomeinencap-sulated packagesrdquo29 Steven Mithen although scarcely a card-carryingevolutionarypsychologistarguedthatthehumanmindevolved itsdistinctive characterpreciselybecausepreviously en-capsulatedmodulesbegantoldquoleakrdquocreatingwhathecallsldquocogni-tivefluidityrdquo30 ItisasthoughthemodulesstoppedmindingtheirownbusinessesandbegantogossipMyapproachinthisbookisnotentirelyatoddswiththisviewinthatIargueacommonprin-ciplemightunderlieanumberofourdistinctiveabilities
Others are also beginning to question the Swiss-army-knifemodelofthehumanmindmorestarklyDavidPremackforex-ampleadoptsanapproachsimilartothatofferedinthisbookRe-viewingtheevidencefordiscontinuitybetweenhumansandotheranimalshewritesldquoAnimalcompetenciesaremainlyadaptationsrestrictedtoasinglegoalHumancompetenciesaredomain-generalandservenumerousgoalsrdquo31Thisineffectreversestheevolution-arypsychologyargumentmdashthemindhasbecomelessratherthanmoremodularThetidemaywellbeturning
In any event it is unlikely that recursion canbe considered amoduleAsweshallseerecursionseemstobeanorganizingprin-cipleinverydifferentspheresofhumanmentalactivityfromlan-guage tomemory tomind readingRecursive thinkingprobablydependsonothermentalattributesOneoftheseiswhathasbeentermedworkingmemorywhichholdsinformationinconscious-nessInordertoembedprocesseswithinprocessesitisnecessarytorememberwhereonehadgottointheearlierprocesswhenanembeddedprocesshasbeencompletedForinstanceinasentencelike My dog who eats bananas often gets sickone must holdthe early part of the sentence (My dog) and link it to the nextpart(often gets sick)DwightWReadhasarguedthatnonhumanprimatesevenourclosestrelativesthechimpanzeeshaveawork-ingmemorythatistoolimitedtoallowthiskindofembedding32
Recursionprobablyalsodependsonanexecutiveprocessthator-
What Is RecuRsIon 1
ganizeswhatistobeembeddedinwhatandthismaydependonthefrontallobesofthebrainTheabilitytoorganizeandcarryoutrecursiveoperationsmaythereforedependonseveralprocesses
AlthoughIdonotembracethemodularviewassumedbyevo-lutionarypsychologistsIamatonewiththeminproposingthata distinctively human mind evolved during the Pleistocene theepochthatstretchedfromaround26millionyearsagotosome12000 years agoHow that happened will be told in the laterchaptersofthisbook
plan of the Book
ThebookisdividedintofourpartsPart1dealswithlanguageAlthoughrecursionisnotlimitedto
languageitismostcommonlyinvokedtoexplainwhyhumanlan-guagediffersfromotherformsofanimalcommunicationanin-sightlargelyattributabletoNoamChomskyChapter2discussesthe nature of languagewithparticular emphasis on the role ofrecursionChapter3thenraisestheage-oldquestionofwhetherotheranimalshaveanythingresemblinghumanlanguageChapter4developstheideathatlanguageevolvedfrommanualgesturesmdashanideathatsuggestsgreaterevolutionarycontinuitybetweenhu-mansandotherprimatesthanthemorecommonassumptionthatlanguageemergedfromvocalcalls
Part2dealswithmentaltimetraveltheabilitytobringtomindeventsremovedfromthepresentinbothtimeandplaceChapter5startswithmemoryanddevelopstheideathatmemoryforspecificepisodesisuniquetohumansChapter6extendsthenotionofepi-sodicmemorytotheimaginingofpossiblefutureeventsleadingtotheconceptoftheselfasexistingthroughtimeThisleadstothenotiondiscussedinchapter7thatlanguageitselfevolvedtoenablepeopletosharetheirmemoriesandplansandsotocom-municateabouteventsthatarenotpresentintheimmediateenvi-ronmentThisleadsalsotofictionmdashthetellingofstoriesthatneednotbebasedonfactbutthatnonethelesshonethecapacitytodealwiththeepisodicexigenciesofhumansociallife
1 chapteR 1
Part3dealsanotherrecursiveaspectofhumanthoughtnamelytheoryofmindmdashortheabilitytounderstandwhatothersarethink-ingorfeelingChapter8introducesmind-readingnotasapsychicphenomenonbutasanaturalabilitytoinferthementalperspec-tivesofotherpeopleThisabilityisagaincriticaltosocialcohesionandcooperationChapter9explainshowtheoryofmindwasalsocriticaltotheemergenceoflanguage
Part4delvesmorespecificallyintothequestionofhowthere-cursivemindevolvedChapter10setsthisquestioninthecontextoftheclassicdebatebetweenCartesiandiscontinuityandDarwin-iancontinuityChapter11examinessomeofthestepsbywhichthe hominins33 after splitting from the line leading to modernchimpanzees and bonobos began to assume human-like attri-butesChapter12thenconsidersthefinalsteptoldquomodernrdquoHomo sapiensthesolesurvivinghomininspeciesmdashdominantmanipula-tiveMachiavellianandcapableofponderingourownnatureandstatusontheplanetThatperhapsistheultimatetriumphoftherecursivemind
Chapter13presentsthefinalsummaryandconclusions
10 chapteR 1
goesIt rained and it rained and it rained Thiscouldgoonfor-evermdashoratleastuntilPigletisdrownedmdashbuttherepetitionsimplyconveys the information that it rained rathera lotcausingPig-letsomeennuiItisnotrecursivebecauseeachadditionofand it rained isnotdrivenbythepreviousoneitissimplyaddedatthediscretionofthewriter
InanyeventrepetitiondoesnotdistinguishhumanactivityfromthatofnonhumananimalsBirdsongforexampleisrelentlesslyrepetitivebuteachrepeatedthemedoesnotembellishorqualifythepreviousoneAtmosttherepetitionmightsignalurgencyorsimplysignalcontinuingpresenceasonemightrepeatedlyknockonadoorinthehopeofarousingsomeoneinsideRepetitionisubiquitousinhumanandanimallifeinactivitiesrangingfromtherepeatedjawmovementsineatingtothecuriouslyrepetitivena-tureofsexualactivityThespidernolessiscapableofrepetitionasinWaltWhitmanrsquosLeaves of Grass15
ANOISELESSpatientspiderImarkrsquodwhereonalittlepromontoryitstoodisolatedMarkrsquodhowtoexplorethevacantvastsurroundingItlaunchrsquodforthfilamentfilamentfilamentoutofitselfEverunreelingthemmdashevertirelesslyspeedingthem
Informationcanalsobeaggregatedinnonrecursivefashionaswhentheshort-storywriterSaki(HHMunro)wroteldquoHungerfatigueanddespairinghopelessnesshadnumbedhisbrainrdquo16Ag-gregationofdifferentphrases similarlycompoundsmeaningad-ditivelyaswhenthehistorianPeterHennessywrote
ThemodelofamodernPrimeMinisterwouldbeakindofgrotesquecompositefreakmdashsomeonewiththededicationtodutyofaPeelthephysicalenergyofaGladstonethedetachmentofaSalisburytheballsofaLloydGeorgetheword-powerofaChurchilltheadministrativegiftsof anAttleethe styleof aMacmillanthemanagerialismof aHeathandthesleeprequirementsofaThatcher17
ThesentenceitselfhasrecursiveelementsbuttheaggregationofphrasestodescribethefreakishcompositeisnotrecursiveinthateachdoesnotcallthenextInsteadtheyareeffectivelyelements
What Is RecuRsIon 11
inalistinsertedtoaddinformationNonhumanspeciesmaywellhaveasimilarabilitytoaccumulateinformationaswhenunder-standingapredatoraslargefierceandwithsharpteethandclaws
Iteration
AslightlymoresubtlevariantonrepetitionandaggregationisitshyerationwhereaprocessisrepeatedbutinthiscasethereisinputfromthepreviousapplicationoftheprocessInthisrespect it islikerecursionandindeedconsideredbymathematicianstobelongtotheclassofldquogeneralrecursivefunctionsrdquoForthemainpurposesofthisbookthoughitdoesnotqualifyastruerecursionbecauseeachoutputisdiscardedonceithasbeenenteredintothenextap-plicationThedictionarydefinitionofrecursionthatIgaveearlierinthischapterwasalsoreallyanexampleofiterationratherthanrecursionbecauseyoujustkeepgoingroundandroundtheloopwithoutanyaddedstructureTheiterationsthereforedonotleadtoaddedcomplexity18
Iterativeproceduresareusedincomputationalmathematicstoarriveat increasinglyaccuratesolutionstoaproblemThebasicideaistostartwithapreliminarysolutionmdashperhapsaguessmdashandthenuseaproceduretocomputeanewsolutionThissolutionisthenusedasthestartingpointforthenextcomputationandthenew solution is then the starting point for the next roundThecycle is repeateduntil the solutions stabilize to someacceptablecriterion19Feedbacksystemsoperateinmuchthesamewaytypi-callyasameansofmaintaininghomeostasisForexampleather-mostatmayinvolveasystemforraisingorloweringtemperatureandthegoalistoachievesomegiventemperatureTheactualtem-peratureisfedintothesystemwhichoperatestoraiseorlowerthe temperatureuntil thedesired is reachedThebody isawashwith feedback systems to maintain homeostasis of temperatureironenergybloodcompositionandsoonThemainregulatoristhehypothalamusinthelimbicsystemofthebrainSuchsystemsagaindonotdifferentiatehumansfromotheranimals
Sometimesthedistinctionbetweenrecursionanditerationmaybeamatterof interpretationIn the infinite loopcreatedby the
1 chapteR 1
parodyofPaul CliffordonemightsaythateachbeginningofthestoryisinitiatedbythepreviousonewhichisthenforgottenTheparodyisbestappreciatedthoughifthestoryisseenasanendlessever deepening whirlpoolwith each segment remaining as partofitIrsquomtoldthestoryworksbestifeachsegmentisspokenwithadifferentaccent
Considertoothislinefromawell-knownchildrenrsquosverse
ThisisthedogthatworriedthecatthatkilledtheratthatatethemaltthatlayinthehousethatJackbuilt
TounderstandthissentenceastrulyrecursiveonemustappreciatethatitdescribesastateofaffairsasacomplexwholeandreferstoparticularcasesofadogcatratmalthouseandfellowcalledJackItisnotsimplythestringingtogetherofadogthatworriedacatacatthatkilledaratandsoforthAyoungchildthoughmightprocessitinthispiecemealwayasasuccessionofunrelatedevents
Recursion and evolutionary psychology
Inemphasizingrecursionasaunifyingconcepttheapproachtakeninthisbookcontrastswiththatadoptedbyso-calledevolutionarypsychologistswhohavearguedthatthemindhasmultiplefacetsThebasictenetsofevolutionarypsychologywerelaidoutinthe1992volumeThe Adapted MindeditedbyJeromeBarkowLedaCosmidesandJohnToobyandpopularizedbyStevenPinkerinhis influential 1997 book How the Mind Works20 Thus Pinkerwritesthatthehumanmindldquoisnotasingleorganbutasystemoforganswhichwecanthinkofaspsychologicalfacultiesormentalmodulesrdquo21 Inexaminingpresent-dayhumanbehaviortheevolu-tionarypsychologistrsquosagendaistodiscoverindependentprocessesasthebasicmodulesandrelatethemtoconditionsthatprevailedin the Pleistocene when humans existed primarily as hunter-gatherersAs Pinker puts it the aim is to carve the mind at itsjointssotospeakandldquoreverse-engineerrdquoitscomponentsormod-ulesbacktotheepochduringwhichthehumanmindwasformed
What Is RecuRsIon 1
In this view the mind is really a collection of miniminds eachbeaveringawayonitsownspecificproblemamongwhicharelan-guageandtheoryofmindThishasbeencalledtheSwiss-army-knifemodelofthemindwithabladeforeverypurpose22
Thedangerwith thisapproach is that itbecomes tooeasy topostulate modules and to tellldquojust sordquostories about how theyevolvedsothatthereisariskofreturningtothenow-abandonedinstinctpsychologyoftheearlytwentiethcentury23 Instinctpsy-chologyperishedunderthesheerweightofnumbersmdashtheauthorof one text counted 1594 instincts that had been attributed toanimals and humans24mdashand evolutionary psychology may alsodrowninaseaofmodulesifnotofmixedmetaphorsPinkersug-geststhatwelikepotatochipsbecausefattyfoodswerenutrition-allyvaluableduringthePleistocenebutscarceenoughthattherewasnodangerofobesitywelike landscapeswithtreesbecausetreesprovidesshadeandescapefromdangerouscarnivoresontheAfrica savanna flowers please us because they are markers foredible fruits nutsor tubers amid the greenery of the savannaandsoonldquoTherearemodulesrdquohewritesldquoforobjectsandforcesforanimatebeingsformindsandfornaturalkindslikeanimalsplantsandmineralsrdquo25
This isnot to say that theSwiss-army-knifemodel iswithoutmeritSomeofthepostulatedmodulesdoprovideinsightintothehumanconditionandarereasonablywellfoundedForexamplefoundationalworkinevolutionarypsychologybyLedaCosmidesprovidedgoodevidenceforaldquocheater-detectionmodulerdquomdashanin-stinctiveability todetect thosewhoflout social conventions fortheirowngain26 Arecentstudysuggests thathumanspossessaldquocategory-specificattentionsystemrdquothatisespeciallyadaptedforattendingtoanimals27 oneoftheauthorsJohnToobyisquotedas sayingldquoEvendullanimals likepigeons recruita surpris-ingamountofattentionmdashasdoturtlesresemblingrocksrdquo28 Thisbookisnotintendedtodenythattherearemanyspecificdisposi-tionsthatshapeourmentalandsociallivesrathermyaimistosuggest that therearedeeperaspectsofhumanthoughtthataregovernedbysimilarprinciplesandthatrecursionisoneofthoseprinciplesmdashandperhapsthemostimportantone
1 chapteR 1
Tobefairtoonotallevolutionarypsychologistshaveinsistedthatmodulesarecompletelyencapsulatedshutofffromanycom-munication with one anotherEven Steven Pinker for examplewrites ldquo[Modules] accomplish specialized functions thanks totheirspecializedstructuresbutdonrsquotnecessarilycomeinencap-sulated packagesrdquo29 Steven Mithen although scarcely a card-carryingevolutionarypsychologistarguedthatthehumanmindevolved itsdistinctive characterpreciselybecausepreviously en-capsulatedmodulesbegantoldquoleakrdquocreatingwhathecallsldquocogni-tivefluidityrdquo30 ItisasthoughthemodulesstoppedmindingtheirownbusinessesandbegantogossipMyapproachinthisbookisnotentirelyatoddswiththisviewinthatIargueacommonprin-ciplemightunderlieanumberofourdistinctiveabilities
Others are also beginning to question the Swiss-army-knifemodelofthehumanmindmorestarklyDavidPremackforex-ampleadoptsanapproachsimilartothatofferedinthisbookRe-viewingtheevidencefordiscontinuitybetweenhumansandotheranimalshewritesldquoAnimalcompetenciesaremainlyadaptationsrestrictedtoasinglegoalHumancompetenciesaredomain-generalandservenumerousgoalsrdquo31Thisineffectreversestheevolution-arypsychologyargumentmdashthemindhasbecomelessratherthanmoremodularThetidemaywellbeturning
In any event it is unlikely that recursion canbe considered amoduleAsweshallseerecursionseemstobeanorganizingprin-cipleinverydifferentspheresofhumanmentalactivityfromlan-guage tomemory tomind readingRecursive thinkingprobablydependsonothermentalattributesOneoftheseiswhathasbeentermedworkingmemorywhichholdsinformationinconscious-nessInordertoembedprocesseswithinprocessesitisnecessarytorememberwhereonehadgottointheearlierprocesswhenanembeddedprocesshasbeencompletedForinstanceinasentencelike My dog who eats bananas often gets sickone must holdthe early part of the sentence (My dog) and link it to the nextpart(often gets sick)DwightWReadhasarguedthatnonhumanprimatesevenourclosestrelativesthechimpanzeeshaveawork-ingmemorythatistoolimitedtoallowthiskindofembedding32
Recursionprobablyalsodependsonanexecutiveprocessthator-
What Is RecuRsIon 1
ganizeswhatistobeembeddedinwhatandthismaydependonthefrontallobesofthebrainTheabilitytoorganizeandcarryoutrecursiveoperationsmaythereforedependonseveralprocesses
AlthoughIdonotembracethemodularviewassumedbyevo-lutionarypsychologistsIamatonewiththeminproposingthata distinctively human mind evolved during the Pleistocene theepochthatstretchedfromaround26millionyearsagotosome12000 years agoHow that happened will be told in the laterchaptersofthisbook
plan of the Book
ThebookisdividedintofourpartsPart1dealswithlanguageAlthoughrecursionisnotlimitedto
languageitismostcommonlyinvokedtoexplainwhyhumanlan-guagediffersfromotherformsofanimalcommunicationanin-sightlargelyattributabletoNoamChomskyChapter2discussesthe nature of languagewithparticular emphasis on the role ofrecursionChapter3thenraisestheage-oldquestionofwhetherotheranimalshaveanythingresemblinghumanlanguageChapter4developstheideathatlanguageevolvedfrommanualgesturesmdashanideathatsuggestsgreaterevolutionarycontinuitybetweenhu-mansandotherprimatesthanthemorecommonassumptionthatlanguageemergedfromvocalcalls
Part2dealswithmentaltimetraveltheabilitytobringtomindeventsremovedfromthepresentinbothtimeandplaceChapter5startswithmemoryanddevelopstheideathatmemoryforspecificepisodesisuniquetohumansChapter6extendsthenotionofepi-sodicmemorytotheimaginingofpossiblefutureeventsleadingtotheconceptoftheselfasexistingthroughtimeThisleadstothenotiondiscussedinchapter7thatlanguageitselfevolvedtoenablepeopletosharetheirmemoriesandplansandsotocom-municateabouteventsthatarenotpresentintheimmediateenvi-ronmentThisleadsalsotofictionmdashthetellingofstoriesthatneednotbebasedonfactbutthatnonethelesshonethecapacitytodealwiththeepisodicexigenciesofhumansociallife
1 chapteR 1
Part3dealsanotherrecursiveaspectofhumanthoughtnamelytheoryofmindmdashortheabilitytounderstandwhatothersarethink-ingorfeelingChapter8introducesmind-readingnotasapsychicphenomenonbutasanaturalabilitytoinferthementalperspec-tivesofotherpeopleThisabilityisagaincriticaltosocialcohesionandcooperationChapter9explainshowtheoryofmindwasalsocriticaltotheemergenceoflanguage
Part4delvesmorespecificallyintothequestionofhowthere-cursivemindevolvedChapter10setsthisquestioninthecontextoftheclassicdebatebetweenCartesiandiscontinuityandDarwin-iancontinuityChapter11examinessomeofthestepsbywhichthe hominins33 after splitting from the line leading to modernchimpanzees and bonobos began to assume human-like attri-butesChapter12thenconsidersthefinalsteptoldquomodernrdquoHomo sapiensthesolesurvivinghomininspeciesmdashdominantmanipula-tiveMachiavellianandcapableofponderingourownnatureandstatusontheplanetThatperhapsistheultimatetriumphoftherecursivemind
Chapter13presentsthefinalsummaryandconclusions
What Is RecuRsIon 11
inalistinsertedtoaddinformationNonhumanspeciesmaywellhaveasimilarabilitytoaccumulateinformationaswhenunder-standingapredatoraslargefierceandwithsharpteethandclaws
Iteration
AslightlymoresubtlevariantonrepetitionandaggregationisitshyerationwhereaprocessisrepeatedbutinthiscasethereisinputfromthepreviousapplicationoftheprocessInthisrespect it islikerecursionandindeedconsideredbymathematicianstobelongtotheclassofldquogeneralrecursivefunctionsrdquoForthemainpurposesofthisbookthoughitdoesnotqualifyastruerecursionbecauseeachoutputisdiscardedonceithasbeenenteredintothenextap-plicationThedictionarydefinitionofrecursionthatIgaveearlierinthischapterwasalsoreallyanexampleofiterationratherthanrecursionbecauseyoujustkeepgoingroundandroundtheloopwithoutanyaddedstructureTheiterationsthereforedonotleadtoaddedcomplexity18
Iterativeproceduresareusedincomputationalmathematicstoarriveat increasinglyaccuratesolutionstoaproblemThebasicideaistostartwithapreliminarysolutionmdashperhapsaguessmdashandthenuseaproceduretocomputeanewsolutionThissolutionisthenusedasthestartingpointforthenextcomputationandthenew solution is then the starting point for the next roundThecycle is repeateduntil the solutions stabilize to someacceptablecriterion19Feedbacksystemsoperateinmuchthesamewaytypi-callyasameansofmaintaininghomeostasisForexampleather-mostatmayinvolveasystemforraisingorloweringtemperatureandthegoalistoachievesomegiventemperatureTheactualtem-peratureisfedintothesystemwhichoperatestoraiseorlowerthe temperatureuntil thedesired is reachedThebody isawashwith feedback systems to maintain homeostasis of temperatureironenergybloodcompositionandsoonThemainregulatoristhehypothalamusinthelimbicsystemofthebrainSuchsystemsagaindonotdifferentiatehumansfromotheranimals
Sometimesthedistinctionbetweenrecursionanditerationmaybeamatterof interpretationIn the infinite loopcreatedby the
1 chapteR 1
parodyofPaul CliffordonemightsaythateachbeginningofthestoryisinitiatedbythepreviousonewhichisthenforgottenTheparodyisbestappreciatedthoughifthestoryisseenasanendlessever deepening whirlpoolwith each segment remaining as partofitIrsquomtoldthestoryworksbestifeachsegmentisspokenwithadifferentaccent
Considertoothislinefromawell-knownchildrenrsquosverse
ThisisthedogthatworriedthecatthatkilledtheratthatatethemaltthatlayinthehousethatJackbuilt
TounderstandthissentenceastrulyrecursiveonemustappreciatethatitdescribesastateofaffairsasacomplexwholeandreferstoparticularcasesofadogcatratmalthouseandfellowcalledJackItisnotsimplythestringingtogetherofadogthatworriedacatacatthatkilledaratandsoforthAyoungchildthoughmightprocessitinthispiecemealwayasasuccessionofunrelatedevents
Recursion and evolutionary psychology
Inemphasizingrecursionasaunifyingconcepttheapproachtakeninthisbookcontrastswiththatadoptedbyso-calledevolutionarypsychologistswhohavearguedthatthemindhasmultiplefacetsThebasictenetsofevolutionarypsychologywerelaidoutinthe1992volumeThe Adapted MindeditedbyJeromeBarkowLedaCosmidesandJohnToobyandpopularizedbyStevenPinkerinhis influential 1997 book How the Mind Works20 Thus Pinkerwritesthatthehumanmindldquoisnotasingleorganbutasystemoforganswhichwecanthinkofaspsychologicalfacultiesormentalmodulesrdquo21 Inexaminingpresent-dayhumanbehaviortheevolu-tionarypsychologistrsquosagendaistodiscoverindependentprocessesasthebasicmodulesandrelatethemtoconditionsthatprevailedin the Pleistocene when humans existed primarily as hunter-gatherersAs Pinker puts it the aim is to carve the mind at itsjointssotospeakandldquoreverse-engineerrdquoitscomponentsormod-ulesbacktotheepochduringwhichthehumanmindwasformed
What Is RecuRsIon 1
In this view the mind is really a collection of miniminds eachbeaveringawayonitsownspecificproblemamongwhicharelan-guageandtheoryofmindThishasbeencalledtheSwiss-army-knifemodelofthemindwithabladeforeverypurpose22
Thedangerwith thisapproach is that itbecomes tooeasy topostulate modules and to tellldquojust sordquostories about how theyevolvedsothatthereisariskofreturningtothenow-abandonedinstinctpsychologyoftheearlytwentiethcentury23 Instinctpsy-chologyperishedunderthesheerweightofnumbersmdashtheauthorof one text counted 1594 instincts that had been attributed toanimals and humans24mdashand evolutionary psychology may alsodrowninaseaofmodulesifnotofmixedmetaphorsPinkersug-geststhatwelikepotatochipsbecausefattyfoodswerenutrition-allyvaluableduringthePleistocenebutscarceenoughthattherewasnodangerofobesitywelike landscapeswithtreesbecausetreesprovidesshadeandescapefromdangerouscarnivoresontheAfrica savanna flowers please us because they are markers foredible fruits nutsor tubers amid the greenery of the savannaandsoonldquoTherearemodulesrdquohewritesldquoforobjectsandforcesforanimatebeingsformindsandfornaturalkindslikeanimalsplantsandmineralsrdquo25
This isnot to say that theSwiss-army-knifemodel iswithoutmeritSomeofthepostulatedmodulesdoprovideinsightintothehumanconditionandarereasonablywellfoundedForexamplefoundationalworkinevolutionarypsychologybyLedaCosmidesprovidedgoodevidenceforaldquocheater-detectionmodulerdquomdashanin-stinctiveability todetect thosewhoflout social conventions fortheirowngain26 Arecentstudysuggests thathumanspossessaldquocategory-specificattentionsystemrdquothatisespeciallyadaptedforattendingtoanimals27 oneoftheauthorsJohnToobyisquotedas sayingldquoEvendullanimals likepigeons recruita surpris-ingamountofattentionmdashasdoturtlesresemblingrocksrdquo28 Thisbookisnotintendedtodenythattherearemanyspecificdisposi-tionsthatshapeourmentalandsociallivesrathermyaimistosuggest that therearedeeperaspectsofhumanthoughtthataregovernedbysimilarprinciplesandthatrecursionisoneofthoseprinciplesmdashandperhapsthemostimportantone
1 chapteR 1
Tobefairtoonotallevolutionarypsychologistshaveinsistedthatmodulesarecompletelyencapsulatedshutofffromanycom-munication with one anotherEven Steven Pinker for examplewrites ldquo[Modules] accomplish specialized functions thanks totheirspecializedstructuresbutdonrsquotnecessarilycomeinencap-sulated packagesrdquo29 Steven Mithen although scarcely a card-carryingevolutionarypsychologistarguedthatthehumanmindevolved itsdistinctive characterpreciselybecausepreviously en-capsulatedmodulesbegantoldquoleakrdquocreatingwhathecallsldquocogni-tivefluidityrdquo30 ItisasthoughthemodulesstoppedmindingtheirownbusinessesandbegantogossipMyapproachinthisbookisnotentirelyatoddswiththisviewinthatIargueacommonprin-ciplemightunderlieanumberofourdistinctiveabilities
Others are also beginning to question the Swiss-army-knifemodelofthehumanmindmorestarklyDavidPremackforex-ampleadoptsanapproachsimilartothatofferedinthisbookRe-viewingtheevidencefordiscontinuitybetweenhumansandotheranimalshewritesldquoAnimalcompetenciesaremainlyadaptationsrestrictedtoasinglegoalHumancompetenciesaredomain-generalandservenumerousgoalsrdquo31Thisineffectreversestheevolution-arypsychologyargumentmdashthemindhasbecomelessratherthanmoremodularThetidemaywellbeturning
In any event it is unlikely that recursion canbe considered amoduleAsweshallseerecursionseemstobeanorganizingprin-cipleinverydifferentspheresofhumanmentalactivityfromlan-guage tomemory tomind readingRecursive thinkingprobablydependsonothermentalattributesOneoftheseiswhathasbeentermedworkingmemorywhichholdsinformationinconscious-nessInordertoembedprocesseswithinprocessesitisnecessarytorememberwhereonehadgottointheearlierprocesswhenanembeddedprocesshasbeencompletedForinstanceinasentencelike My dog who eats bananas often gets sickone must holdthe early part of the sentence (My dog) and link it to the nextpart(often gets sick)DwightWReadhasarguedthatnonhumanprimatesevenourclosestrelativesthechimpanzeeshaveawork-ingmemorythatistoolimitedtoallowthiskindofembedding32
Recursionprobablyalsodependsonanexecutiveprocessthator-
What Is RecuRsIon 1
ganizeswhatistobeembeddedinwhatandthismaydependonthefrontallobesofthebrainTheabilitytoorganizeandcarryoutrecursiveoperationsmaythereforedependonseveralprocesses
AlthoughIdonotembracethemodularviewassumedbyevo-lutionarypsychologistsIamatonewiththeminproposingthata distinctively human mind evolved during the Pleistocene theepochthatstretchedfromaround26millionyearsagotosome12000 years agoHow that happened will be told in the laterchaptersofthisbook
plan of the Book
ThebookisdividedintofourpartsPart1dealswithlanguageAlthoughrecursionisnotlimitedto
languageitismostcommonlyinvokedtoexplainwhyhumanlan-guagediffersfromotherformsofanimalcommunicationanin-sightlargelyattributabletoNoamChomskyChapter2discussesthe nature of languagewithparticular emphasis on the role ofrecursionChapter3thenraisestheage-oldquestionofwhetherotheranimalshaveanythingresemblinghumanlanguageChapter4developstheideathatlanguageevolvedfrommanualgesturesmdashanideathatsuggestsgreaterevolutionarycontinuitybetweenhu-mansandotherprimatesthanthemorecommonassumptionthatlanguageemergedfromvocalcalls
Part2dealswithmentaltimetraveltheabilitytobringtomindeventsremovedfromthepresentinbothtimeandplaceChapter5startswithmemoryanddevelopstheideathatmemoryforspecificepisodesisuniquetohumansChapter6extendsthenotionofepi-sodicmemorytotheimaginingofpossiblefutureeventsleadingtotheconceptoftheselfasexistingthroughtimeThisleadstothenotiondiscussedinchapter7thatlanguageitselfevolvedtoenablepeopletosharetheirmemoriesandplansandsotocom-municateabouteventsthatarenotpresentintheimmediateenvi-ronmentThisleadsalsotofictionmdashthetellingofstoriesthatneednotbebasedonfactbutthatnonethelesshonethecapacitytodealwiththeepisodicexigenciesofhumansociallife
1 chapteR 1
Part3dealsanotherrecursiveaspectofhumanthoughtnamelytheoryofmindmdashortheabilitytounderstandwhatothersarethink-ingorfeelingChapter8introducesmind-readingnotasapsychicphenomenonbutasanaturalabilitytoinferthementalperspec-tivesofotherpeopleThisabilityisagaincriticaltosocialcohesionandcooperationChapter9explainshowtheoryofmindwasalsocriticaltotheemergenceoflanguage
Part4delvesmorespecificallyintothequestionofhowthere-cursivemindevolvedChapter10setsthisquestioninthecontextoftheclassicdebatebetweenCartesiandiscontinuityandDarwin-iancontinuityChapter11examinessomeofthestepsbywhichthe hominins33 after splitting from the line leading to modernchimpanzees and bonobos began to assume human-like attri-butesChapter12thenconsidersthefinalsteptoldquomodernrdquoHomo sapiensthesolesurvivinghomininspeciesmdashdominantmanipula-tiveMachiavellianandcapableofponderingourownnatureandstatusontheplanetThatperhapsistheultimatetriumphoftherecursivemind
Chapter13presentsthefinalsummaryandconclusions
1 chapteR 1
parodyofPaul CliffordonemightsaythateachbeginningofthestoryisinitiatedbythepreviousonewhichisthenforgottenTheparodyisbestappreciatedthoughifthestoryisseenasanendlessever deepening whirlpoolwith each segment remaining as partofitIrsquomtoldthestoryworksbestifeachsegmentisspokenwithadifferentaccent
Considertoothislinefromawell-knownchildrenrsquosverse
ThisisthedogthatworriedthecatthatkilledtheratthatatethemaltthatlayinthehousethatJackbuilt
TounderstandthissentenceastrulyrecursiveonemustappreciatethatitdescribesastateofaffairsasacomplexwholeandreferstoparticularcasesofadogcatratmalthouseandfellowcalledJackItisnotsimplythestringingtogetherofadogthatworriedacatacatthatkilledaratandsoforthAyoungchildthoughmightprocessitinthispiecemealwayasasuccessionofunrelatedevents
Recursion and evolutionary psychology
Inemphasizingrecursionasaunifyingconcepttheapproachtakeninthisbookcontrastswiththatadoptedbyso-calledevolutionarypsychologistswhohavearguedthatthemindhasmultiplefacetsThebasictenetsofevolutionarypsychologywerelaidoutinthe1992volumeThe Adapted MindeditedbyJeromeBarkowLedaCosmidesandJohnToobyandpopularizedbyStevenPinkerinhis influential 1997 book How the Mind Works20 Thus Pinkerwritesthatthehumanmindldquoisnotasingleorganbutasystemoforganswhichwecanthinkofaspsychologicalfacultiesormentalmodulesrdquo21 Inexaminingpresent-dayhumanbehaviortheevolu-tionarypsychologistrsquosagendaistodiscoverindependentprocessesasthebasicmodulesandrelatethemtoconditionsthatprevailedin the Pleistocene when humans existed primarily as hunter-gatherersAs Pinker puts it the aim is to carve the mind at itsjointssotospeakandldquoreverse-engineerrdquoitscomponentsormod-ulesbacktotheepochduringwhichthehumanmindwasformed
What Is RecuRsIon 1
In this view the mind is really a collection of miniminds eachbeaveringawayonitsownspecificproblemamongwhicharelan-guageandtheoryofmindThishasbeencalledtheSwiss-army-knifemodelofthemindwithabladeforeverypurpose22
Thedangerwith thisapproach is that itbecomes tooeasy topostulate modules and to tellldquojust sordquostories about how theyevolvedsothatthereisariskofreturningtothenow-abandonedinstinctpsychologyoftheearlytwentiethcentury23 Instinctpsy-chologyperishedunderthesheerweightofnumbersmdashtheauthorof one text counted 1594 instincts that had been attributed toanimals and humans24mdashand evolutionary psychology may alsodrowninaseaofmodulesifnotofmixedmetaphorsPinkersug-geststhatwelikepotatochipsbecausefattyfoodswerenutrition-allyvaluableduringthePleistocenebutscarceenoughthattherewasnodangerofobesitywelike landscapeswithtreesbecausetreesprovidesshadeandescapefromdangerouscarnivoresontheAfrica savanna flowers please us because they are markers foredible fruits nutsor tubers amid the greenery of the savannaandsoonldquoTherearemodulesrdquohewritesldquoforobjectsandforcesforanimatebeingsformindsandfornaturalkindslikeanimalsplantsandmineralsrdquo25
This isnot to say that theSwiss-army-knifemodel iswithoutmeritSomeofthepostulatedmodulesdoprovideinsightintothehumanconditionandarereasonablywellfoundedForexamplefoundationalworkinevolutionarypsychologybyLedaCosmidesprovidedgoodevidenceforaldquocheater-detectionmodulerdquomdashanin-stinctiveability todetect thosewhoflout social conventions fortheirowngain26 Arecentstudysuggests thathumanspossessaldquocategory-specificattentionsystemrdquothatisespeciallyadaptedforattendingtoanimals27 oneoftheauthorsJohnToobyisquotedas sayingldquoEvendullanimals likepigeons recruita surpris-ingamountofattentionmdashasdoturtlesresemblingrocksrdquo28 Thisbookisnotintendedtodenythattherearemanyspecificdisposi-tionsthatshapeourmentalandsociallivesrathermyaimistosuggest that therearedeeperaspectsofhumanthoughtthataregovernedbysimilarprinciplesandthatrecursionisoneofthoseprinciplesmdashandperhapsthemostimportantone
1 chapteR 1
Tobefairtoonotallevolutionarypsychologistshaveinsistedthatmodulesarecompletelyencapsulatedshutofffromanycom-munication with one anotherEven Steven Pinker for examplewrites ldquo[Modules] accomplish specialized functions thanks totheirspecializedstructuresbutdonrsquotnecessarilycomeinencap-sulated packagesrdquo29 Steven Mithen although scarcely a card-carryingevolutionarypsychologistarguedthatthehumanmindevolved itsdistinctive characterpreciselybecausepreviously en-capsulatedmodulesbegantoldquoleakrdquocreatingwhathecallsldquocogni-tivefluidityrdquo30 ItisasthoughthemodulesstoppedmindingtheirownbusinessesandbegantogossipMyapproachinthisbookisnotentirelyatoddswiththisviewinthatIargueacommonprin-ciplemightunderlieanumberofourdistinctiveabilities
Others are also beginning to question the Swiss-army-knifemodelofthehumanmindmorestarklyDavidPremackforex-ampleadoptsanapproachsimilartothatofferedinthisbookRe-viewingtheevidencefordiscontinuitybetweenhumansandotheranimalshewritesldquoAnimalcompetenciesaremainlyadaptationsrestrictedtoasinglegoalHumancompetenciesaredomain-generalandservenumerousgoalsrdquo31Thisineffectreversestheevolution-arypsychologyargumentmdashthemindhasbecomelessratherthanmoremodularThetidemaywellbeturning
In any event it is unlikely that recursion canbe considered amoduleAsweshallseerecursionseemstobeanorganizingprin-cipleinverydifferentspheresofhumanmentalactivityfromlan-guage tomemory tomind readingRecursive thinkingprobablydependsonothermentalattributesOneoftheseiswhathasbeentermedworkingmemorywhichholdsinformationinconscious-nessInordertoembedprocesseswithinprocessesitisnecessarytorememberwhereonehadgottointheearlierprocesswhenanembeddedprocesshasbeencompletedForinstanceinasentencelike My dog who eats bananas often gets sickone must holdthe early part of the sentence (My dog) and link it to the nextpart(often gets sick)DwightWReadhasarguedthatnonhumanprimatesevenourclosestrelativesthechimpanzeeshaveawork-ingmemorythatistoolimitedtoallowthiskindofembedding32
Recursionprobablyalsodependsonanexecutiveprocessthator-
What Is RecuRsIon 1
ganizeswhatistobeembeddedinwhatandthismaydependonthefrontallobesofthebrainTheabilitytoorganizeandcarryoutrecursiveoperationsmaythereforedependonseveralprocesses
AlthoughIdonotembracethemodularviewassumedbyevo-lutionarypsychologistsIamatonewiththeminproposingthata distinctively human mind evolved during the Pleistocene theepochthatstretchedfromaround26millionyearsagotosome12000 years agoHow that happened will be told in the laterchaptersofthisbook
plan of the Book
ThebookisdividedintofourpartsPart1dealswithlanguageAlthoughrecursionisnotlimitedto
languageitismostcommonlyinvokedtoexplainwhyhumanlan-guagediffersfromotherformsofanimalcommunicationanin-sightlargelyattributabletoNoamChomskyChapter2discussesthe nature of languagewithparticular emphasis on the role ofrecursionChapter3thenraisestheage-oldquestionofwhetherotheranimalshaveanythingresemblinghumanlanguageChapter4developstheideathatlanguageevolvedfrommanualgesturesmdashanideathatsuggestsgreaterevolutionarycontinuitybetweenhu-mansandotherprimatesthanthemorecommonassumptionthatlanguageemergedfromvocalcalls
Part2dealswithmentaltimetraveltheabilitytobringtomindeventsremovedfromthepresentinbothtimeandplaceChapter5startswithmemoryanddevelopstheideathatmemoryforspecificepisodesisuniquetohumansChapter6extendsthenotionofepi-sodicmemorytotheimaginingofpossiblefutureeventsleadingtotheconceptoftheselfasexistingthroughtimeThisleadstothenotiondiscussedinchapter7thatlanguageitselfevolvedtoenablepeopletosharetheirmemoriesandplansandsotocom-municateabouteventsthatarenotpresentintheimmediateenvi-ronmentThisleadsalsotofictionmdashthetellingofstoriesthatneednotbebasedonfactbutthatnonethelesshonethecapacitytodealwiththeepisodicexigenciesofhumansociallife
1 chapteR 1
Part3dealsanotherrecursiveaspectofhumanthoughtnamelytheoryofmindmdashortheabilitytounderstandwhatothersarethink-ingorfeelingChapter8introducesmind-readingnotasapsychicphenomenonbutasanaturalabilitytoinferthementalperspec-tivesofotherpeopleThisabilityisagaincriticaltosocialcohesionandcooperationChapter9explainshowtheoryofmindwasalsocriticaltotheemergenceoflanguage
Part4delvesmorespecificallyintothequestionofhowthere-cursivemindevolvedChapter10setsthisquestioninthecontextoftheclassicdebatebetweenCartesiandiscontinuityandDarwin-iancontinuityChapter11examinessomeofthestepsbywhichthe hominins33 after splitting from the line leading to modernchimpanzees and bonobos began to assume human-like attri-butesChapter12thenconsidersthefinalsteptoldquomodernrdquoHomo sapiensthesolesurvivinghomininspeciesmdashdominantmanipula-tiveMachiavellianandcapableofponderingourownnatureandstatusontheplanetThatperhapsistheultimatetriumphoftherecursivemind
Chapter13presentsthefinalsummaryandconclusions
What Is RecuRsIon 1
In this view the mind is really a collection of miniminds eachbeaveringawayonitsownspecificproblemamongwhicharelan-guageandtheoryofmindThishasbeencalledtheSwiss-army-knifemodelofthemindwithabladeforeverypurpose22
Thedangerwith thisapproach is that itbecomes tooeasy topostulate modules and to tellldquojust sordquostories about how theyevolvedsothatthereisariskofreturningtothenow-abandonedinstinctpsychologyoftheearlytwentiethcentury23 Instinctpsy-chologyperishedunderthesheerweightofnumbersmdashtheauthorof one text counted 1594 instincts that had been attributed toanimals and humans24mdashand evolutionary psychology may alsodrowninaseaofmodulesifnotofmixedmetaphorsPinkersug-geststhatwelikepotatochipsbecausefattyfoodswerenutrition-allyvaluableduringthePleistocenebutscarceenoughthattherewasnodangerofobesitywelike landscapeswithtreesbecausetreesprovidesshadeandescapefromdangerouscarnivoresontheAfrica savanna flowers please us because they are markers foredible fruits nutsor tubers amid the greenery of the savannaandsoonldquoTherearemodulesrdquohewritesldquoforobjectsandforcesforanimatebeingsformindsandfornaturalkindslikeanimalsplantsandmineralsrdquo25
This isnot to say that theSwiss-army-knifemodel iswithoutmeritSomeofthepostulatedmodulesdoprovideinsightintothehumanconditionandarereasonablywellfoundedForexamplefoundationalworkinevolutionarypsychologybyLedaCosmidesprovidedgoodevidenceforaldquocheater-detectionmodulerdquomdashanin-stinctiveability todetect thosewhoflout social conventions fortheirowngain26 Arecentstudysuggests thathumanspossessaldquocategory-specificattentionsystemrdquothatisespeciallyadaptedforattendingtoanimals27 oneoftheauthorsJohnToobyisquotedas sayingldquoEvendullanimals likepigeons recruita surpris-ingamountofattentionmdashasdoturtlesresemblingrocksrdquo28 Thisbookisnotintendedtodenythattherearemanyspecificdisposi-tionsthatshapeourmentalandsociallivesrathermyaimistosuggest that therearedeeperaspectsofhumanthoughtthataregovernedbysimilarprinciplesandthatrecursionisoneofthoseprinciplesmdashandperhapsthemostimportantone
1 chapteR 1
Tobefairtoonotallevolutionarypsychologistshaveinsistedthatmodulesarecompletelyencapsulatedshutofffromanycom-munication with one anotherEven Steven Pinker for examplewrites ldquo[Modules] accomplish specialized functions thanks totheirspecializedstructuresbutdonrsquotnecessarilycomeinencap-sulated packagesrdquo29 Steven Mithen although scarcely a card-carryingevolutionarypsychologistarguedthatthehumanmindevolved itsdistinctive characterpreciselybecausepreviously en-capsulatedmodulesbegantoldquoleakrdquocreatingwhathecallsldquocogni-tivefluidityrdquo30 ItisasthoughthemodulesstoppedmindingtheirownbusinessesandbegantogossipMyapproachinthisbookisnotentirelyatoddswiththisviewinthatIargueacommonprin-ciplemightunderlieanumberofourdistinctiveabilities
Others are also beginning to question the Swiss-army-knifemodelofthehumanmindmorestarklyDavidPremackforex-ampleadoptsanapproachsimilartothatofferedinthisbookRe-viewingtheevidencefordiscontinuitybetweenhumansandotheranimalshewritesldquoAnimalcompetenciesaremainlyadaptationsrestrictedtoasinglegoalHumancompetenciesaredomain-generalandservenumerousgoalsrdquo31Thisineffectreversestheevolution-arypsychologyargumentmdashthemindhasbecomelessratherthanmoremodularThetidemaywellbeturning
In any event it is unlikely that recursion canbe considered amoduleAsweshallseerecursionseemstobeanorganizingprin-cipleinverydifferentspheresofhumanmentalactivityfromlan-guage tomemory tomind readingRecursive thinkingprobablydependsonothermentalattributesOneoftheseiswhathasbeentermedworkingmemorywhichholdsinformationinconscious-nessInordertoembedprocesseswithinprocessesitisnecessarytorememberwhereonehadgottointheearlierprocesswhenanembeddedprocesshasbeencompletedForinstanceinasentencelike My dog who eats bananas often gets sickone must holdthe early part of the sentence (My dog) and link it to the nextpart(often gets sick)DwightWReadhasarguedthatnonhumanprimatesevenourclosestrelativesthechimpanzeeshaveawork-ingmemorythatistoolimitedtoallowthiskindofembedding32
Recursionprobablyalsodependsonanexecutiveprocessthator-
What Is RecuRsIon 1
ganizeswhatistobeembeddedinwhatandthismaydependonthefrontallobesofthebrainTheabilitytoorganizeandcarryoutrecursiveoperationsmaythereforedependonseveralprocesses
AlthoughIdonotembracethemodularviewassumedbyevo-lutionarypsychologistsIamatonewiththeminproposingthata distinctively human mind evolved during the Pleistocene theepochthatstretchedfromaround26millionyearsagotosome12000 years agoHow that happened will be told in the laterchaptersofthisbook
plan of the Book
ThebookisdividedintofourpartsPart1dealswithlanguageAlthoughrecursionisnotlimitedto
languageitismostcommonlyinvokedtoexplainwhyhumanlan-guagediffersfromotherformsofanimalcommunicationanin-sightlargelyattributabletoNoamChomskyChapter2discussesthe nature of languagewithparticular emphasis on the role ofrecursionChapter3thenraisestheage-oldquestionofwhetherotheranimalshaveanythingresemblinghumanlanguageChapter4developstheideathatlanguageevolvedfrommanualgesturesmdashanideathatsuggestsgreaterevolutionarycontinuitybetweenhu-mansandotherprimatesthanthemorecommonassumptionthatlanguageemergedfromvocalcalls
Part2dealswithmentaltimetraveltheabilitytobringtomindeventsremovedfromthepresentinbothtimeandplaceChapter5startswithmemoryanddevelopstheideathatmemoryforspecificepisodesisuniquetohumansChapter6extendsthenotionofepi-sodicmemorytotheimaginingofpossiblefutureeventsleadingtotheconceptoftheselfasexistingthroughtimeThisleadstothenotiondiscussedinchapter7thatlanguageitselfevolvedtoenablepeopletosharetheirmemoriesandplansandsotocom-municateabouteventsthatarenotpresentintheimmediateenvi-ronmentThisleadsalsotofictionmdashthetellingofstoriesthatneednotbebasedonfactbutthatnonethelesshonethecapacitytodealwiththeepisodicexigenciesofhumansociallife
1 chapteR 1
Part3dealsanotherrecursiveaspectofhumanthoughtnamelytheoryofmindmdashortheabilitytounderstandwhatothersarethink-ingorfeelingChapter8introducesmind-readingnotasapsychicphenomenonbutasanaturalabilitytoinferthementalperspec-tivesofotherpeopleThisabilityisagaincriticaltosocialcohesionandcooperationChapter9explainshowtheoryofmindwasalsocriticaltotheemergenceoflanguage
Part4delvesmorespecificallyintothequestionofhowthere-cursivemindevolvedChapter10setsthisquestioninthecontextoftheclassicdebatebetweenCartesiandiscontinuityandDarwin-iancontinuityChapter11examinessomeofthestepsbywhichthe hominins33 after splitting from the line leading to modernchimpanzees and bonobos began to assume human-like attri-butesChapter12thenconsidersthefinalsteptoldquomodernrdquoHomo sapiensthesolesurvivinghomininspeciesmdashdominantmanipula-tiveMachiavellianandcapableofponderingourownnatureandstatusontheplanetThatperhapsistheultimatetriumphoftherecursivemind
Chapter13presentsthefinalsummaryandconclusions
1 chapteR 1
Tobefairtoonotallevolutionarypsychologistshaveinsistedthatmodulesarecompletelyencapsulatedshutofffromanycom-munication with one anotherEven Steven Pinker for examplewrites ldquo[Modules] accomplish specialized functions thanks totheirspecializedstructuresbutdonrsquotnecessarilycomeinencap-sulated packagesrdquo29 Steven Mithen although scarcely a card-carryingevolutionarypsychologistarguedthatthehumanmindevolved itsdistinctive characterpreciselybecausepreviously en-capsulatedmodulesbegantoldquoleakrdquocreatingwhathecallsldquocogni-tivefluidityrdquo30 ItisasthoughthemodulesstoppedmindingtheirownbusinessesandbegantogossipMyapproachinthisbookisnotentirelyatoddswiththisviewinthatIargueacommonprin-ciplemightunderlieanumberofourdistinctiveabilities
Others are also beginning to question the Swiss-army-knifemodelofthehumanmindmorestarklyDavidPremackforex-ampleadoptsanapproachsimilartothatofferedinthisbookRe-viewingtheevidencefordiscontinuitybetweenhumansandotheranimalshewritesldquoAnimalcompetenciesaremainlyadaptationsrestrictedtoasinglegoalHumancompetenciesaredomain-generalandservenumerousgoalsrdquo31Thisineffectreversestheevolution-arypsychologyargumentmdashthemindhasbecomelessratherthanmoremodularThetidemaywellbeturning
In any event it is unlikely that recursion canbe considered amoduleAsweshallseerecursionseemstobeanorganizingprin-cipleinverydifferentspheresofhumanmentalactivityfromlan-guage tomemory tomind readingRecursive thinkingprobablydependsonothermentalattributesOneoftheseiswhathasbeentermedworkingmemorywhichholdsinformationinconscious-nessInordertoembedprocesseswithinprocessesitisnecessarytorememberwhereonehadgottointheearlierprocesswhenanembeddedprocesshasbeencompletedForinstanceinasentencelike My dog who eats bananas often gets sickone must holdthe early part of the sentence (My dog) and link it to the nextpart(often gets sick)DwightWReadhasarguedthatnonhumanprimatesevenourclosestrelativesthechimpanzeeshaveawork-ingmemorythatistoolimitedtoallowthiskindofembedding32
Recursionprobablyalsodependsonanexecutiveprocessthator-
What Is RecuRsIon 1
ganizeswhatistobeembeddedinwhatandthismaydependonthefrontallobesofthebrainTheabilitytoorganizeandcarryoutrecursiveoperationsmaythereforedependonseveralprocesses
AlthoughIdonotembracethemodularviewassumedbyevo-lutionarypsychologistsIamatonewiththeminproposingthata distinctively human mind evolved during the Pleistocene theepochthatstretchedfromaround26millionyearsagotosome12000 years agoHow that happened will be told in the laterchaptersofthisbook
plan of the Book
ThebookisdividedintofourpartsPart1dealswithlanguageAlthoughrecursionisnotlimitedto
languageitismostcommonlyinvokedtoexplainwhyhumanlan-guagediffersfromotherformsofanimalcommunicationanin-sightlargelyattributabletoNoamChomskyChapter2discussesthe nature of languagewithparticular emphasis on the role ofrecursionChapter3thenraisestheage-oldquestionofwhetherotheranimalshaveanythingresemblinghumanlanguageChapter4developstheideathatlanguageevolvedfrommanualgesturesmdashanideathatsuggestsgreaterevolutionarycontinuitybetweenhu-mansandotherprimatesthanthemorecommonassumptionthatlanguageemergedfromvocalcalls
Part2dealswithmentaltimetraveltheabilitytobringtomindeventsremovedfromthepresentinbothtimeandplaceChapter5startswithmemoryanddevelopstheideathatmemoryforspecificepisodesisuniquetohumansChapter6extendsthenotionofepi-sodicmemorytotheimaginingofpossiblefutureeventsleadingtotheconceptoftheselfasexistingthroughtimeThisleadstothenotiondiscussedinchapter7thatlanguageitselfevolvedtoenablepeopletosharetheirmemoriesandplansandsotocom-municateabouteventsthatarenotpresentintheimmediateenvi-ronmentThisleadsalsotofictionmdashthetellingofstoriesthatneednotbebasedonfactbutthatnonethelesshonethecapacitytodealwiththeepisodicexigenciesofhumansociallife
1 chapteR 1
Part3dealsanotherrecursiveaspectofhumanthoughtnamelytheoryofmindmdashortheabilitytounderstandwhatothersarethink-ingorfeelingChapter8introducesmind-readingnotasapsychicphenomenonbutasanaturalabilitytoinferthementalperspec-tivesofotherpeopleThisabilityisagaincriticaltosocialcohesionandcooperationChapter9explainshowtheoryofmindwasalsocriticaltotheemergenceoflanguage
Part4delvesmorespecificallyintothequestionofhowthere-cursivemindevolvedChapter10setsthisquestioninthecontextoftheclassicdebatebetweenCartesiandiscontinuityandDarwin-iancontinuityChapter11examinessomeofthestepsbywhichthe hominins33 after splitting from the line leading to modernchimpanzees and bonobos began to assume human-like attri-butesChapter12thenconsidersthefinalsteptoldquomodernrdquoHomo sapiensthesolesurvivinghomininspeciesmdashdominantmanipula-tiveMachiavellianandcapableofponderingourownnatureandstatusontheplanetThatperhapsistheultimatetriumphoftherecursivemind
Chapter13presentsthefinalsummaryandconclusions
What Is RecuRsIon 1
ganizeswhatistobeembeddedinwhatandthismaydependonthefrontallobesofthebrainTheabilitytoorganizeandcarryoutrecursiveoperationsmaythereforedependonseveralprocesses
AlthoughIdonotembracethemodularviewassumedbyevo-lutionarypsychologistsIamatonewiththeminproposingthata distinctively human mind evolved during the Pleistocene theepochthatstretchedfromaround26millionyearsagotosome12000 years agoHow that happened will be told in the laterchaptersofthisbook
plan of the Book
ThebookisdividedintofourpartsPart1dealswithlanguageAlthoughrecursionisnotlimitedto
languageitismostcommonlyinvokedtoexplainwhyhumanlan-guagediffersfromotherformsofanimalcommunicationanin-sightlargelyattributabletoNoamChomskyChapter2discussesthe nature of languagewithparticular emphasis on the role ofrecursionChapter3thenraisestheage-oldquestionofwhetherotheranimalshaveanythingresemblinghumanlanguageChapter4developstheideathatlanguageevolvedfrommanualgesturesmdashanideathatsuggestsgreaterevolutionarycontinuitybetweenhu-mansandotherprimatesthanthemorecommonassumptionthatlanguageemergedfromvocalcalls
Part2dealswithmentaltimetraveltheabilitytobringtomindeventsremovedfromthepresentinbothtimeandplaceChapter5startswithmemoryanddevelopstheideathatmemoryforspecificepisodesisuniquetohumansChapter6extendsthenotionofepi-sodicmemorytotheimaginingofpossiblefutureeventsleadingtotheconceptoftheselfasexistingthroughtimeThisleadstothenotiondiscussedinchapter7thatlanguageitselfevolvedtoenablepeopletosharetheirmemoriesandplansandsotocom-municateabouteventsthatarenotpresentintheimmediateenvi-ronmentThisleadsalsotofictionmdashthetellingofstoriesthatneednotbebasedonfactbutthatnonethelesshonethecapacitytodealwiththeepisodicexigenciesofhumansociallife
1 chapteR 1
Part3dealsanotherrecursiveaspectofhumanthoughtnamelytheoryofmindmdashortheabilitytounderstandwhatothersarethink-ingorfeelingChapter8introducesmind-readingnotasapsychicphenomenonbutasanaturalabilitytoinferthementalperspec-tivesofotherpeopleThisabilityisagaincriticaltosocialcohesionandcooperationChapter9explainshowtheoryofmindwasalsocriticaltotheemergenceoflanguage
Part4delvesmorespecificallyintothequestionofhowthere-cursivemindevolvedChapter10setsthisquestioninthecontextoftheclassicdebatebetweenCartesiandiscontinuityandDarwin-iancontinuityChapter11examinessomeofthestepsbywhichthe hominins33 after splitting from the line leading to modernchimpanzees and bonobos began to assume human-like attri-butesChapter12thenconsidersthefinalsteptoldquomodernrdquoHomo sapiensthesolesurvivinghomininspeciesmdashdominantmanipula-tiveMachiavellianandcapableofponderingourownnatureandstatusontheplanetThatperhapsistheultimatetriumphoftherecursivemind
Chapter13presentsthefinalsummaryandconclusions
1 chapteR 1
Part3dealsanotherrecursiveaspectofhumanthoughtnamelytheoryofmindmdashortheabilitytounderstandwhatothersarethink-ingorfeelingChapter8introducesmind-readingnotasapsychicphenomenonbutasanaturalabilitytoinferthementalperspec-tivesofotherpeopleThisabilityisagaincriticaltosocialcohesionandcooperationChapter9explainshowtheoryofmindwasalsocriticaltotheemergenceoflanguage
Part4delvesmorespecificallyintothequestionofhowthere-cursivemindevolvedChapter10setsthisquestioninthecontextoftheclassicdebatebetweenCartesiandiscontinuityandDarwin-iancontinuityChapter11examinessomeofthestepsbywhichthe hominins33 after splitting from the line leading to modernchimpanzees and bonobos began to assume human-like attri-butesChapter12thenconsidersthefinalsteptoldquomodernrdquoHomo sapiensthesolesurvivinghomininspeciesmdashdominantmanipula-tiveMachiavellianandcapableofponderingourownnatureandstatusontheplanetThatperhapsistheultimatetriumphoftherecursivemind
Chapter13presentsthefinalsummaryandconclusions