Analog Magnitude Representations

7/30/2019 Analog Magnitude Representations

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AnalogueMagnitudeRepresentations:APhilosophicalIntroduction

JacobBeck

YorkUniversity,Toronto

[email protected]

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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