analog magnitude representations
TRANSCRIPT
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AnalogueMagnitudeRepresentations:APhilosophicalIntroduction
JacobBeck
YorkUniversity,Toronto
Empiricaldiscussionsofmentalrepresentationpositawidevarietyofrepresentational
kinds.Someofthesekinds,suchasthesententialrepresentationsunderlyinglanguageuse
andthepictorialrepresentationsofvisualimagery,arethoroughlyfamiliartophilosophers.
Others,suchascognitivemaps,aresomewhatlessfamiliar.Stillothershavereceived
almostnoattentionatall.Includedinthislattercategoryareanaloguemagnitude
representations(AMRs).1
AMRsareprimitiverepresentationsofspatial,temporal,numerical,andrelated
magnitudes.Theyareprimitivebecausetheyrepresentmagnitudeswithoutpresupposing
theabilitytorepresentanyunitsofmeasurementormathematicallydefinedsystemof
numbers.AMRsarealsoprimitiveontogeneticallyandphylogenetically.Theyarepresent
insix-month-oldhumaninfants,awidevarietyofmammals,manybirds,andatleastsome
fish.Inhumanadultswithaformaleducation,AMRsexistalongsideculturallyacquired
representationsofspace,time,andnumber.AMRsareanalogueinaspecialsensetobe
explainedinSection2.
1WhileAMRsarerarelymentionedinthephilosophicalliterature,thereareafewnotableexceptions.(LaurenceandMargolis[2005])discussesAMRsinthecontextofassessingtheinfluenceofnaturallanguageontheacquisitionofnaturalnumberconcepts;(Pietroskietal.[2009])appealstoAMRstoanalysethesemanticsofthewordmost;(Burge[2010])discusseshowAMRsofnumerosityarerepresentedinperception;and(Beck[2012])arguesthatAMRsserveasanexampleofcognitiverepresentationsthatarenotsystematicallyrecombinableandthushavenonconceptualcontent.SofarasIknow,however,thispapermarksthefirstattempttoprovideageneral,philosophicallyorientedanalysisofAMRs.
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Thenatureofarepresentationalkindisdeterminedbyitsformat,itscontent,and
thecomputationsitsupports.Specifyingthenatureofanygivenrepresentationalkindis
alwayscontroversial,butwehaveatleastaroughgraspofthenaturesofsome
representationalkinds.Thus,toafirstapproximation:therepresentationsunderlying
languageusehaveasentence-likeformat,topic-neutralcontents,andsupportsyntacticand
logicalcomputations;therepresentationsunderlyingvisualimageryhaveapicture-like
format,representvisiblepropertiesaslaidoutinegocentricspace,andsupport
computationssuchasrotationandscanning;andcognitivemapshaveageometricformat,
representthelocationsofentitiesinallocentricspace,andsupportcomputationsrelatedto
navigationsuchaslocalizationandpath-planning.2AnanalysisofthenatureofAMRs
shouldlikewiseprovideinsightintotheirformat,content,andcomputations.
AMRsareoverdueforanalysis.Thereisnowconsiderableevidencethattheyexist
andplayanimportantroleinthecognitivelivesofawiderangeofanimals,including
humans.Giventheirrelevancetotherepresentationofcategoriessuchasspace,time,and
numberthathavelongfeaturedcentrallyinphilosophicaldiscussionsofcognition,AMRs
oughttobeofintrinsicinteresttophilosophers.Theyshouldalsointerestphilosophers
becausetheybearuponarangeoffoundationalissuesinthephilosophyofmind,including
howmentalrepresentationsarerealizedinthebrain,theproperanalysisofanalogue
representation,therequirementsonrepresentationalcontent,theexistenceof
nonconceptualcontent,andthenatureofanimalcognition.Thispaperaimstointroduce
AMRstoaphilosophicalaudiencebyreviewingevidencefortheirexistence(1)and
2(Fodor[1975])and(Block[1983])containclassicphilosophicaldiscussionsofthenatureoflinguisticandimagisticrepresentations,respectively.(Rescorla[2009])containsanexcellentphilosophicalanalysisofcognitivemaps.
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carefullyanalysingtheirformat(2),theircontent(3),andthecomputationstheysupport
(4).
1.Background
1.1EvidenceofAMRs
Oneinterestinglessontoemergefromthestudyofcognitionoverthepastfewdecadesis
thateducatedhumanadultsarenotaloneinrepresentingmagnitudessuchasdistance,
area,duration,numerosity,3andrate.Awiderangeoforganisms,includingfish,birds,
lowermammalssuchasrats,non-humanprimates,humanchildrenincludingpre-
linguisticinfants,andhumanadultsfromallculturesandeducationalbackgrounds
representsuchmagnitudesaswell(forreviewsseeGallistel[1990],Dehaene[2011],and
Walsh[2003]).Ibeginbyrehearsingafewhighlightsfromthisliterature.
WhenGodinandKeenleyside([1984])fedaschoolofcichlidfishatdifferingrates
fromthreeseparatetubes,thefishquicklyapportionedthemselvesinratiosthatmatched
thefeedingrates,evenbeforemanyofthemhadhadachancetoberewardedbytasting
thefood.Thus,merelyseeingonetubereleasefoodmorselsattwicetherateofanother
wassufficienttoleadtwiceasmanyfishtocongregateinfrontofthefirsttube.Anatural
explanationisthatthefishrepresenttherateatwhichfoodisdispersedfromeachtube
andadjusttheirpositionsaccordingly.Fromabiologicalperspective,thisprobability
matchingbehaviourmakessensesinceitisevolutionarilystable.Iffishgenerallyfedonly
atthesourcewiththegreatestpayoffthenanonconformistfishthatinsteadfedatasource
3Numerosityisanumber-likeproperty.Idiscussthedifferencebetweennumberandnumerosityin3.2.
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withaslightlylowerpayoutbutnocompetitorspresentwouldrecovermorefood.Natural
selectionthusencouragesgroupforagerstoadoptaprobability-matchingstrategy.
Probabilitymatchingiswidelyobservedthroughouttheanimalkingdom.Ina
similarexperimentthatwasperformedonducksbyHarper([1982]),twoexperimenters
tossedmorselsofbreadtotheducksattwodifferentrates.Withinaminute,theducks
dividedthemselvesinproportiontothoserates.Butwhenoneexperimentertossed
morselsthatweretwicethesizeofthemorselstossedbytheotherexperimenter,theducks
alteredtheirstrategy,andwithinfiveminutesrepositionedthemselvesinproportiontothe
productofthemorselsizeandfeedingrate.Gallistel([1990],p.358)summarizesthis
findingasfollows.
Thisresultsuggeststhatbirdsaccuratelyrepresentrates,thattheyaccuratelyrepresent
morselmagnitudes,andthattheycanmultiplytherepresentationofmorselsperunittime
bytherepresentationofmorselmagnitudetocomputetheinternalvariablesthatdetermine
therelativelikelihoodoftheirchoosingoneforagingpatchovertheother.
Inotherwords,Gallistelinterpretstheseresultsasshowingthatducksnotonlyrepresent
magnitudessuchasrateandphysicalsize,butalsoperformcomputationssuchas
multiplicationoverthoserepresentations.
Laboratorystudiesofratspaintasimilarpicture.ChurchandMeck([1984])trained
ratstopressoneleverinresponsetoatwo-secondsequenceoftwotonesandasecond
leverinresponsetoaneight-secondsequenceofeighttones.Theythenvariedeitherthe
durationofthetoneswhileholdingnumberconstant,orthenumberofthetoneswhile
holdingdurationconstant.Inbothcases,theratsgeneralizedfromthetrainingexperiment.
Forexample,whennumberwasheldconstantatfourtones,theypressedthefirstleverin
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responsetoatwo-orthree-secondsequenceandthesecondleverinresponsetoafive-,
six-,oreight-secondsequence,thusshowingthattheyrepresentedthetotaldurationof
eachtonesequence.Butwhendurationwasheldconstantatfourseconds,theypressedthe
firstleverinresponsetotwoorthreetonesandthesecondleverinresponsetofive,six,or
eighttones,thusshowingthattheyalsorepresentedhowmanytoneswereineach
sequence.Theratsalsoimmediatelygeneralizedtheirlearningtonewstimulipresentedto
newsensorymodalities.Forexample,whenpresentedwithatwo-oreight-secondlight
sequence,theypressedthesamelevertheyweretrainedtopressinresponsetoatwo-or
eight-secondtone;andwhenpresentedwithtwooreightflashesoflight,theratspressed
thesamelevertheyweretrainedtopressinresponsetotwooreighttones.Thesecross-
modaltransferexperimentsprovideevidencethattheratsrepresentabstractdurations
andnumericalinformationandnotjustmodality-specificstimuli.
Whileitwillhardlybenewstothereaderthathumanadultskeeptrackofdurations,
distances,numerosities,andothermagnitudes,itmaycomeasasurprisethathuman
infantsdothesame,suggestingthatsuchabilitiesareinnate.XuandSpelke([2000])
presentedsix-month-oldinfantswithadisplayof8or16dotsuntiltheyhabituatedtothe
display.Theinfantswerethenpresentedwithatestdisplayofeither8or16dots.Infants
whowerepresentedwithatestdisplaythathadanovelnumberofdotslooked
significantlylongerthaninfantswhowerepresentedwithatestdisplaythathadthesame
numberofdotsasthehabituationdisplay.Asinfantsareknowntoapportiontheir
attentiontostimulitheydeemnovel,andasothervariableswerecontrolledfor(suchas
totaldotareaanddotdensity),thisfindingsuggeststhatinfantscanmakediscriminations
basedonnumericalinformation.Otherdishabituationstudiessuggestthatinfantsare
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likewisesensitivetoduration(VanMarleandWynn[2006])andarea(Brannonetal.
[2006]).
1.2WebersLaw
Apeculiarfeatureofmagnitudediscriminationsdeservesemphasis:theyobeyWebers
Law,whichholdsthattheabilitytodiscriminatetwomagnitudesisdeterminedbytheir
ratio.Astheratiooftwomagnitudesapproaches1:1theybecomehardertodiscriminate,
andbeyondacertainthresholddeterminedbythesubjectsWeberconstanttheycannot
bediscriminatedatall.Asanillustrationofthisratiosensitivity,consideragainthe
experimentsofChurchandMeck([1984])inwhichtheratsweretrainedtopressthefirst
leverinresponsetoatwo-secondsequenceoftwotonesandthesecondleverinresponse
toaneight-secondsequenceofeighttones.Whenpresentedwithafive-secondsequenceof
fivetones,theratstendedtopressthesecondlever,foralthoughfiveisequidistant
betweentwoandeight,theratioof5:2isgreaterthantheratioof8:5.Whentheywere
presentedwithafour-secondsequenceoffourtones,however,theratsfavouredneither
lever.Since8:4=4:2,fourwasthesubjectivemidpointfortheratsbetweeneightandtwo.
Whenexplicitcountingisnotpossible,themagnitudediscriminationsofhuman
adultsalsoexhibittheratiosensitivityassociatedwithWebersLaw.Forexample,iftwo
displaysofdotsareflashedtooquicklyforyoutoseriallycountthedotsoneach,your
probabilityofcorrectlyguessingwhichdisplayhasmoredotswilldecreaseastheratio
approaches1:1,andbeyondacertainratio(roughly7:8),yourguesseswillbeatchance
(Barthetal.[2003]).Similarly,incultureswhereexplicitcountingisabsentandthereisno
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culturallyacquiredintegerlist,magnitudediscriminationsarealmost4alwayssubjectto
WebersLaw(Gordon[2005];Picaetal.[2005]).Theseresultsareconsistentwiththe
hypothesisthatmosteducatedhumanadultshavetwosystemsforrepresenting
magnitudes:aprimitiveanaloguemagnitudesystemthatissharedwithawiderangeof
otherorganisms;andauniquelyhumanmagnitudesystemthatdependsuponthe
acquisitionofconceptsofnaturalnumbersandunitsofmeasurement.
1.3ScepticismaboutAMRs
Itisnaturaltowonderwhetherthereisasimplerexplanationoftheresultsreviewedin
thissectionthatdoesnotappealtoAMRsanexplanationthatappealstoless
sophisticatedrepresentationsorperhapstonorepresentationsatall.Afterall,researchers
havebeenleddownthegardenpathbefore.Intheearly20thCentury,manywere
convincedthatthehorseCleverHanscouldperformarithmeticsincehewouldstomphis
hooftheappropriatenumberoftimesinresponsetoproblemsposedbyonlookers.Butthe
psychologistOskarPfungst([1965])demonstratedthattherewasasimplerexplanation:
Hanswaspickinguponsmallunintentionalmovementsinhisaudiencethatindicated
whentostartandstopstomping.Similarly,whilethewaggledanceofhoneybeesisoften
takentorepresentthedistancetoafoodsource,thereisevidencethatthedancecorrelates
lesswithdistancethanwithopticflowtheamountanimagemovesovertheretina
suggestingthatbeesdontrepresentdistanceassuchafterall(Eschetal.[2001]).These
4Whyalmostalways?Tworeasons:(1)discriminationerrorsarelesscommonthanWebersLawpredictsforsetswithfourorfewermembers(thisisonefindinginsupportoftheexistenceofaseparatesmallnumberorobjectfilesystem);and(2)thereissomeevidencethatwhenitemscanbelinedupandputinone-to-onecorrespondence,participantscanbeinducedtorelyonHumesprincipletodeterminewhethertwosetsareequinumerous.
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examplesshowthatcareneedstobetakenbeforerepresentationsofmagnitudesare
posited.Alternativehypothesesneedtobeconsideredandruledout.
InthecaseofthecontemporarystudyofAMRs,experimentershavetakengreatcare
toavoidobserverexpectancyeffects.ManyofthetechniquespioneeredbyPfungsttotest
Hansarenowroutinelyappliedincomparativepsychology.Forexample,experimental
assistantsareoftenkeptblindaboutcrucialaspectsoftheexperimenttopreventthem
frominadvertentlysignallingtheanimalswithwhichtheyinteract.StudiesofAMRsalso
routinelycontrolforotheralternativehypotheses.Forinstance,ChurchandMeckscross-
modaltransferexperimentswereinpartdesignedtotestwhetheranimalsrepresent
simplesensoryproperties(suchasretinalareaortotalillumination)ratherthanabstract
propertiessuchasnumerosityandduration.Thefactthatratsareabletotrackmagnitudes
acrossdisparatetypesofstimulipresentedtodisparatesensorymodalitiessuggeststhat
ratsrepresentmorethansensoryproperties.Theoverallbodyofempiricalevidencethus
stronglysuggestsevenifitdoesnotapodicticallyprovethatmagnitudesare
represented.Moreover,whiletherearecomprehensiveandmathematicallyrigorous
explanationsofthefindingsdiscussedinthissectionthatappealtoAMRs(e.g.,Gallistel
[1990]),tothebestofmyknowledgetherearenocorrespondingexplanationsofthese
resultsinnon-representationalterms,orintermsofrepresentationsofnon-magnitudes.I
thustaketheexistenceofAMRsasareasonable,yetdefeasible,empiricalhypothesis.5
5SomephilosophersmaywonderhowtheoriesthatpositAMRsrelatetotheoriesofthemindthatarentsquarelyinthesymbolicmould,suchasbehaviourism,connectionism,anddynamicalsystemstheory.TheoriesthatpositAMRsareincompatiblewithbehaviourismsincetheypositinternalrepresentationsthatarenotgovernedbyprinciplesofassociationthatcharacterizeclassicalorinstrumentalconditioning.Infact,(Gallistel[1990])and(GallistelandGibbon[2002])arguethatmanyphenomenathathavetraditionallybeenexplainedthroughconditioningarebetterexplainedbyarithmeticcomputationsoverAMRs.Bycontrast,becauserepresentationscanberealizedinconnectionistnetworks(eveniftheyareoftendistributedacrossthosenetworks),thereisnoconflictbetweenconnectionismandtheexistenceofAMRs.Someresearchers
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2.Format
2.1Careysanalogy
MostresearchersagreethattheratiosensitivityassociatedwithWebersLawisevidence
thatmagnituderepresentationshaveaspecificformat:theyinvolvesomeneuralentitythat
isadirectanalogueofthemagnitudeitrepresents(henceAMR).Astheratiooftwo
magnitudesapproaches1:1,theneuralentitiesbecomeincreasinglysimilarandthus,
assumingthereissomenoiseinthesystem,hardertodiscriminate.Toexplainwhy
researchersbelieveAMRshaveananalogueformat,SusanCarey([2009],p.118)provides
ahelpfulanalogytothefollowingexternalsystemofanaloguenumberrepresentations.
number symbol
1: __
2: ____
3: ______
4: ________
7: ______________
8: ________________
Inthissystem,linelengthisadirectanalogueofnumber.Thegreaterthenumber
represented,thelongertheline.Supposeourbrainsdeploymagnituderepresentationsthat
arelikewiseanalogue.Thenjustasitisharderforustovisuallydiscriminate______________
from________________than__from____,wewouldexpectourbrainstofinditharderto
haveevendevelopedconnectionistmodelsfortheimplementationofAMRs(ChurchandBroadbent[1990];DehaeneandChangeux[1993]).Inprinciple,AMRsarealsocompatiblewithdynamicalsystemstheorysinceasystemsbeingdescribabledynamicallyneednotprecludeitsbeingdescribableintermsofrepresentationsandcomputations.Tothebestofmyknowledge,however,noonehasdevelopeddynamicalmodelsofthesortsofbehavioursthatAMRsarepositedtoexplain.
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insidetheintraparietalsulcusthatcorrelateswithmagnitudediscriminationsinhumans
andotherprimatesregardlessofthemodalitythroughwhichthemagnitudeisperceivedor
represented(Dehaeneetal.[2003]).Moreover,theactivationinthisareaexhibitsitsown
typeofratiosensitivity:astheratiooftwomagnitudesbeingcomparedapproaches1:1,the
activationincreases(Pineletal.[2001]).Infact,researchershavefoundneuronsinthe
intraparietalsulcusofmonkeysthataretunedtospecificmagnitudes,withbell-shaped
activationfunctionsoffixedvariancewhenplottedonalogarithmicaxis(NiederandMiller
[2003];NiederandMiller[2004]).Theseactivationfunctionsareexactlywhatonewould
predictgivenWebersLaw;astheratiooftwomagnitudesapproaches1:1,theactivation
patternsoftheneuronscorrespondingtothosemagnitudesbecomehardertotellapart.
Theactivationpatternsoftheseneuronsthusmirrorthebehaviouralpatternsassociated
withmagnitudediscriminations.Whilemanyquestionsstillexistabouthowtheseneurons
arecoordinatedwithinlargerpopulations,functionalaccountsofAMRsareincreasingly
beingintegratedwithneuroscience.AMRsshouldthusserveaswelcometargetsfor
philosophersinterestingintheorizingabouttheneuralrealizationofmental
representations.
2.3Analoguerepresentation
Second,thelinesegments(suchas______)thatCareyanalogizestoAMRsarecontinuous,
andthusanalogueinGoodmans([1976])senseofbeingdense(betweenanytwo
representedvaluesthereisalwaysathirdrepresentedvalue).Butitisanopenquestion
whetherAMRsthemselvesarecontinuousordensesincediscretesymbolscouldlikewise
giverisetoratiosensitivityinmagnitudediscriminations.Toseethis,noticethatthe
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followingexternalmagnitudesystemwouldhaveservedjustaswelltoillustrateCareys
point.
number symbol
1: |
2: ||
3: |||
4: ||||
7: |||||||
8: ||||||||
Althoughtheserepresentationsarediscrete,theytooareadirectanalogueofthenumbers
theyrepresent,andthusbecomehardertodiscriminateastheirratioapproaches1:1.
Indeed,asCareyherselfobserves,AMRscouldberepresentedbythetotalnumberof
neuronsfiringwithinagivenpopulation.Sinceactionpotentialsareallornoneneurons
eitherfireortheydonttherewouldthenbenointermediaterepresentationbetweena
networkofnneuronsfiringandanetworkofn+1neuronsfiring.
How,then,shouldtherelevantsenseofanaloguerepresentationbespecifiedifnot
intermsofcontinuity?Careyoffersthefollowingsuggestion.
Iconicrepresentationsareanalogue;roughly,thepartsoftherepresentationcorrespondto
thepartsoftheentitiesrepresented.Apictureofatigerisaniconicrepresentation;the
wordtigerisnot.Theheadinthepicturerepresentstheheadofthetiger;thetailinthe
picturerepresentsthetail.Thetintigerdoesnotrepresentanypartofthetiger.([2009],
458)
Careyssuggestion,inotherwords,isthaticonicrepresentationssuchaspicturesare
analoguebecausetheyhavepartsthatcorrespondtothepartsoftheentitythatis
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represented,andthatrepresentationssuchasthewordtigerarenotanaloguebecause
theydonothavepartsthatcorrespondtothepartsoftheentityrepresented.Careythus
seemstoendorsewhatIwillcallthePartAnalogueThesis(PAT):Risananalogue
representationofXifandonlyifthepartsofRrepresentthepartsofX.ApplyingPATto
AMRs,Carey([2009],p.458)illustrateswhysheholdsthatAMRsareanalogue.
Analoguerepresentationsofnumberrepresentaswouldanumberlinetherepresentation
oftwo(____)isaquantitythatissmallerthanandiscontainedintherepresentationfor
three(______).
Thus,CareymaintainsthatAMRsareanaloguebecausetheyobeyPAT;thepartsofanAMR
representpartofwhattheAMRasawholerepresents.
Notice,however,thatPATismissingaquantifier.Musteverypart,oronlysome
parts,ofRrepresentpartofX?Ifwechoosetheexistentialquantifier,PATbecomestoo
permissive.ThesentenceBillistallrepresentsanindividual,Bill,andthepropertyof
beingtall.SoBillrepresentspartofwhatthatsentenceasawholerepresents,andPAT
failstodistinguishAMRsfromsentences.UsingauniversalquantifierdoesntmakePAT
anymoreplausible.IfeverypartofRhastorepresentpartofX,PATbecomestoo
restrictive.RecallCareyssuggestionthatAMRsmightberealizedbyapopulationof
neuronsfiring.Certainlyapartofoneofthoseneuronsdoesntrepresentanythingallonits
own.Onlytheadditionofeachneuron(orperhapstheadditionofafamilyornetworkof
neurons)indicatesanewmagnitude.Moreover,thereisnothingincoherentintheideathatgreatermagnitudesmightberepresentedbysmallerpopulationsofneuronsorlower
neuralfiringrates.Inthatcase,however,apartofarepresentationofamagnitudeof(say)
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twentywouldnotrepresentpartoftwenty;itwouldrepresentmorethantwenty.PATthus
failstocapturethesenseinwhichAMRsareanalogue.
Maley([2011])providesamorepromisingaccountofanaloguerepresentationthat
canbeappliedtoAMRs.Modifyinghisanalysisonlyslightly,wecansaythata
representationRofarepresentedmagnitudeMisanalogueifandonlyif:(1)thereissome
propertyPofRsuchthatthequantityoramountofPdeterminesM;and(2)asMincreases
ordecreasesbyanamountd,PincreasesordecreasesasamonotonicfunctionofM+dor
Md.6Thus,inorderforanAMRthatrepresentsamagnitude,M,tobeanalogue,therehas
tobesomeproperty,P,oftheAMRthatincreasesordecreasesmonotonicallywithM.For
example,considerthreeAMRsthatrepresentfour,five,andsixflashesoflightcallthem
AMR4,AMR5,andAMR6,respectively.AccordingtoMaleysaccount,iftheserepresentations
aretocountasgenuinelyanalogue,thenthereneedstobesomepropertypresumably
someneuralpropertythatbelongstoeachandthatmonotonicallyincreasesordecreases
fromAMR4toAMR5toAMR6.Thatpropertycouldbetherateatwhichapopulationof
neuronsfires,thenumberofneuronsthatfire,orsomethingelseentirely;butsomesuch
propertyhastoexistthatdetermineswhichmagnitudeeachoftheAMRsrepresent.
Noticethatthisaccountdoesnotrequireanaloguerepresentationstobecontinuous
ordense.EvenifPturnedouttobeanon-continuouspropertysuchasthenumberof
neuronsfiring,themagnituderepresentedbyRwouldbeamonotonicfunctionofthe
6ThisanalysisdiffersfromMaleys([2011],p.123)intwosignificantrespects.First,Maleysaccountiscouchedintermsofrepresentationsofnumberswhereasthepresentaccountiscouchedintermsofrepresentationsofmagnitudes.ObviouslythisalterationisnecessaryifwewanttoapplytheaccounttoAMRssincetheyrepresentnon-numericalmagnitudessuchasdurationanddistance.Second,Maleysaccountappealstoalinearfunctionratherthanamonotonicfunction.SinceAMRsarestandardlyinterpretedaslogarithmicallycompressed(Dehaene[2003]),thisamendmentiscrucial.Tohiscredit,Maley([2011],p.123,n.2)anticipatestheamendment.
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quantityoramountofPandwouldthuscountasanalogue.Additionally,becauseMaleys
accountavoidstheproblematicnotionofapart,itavoidsthedifficultiesthatbefallPAT.
TheanaloguenatureofAMRsmakesthemunlikethesymbolsfamiliarfrommodern
digitalcomputerssincethereisnopropertyofsuchsymbolsthatmonotonicallyincreases
ordecreaseswiththemagnitudestheydetermine.Forexample,althoughthebinary
symbols00,01,10,and11representmagnitudesthatcanbeorderedfromleastto
greatest,thereisnopropertyofthesymbolsthemselvesthatdeterminesthosemagnitudes
andincreasesordecreasesmonotonicallywiththem.
TherelationshipbetweentheanaloguenatureofAMRsandthelanguageofthought
hypothesisislessclear,primarilybecauseproponentsofthelanguageofthought
hypothesisarerarelyexplicitaboutthepreciseformatthatrepresentationsinthelanguage
ofthoughtmusthave.Whileeveryoneagreesthatrepresentationsinthelanguageof
thoughtmustbelanguage-like,paradigmaticlanguageshavearangeofproperties,andit
isunclearwhichofthesepropertiesrepresentationsinthelanguageofthoughtneedto
possess.Iflanguage-likerepresentationsmerelyneedtobecompositionalandsupport
accuracyconditions,thereisnothingstoppingthemfrombeinganalogue.However,itis
alsoafamiliarpropertyofprototypicallanguagesthattheirrepresentationsarenot
analogue.Likethebinarysymbolsofdigitalcomputers,naturallanguagewordssuchas
threeandfourdonothavesomepropertythatmonotonicallyincreasesordecreases
withwhattheyrepresent.WhethertheanalogueformatofAMRsputsthematoddswith
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thelanguageofthoughthypothesisthusultimatelydependsonthevexedquestionofhow
language-likerepresentationsinthelanguageofthoughtmustbe.7
2.4AMRcomponents
ThereisathirdwayinwhichCareysanalysismaybemisleading.Thereisnothinginthe
linelengths(suchas______)thatsheanalogizestoAMRstoindicatethattheycorrespond
tonumberasopposedtodistance,duration,rate,oranyothermagnitude.Butthebrain
clearlyhassomemeansofdistinguishinganAMRofsevenindividualsfromanAMRof
sevenmetresorsevenseconds.Additionally,AMRsdifferintheirobjects.AnAMRofa
seven-secondlightdiffersfromanAMRofaseven-secondtone.Thus,amoreperspicuous
depictionofAMRswouldincludearepresentationnotjustofthemagnitudessize,butalso
whatwemightcallitsmodeandobject.WecanthusdepicteachAMRasanorderedtriplet
thatincludesasize,mode,andobjectcomponent.Forexample,wecoulddepictanAMRof
fourfishas{________,NUMEROSITY,FISH},anAMRofathree-secondtoneas{______,DURATION,
TONE},etc.NoticethatalthoughtheratiosensitivityassociatedwithWebersLawis
evidencethatthesizecomponentofAMRsisanalogue,itdoesnotprovideevidencethat
themodeorobjectcomponentsareanalogue.Infact,giventhattheanalysisofanalogue
representationinspiredbyMaleyisdefinedtoapplyonlytorepresentationsofmagnitudes,
themodeandobjectcomponentsconsideredontheirowncannotbeanalogueaccordingto
thatanalysis.Magnitudesarethingsthatcanbequantitativelyrelated;yetmodesand
objectsassuchcannotbequantitativelyrelated.ThequestionsIsdurationmoreorless
thandistance?andArelightflashesgreaterorfewerthantones?donotmakesense.Itis
7Inotherwork(Beck[2012],[forthcoming]),IhavearguedthatthereisafurtherrespectinwhichAMRsareunlikeparadigmaticlinguisticrepresentations:theylackthesystematicrecombinabilityofsuchrepresentations.Ireturntothisissuein3.3.
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onlythesizecomponentofAMRs,orAMRsconsideredasawhole,thatcanbe
quantitativelyrelated,andthusanalogueaccordingtotheanalysisinspiredbyMaley.
IsthesizecomponentofAMRstrulyindependentofthemodeandobject
components?Orisitanabstraction,imposedbyusastheorists,onwhatinrealityisan
undifferentiated,homogenousrepresentation?Althoughthesequestionshavenotbeen
definitivelyresolved,theycanbe,andhavebeen,subjectedtoempiricalinvestigation.The
basicstrategyistolookforevidenceofacommonneuralmechanismacrossAMRsthat
differintheirmodeandobjectcomponents.Anysuchcommonmechanismwouldthen
presumablybeattributabletothesizecomponent,suggestingthatthesizecomponenttruly
isindependentofthemodeandobjectcomponents.Thereareseveralfindingsthathave
beencitedinsupportofacommonmechanism.
Althoughratiosensitivityvariesacrossspeciesandages,thereisevidencethatthedegreeofsensitivityisthesameforallmodeswithinagivenspeciesatagivenage.
Thus,theratiosatwhichmatureratssuccessfullydiscriminatedurationsarethe
sameastheratiosatwhichtheysuccessfullydiscriminatenumerosities(Meckand
Church[1983]),andsix-month-oldinfantsshowthesameratiosensitivityforarea,
numerosity,andduration(Feigenson[2007]).
Attemptstointerveneontherepresentationofsizeinonemodetendtohavesimilareffectsontherepresentationofsizeinallmodes.Forexample,when
methamphetamineisadministeredtorats,thereisanidenticalacceleratedshiftin
therepresentationofbothnumerosityandduration(MeckandChurch[1983]).
Behaviouraltasksthattapintoonemodetendtointerferewithperformanceontasksthattapintoothermodes.Forexample,adultsarefasteratcomparingtwo
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digitswhenthegreaterdigitisinalargerfontcomparedtothelesserdigitand
slowerwhenthegreaterdigitisinasmallerfontcomparedtothelesserdigit(Henik
andTzelgov[1982]).Likewise,adultsjudgethedurationofpresentationofonedigit
tobelongerthanthe(equal)durationofpresentationofaseconddigitwhenthe
firstdigitisgreaterthanthesecond(Oliverietal.[2008]).8
Associationswithinonemodetendtogeneralizetoothermodes,eveninyounginfants.Forexample,if9-month-oldinfantsareshownthatblackstripedrectangles
arelargerthanwhitedottedrectangles,theyexpecttheblackstripedrectanglesto
bemorenumerousandtolastforalongerdurationaswell(LourencoandLongo
[2010]).
Evidencefrombothbraindeficitsandbrainimagingpointtooverlappinglociformagnituderepresentationsofvariousmodesintheinferiorparietalcortexof
humansandotherprimates(Walsh[2003];Jacob,Vallentin,andNieder[2012]).
Tobesure,thesevariouslinesofevidencearehardlyconclusive.Otherexplanationsofthe
resultsarepossible.PerhapstheratiosensitivityofAMRsofdifferentmodesallimproveat
thesamerateacrossontogenynotbecausetheyshareacommonsizemechanism,but
becauseorganismstendtousetheirAMRsofdistance,duration,numerosity,etc.equally
often,leadingthemalltobeindependentlyhonedatthesamerate.Similarly,otherresults
mightbepickingupnotonacommonsizemechanism,butonhomogenousAMRsthatare
instantiatedinoverlappingoradjacentneuralnetworks.Forexample,ifAMRsofduration
andnumerositywereembeddedinsufficientlyproximalneuralnetworks,tasksthat8Interestingly,noteverymagnitudeshowsthesameinterferenceeffects.Forexample,althoughloudnessdiscriminationsaresubjecttoWebersLaw,loudnessdoesnotinterferewithdistancediscriminationsinthewaythatdurationdoes(SrinivasanandCarey[2010]),suggestingthatrepresentationsofloudnessdonotrecruitthesamesizecomponentthatAMRsofdistance,duration,andnumerosityseeminglydrawupon.
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fish,birds,lowermammals,andevenhumaninfantsrepresentmagnitudes.Consequently,
demandingtheoriessituneasilywiththeempiricalevidence,reviewedinSection1,that
AMRsdorealworkexplainingthebehavioursoftheseanimals,suchaswhyaratwillpress
leverAratherthanleverBfollowingthepresentationofaparticularstimulus.AlthoughI
cannotdefendtheopinionhere,IalsoagreewithBurge([2010])thatthearguments
advancedinfavourofdemandingtheoriesofrepresentationalcontentarenotcompelling.I
amthusinclinedtojoinBurgeinrejectingsuchtheoriesasunmotivatedandempirically
dubious.10
Attheotherextremesitdeflationarytheoriesofrepresentation,whichseekto
reducethenotionofrepresentationalcontenttoothernotions,suchasinformationand
learning(Dretske[1981]),biologicalfunction(Millikan[1989]),orfunctioning
isomorphism(Gallistel[1990]).Becausethesereductiveanalysesarerelativelyeasyto
satisfy,representationsinthedeflationarysensearerelativelyeasytocomeby.Allofthese
theoriesarethuslikelytocountAMRsashavingrepresentationalcontent,evenin
relativelysimpleorganisms.Gallistel([1990])istheonedeflationarytheoristIamawareof
whoexplicitlydiscussesAMRs.AccordingtoGallistel([1990],p.15),
Thebrainissaidtorepresentanaspectoftheenvironmentwhenthereisafunctioning
isomorphismbetweensomeaspectoftheenvironmentandabrainprocessthatadaptsthe
animalsbehaviourtoit.
10SomeproponentsofdemandingtheoriesofrepresentationmightallowthatAMRsarerepresentationaleveninloweranimalsprovidedthattheyaretreatedassub-individualstates.Some subsystemoftheratsrepresentsmagnitudesbut theratdoesnotrepresentsmagnitudes.Yetgiventhatitisthesurprisinglyintelligentbehaviouroftheratthatneedstobeexplained,includingtheratsabilitytolearn,thisreplystrikesmeasimplausible.Suchsophisticatedbehaviours,itseemstome,arebestexplainedintermsofmentalstatesthatbelongtorats.
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ThesedifficultiesalsospelltroubleforGallistelandGelmans([2000])proposalthat
thecontentsofAMRsarebestcharacterizedbyrealnumbers.AsBurge([2010],p.481)
observes,theintegersareasubsetoftherealnumbers,soifAMRscannotrepresent
integersnorcantheyrepresenttherealnumbers.GallistelandGelmanreasonthatinsofar
astheformatofthesizecomponentofAMRsisanalogue,itmustbecontinuous,andthus
thatthesizecomponentofAMRsmustrepresentrealnumbers,whicharelikewise
continuous.Butthislineofreasoningisproblematic.AswesawinSection2.3,itisa
mistaketosupposethatAMRsmustbecontinuousjustbecausetheyareanalogue.Webers
Lawcanbeexplainedbydiscretesymbolsthatareadirectanalogueofthemagnitudes
represented.Moreover,asLaurenceandMargolis([2005])emphasize,GallistelandGelman
seemtobeseducedbyaspuriousformat-contentconflation.Justasdiscrete
representationalvehiclessuchasor2canstandforrealnumbers,continuous
representationalvehiclescansurelystandforsomethingotherthanrealnumbers.Thus,
evenifAMRsweretohaveacontinuousformat,itsimplywouldntfollowthattheymust
representtherealnumbers.
RatherthaninterpretingAMRsintermsofintegersorrealnumbers,Carey([2009],
p.135)proposestointerpretthemasapproximatecardinalvalues.Forexample,anAMR
mightrepresentapproximatelysevenlightflashesorapproximatelysevenseconds.Butof
courseonecannotrepresentapproximatelysevenwithoutbeingabletorepresentseven,
andsoCareyssuggestionfaresnobetterthanGallistelandGelmans.Moreover,Careys
suggestionfailstospecifyhowapproximateAMRsare.Whileonemightreachforgreater
specificitybyappealingtoWebersLaw,WebersLawisitselfdefinedintermsofratios
amongintegers,whichweweretryingtoavoid.
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Burge([2010],pp.4823)recommendslookingtoEudoxustheoryofpure
magnitudestoarticulatethecontentsofAMRs.12Eudoxusdevelopedhistheoryofpure
magnitudesinordertohandleratiosthatbedevilledthePythagoreans.Becausethe
Pythagoreansmaintainedthatwholenumbersarethebasisofallratios,theycouldnot
expressincommensurableratiossuchasthatbetweenthesideofasquareanditsdiagonal.
(Inourterminology,theycouldnotexpressirrationalnumberssuchas2.)Eudoxus
respondedbydefiningratiosandproportionswithoutappealingtonumbers.Instead,he
definedratiosintermsofsizerelationsamonghomogenousmagnitudesthemselves,and
proportionsintermsofcomparativesizerelationsofratios.13Thus,twolinelengthscan
enterintooneratio;twoweightscanenterintoanotherratio;andthenthosetworatios
canbecomparedandfoundtobeproportionalornot.Discretemagnitudessuchaswhole
numberscanenterintoratiosandproportionsaswell,andareunderstoodinthesameway
asratiosandproportionsthatinvolvecontinuousmagnitudessuchaslinelengthorweight.
Thus,thekeyfeatureofEudoxustheory,accordingtoBurge,isthatitsconceptofpure
magnitudedoesnotdifferentiatebetweencontinuousanddiscretemagnitudes.Pure
magnitudesarelikethesizecomponentofAMRsinthattheycannotbeexpressedusing
numbers.ThisleadsBurgetohypothesizethatAMRsrefertopuremagnitudes.
Burgeshypothesisissuggestive,andseemslikeastepintherightdirection.Itis
plausiblethatthesizecomponentsofAMRsdonotrefertonumbers,whichmakespure
magnitudesinvitingcandidatestoserveastheircontents.Nevertheless,Eudoxeanpure
12EudoxustheoryofpuremagnitudesisdescribedbyEuclid([1956],book5).Iamindebtedto(Sutherland[2006])and(Stein[1990])forexposition.
13Eudoxusprincipalinsightwasthatsuchcomparisonscouldbespelledoutintermsof equimultiples.See(Euclid[1956],book5,definition5;Sutherland[2006],pp.5367;andStein[1990],pp.1669).
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predictsthatwhenthemodelisrunninginitsnumerositymodethetimeittakesto
representasetisproportionaltothesetssize(Carey[2009],pp.1323).Morerecent
models,whichoperateinparallel,arenotobviouslycommittedtoanyunits.Forexample,
accordingtoChurchandBroadbents([1990])model,durationsareassociatedwiththe
phasesofasetofneuraloscillatorsofdifferingperiods.Thus,agivendurationwilltypically
beassociatedwithseveraloscillatorswhoseindividualperiodstotaltothedurationbeing
measured(ignoringerror).Asaresult,thereisnounitinthismodelbywhichatotal
durationismeasured.Ofcourse,weastheoristsmightuseseconds,minutes,orsomeother
unitoftimetodescribeatotalduration,butsuchunitsdonotplayafunctionalroleinthe
modelitself.ToborrowahelpfulphrasefromChristopherPeacocke([1986]),AMRsmight
thusbecharacterizedasunitfree.14
TurningfinallytotheobjectcomponentofAMRs,itisnotablethatAMRsarenot
restrictedtorepresentingobjectsfromonlyonemodality.Forexample,AMRscan
representvisualobjectssuchaslightflashesordotsonascreenaswellasauditoryobjects
suchastones.NorareAMRsrestrictedtoattributingmagnitudestospatiallywell-defined
objects.AMRscanalsoassignmagnitudestoabstractevents,suchasasequenceofrabbit
jumps(WoodandSpelke[2005]).Infact,thereisnoevidenceIamawareofsuggestingthat
theobjectsofAMRshaveanyrestrictionsbeyondthoseimposedbythebroader
representationalcapacitiesofthesubject.15
14Peacockeintroducesthistermtocharacterizethecontentofperceptualexperiencesofmagnitudes,aswhenyouvisuallyexperiencethelengthofapianowithoutexperiencingitinfeetormetres.IhypothesizethatthephenomenonPeacockeisolatesisgroundedinAMRsi.e.,thatourconsciousexperienceshaveaunit-freecharacterbecausethoseexperiencesaregeneratedbyAMRswhicharethemselvesunitfree.
15Burge([2010],p.472)suggeststhatwhenAMRsareusedinperceptiontheyarelimitedtorepresentingperceivableentities,buthedoesnotconsiderwhetherAMRsaresolimitedwhentheyareusedoutsideofperceptione.g.,whenapersonconsiderstheabstractquestionwhethertheunionoftwosetswith35and
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3.3WhatcontenttypesdoAMRshave?
WehavejustbeenconsideringwhatAMRsrepresent.Thereisafurtherquestion,however,
concerninghowAMRsrepresentwhattheydo.WhattypeofcontentsdoAMRshave?
AsIunderstandthem,mentalcontentshavethefunctionofcapturinghowthings
arefromtheperspectiveofthethinker.Theyfulfilthisfunction,inpart,bymarkinga
thinkersmentalabilities.Whereamentalstateistheresultoftwoseparateabilities,many
philosophershavethereforewantedtouseacontentthatisstructuredfromtwoelements
tocaptureit.Forexample,ifapersonwhocanthinkthatAmydiedrichandthatAmydied
poorwouldexerciseasingleabilitytheabilitytothinkaboutAmyinthecourseof
thinkingeachthought,wecanmarkthatfactbyattributingacommonelementthe
conceptAmytothecontentofeachofthosethoughts(Evans[1982],pp.100-5).Sincethe
conceptualthoughtsofhumanbeingsseemtobestructuredfromdiscreteabilitiesinthis
sense,manyphilosophershavewantedtousestructuredcontentstocapturethem.
WehaveseenreasontothinkthatAMRsarealsostructuredfromdiscreteabilities.
Forexample,theabilitytorepresentsevenflashesoflightseemstosharesomethingin
commonwiththeabilitytorepresentasequenceofseventones.Moregenerally,theability
todeployAMRsseemstodecomposeintoabilitiestorepresentasize,amode,andan
object.Ifthatisright,thenwehavereasontoviewthecontentsofAMRsasstructured.
Anotherimportantdimensionalongwhichmentalcontentscandifferistheir
finenessofgrain.Anargumentoftenadvancedinfavouroffine-grainedcontentsisthat
theycanaccommodatedistinctmodesofpresentationofthesameentity.Forexample,a
24members,respectively,wouldhavegreaterorfewerthan90totalmembers.Thereisevidence,however,thatAMRsareinvolvedinsuchabstractcomparisonswhentheobjectsofcomparisoncannotbeperceived(Dehaene[2011],p.239ff.).IthusseenoreasontodenythatAMRscanrepresentnon-perceivableentitiesoutsideofperception.
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personmighthavetheabilitytothinkaboutVenusundertheHesperusmodeof
presentationwhilelackingtheabilitytothinkaboutVenusunderthePhosphorusmodeof
presentation.ThisconsiderationappliestoAMRs.Theabilitytothinkaboutagiven
magnitudeusingnumbersandunitsdiffersinkindfromtheabilitytothinkaboutitusing
AMRs.Forexample,theabilitytorepresentaten-seconddurationastensecondsisquite
differentfromtheabilitytorepresentitintheunit-freemanorassociatedwithAMRs.
ThereisthusreasontoviewAMRcontentsascomposedfrommodesofpresentation.
Theideathatcontentsarestructuredfrommodesofpresentationislikelytobring
tomindFregeanThoughts,whicharestructuredfromsenses.Infact,byclaimingthatAMR
contentsarestructuredfrommodesofpresentationitmayseemthatIhaveidentifiedthe
contentsofAMRswithFregeanThoughts.Itwouldbetoohastytodrawthatconclusion,
however,andfortworeasons.
First,ifpropertiesareindividuatedfinelyenough,itmaybepossibletoexplainthe
differencebetweenrepresentingaten-seconddurationastensecondsandrepresentingit
intheunit-freemanorassociatedwithAMRsintermsoftherepresentationofdistinct
properties.Thus,justassomearguethatthedifferencebetweentheHesperusand
Phosphorusmodesofpresentationboilsdowntothedifferencebetweenrepresentingthe
propertiesvisibleintheeveningandvisibleinthemorning,itisopentosomeonetoargue
thatthedifferencebetweenunit-ladenandunit-freemodesofpresentationboilsdownto
whetherintegerandunitpropertiesarerepresented.Inthatcase,however,therewouldbe
noneedtoappealtoFregeansenses,whicharesupposedtobedistinctfromrepresented
properties.
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Second,atleastastypicallyconceived,FregeanThoughtsandtheircomponents,
senses,aremeanttomarkaparticulartypeofabilitythatIllcallaconceptualability.One
featureofconceptualabilitiesisthattheyaresystematicallyrecombinable.Theconceptual
abilitytothinkthataisFandthatbisGentailstheconceptualabilitytothinkthataisG
andthatbisF.Inotherwords,conceptualabilitiesobeywhatEvans([1982])callsthe
GeneralityConstraint,whichholdsthattheconceptualthoughtsonecanthinkareclosed
underallmeaningfulrecombinationsoftheconstituentsofthesentencesthatbestexpress
them.Itisquestionable,however,whetherAMRsobeytheGeneralityConstraint.Ihave
arguedelsewhere(Beck[2012],[forthcoming])thatbecauseofWebersLawathinker
usingAMRscanhavetheabilitytorepresentthatamagnitudeof9islessthanamagnitude
of18,andthatamagnitudeof10islessthanamagnitudeof20,butnotthatamagnitudeof
9islessthanamagnitudeof10northatamagnitudeof18islessthanamagnitudeof20.
Moreover,becauseWebersLawistraceabletotheanalogueformatofAMRsthemselves,I
arguedthatthisfailureofrecombinabilityisafailureofrepresentationalcompetenceand
notmerelyafailureofdiscriminativeperformance.ItisinthenatureofAMRsthattheyare
unabletorepresentthatonemagnitudeislessthananotherwhentheirratioexceedsa
certainthreshold.Ifthatisright,thesortsofrepresentationalabilitiesthatunderlieAMRs
differinkindfromconceptualabilities.Theydonotexhibitthesameunfettered
recombinability.WethushavereasontodistinguishthecontentsofAMRsfromFregean
Thoughtsevenifbothtypesofcontentarestructuredfrommodesofpresentation.We
shouldconcludethatAMRshaveasuigeneristypeofnonconceptualcontentinstead.
4.Computations
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SupposingthatAMRsexist,thequestionarisesofwhatorganismscandowiththem.What
sortsofcomputationsdotheysupport?
4.1Arithmeticcomputation
Aboveall,AMRsareassociatedwitharithmeticcomputations,includingcomparison,
addition,subtraction,multiplication,anddivision.Recall,forexample,theabilityofducks
topositionthemselvesinproportiontotherateatwhichexperimenterstossmorselsof
bread.Thisabilityembodiesacapacityfordivisionsincetheabilitytorepresentrate
plausiblyderivesfromtheabilitytodividerepresentationsofnumerositiesby
representationsofdurations(Gallistel[1990],pp.351383).Italsoembodiesanabilityto
comparetworatesandcalculatehowmuchgreateroneisthantheother.Recallaswellthat
theducksalteredtheirstrategywhenoneexperimentertossedmorselsthatweretwicethe
sizeofthosetossedbytheother,therebyexemplifyinganabilitytomultiplymorselsizeby
feedingrate.
IfAMRssupportarithmeticcomputationsandarepartoftheinnatecognitive
hardwareofhumanbeings,thenhumanchildrenshouldexhibitaprimitiveabilityto
engageinarithmeticpriortobeingtrainedinit.Barthetal.([2006])setouttotestthis
prediction.Usinganimateddisplaysinwhichsetsofcoloureddotsmovebehindoremerge
fromanoccluder,theytestedtheabilityofpre-schoolchildrentocompare,add,and
subtractsetsofdots.Forexample,thechildrenmightseeasetof25bluedotsmovebehind
anoccluder,thensee25morebluedotsjointhembehindtheoccluder,thensee30reddots
andbeaskedwhethertherearemore(occluded)bluedotsor(unoccluded)reddots.
Childrensucceededoncomparison,addition,andsubtractiontasksevenaftercontrolling
fornon-numericalvariablessuchasdotcircumference,area,anddensity.Morerecently,
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McCrinkandSpelke([2010])usedasimilarparadigmtoshowthatchildrencouldsucceed
onnon-symbolicmultiplicationtasksbeforebeingschooledinmultiplicationordivision.
BothsetsofresultssuggestthatAMRssupportaprimitivetypeofarithmeticthatdoesnot
dependuponformalmathematicaltraining.16
4.2Practicaldeliberation
TofullyappreciatethecomputationalpowerofAMRs,itisessentialtonoticethattheycan
beusednotonlytorepresenthowtheworldis,butalsohowthecognizerwouldlikeitto
be.Thatis,AMRscanplayadesire-likeroleinadditiontoabelief-likerole.Considerthe
long-tailedhummingbird,whichforagesbyrecoveringnectarfromavarietyofwidely
dispersedsources,oftenflyingatleasthalfakilometreforanyonefeeding.Becausethe
birdistoosmalltostoremuchenergy,andconsumesenergyratherquickly,thereis
considerablepressureforittooptimizeitsfrequentforagingrunstofindanintervalthat
islongenoughfortheharvesttohavereplenishedsincethepreviousvisit,butnotsolong
thatthebountyislikelytohavebeenpilferedbyacompetitor.Thatinturnrequiresthe
birdtorepresenttherateatwhichvariousnectarsourcesreplenishafterdepletionandthe
temporalintervalsbetweenvisits.Itcanthencomputeanestimationoftheamountof
nectarthatiscurrentlyateachsource,andusethatvalueasaproxyfortheutilityof
visitingeachsource.Incontrolledenvironmentsusingartificialflowersthatarefilledwith
sugarwateratintervalssetbytheexperimenter,itcanbeshownthatbirdswill,infact,
optimizetheirvisits(Gill[1988]).
16ArithmeticcomputationsoverAMRsarediscussedatlengthin(Gallistel[1990]).Seealso(Brannonetal.[2001];Flombaumetal.[2005];McCrinkandWynn[2004];andBeranandBeran[2004]).
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OtherexamplesofhowAMRsmightencodedesiresorutilitiesarenothardto
dreamup.ArobinmightuseitsAMRoftherateofreturnofwormsinagivenfieldasa
proxyforthedesirabilityofforagingthere,amonkeymightuseitsAMRofthenumerosity
ofpredatorsstalkingittoestimatetheutilityofretreat,andahumanchildmightuseits
AMRofthesizeoftwopiecesofcakeasameasureofthedesirabilityofeach.Theabilityto
useAMRsinthiswayisimportant,asitopensupthepossibilityofembeddinganentire
processofpracticaldeliberationoverAMRswithinaformaldecision-theoreticframework,
suchasexpectedutilitytheory,wherebyananimalcalculatestheexpectedutilityofeachof
arangeofactionsandthenchoosestheactionwiththemaximumexpectedutility.
Toseehowthismightworkforasimplifiedcase,supposethatarobinisdeciding
betweentwoactions:foragingforwormsinthefield(A1)orforagingforberriesinthe
forest(A2).Andsupposethattheworldcanbeinoneoftwopossiblestates:raining(S1)or
notraining(S2),whererainincreasestheprevalenceofwormsinthefieldbuthasno
immediateeffectonhowmanyberriesareavailableintheforest.Assumingthatrobins
valuewormsandberriesonapar,theycouldcalculatetheexpectedutilityofeachactionas
follows(whereu(A|S)isthedesirabilitythattherobinassignstoactionAgiventhatthe
worldisinS,andprob(S|A)isthesubjectiveprobabilitythattheworldwillbeinSgiven
thatactionAisperformed).
EU(A1)=[u(A1|S1)xprob(S1|A1)]+[u(A1|S2)xprob(S2|A1)]
EU(A2)=[u(A2|S1)xprob(S1|A2)]+[u(A2|S2)xprob(S2|A2)]
Aswevealreadyseen,u(A|S)canbecalculatedbytherobinsAMRoftherateofreturnof
wormsinthefieldandberriesintheforestduringrain,andnorain,inthepast.Buthow
willtherobincalculatethevaluesofprob(S|A),i.e.theprobabilityofrain?Onepossibilityis
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spiteoftherelevanceofAMRstoahostofcentralissuesinthephilosophyofmind
concerningneuralrealization,analoguerepresentation,representationalcontent,
nonconceptualcontent,andanimalcognition.InthispaperIhavesoughttotakesomefirst
stepstowardsredressingthisimbalancebydrawingattentiontoAMRsandanalysingtheir
format,theircontent,andthecomputationstheysupport.AlthoughmyanalysisofAMRs
hasonlyscratchedthesurface,Ihopethatitwillassistphilosopherswhowanttogive
AMRsamorecentralplacewhentheytheorizeaboutthemind.
Acknowledgements
IpresentedmaterialfromthispaperattheannualmeetingoftheSouthernSocietyfor
PhilosophyandPsychologyinAustin,TexasinFebruary2013.Thankstothosewhobraved
an8amstarttoprovidemewithfeedback.IamalsoindebtedtomycolleaguesBrianHuss,
HenryJackman,JoshMugg,andespeciallyKristinAndrewsforvaluablecommentson
drafts,andtotwoanonymousrefereesforthisjournalwhosehelpfulsuggestionsspawned
substantialimprovements.
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