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Promoting Maths to the General Public
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Subject: Psychology,CognitivePsychology,EducationalPsychology
OnlinePublicationDate: Feb2015
DOI: 10.1093/oxfordhb/9780199642342.013.47
PromotingMathstotheGeneralPublic ChrisJ.BuddTheOxfordHandbookofNumericalCognition(Forthcoming)EditedbyRoiCohenKadoshandAnnDowker
OxfordHandbooksOnline
AbstractandKeywords
InthischapterIwilladdresstheissuethatwhilstmathematicsisvitaltoallofourlives,playingavitalroleinmoderntechnologyandevenhelpingustounderstandthebrain,itisoftenperceivedtobeadry,boring,anduselesssubject.Thechapterwillexplorevariouswaysthatmathematicscanbepresentedtothegeneralpublicinawaythatmakesitseemtobeexcitingandrelevant,andcapturesitsessencewithoutdumbingitdown.Inparticularitwilllookatstrategiesthathavebeenshowntoworkwellwiththepublicincludingtheuseofcarefulreallifeexamplesandrelatingmathstopeople,handsonmathsatsciencefairs,andmathsinthemediaandontheInternet.Thechapterincludessomecasestudiesofwhatdoesanddoesnotworkinthefieldofmathscommunication.Keywords:Mathematics,popularization,formulae,sciencefairs,masterclasses
Whatsitallabout?Mathematicsisallaroundus,itplaysavitalroleinmuchofmoderntechnologyfromGoogletotheInternetandfromspacetraveltothemobilephone.Itiscentraltoeveryschoolstudentseducation,andanyoneneedingtogetamortgage,buyacar,sortouttheirhouseholdbills,orjustunderstandthevastamountofinformationnowthrownatthem,needstoknowsomemaths.Mathsisevenusedtohelpusunderstand,andimage,thecomplexnetworksandpatternsinthebrainandmanyoftheprocessesofperception.However,liketheairaroundus,theimportanceofmathematicsisofteninvisibleandpoorlyunderstood,andasaresultmanypeopleareleftunawareofthevitalrolethatitcould,anddoes,playintheirlives.Inanincreasinglytechnologyandinformationdrivenworldthisispotentiallyamajorproblem.
However,wehavetobehonest,mathematicsanditsrelevance,isadifficultsubjecttocommunicatetothegeneralpublic.Itcertainlydoesnthavetheinstantappealofsexandviolencethatwefindinotherareas(althoughitdoeshaveapplicationstothese)andthereisaproudculturaltraditionintheUKthatitisgoodtobebadatmaths.ForexamplewhenIappearedonceontheOneShow,bothpresenterswereverykeentotellmethattheywererubbishatmathsandthatitdidntseemtohavedonethemanyharm!(Idowonderwhethertheywouldhavesaidthesametoafamousauthor,artist,oractor.)Mathsisalsoperceivedasadrysubjectwithoutanyapplications(thisisalsoveryuntrueandIwilldiscussthislater)andthisperceptiondoesputalotofschoolstudents(andindeedtheirteachers)off.Finally,and(perhapsthisiswhatmakesitespeciallyhardtocommunicate),mathsisalinearsubject,andalotofbackgroundknowledge,andindeedinvestmentoftime,isrequiredofanyaudiencetowhomyoumightwanttocommunicateitsbeautyandeffectiveness.Forexample,oneofthemostimportantwaythatmathsaffectsallofourlivesisthroughtheapplicationofthemethodsofcalculus.Butveryfewpeoplehaveheardofcalculus,andthosethathavearegenerallyscaredbytheveryname.Italsotakestimeandenergytocommunicatemathswelland(tobehonest),mostmathematiciansarenotborncommunicators(infactrathertheopposite).However,it
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isapleasuretosaythattherearesomegiftedmathscommunicatorsouttherewhoaremakingaverypositiveimpact,aswellasuniversitycoursesteachingmathscommunicationskills.Indeed,thepopularizationofmathematicshasbecomeanincreasinglyrespectableandwidespreadactivity,andIwilldescribesomeofthisworkinthischapter.
Sowhydowebothercommunicatingmathsinthefirstplace,andwhatwehopetoachievewhenweattempttocommunicatemathstoanyaudience,whetheritisaprimaryschoolclass,bouncingoffthewallswithenthusiasm,oraboredclassofteenagersonthelastlessonoftheafternoon?Well,thereasonisthatmathsisinsanelyimportanttoeveryonesliveswhethertheyrealizeitdirectly(forexamplethroughtryingtounderstandwhatamortgagepercentageonanAPRactuallymeans)orindirectlythroughthevitalrolethatmathsplaysintheInternet,Google,andmobilephonestonameonlythreetechnologiesthatrelyonmaths.Moderntechnologyisanincreasinglymathematicaltechnologyandunlessweinspirethenextgenerationthenwewillrapidlyfallbehindourcompetitors. However,whencommunicatingmathswealwayshavetotreadanarrowlinebetweenboringouraudiencewithtechnicalitiesatoneend,andwateringmathsdowntotheextentofdumbingdownthemessageattheother.Ideally,incommunicatingmathswewanttogetthemessageacrossthatmathsisimportant,fun,beautiful,powerful,challenging,allaroundusandcentraltocivilization,toentertainandinspireouraudienceandtoleavetheaudiencewantingtolearnmoremaths(andmoreaboutmaths)inthefuture,andnottobeputoffitforlife.Ratherthandumbingdownmaths,publicengagementshouldbeaboutmakingmathematicscomealivetopeople.Thisiscertainlyatallorder,butisitpossible?WhiletheansweriscertainlyYES,thereareanumberofpitfallstotraptheunwaryalongtheway.
InthischapterIwillexploresomeofthereasonsthatmathshasabadimageand/orisdifficulttocommunicatetothegeneralpublic.Iwillthendiscusssomegeneraltechniqueswhichhaveworkedformyself,andothers,inthecontextofcommunicatingmathstoageneralaudience.Iwillthengoontodescribesomeinitiativeswhicharecurrentlyunderwaytodothis.FinallyIwillgivesomecasestudiesofwhatworksandwhatdoesnot.
Whatstheproblemwithmaths?Letsbehonest,wedohaveaprobleminconveyingthejoyandbeautyofmathematicstoalayaudience,andmathshasaterriblepopularimage.Alotofimportantmathsisbuiltonconceptswellbeyondwhatageneralaudiencehasstudied.Alsomathematicalnotationcanbecompletelybaffling,evenforothermathematiciansworkinginadifferentfield.Hereforexampleisashortquotefromapaper,authoredbymyself,abouttheequationsdescribingthe(onthefaceofitveryinteresting)mathematicsrelatedtohowthingscombustandthenexplode:
Thisquoteismeaninglesstoanyotherthanahighlyspecialistaudience.Tryingtotalkabout(sayinthisexample)thedetailedtheoryandprocessesinvolvedinsolvingdifferentialequationswithanaudiencewhich(ingeneral)doesntknowanycalculus,isawasteofeveryonestimeandenergy.Asaresultitisextremelyeasytokilloffevenaquiteknowledgeableaudiencewhengivingamathspresentationoreventalkingaboutmathsingeneral.Thesameproblemextendstoalllevelsofsociety.Mathsisperceivedbythegreatermajorityofthecountryasaboring,uncreative,irrelevantsubject,onlyfor(white,male)geeks.Allmathematiciansknowthistobeuntrue.Mathsisanextraordinarilycreativesubject,withmathematicalideastakinguswellbeyondourimagination.Itisalsoasubjectwithlimitlessapplicationswithoutwhichthemodernworldwouldsimplynotfunction.Notbeingabletodomaths(oratleastbeingnumerate)coststheUKanestimated2.4BeveryyearaccordingtoarecentConfederationofBritishIndustryreport(CBIReport,2010).Uniquelyamongstall(abstract)subjects,mathematiciansandmathematicsteachersareaskedtojustifywhytheirsubjectisuseful.Notonlyisthisunfair(whyismathsaskedtojustifyitselfinthisway,andnotmusicorhistory),itisalsoridiculousgiventhatwithoutmathstheworldwouldstarve,wewouldhavenomobilephonesandtheInternetwouldnotfunction.
Ihavethoughtveryhardaboutwhythepopularimageandperceptionofmathsissodifferentfromrealityandwhyitisculturallyfinetosaythatyouarebadatmaths.Therearemanypossiblereasonsforthis.
1
Let = f () . AweaksolutionofthisPDFsatisfiestheidentity dx = f ()dx () .H 10Assumethatf () growssub criticallyitisclearfromSobolevembeddingthat H 10
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Firstly,theobvious.Mathsisreallyhard,andnoteveryonecandoit.Fairenough.However,soislearningaforeignlanguageortakingafreekick,orplayingamusicalinstrument,andnoneofthesecarrythesamestigmathatmathsdoes.
Secondly,mathsisoftentaughtinaveryabstractwayatschoolwithlittleemphasisonitsextraordinaryrangeofapplications.ThiscaneasilyturnanaveragestudentoffwhatstheuseofthisMissisanoftenheardquestiontoteachers.Dontgetmewrong,Imallinfavorofmathsbeingtaughtasanabstractsubjectinitsownright.Itistheabstractnessofmathsthatunderliesitsrealpower,andevenquiteyoungstudentscanbecaptivatedbythepuzzlesandpatternsinmaths.However,Iamalsostronglyinfavorofallteachingofmathsbeinginfusedwithexamplesandapplications.Mathematiciansoftengomuchtoofaringlorifyingintheuselessnessoftheirsubject(witnesstheoftenquotedremarksbyHardyinAMathematiciansApology(Hardy1940)seeforexamplehisconcludingremarkinthatbook, whichwascertainlynottrue,givenHardyshugeimpactonmanyfieldsofscience).Howeverthisissheernonsense.Nothinginmathsiseveruseless.Ithinkthatitisthedutyofallmathematicianstounderstand,andconvey,theimportanceandapplicationsofthesubjecttoasbroadanaudienceaspossible,andtoteachersinparticular.
Thirdly,wehavestructuralproblemsinthewaythatweteachmathsinEnglishschools(lesssoinScotland).MostUKstudentsgiveupmathsattheageoffifteenorsixteenandneverseeitagain.Thesestudentsincludefutureleadersingovernmentandinthemedia.Whatmakesthisworseisthatthehugemajorityofprimaryschoolteachersalsofallintothiscategory.Theresultisthatprimarylevelmathsistaughtbyteacherswhoareoftennotveryconfidentinitthemselves,andwhocertainlycannotchallengethebrightestpupils.Theycertainlycannotappreciateitscreativeandusefulaspects.(IndeedwhenIwasatprimaryschoolinthe1960smathslessonswereactuallybannedbytheheadmistressasnotbeingcreative.)Studentsatschoolarethusbeingputoffmathsfartooearly,andaregivennoincentivetotakeitonpastGCSE.Evenscientists(suchaspsychologists!)whoneedmathematics(andespeciallystatistics)aregivingupmathsfartooearly.Perhapsmostseriouslyofall,thoseingovernmentorpositionsofpower,maythemselveshavehadnoexposuretomathsaftertheageof15,andindeedthereisawoefullackofMPswithanyformofscientifictraining.Howarethesepolicymakersthenabletocopewiththecomplexmathematicalissueswhicharise(forexample)intheproblemsassociatedwithclimatechange(seetheexampleattheendofthischapter).Weurgentlyneedtorectifythissituation,andthesolutionisforeverystudenttostudysomeformofmathsuptotheageof18,withdifferentpathwaysforstudentswithdifferentabilitiesandmotivation.(SeetheReportonMathematicalPathwayspost16(ACMEReport2011andalsoVorderman2011).
Finally,andIknowthatthisisasoftandobvioustarget,butIreallydoblamethemedia.Withnotable(andglorious)exceptions,mathshardlyevermakesitontoTV,theradio,orthepapers.Whenitdoesitisofteneitherextremelywrong(suchasthereportintheDailyExpressaboutthechanceofgettingsixdouble-yokedeggsinonebox)oritistreatedasacompletejoke(thelocalTVreportsofthehugeInternationalConferenceinIndustrialandAppliedMathematicsatVancouverin2011areagoodexampleofthis,see).
Sadlythistypeofreportistheruleratherthantheexception,orisgivensuchlittleairtimethatifyoublinkthenyoumissit.Contrastthiswiththeacresoftimegiventotheartsoreventonaturalhistory,andthereverencethatisgiventoafamousauthorwhentheyappearonthemedia.Partofthiscanbeexplainedbytheignoranceofthereporters(againafeatureofthestoppingofmathematicsattheageof15),butnothingIfeelcanexcusetheantagonisticwayinwhichreporterstreatbothmathematiciansandmathematics.Ihaveoftenbeenfacedbyaninterviewerwhohassaidthattheycouldntdomathswhentheywereatschool,ortheyneverusemathsinreallife,andthattheyhavedonereallywell.TowhichmyanswersarethattheyarenotatschoolanymoreandthatiftheycanunderstandtheirmortgageorinflationorAPRwithoutmathsthentheyaredoingwell.Worstofallarethosejournaliststhataskyoutoughmentalarithmeticquestionsliveonairtomakeyoulookafool(believemeyourmindturnstojellyinthissituation).Itisclearlyvitaltoworkwiththemedia(seelater),butthemediaalsoneedstoputitsownhouseinordertoundothedamagethatithasdonetothepublicsperceptionofmathematics.
Howcanmathsbegivenabetterimage?Aswithallthingsthereisnoonesolutiontotheproblemofhowtocommunicatetothebroaderpublicthatmaths
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isnttheirrelevantandscarymonsterthatthey(andthemedia)oftenmakeitouttobe.Manydifferentmathspresentershaveadopteddifferent(andequallysuccessful)styles.HoweversometechniquesthatIhavefoundtohaveworkedwithmanyaudiences(bothyoungandold)includethefollowing.
Startingwithanapplicationofmathsrelevanttothelivesoftheaudience,forexampleGoogle,iPods,crimefighting,music,codebreaking,dancing(yes,dancing).Hookthemwiththisandthenshow,anddevelop,themathsinvolved(suchasintheexamplesabove,networktheory,matrixtheory,andgrouptheory).Sciencepresenterscanoftenbeaccusedofdumbingdowntheirsubject,anditiscertainlytruethatitisimpossibletopresenthigherlevelmathstoageneralaudienceforthereasonsdiscussedabove.However,agoodapplicationcanoftenleadtomanyfascinatingmathematicalinvestigationsBeingproudnotdefensiveofthesubject.MathsreallyDOESmakeadifferencetotheworld.Ifmathematicianscantbeproudandpassionateofitthenwhowillbe?Beverypositivewhenaskedbyanyinterviewerwhatsthepointofthat.Showingtheaudiencethesurpriseandwonderofmathematics.Itisthecounter-intuitivesideofmaths,oftenfoundinpuzzlesortricks,thatoftengrabsattention,andcanbeusedtorevealsomeofthebeautyofmaths.Thepubliclovespuzzles,witnessthesuccessofSudoku,andmanyofthese(suchasGriddler,KillerSudoku,andproblemsincodebreaking)haveastrongmathematicalbasis.(ThosethatsaythatSudokuhasnothingtodowithmathssimplydontunderstandwhatmathsreallyisallabout!)Therearealsomanylinksbetweenmathsandmagic(asweshallseelater);manygoodmagictricksarebasedontheorems(suchasfixedpointtheoremsincardshufflingandnumbertheoremsinmind-readingtricks).Indeedagoodmathematicaltheoremitselfhasmanyoftheaspectsofamagictrickaboutit,inthatitisamazing,surprising,remarkable,andwhentheproofisrevealed,youbecomepartofthemagictoo.Linkingmathstorealpeople.Manyofourpotentialaudiencesthinkthatmathseithercomesoutofabook,orwascarvedinstonesomewhere.Nothingcouldbefurtherfromthetruth.Oneoftheproblemswiththeimageofmathsintheeyesofthegeneralpublicisthatitdoesnotseemtoconnecttopeople.IndeedarecentletterinOxfordToday()theOxfordalumnimagazine(whichreallyshouldhaveknownbetter!)saidthatthehumanitieswereaboutpeopleandthatsciencewasaboutthings(andthatasaconsequencethehumanitiesweremoreimportant).Whatrubbish!Allmathsatsomepointwascreatedbyarealperson,oftenwithalotofemotionalstruggleinvolvedorwithargumentandpassion.NoonewhohasseenAndrewWilesovercomewithemotionatthestartoftheBBCfilmFermatsLastTheoremproducedbySimonSinghanddescribedinhiswonderfulbook(Singh1997),canfailtobemovedwhenhedescribesthemomentthathecompletedhisproof.AlsostoriessuchasthelifeandviolentdeathofGalois,therecentsolutionofthePoincareConjecturebyabrilliant,butverysecretiveRussianmathematician,oreventhefamouspunchupsurroundingthesolutionofthecubicequationorthefactorizationofmatricesonacomputer,cannotfailtomoveeventhemoststony-facedofaudiences.Notbeingafraidtoshowyouraudiencearealequation.StephenHawkingfamouslyclaimedthatthevalueofamathsbookdiminisheswitheveryformula.Thisispartlytrueasmyearlierexampleshowed.Thereare,however,manyexceptionstothis.Evenanaudiencethatlacksmathematicaltrainingcanappreciatetheeleganceofaformulathatcanconveybigideassoconcisely.Someformulaeindeedhaveaneternalqualitythatveryfewotheraspectsofhumanendeavorcaneverachieve.Mindyou,itmaybeagoodideatowarnyouraudienceinadvancethataformulaiscomingsothattheycanbracethemselves.Soheregoes:
Isntthatsheermagic.Youcaneasilyspendanentirelecture,orpopulararticle,talkingaboutthatformulaalone.IfIameveraskedtodefinemathematicsthenthatismyanswer.Anyonewhodoesnotappreciatethatformulasimplyhasnosoul!Youcanfindoutmoreinmyarticle(Budd2013).Whole(andbestselling)books(Nahin2006)havebeenwrittenonarguablythemostimportantandbeautifulformulaofalltime
whichwasdiscoveredbyEulerandliesbehindthetechnologyofthemobilephoneandalsotheelectricitysupplyindustry.Formorefabulousformulaeseethebook17EquationsthatChangedtheWorld,byIanStewart(2012).
= 1 + + + 413
15
17
19
111
113
= 1ei
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Aboveall,beextremelyenthusiastic.Ifyouenjoyyourselfthenthereisagoodchancethatyouraudiencewilltoo.
So,whatsgoingon?AsIsaidearlier,wehaveseenarapidincreaseintheamountofworkbeingdonetopopularizemaths.Partlythisisadirectresultoftherealizationthatwedoneedtojustifytheamountofmoneybeingspentonmaths,andtoincreasethenumberofstudentsbothstudyingmathsandalsousingitintheirworkinglives.Ialsoliketothinkthatmorepeoplearepopularizingmathsbecauseitisanexcitingthingtodowhichbringsitsownrewards,inmuchthesamewaythatplayinganinstrumentoractinginaplaydoes.Mathscommunicationactivitiesrangefromhighprofileworkwiththemedia,towritingbooksandarticles,runningweb-basedactivities,publiclectures,engagingwithschools,busking,standupevents,outreachbyundergraduates,andsciencefairs.Inalltheseactivitieswearetryingtoreachthreegroups;youngpeople,thegeneralpublic,andthosewhocontrolthepursestrings.
TheMedia.AsIdescribedabove,themediaisaveryhardnuttocrack,withalotofresistancetoputtinggoodmathsinthespotlight.However,havingsaidthatweareveryfortunatetohaveanumberofhighprofilemathematicianscurrentlyworkingwiththemediaingeneralandTV/radioinparticular.OftheseImentioninparticularIanStewart,SimonSingh,MattParker,MarcusduSautoy,andSirDavidSpiegelhalter,buttherearemanyothers.TherecentBBC4seriesbyMarcusduSautoyonthehistoryofmathswasatriumphandhopefullytheDVDversionofthiswillendupinmanyschools)andwemustntalsoforgetthepioneeringworkofSirChristopherZeemanandRobinWilson.MarcusduSautoy,Matt,andSteveHumble(akaDrMaths)alsoshowusallhowitcanbedone,bywritingregularcolumnsforthenewspapers.Itishardtounderestimatetheimpactofthismediawork,withitsabilitytoreachmillions,althoughitisalongwaytogobeforemathsisaspopularinthemediaascooking,gardeningandevenarchaeology.
PopularBooks.IanStewart,RobinWilson,SimonSingh,andMarcusduSautoyarealsowell-knownfortheirpopularmathsbooksandareinexcellentcompanywithJohnBarrow,DavidAcheson,andRobEastaway,butIthinkthemostpopularmathsauthorbyquiteawidemarginisKjartanPoskitt.IfyouhaventreadanyofhisMurderousMathsseriesthendoso.Theyareobstensivelyaimedatrelativelyyoungpeopleandarefullofcartoons,buteverytimeIreadthemIlearnsomethingnew.Certainlymysonhaslearnt(andbecomeveryenthusiasticaboutmaths)fromdevouringmanyofthesebooks.
TheInternet.Mathematics,asahighlyvisualsubject,isverywell-suitedtobeingpresentedontheInternetandthisgivesusaverypowerfultoolfornotonlybringingmathsintopeopleshomesbutalsobeingabletohaveadialoguebetweenthemandexperiencedmathematiciansviablogsitesandsocialmedia.The(Cambridge-based)MathematicsMillenniumProject(theMMP)hasproducedatrulywonderfulsetofInternetresourcesthroughtheNRICHandPLUSwebsitesandtheSTIMULUSinteractiveproject.Dohavealookattheseifyouhavetime.IhavepersonallyfoundthePLUSwebsitetobeareallyfantasticwayofpublishingpopulararticleswhichreachaverylargeaudience.TheCombinedmathematicalSocieties(CMS)havealsosetuptheMathsCareerswebsite,,showcasingthecareersavailabletomathematicians.ImustntalsoforgettheverypopularCipherChallengewebsiterunbytheUniversityofSouthampton.
Directengagementwiththepublic.Thereisnosubstituteforgoingintoschoolsorengagingdirectlywiththepublic.Anumberofmechanismsexisttolinkprofessionalmathematicianstoschools,ofwhichthemostprominentaretheRoyalInstitutionMathematicsMasterclasses.Iambiasedhere,asIamthechairofmathsattheRoyalInstitution,butthemasterclasseshaveanenormousimpact.Everyweekmanyschoolsinover50regionsaroundthecountrywillsendyoungpeopletotake
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partinSaturdaymorningmasterclassesontopicsasvariousasthemathsofdeepseadivingtotheFibbonaccisequence.Thesemasterclassesareoftenrun(andarebasedin)theuniversitylocaltotheregionandareareallygoodwayforuniversitystafftoengagewithyoungpeople.Ofcourseitisimpossibletogettoeveryschoolinthecountryanditismuchmoreefficienttobringlotsofschoolstoreallygoodevents.OnewaytodothisisthroughtheLMSPopularLectures,theTrainingPartnershipLectures,andtheMathsInspirationseries().Thelatter(ofwhichImproudtobeapart)arerunbyRobEastawayanddelivermathslecturesinatheatresetting,oftenwithaveryinteractivequestionandanswersession.ArecentdevelopmenthasbeenthegrowthofMathsBusking().Thisisreallybuskingwheremathsitselfisthegimmickandreachesouttoanewaudiencewhowouldotherwisenotengagewithmathematicsormathematicians.CloselyrelatedarevariousstandupshowslinkedtomathssuchastheFestivaloftheSpokenNerdorYourDaysareNumbered.Theselinkmathstocomedyandreachouttoaverynon-traditionalmathsaudience,appearing,forexample,attheEdinburghFringe.
Sciencefairsareapopularwayofcommunicatingsciencetothepublic.Examplesrangefromthehuge,suchastheBritishScienceAssociationannualfestival,theBigBangFair,andtheCheltenhamfestivalofscience,tosmallerlocalactivitiessuchasBathTapsIntoScienceandMathsintheMalls(Newcastle).Ivisitandtakepartinalotofsciencefairsanditisfairtosaythatingeneralmathshastraditionallybeenverymuchunder-represented.Amongstthevastnumberoftalks/showsonbiology,astronomy,archaeology,andpsychologyyoumaybeluckytofindonetalkonmaths.Theproblemswereferredtoearlierofaresistancetocommunicatemathsinthemediaoftenseemtoextendtosciencecommunicatorsaswell.Fortunatelythingsareimproving,andthemathssectionoftheBritishScienceAssociationhasinrecentyearsbeenveryactiveinensuringthattheannualfestivaloftheBSAhasastrongmathspresence.Similarly,themathscontributiontowardstheBigBanghasgrownsignificantly,withtheIMArunninglargeeventssince2011,attendedbyapproximately50000participants.Hopefullymathematicswillhaveasimilarhighprofilepresenceatfuturesuchevents.Indeed2014marksthelaunchoftheveryfirstFestivalofMathematicsintheUK.Arelatedtopicisthepresenceofmathematicsexhibitsinsciencemuseums.ItissadtosaythatthemathsgalleryintheScienceMuseum,London,isveryoldandisfarfromsatisfactoryasanexhibitionofmodernmathematics.Fortunatelyitisnowinaprocessofredesign.SimilarlythegreatermajorityofexhibitsinsciencemuseumsaroundtheUKhavenomathsinthematall.Thereseemstobeasurprisingreluctancefrommuseumorganizerstoincludemathsintheirexhibits.However,ourexperienceofputtingmathsintosciencefairsshowsthatmathscanbepresentedinanexcitingandhandsonway,well-suitedtoamuseumexhibition.ItiscertainlymuchcheapertodisplaymathsthanmostotherexamplesofSTEM(Science,Technology,andMathematics)disciplines.ThesituationisratherbetterinGermanywheretheyhavetheMathemtikum()whichcontainsmanyhands-onmathsexhibitsaswellasorganizingpopularmathslectures,andinNewYorkwiththeMuseumofMaths.PlansareunderwaytocreateMathsWorldUKwhichwillbeaUK-basedmuseumofmaths.
MathsCommunicators.Finally,myfavoriteformofoutreachareambassadorschemesinwhichundergraduatesgointothecommunitytotalkaboutmathematics.Theycandothisfordegreecredit(asintheUndergraduateAmbassadorScheme()ortheBathMathsCommunicatorsscheme),forpaymentasintheStudentAssociateScheme,ortheycanactasvolunteerssuchasintheCambridgeSTIMULUSprogrammewhichencouragesundergraduatestoworkwithschoolstudentsthroughtheInternet.Theundergraduatescanbemainlybasedinschools,orcanhaveabroaderspectrumofactivities.WhateverthemechanismStudentAmbassadorSchemeshavebeenidentifiedasoneofthemosteffectiveactivitiesintermsofWideningParticipationandOutreach.TheycombinetheenthusiasmandcreativebrillianceofthepoolofmathsundergraduatesthatwehaveintheUK,withtheveryneednoonlytocommunicatemathsbuttoteachtheseundergraduatescommunicationskillswhichwillbeinvaluablefortheirsubsequentcareers.Everybodywinsinthisarrangement.Thestudentsoftendescribethesecoursesasthebestthingthattheydointhedegree,andtheycreatealastinglegacyofresourcesandalastingimpressionamongsttheyoungpeopleandgeneralpublicwhotheyworkwith.TherecentIMAreportonMathsStudentambassadorCasestudies()givesdetailsonanumberoftheseschemes.
Whatdoesntwork
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Irepeatthefactthatmathscanremainhardtocommunicate,anditisveryeasytofallintoanumberoftraps.Forthesakeofabalancedchapter(andtowarntheunwary)hereareafewexamplesofthese.
Toomuchortoolittle.Wehavealreadyseenanexampleofwheretoomuchmathsinatalkcanblowyouraudienceaway.Itisincrediblyeasytobetootechnicalinatalk,toassumetoomuchknowledgeandtofailtodefineyournotation.Weveallbeenthere,eitheronthegivingorthereceivingend.Thekeytowhatlevelofmathematicstoincludeistofindoutaboutyouraudienceinadvance.Inthecaseofschoolaudiencesthisisrelativelyeasyknowingtheyeargroupandwhetheryouaretalkingtotoporbottomsetsshouldgiveyouagoodideaofhowmuchmathstheyarelikelytoknow.YettoooftenIhaveseenspeakersstandinginfrontofamixedGCSEgrouptalkingabouttopicslikedotproductsanddifferentiationandassumingthattheseconceptswillbefamiliar.Itisequallydangeroustoputintoolittlemathsandtowaterdownthemathematicalcontentsothatitbecomescompletelyinvisible,or(asisoftenthecase)totalkonlyaboutarithmeticandtomissoutmathsalltogether.Withafewnotableexceptions,mostproducersandpresentersinthemedia,thinkthatanymathsistoomuchmathsandthattheiraudiencecannotbeexpectedtocopewithitatall.Butthisonlyhighlightstherealchallengeofpresentingmathsinthemediawheretimeandproductionconstraintsmakeitveryhardindeedtopresentamathematicalargument.InhisRoyalInstitutionChristmasLecturesin1978,ProfChristopherZeemanspent12minutesprovingthatthesquarerootoftwowasirrational.Itishardtothinkofanymainstreamprimetimebroadcasttodaywhereamathematicalideacouldbeinvestigatedinsuchdepth.Acoupleofminuteswouldprobablybethelimit,fartooshortatimetobuildaproof.Perhapsatsomepointinthefuturethiswillchange,butforthetimebeing,mathscommunicatorshavetoacceptthattelevisionisaverylimitedmediumfordealingwithmanyaccessiblemathematicalideas.
Thecurseoftheformula.AsIhavesaid,oneofthewaysofengagingaudiencesinmathsisbyrelatingittoeverydaylifeanddonecorrectlythiscanbeveryeffective.Thiscan,however,betakentoofar.Takingatopicthatisofgeneralinterestromance,forexampleandattemptingtomathematizeitinthehopethattheinterestofthetopicwillruboffonthemaths,canbackfirebadly.Muchofthemathsthatgetsreportedinthepressislikethis.Althoughwelovetheuseofformulaewhentheyarerelevant,theuseofirrelevantformulaeinatalkoranarticlecanmakemathsappeartrivial.Forexample,IwasoncerungupbythepressjustbeforeChristmasandaskedfortheformulaforthebestwaytostackafridgefortheChristmasDinner.Thecorrectanswertothisquestionisthatthereisnosuchformula,andanevenbetteransweristhatifanyonewasabletocomeupwithonetheywould(bytheprocessofsolvingtheNP-hardKnapsackproblem)pocket$1000000fromtheClayFoundation.Howeverthejournalistconcernedseemeddisappointedwiththeanswer.Nosuchreluctancehowevergotinthewayofthepersonthatcameupwith
Whichisapparentlytheformulafortheperfectkiss.AllIcansayis:whateveryoudo,dontdropyourbrackets.Forthemathematiciancollaboratingwiththepressthismightseemlikeagreatopportunitytogetmathsintothepubliceye.Tothejournalistandthereadingpublic,however,moreoftenitissimplyachancetodemonstratetheirrelevanceoftheworkdonebyboffins.Suchthingsarebestavoided.
AndwhatdoesworkIwillconcludethischapterwithsomeexamplesoftopicsthatcontainhigherlevelofmathsinthemthanmightbeanticipatedandcommunicatemathsinaveryeffectiveway.Moreexamplesofcasestudiescanbefoundinmyarticle(BuddandEastaway2010),oronmywebsite,oronthePlusmathswebsite.
Example1.AspergersSyndrome.InthebookTheCuriousIncidentoftheDogintheNight-timebyMarkHaddon(2004)thereaderwasinvitedtofindanexampleofaright-angledtriangleinwhichitssidescouldnotbewrittenintheformn +1,n -1and2n
K = F (T +C) LS
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(wheren>1).Onthefaceofitthiswasquiteahighlevelofmathematicsforapopularbook(whichhasnowbeenturnedintoaplay).TheCuriousIncidentisabookaboutAspergersSyndrome,writtenfromapersonalperspective.Millionsofpeoplehavereadthisbook,andmanyofthese(whoarenotinanysensemathematicians)havereadthispartofitandhaveactuallyenjoyed,andlearnedsomething,fromthis.Thereasonthisworkedwastwofold.First,themathswasputintothecontextofahumanstory,whichmadeiteasierforthereadertoempathizewithit.Thesecondwasthattheauthorusedacleverdevicewherebyheallowedtheleadcharactertospeakformaths,whilehisfriendspokeforthebaffledunmathematicalreader.Asaresult,Haddon(akeenmathematician)managedtosneakalotofmathsintothebookwithoutcomingacrossasageekhimself.
Example2.MathsMagic.Everyone(wellnearlyeveryone)likesthemysteryandsurprisethatisassociatedwithmagic.Toamathematician,mathematicshasthesamequalities,buttheyarelesswellappreciatedbythegeneralpublic.Onewaytobringthemtogetheristodevisemagictricksbasedonmaths.Ihavealreadyalludedtosomeofthese.Thegeneralideaistotranslatesomeamazingmathematicaltheoremintoasituationwhicheveryonecanappreciateandenjoy.Thesemayinvolvecards,orropes,orevenmind-reading.Asanexample,itisawell-knowntheoremthatifanynumberismultipliedbynine,thenthesumofthedigitsoftheanswerisitselfamultipleofnine.Similarly,ifyoutakeanynumberandsubtractfromitthesumofitsdigitsthenyougetamultipleofnine.Putlikethistheseresultssoundratherboring,butinthecontextofamagictricktheyarewonderfulambassadorsformathematics.Thefirstleadstoalovelymind-readingtrick.Askyouraudiencetothinkofawholenumberbetweenoneandnineandthenmultiplyitbynine.Theyshouldthensumthedigitsandsubtractfivefromtheiranswer.IftheyhaveaonetheyshouldthinkA,twothinkB,threethinkC,etc.Nowtakethelettertheyhaveandthinkofacountrybeginningwiththatletter.Takethelastletterofthatcountryandthinkofananimalbeginningwiththatletter.Nowtakethelastletteroftheanimalandthinkofacolorbeginningwiththatletter.Gotthat.WellhopefullyyouarenowallthinkingofanOrangeKangeroofromDenmark.
Thereasonthatthistrickworks,isthatfromthefirstoftheabovetheorems,thesumofthedigitsofthenumberthattheygetmustbenine.Subtractfivetogivefour,andtherestisforced.ThistrickworksnearlyeverytimeandIwasdelightedtoonceuseitforagroupofblindstudents,wholovedanythingtodowithmentalarithmetic.Forasecondtrick,takeapackofcardsandputtheJokerinascardnumbernine.Askavolunteerforanumberbetween10and19anddealputthatnumberofcardsfromthetop.Pickthisnewpackupandaskforthesumofthedigitsofthevolunteersnumber.Dealthatnumberofcardsfromthetop.Thenturnoverthenextcard.ItwillalwaysbetheJoker.Thisisbecauseifyoutakeanynumberbetween10and19andsubtractthesumofthedigitsthenyoualwaysgetnine.
Withacollectionofmagictricksyoucanintroducemanymathematicalconcepts,fromprimaryagemathstoadvancedleveluniversitymaths.Thebestwastodothis,istofirstshowthetrick,thenexplainthemathsbehindit,thengettheaudiencetopracticethetrick,andthen(andbestofall)getthemtodevisenewtricksusingthemathsthattheyhavejustlearned.Youneverknewthatmathscouldbesomuchfun!
Example3.HowMathsWontheBattleofBritain.Itmaybeunlikelytothinkofmathematiciansasheroes,butwithouttheworkofteamsofmathematicianstheAllieswouldprobablyhavelosttheSecondWorldWar.Partofthisstoryiswell-known.Theextraordinaryworkofthemathematicalcodebreakers,especiallyAlanTuringandBillTutte,atBletchleyParkhasbeenthesubjectofmanydocumentariesandbooks(andthisisoneareawherethemediahasgotitright).ThishasbeendescribedverywellintheCodeBook(also)bySimonSingh(1999).However,mathematicsplayedanequallyvitalroleintheBattleofBritainandbeyond.OneofthemainproblemsfacedbytheRAFduringtheBattleofBritainwasthatofdetectingtheincomingbombersandinguidingthedefendingfighterstomeetthem.TheproceduresetupbyAirViceMarshallDowdingtodothis,wastocollectasmuchdataaspossibleaboutthelikelylocationoftheaircraftfromanumberofsources,suchasradarstationsandtheRoyalObserverCorps,andtothenpassthistotheFilterRoomwhereitwascombinedtofindtheactualaircraftposition.TheFilterRoomwasstaffedbymathematicianswhosjobwastodeterminethelocationoftheaircraftbyusingacombinationof(three-dimensional)trigonometrytopredicttheirheight,number,andlocationfromtheirpreviousknownlocations,combinedwithastatisticalassessmentoftheirmostlikelypositiongiventhelessthanreliabledatacomingfromtheradarstationsandothersources.Once
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thelocationoftheaircraftwasknownfurthertrigonometrywasrequiredtoguidethefightersonthecorrectinterceptionpath(usingaflightdirectionoftencalledtheTizzyangleafterthescientificcivilservantTizard).AnexcellentaccountofthisandrelatedapplicationsofmathsisgiveninKorner(1996).Inaclassroomsettingthismakesforafascinatingandinteractiveworkshopinwhichtheconditionsinthefilterroomarerecreatedandthestudentshavetodothesamecalculationsunderextremetimepressure.Oneoftherealsecretstopopularizingmathsistogettheaudiencereallyinvolvedinahands-onmanner!(ItisworthsayingthatthesameideasofcomparingpredictionswithunreliabledatatodeterminewhatisactuallygoingonareusedtodaybothinAirTrafficControl,meteorologyandrobotics.)
Whilstitmightbethoughtthatthisisarathermaleorientedviewofappliedmathematics,itiswellworthsayingthatthemajorityofthemathematiciansemployedinthefilterroomswererelativelyyoungwomenintheWAAF,oftenrecruiteddirectlyfromschoolfortheirmathematicalabilities.Inaremarkablebook,EileenYounghusband(2011)recountshowshehadtodocomplexthree-dimensionaltrigonometricunderextremepressure,bothintimeandalsoknowinghowmanylivesdependedonhergettingthecalculationsright.AftertheBattleofBritainshegraduatedtotheevenharderproblemoftrackingtheV2rocketsbeingfiredatBrussels.WhenItellthisstorytoteenagers,theygetincrediblyinvolvedandthereisnotadryeyeinthehouse.Noonecaneveraccusetrigonometryofnotbeingusefulorinteresting!
Example4.WeatherandClimate.Oneofthemostimportantchallengesfacingthehumanraceisthatofclimatechange.Itisdescribedallthetimeinthemediaandyoungpeopleespeciallyareveryinvolvedwithissuesrelatedtoit.Thedebatesaboutclimatechangeareveryheated.Fromtheperspectiveofpromotingmathematics,climatechangegivesaperfectexampleofhowpowerfulmathematicscanbebroughttobearonavitallyimportantproblem,andinparticulargivespresentersachancetotalkaboutthewaythatequationscannotonlymodeltheworld,butareusedtomakepredictionsaboutit.Muchofthemathematicalmodelingprocesscanbedescribedandexplainedthroughtheexampleofpredictingtheclimateandtheaudienceledthroughthebasicstepsof:
(1)Makinglotsofobservationsofpressure,temperature,windspeed,moisture,etc.(2)Writingdownthe(partialdifferential)equations,whichtellyouhowthesevariablesarerelated.(3)Solvingtheequationsonacomputer.(4)Constantlyupdatingandcheckingthecomputersimulationswithnewdata.(5)Assessingthereliabilityoftheprediction.(6)Informingpolicybodiesabouttheresultsofthesimulations.
Thereareplentyofmathematicsandhumanelementstothisstory,startingfromEulersderivationofthefirstlawsoffluidmotion,theworkofthemathematiciansNavierandStokesonfluidsorKelvininthermodynamics(thelatterwasarealcharacter),thepioneeringworkofRichardson(anothergreatcharacter)innumericalweatherforecasting,andthemoderndayachievementsandworkofclimatechangescientistsandmeteorologists.However,therealclimaxoftalkingabouttheclimateshouldbethemathsitselfwhichcomesacrosswellasbeinganimpartialfactorinthedebate,farremovedfromthehotairofthepoliticians.Asasimpleexample,ifTisthetemperatureoftheEarth,eisitsemmisivity(whichdecreasesasthecarbondioxidelevelsintheatmosphereincrease),aisitsalbedo(whichdecreasesastheicemelts),andSistheenergyfromthesun(whichisaboutkWperm onaverage)then:
ThisformulacanbesolvedusingtechniquestaughtinAlevelmathematics,andallowsyoutocalculatetheaveragetemperatureoftheEarth.Thenicethingaboutthisformulaisthatunliketheformulafortheperfectkiss,thisonecanbeeasilycheckedagainstactualdata.FromtheperspectiveofclimatescienceitstrueimportanceisthatitclearlyshowstheeffectsontheEarthstemperature(andthereforeontherestoftheclimate)ofreducingtheemmisivitye(byincreasingtheamountofCarbonDioxideintheatmosphere)orofreducingthealbedoa(byreducingthesizeoftheicesheets.Thisleadstoafrighteningprediction.Thehotteritisthelessicewehaveastheicesheetsmelt.Asaconsequencethealbedo,a,decreases,sotheEarthreflectslessoftheSunsradiation.OurformulathenpredictsthattheEarthwillgethotter,andsomoreicemeltsandthecyclecontinues.Thuswecanseethepossibleeffectsofapositivefeedbackloopleadingtotheclimatespiralingoutofcontrol.Thisis
2
e = (1 a)ST 4
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somethingthatanyaudiencecanconnectwith,andleadstofiercedebates!Itmaycomeasasurprise,butIhavealwaysfoundthataudiencesgenerallyliketheunveilingofthisequation,andseeinghowitcanbeusedtomakepredictions.Atalkaboutmathematicscanbeexactlythat,i.e.aboutmathematics.Iftheaudiencegainstheimpressionthatmathsisimportant,andthattheworldreallycanbedescribedintermsofmathematicalequationsandthatalotofmathematicshastobe(andstillisbeingdone)tomakesenseoftheseequations,thenthetalktoacertainextenthasachieveditspurpose.Talksonclimatechangeoftenleadtoafuriousemail(andother)correspondence,whichgoesagainsttheimplicitassumptioninthemediathatnooneisreallyinterestedinamathematicalproblem.Atanotherlevel,climatechangeisexactlythesortofareawheremathematiciansandpolicymakersneedtocommunicatewitheachotherasclearlyaspossible,witheachsideunderstandingthelanguage(andmodusoperandi)oftheother.
Example5.MathsandArt.
ClicktoviewlargerFigure1 (a)ACircularCelticKnot.(b)TheChasedChickenSonapattern.
Oneoftheaspectsofmathematicswhichtendstoputpeopleoffisthatitisperceivedasadrysubject,farremovedfromcreativesubjectssuchasartandmusic.Ofcoursethisisnonsense,asmathsisascreativeasubjectasitispossibletoget(Ispendmylifecreatingnewmathematics),butitisworthmakingveryexplicitthewonderfullinksbetweenmathematicsandart.(Whenfacedwiththequestion:ismathsanartorascience?ThecorrectanswerissimplyYes.)Someoftheselinksrunverydeep,forexamplethemusicalscaleistheproductofmanycenturiesofmathematicalthought(startedbyPythagoras).Thesubjectoforigamiwasformanyyearstreatedsimplyasanartform.However,workingoutthefoldingpatterntocreateathree-dimensionalobject(suchasabeetle)fromasinglesheetofpaperisfundamentallyamathematicalproblem.ThiswasrealizedrecentlybyRobertLangamongstothers,andthefusionofmathematicswithOrigamileadstosublimeartisticcreations.AnotherareawhereartmeetsmathsinamulticulturalsettingisinCelticKnotsandtherelatedSonadrawingsfromAfrica.ExamplesofbothoftheseareillustratedinFigure1a,b,withFigure1ashowingacircularCelticKnotcreatedbyaschoolstudent,andFigure1baSonadesigncalledtheChasedChicken.
CelticKnotsaredrawnonagridaccordingtocertainrules.Theserulescanbetranslatedintoalgebraicstructuresandmanipulatedusingmathematics.Bydoingthis,studentscanexplorevariouscombinationsoftherules,andthenturnthemintopatternsofart.Thisisanincrediblypowerfulexperienceforthemastheyseethedirectrelationbetweenquitedeepsymmetrypatternsinmathematicsandbeautifulartwork.UsuallywhenIdoCelticArtworkshopsIhavetwosessions,onewhereIdescribethemathsandthenIwaitforamonthwhilstthestudentsworkwithanartdepartment.Bydoingthistheylearnbothmathsandartatthesametime.AsIsaid,averypowerfulexperienceallround.Anicespin-offistherelatedquestionofinvestigatingAfricanSonapatterns.MathematicallytheseareverysimilartoCelticKnots,andinfacttheideasbehindthempredatethoseofCelticKnots.AnexcellentaccountofthesepatternsalongwithmanyotherexamplesofthefusionofAfricanmathematicsandart,isgiveninGerdes(1999).DoingaworkshoponCelticKnotsandSonapatterns,demonstratesthefactthatmathsisnotacreationoftheWesternWorld,butisatrulyinternationalandmulti-culturalactivity.
AndfinallyIhopethatIhavedemonstratedinthischapterthatalthoughmathsishardandhasaterriblepublicimage,itisasubjectthatcanbepresentedinaveryengagingandhandsonwaytothegeneralpublic.Indeeditcanbeusedtobringmanyideastogetherfromarttoengineeringandfrommusictomulti-culturalism.Bydoingso,everyonecanbothenjoy,andseetherelevance,ofmaths.Thereisstillalongwaytogobeforemathshasthesamepopularity(andimage)onthemediaas(say)cookingorgardening(orevenastronomyorarchaeology),butsignificant
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progressisbeingmade(asmathematicianssayithasapositivegradient)andIamveryoptimisticthatintenyearstime,mathswillhaveaverymuchbetterpublicimagethanitdoesatthepresent.
ReferencesACMEReport(2011).Mathematicalneedsintheworkplaceandinhighereducation..
Budd,C.J.(2013).Howtoaddupquickly.PlusMathsMagazine,.
Budd,C.J.andEastaway,R.(2010).Howmuchmathsistoomuchmaths?MathsToday,(ThejournaloftheIMA),October2010.
CBIReport.(2010).Makingitalladdup..
Gerdes,P.(1999).GeometryfromAfrica.TheMathematicalAssociationofAmerica.
Haddon,M.(2004).TheCuriousIncidentoftheDogintheNighttime.Vintage,London.
Hardy,G.H.(1940).AMathematiciansApology.Cambridge:CambridgeUniversityPress.
Korner,T.W.(1996).ThePleasuresofCounting.Cambridge:CambridgeUniversityPress.
Nahin,P.J.(2006).DrEulersFabulousFormula.NewJersey,PrincetonUniversityPress.
Poskitt,K.MurderousMaths(series).Scholastic,London..
Singh,S.(1997),FermatsLastTheorem.4thEstate,London.
Singh,S.(1999).TheCodeBook.Doubleday,NewYork.
Stewart,I.(2012).SeventeenEquationsthatChangedtheWorld.ProfileBooks,London.
Vorderman,C.,Porkess,R.,Budd,C.,Dunne,R.,andHart,P.(2011).Aworldclassmathematicseducationforallouryoungpeople.
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Subject: Psychology,CognitivePsychologyOnlinePublicationDate: Aug2014
DOI: 10.1093/oxfordhb/9780199642342.013.039
PhilosophyofNumber MarcusGiaquintoTheOxfordHandbookofNumericalCognition(Forthcoming)EditedbyRoiCohenKadoshandAnnDowker
OxfordHandbooksOnline
AbstractandKeywords
Therearemanykindsofnumber.Thischapterconcentratesonfinitecardinalnumbers,astheyhaveabasicroleinourthinking.Numberscannotbeseen,heard,touched,tasted,orsmelled;theydonotemitorreflectsignals;theyleavenotraces.Sowhatkindofthingarethey?Howcanwehaveknowledgeofthem?Theaimofthischapteristopresentandassessthemainanswerstothesequestionsclassicalandneo-classical,nominalism,mentalism,fictionalism,logicism,andtheset-sizeview.Allviewsaredisputed,includingtheviewIwillarguefor,theset-sizeview.Thefinalsectionrelatesthefinitecardinalnumberstothenaturalnumbers.Keywords:cardinalnumber,nominalism,mentalism,fictionalism,logicism,set-size,structureofnaturalnumbers
Therearemanykindsofnumber:naturalnumbers,integers,rationalnumbers,realnumbers,complexnumbersandothers.Moreover,thesystemofnaturalnumbersisinstantiatedbyboththefinitecardinalnumbersandthefiniteordinalnumbers.Wecannotdealproperlywithallofthesenumberkindshere.Thischapterconcentratesonthefinitecardinalnumbers.ThesearethenumberswhichareanswerstoquestionsoftheformHowmanyFsarethere?Inwhatfollows,anunqualifieduseofthewordnumberabbreviatescardinalnumber.
Numberscannotbeseen,heard,touched,tasted,orsmelled;theydonotemitorreflectsignals;theyleavenotraces.Sowhatkindofthingsarethey?Howcanwehaveknowledgeofthem?Thesearethecentralphilosophicalquestionsaboutnumbers.Plausiblecombinationsofanswershaveprovedelusive.Theaimofthischapteristopresentandassessthemainviewsclassicalandneo-classical,nominalism,mentalism,fictionalism,logicism,andtheset-sizeview.Allviewsaredisputed,includingtheviewIwillarguefortheset-sizeview.Thefinalsectionrelatesfinitecardinalnumberstonaturalnumbers.
TheClassicalView:MultitudesofUnitsAtthestartofBookVIIofEuclidsElements,havingdefinedaunittobeasingleindividualthing,anumber(arithmos)isdefinedthus:
Anumberisamultitudeofunits.
(Euclid2002,p.157,BookVII,definitions1and2).Onthisview,anypairofitemsisa2andsotherearemany2s;anytrioisa3andsotherearemany3s.Ingeneral,anypluralityofkthingsisakandtherearemanyks.Therewasnonotionofzero;a1isaunit,notapluralityofunitsandthereforenotanumber.Weretainacorrespondinguseofthewordnumber,aswhenwesaythatanumberofauthorswerelatewiththeirsubmissions.Wecanapplyarithmeticperfectlywelltakingnumberstobepluralities:asthenumberofauthorswhowerelateisa9andthenumberwhowereontimeisa14,thenumberofauthorsintotalisa23.Multiplicationentailsthatpluralitiesthemselvesmayconstituteunits,as6multipliedby3istheaggregateofatrioof6s;thepotentialforconfusion
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seemsnottohavebeenaproblem.
Whilethetruthsofarithmeticarespecificnumericalfacts(suchas:1919=361),numbertheoryconsistsofgeneraltruthsaboutnumbers(suchas:thereisnogreatestprimenumber)andproofsofthosetruths.AtraditionofnumbertheorywasoneoftheimpressiveintellectualachievementsoftheancientGreekspeakingworld.Itwasstudiedforitsownsakeandregardedasabodyofunqualifiedandunchangingtruths.ManyareprovedinEuclidsElements(2002),BooksVIIIX.Here,forexample,isProposition30ofBookVII:
Ifaprimenumbermeasures[divideswithoutremainder]theproductoftwonumbers,italsomeasuresoneofthosetwonumbers.
Numbertheory,incontrasttoappliedarithmetic,seemsnottobeaboutpluralitiesoftheordinarythingswecount(suchassheep,votesorchimes).Onthecontrary,themathematiciansseemedtohaveinmindpluralitiesofunitswithoutdistinguishingcharacteristics.Plato(1997a,56d)proposedexactlythis:thenumbersofnumbertheoryarepluralitiesofpureunits,whereapureunitisasingleentitylackinganycharacteristicdistinguishingitfromanyotherpureunit.Thisaccounthastwoadvantages.Itavoidsaproblemabouttheperceivedinequalityofordinaryunits:Howcouldoneshipbeequaltooneplank?Anditavoidsmakingthesubjectmatterofnumbertheorycontingent.
ButtherearealsoseveralproblemswithPlatosproposal.Pureunitsaremysterious.Howcantherebetwoormoreentitieswhoseonlydifferenceisthattheyaredifferent?Twodistinctthingsmayhaveallthesamequalities.Butwouldtheynothavetobeindifferentpositions?Apureunit,however,lacksposition.Andtherearefurtherquestions.Whatistheoriginofpureunits?Aretheyinternalorexternaltothemind?Isthereaninexhaustiblesupplyofthem?Howcanweknowofthem?Howcanwehaveacognitivegraspofapluralityofthemiftheyareindistinguishablefromeachother?Plato(1997b,526a)saysthatthesepurenumberscanbegraspedonlyinthought,butdoesnotelaborate.
AlthoughPlatosaccounthasanechoinamentalistviewputforwardbyGeorgCantorovertwomillennialater,itwastoofraughtwithdifficultiestohavemuchstayingpower.Whiletheclassicalviewcontinuedtobeacceptedforappliedarithmeticbysomelaterthinkers,otheraccountsofnumberinpurearithmeticandnumbertheoryweresought,anditistothesethatwenowturn.
NominalismThedecimalplacesystemofnumerals,originatinginIndia,reachedEuropeinthe13thcenturyviaArabmathematicians.Bythe17thcentury,symbol-manipulationalgorithmsusingthedecimalplacesystemhadsupersededcalculationbyabacus.ThiswasthebackdropfortheviewproposedbythephilosopherGeorgeBerkeley(Berkeley,1956,p.25;Berkeley,1989,entry763)thatthenumbersofpurearithmetic(i.e.whennotappliedtopluralitiesofphysicalobjects)arenothingbutnames,meaningthat,forexample,thenumber26isnothingoverandabovethenumeral26.ThemathematicianDavidHilbert(1967,p.377)suggestedthatanumberisahorizontalstringofshortverticalstrokes;arabicnumeralsabbreviatethecorrespondingstrings,e.g.3isshortforIII.Morerecently,thephilosopherSaulKripkehassuggested,inunpublishedlectures,thatnumbersarenumeralsinaplacesystemofnumerals.Theseviewsareversionsofnominalism,bywhichIrefertotheidentificationofnumberswithnumerals.Thisistobesharplydistinguishedfromtheclaimthatthereisnothingabstract,alsosometimescallednominalism.Anumeral,asopposedtoitsparticularoccurrences,hastobeabstract,beingatypeofmark.
Whythinkthatnumbersarenumerals?Berkeley(1956,p.25;1989,entry761)notedthatlargenumberswithintherangeofperformablecalculationsdefyprecisesensoryrepresentation.So,whenwethinkof201,whatispresenttothemindisnotarepresentationof201items,butjustthenumeral.Althoughempiricalstudiesindicatethatwecannot,withoutcounting,telltheprecisenumberofanylargecollectionofthingspresentedtous,itdoesnotfollowthatourideaofalargenumberisjustarepresentationofitsnumeral.Analternativeisthatwehaveadescriptivewayofmentallydesignatingthenumberintermsofsmallernumbers,suchastwotensoften,plusone;thenumbertenisknownasthenumberofonesfingersandthenumbersoneandtwohavepreciserepresentationsinwhatcognitivescientistshavecalledthenumbersense,whichIsaymoreaboutinthesectionNumbersasSet-sizes.
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Berkeleysotherreasonfornominalismisthatwhenwemakearithmeticalcalculationsweseekananswergivenbyanumeralinthedecimalplacesystem(1989,entry766).Ananswerinanyotherformatisnotwhatiswanted(e.g.abacusdisplay,Romanorbinarynumerals).Thatisright.IfyouaskWhatiseighttothepowerofsixIcananswerrightawaythatitis1,000,000inbase8notationatrivialandunhelpfulanswer.Wewantanswersinthedecimalplaceformat.Isthatfactbestexplained,however,byclaimingthatnumbersinarithmeticarethedecimalplacenumerals?Hereisanalternativeexplanation.Wewantanswersinthedecimalplaceformatbecause(a)wewanttobeabletouseanswersasinputsforothercalculations,andourcalculationalgorithmsrequireinputsindecimalplaceformat;(b)oursenseofnumbersizeistiedtothenumeralswearemostfamiliarwith,thearabicdecimalplacenumerals.Considerthefollowingnumberherepresentedinbinarynotation:1010101.Isitlargerorsmallerthanseventy?Youwillprobablyhavetoconvertthisintoaverbalnumberexpressionordecimalnotationinordertobesureoftheanswer,butifthenumberispresentedasadecimalnumeralyouwillknowimmediatelythatitislargerthanseventy:itis85.Assumingthatoursenseofnumbersizeiswell-linkedtoourverbalnumberexpressions,thisisevidencethatournumbersenseismorestronglyassociatedwithdecimalthanwithbinarynumerals,eventhoughweunderstandboth.
Ameritofnominalismisthatitsaysclearlywhatnumbersareandbringsthemwithintheboundsofhumancognition.Butallversionsofnominalismfaceseriousobjections.AdecisiveobjectiontoBerkeleysnominalismisthatthesamecommoncoreofarithmeticalinformationcanbeexpressedusingdifferentnumeralsystemsorevenordinarywords:XIIetIXfitXXI;12+9=21;1100+100=10101;twelveplusnineistwenty-one.ThiscanbemetbyHilbertsprescription(1967)thatthenumeralsofcustomarysystemsberegardedasabbreviationsforrowsofshortverticalstrokes,butthisisunconvincing.Whyrowsofstrokes,asopposedtocolumnsofdots?Anychoiceofcanonicalnumeralswillbearbitrarywecanhavenogoodreasonforthinkingthatthechosensymbolsarewhatmathematiciansarereallyreferringto.
Asecondobjection,decisiveagainstanyversionofnominalismwhenextendedtotheobjectsofnumbertheory,isthattruthsofnumbertheoryareindependentofnumeralsystems.Considerthetheoremthatanypluralnumberisaprimeorproductofprimes.Thisisaconsequenceofthefactthatthereisnoinfinitedecreasingsequenceofsmallerpositivenumbers,sothatifonefactorsanon-primeintotwosmallernumbersandcontinuesfactoringthefactors,theprocessisboundtoterminateafterfinitelymanystepsinprimes.
MentalismMentalismistheviewthatanumberisamentalentity,aninnatelysuppliedrepresentationoraproductofintellectualactivity.Withinthementalistcampviewsdiverge.ThemathematicianGeorgCantor(1955,p.86)claimedthatthenumberofthingsinagivenclassisanimageormentalprojectionthatresultswhenweabstractfromthenatureofmembersoftheclassandtheorderinwhichtheyaregiven.ThemathematicianLuitzenBrouwer(1983,p.80),founderofintuitionistphilosophyofmathematics,regardednumbersasresultingfromthementalsplittingofanexperienceofatemporalperiodintotwo,andthesimultaneousrepresentationofthisintoarememberedthenandacurrentnowasconstructionsofthefirsttwonumbers1and2.Thenwiththepassageoftime,whatwasthenowextendsintoanewthenandnow,togiveus3,andsoon.Analternativeideaisthatnumbershavetheirorigininthepracticeofcounting.Morerecently,thecognitivescientistStanislasDehaene(1997)hassuggestedthatnumbersarejustourmentalnumberrepresentations,aninternalversionofnominalism.
Theadvantageofmentalismisthattheknowledgeofnumbersbecomeslessmysterious.AsDehaene(1997,p.242)putsit,Iftheseobjectsarerealbutimmaterial,inwhatextrasensorywaysdoesamathematicianperceivethem?Butifnumbersarejustmentalitems,theymaybeknowablebyinnerawarenessandreflection.
Letusputasidethequestionoftheplausibilityofthevariouscognitivehypothesesproposedbymentalists.Thebigproblemforanyversionofmentalismisthatonlyfinitelymanybrainstateshaveactuallybeenrealised;hence,thereareonlyfinitelymanymentalentities,whetherinnatelygivenorproducedbyintellectualactivityoracombinationofthetwo.Sotheideathatnumbersarementalentitiesconflictswiththefactthat,foranynumber,thereisayetgreaternumber.Howcanweknowthisfact?Therearemanyways.Forexample,anynumbernisthenumberofprecedingnumbers,aswestartwith0;sothenumberofnumbersuptoandincludingnisgreaterthannbyone.
Therearetworesponsestothis.Oneisstrictfinitism,theviewthatdespiteacceptednumbertheory,thenumbers
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runoutatsomefinitepoint.Thisviewhaslittleplausibilityandcommandsscantsupport,althoughithasbeeninvestigatedbysomelogicians.Theotherresponseistoconcedeandweakentheclaim:numbersarepossiblementalentities.ThephilosopherMichaelDummett(1977,p.58)hassuggestedthatanumbernisthepossibilityofcountingupton.Theimmediateproblemwiththisresponseisthatwhatwecouldmentallyrepresentorconstruct,aswellaswhatweactuallydomentallyrepresentorconstruct,hasfinitelimitations.Thereasonisthatthereareonlyfinitelymanypossiblebrainstatestakeanupperboundonthenumberofneuronsinahumanbrainandmultiplyitbyanupperboundonthenumberofpossiblestatesofaneuron;theresultwillbeafiniteupperboundonthenumberofpossiblehumanbrainstates.Sothenumbersoutstripourpossiblementalconstructions.Onemightseektoescapethisbyholdingthattherecouldbeevermorepowerfulminds,sothatanylimitationononepossiblemindcouldbesurpassedbyanother.Butthisisjustametaphysicalspeculation.Howdoweknowthattherecouldbesuchanintellectualhierarchy?Itfalsifiesourrealepistemicsituationourknowledgethattherearemorethanfinitelymanynumbersdoesnotdependonourknowingthatthismetaphysicalspeculationistrue.
FictionalismHavingreviewedseveralanswerstothequestionWhatkindofthingsarecardinalnumbers?andfoundthemwanting,whatoptionsareleft?Oneanswerisfictionalism:therearenonumbers,andsoacceptedarithmeticalclaimssuchas2+3=5or3isprimeareuntrue,astheyentailthattherearenumbers;butacceptedarithmeticisusefulandmathematicalpracticeshouldcontinueasifitweretrue.
Howmightonereachthisdesperateconclusion?Hereishowthemainlineofthoughtgoes.Numbersarenotmaterialormental.Ifnumbersarenotmaterialormental,theymustbeabstract.Butifabstract,theymustbeunknowable,itisargued,asabstractaareunperceivable,leavenotraces,anddonotinfluencethebehaviourofperceptiblethings.Soournumeralsdonotrefertoanything.
Overrecentdecadesfictionalismhasbeenadvocatedbyseveralphilosophersandtakenveryseriouslybyothers(Balaguer,2011;Field,1980;Leng,2010;Yablo,2005).Butitinvolvesaseriousmethodologicalflaw.Optingforonephilosophicalsolutionoverothersmaybefineifoneisdenyingnothingbutabunchofotherphilosophicalviews,butnotifoneisdenyingbothrivalphilosophicalviewsandpropositionsofindependentstandingthataregenerallyregardedbyrationalthinkersasamongthemostcertainthingsthatweknow.Nometaphysicalorepistemologicaldoctrinehasgreaterrationalcredibilitythanbasicarithmetic.Ourconfidenceinbasicarithmeticisnotanarticleoffaith;ourbeliefthat2+3=5,forexample,iswellsupportedbyourcountingexperience.InthesectionstocomeIwillarguethattherearecrediblenon-fictionalistresponsestoourquestionsaboutnumberandIwillpinpointanerrorthatmaypreventfictionalistsfromappreciatingthis.
Neo-classicalViewsTheclassicalviewthatcardinalnumbersaremultitudesofunitswastakenupbythephilosopherJohnStuartMill,withonemodification.Thechangeisthatunits,orones,countasnumberstoo.Heavoidsproblemsaboutpurenumbersbydenyingthattherearesuchthingsandheavoidsdenyingarithmeticaltheoremsbyconstruingthemasgeneralstatements(Mill,1974,II.vi.2):
Allnumbersmustbenumbersofsomething;therearenosuchthingsasnumbersintheabstract.Tenmustmeantenbodies,ortensounds,ortenbeatingsofthepulse.Butthoughnumbersmustbenumbersofsomething,theymaybenumbersofanything.Propositions,therefore,concerningnumbers,havetheremarkablepeculiaritythattheyarepropositionsconcerningallthingswhatever.
Outsideofanyapplication,anequationsuchas3=2+1meansthatanyparcelofthreethingscanbearrangedtoformoneparceloftwothingsandoneotherthing.Morecomplicatedequations,suchasthecubeof12is1728canbedealtwithinasimilarway(Mill,1974,III,xxiv,5).Onthisreadingnumericaltermsdonotdesignatepurenumbers,butmerelysignifydifferingwaysinwhichapluralityofthingscanbearranged.Millalsoheldthatthethingsinquestion,theunits,areperceptible;thus,theequationsofarithmeticaregeneralclaimswithempiricalcontent.
AmajorproblemwithMillsaccountofequationsisthatithasrestrictedapplication.ReferringtoMillsclaimthat3
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=2+1meansthatanyparcelofthreethingscanbeseparatedintoaparceloftwothingsandoneotherthing,thelogicianGottlobFrege(1980,p.9)says:
Whatamercy,then,thatnoteverythingintheworldisnaileddown;forifitwere,weshouldnotbeabletobringoffthisseparationand2+1wouldnotbe3!
Wecertainlywanttobeabletoapplyarithmetictothingsthatcannotbere-arranged,suchaslunareclipsesorsolutionsofanequation.AnotherproblemforMill:Whatfactaboutseparatingandre-arrangingobjectsisexpressedbytheequation2 =1?
Thewayoutoftheseproblemsistothrowoffempiricistconstraintsandunderstandarithmeticasabodyofgeneraltruthsaboutsetsofanykind(including1-memberedsetsandtheemptyset),andtointerpretnumericalequationsintermof1-1correlations,asisdoneinstandardsettheory.PreciselythisisproposedbythemathematicallogicianJohnMayberry(2000).Onthisview3=2+1meansthatthereisa1-1correlationbetweenany3-memberedsetandtheunionofanypairsetwithanysingle-memberedsetnotincludedinthepairset;23=6meansthatthereisa1-1correlationbetweenthesetofunitsinanypairofdisjointtriplesandanysextet;even2=1comesoutright,thoughthisisnotimmediatelyobvious.Infact,thewholeofcardinalarithmeticispreservedthisway.
AnycardinalnumberonMayberrysviewisasetandanysetisacardinalnumber.Accordingly,foranynumberkapartfromzero,therearemanyks:manypairs,manytrios,andsoon.Thereareseveraladvantagestothisaccount:
(1)Fregesobjectionsdonotapply:theaccountcatersfornumbersofeclipsesandsolutionsaswellasfornumbersofapplesandpebbles.(2)Itallowsforthearithmeticof0and1,asin2 =1.(3)Theproblemofdeterminingwhichparticularmathematicalobjectisnamedbyanumeral,say5,doesnotarise,asanyquintetisa5.(4)Ifnumbersareabstractobjects,weneedtoexplainhowwecanknowofthem;ifnumbersaresets,wecanknowofthembyknowingsets.
Inmanycasesknowingaset,suchasthesetofyoursiblings,isfarlessmysteriousthanknowinganabstractdenizenofsomespecialmathematicalrealm.
ButMayberrysaccountrunsintodifficultywithgeneraltheoremsaboutcardinalnumbers,preciselybecauseitallowsthattherearemanynumbersofeachsize.Forexample,itisatheoremthatanynumberhasauniquesuccessor,butanygivenpaircanbeextendedtomanydifferenttrios,byaddingdifferentobjectstothepair.Anothertheoremisthatexactlyonepositivenumberhasasquarethatisequaltoitsdouble.Thatnumberis2,butontheneo-classicalaccounttherearemany2s.Moreover,wecountnumbersthemselves.Forexample,wesaythatthereareexactlyfourprimeslessthan10,namely,2,3,5,and7.Thismakesnosenseunlessthereisjustonenumberpernumeral.Thepossibilityofenumeratingnumbersgivesrisetoimportantfunctionsinnumbertheory,suchasthenumberofprimeslessthanorequalton.Sotheaccountstandsinprimafacieconflictwithnumbertheory.
Theneo-classicistcanmaketwokindsofresponse,concessive,oraggressive.Theconcessionsaysthattheaccountdoesnotapplytonumbertheoryingeneral,onlytoarithmeticalequations.WhileIagreethatmodernnumbertheoryisnottobeconstruedasatheoryoffinitecardinalnumbers(asIwillexplaininthefinalsection),thetheoremsofnumbertheorysurelyapplytothefinitecardinalnumbers.Sotheconcessiondoesnotsavetheneo-classicalviewfromconflictwithnumbertheory.Theaggressiveresponseistoclaimthatpropositionsofnumbertheorymustbeinterpretedtohaveahiddenprefix,meaning:inanyomega-sequenceofnumbers(sets)startingwiththeemptyset,eachlatersetextendingitspredecessorbyonemember...,whereanomega-sequenceisasequenceofelementsconformingtotheDedekindaxioms(giveninthesectionTheFiniteCardinalNumbersandtheNaturalNumbers).Whilethatwouldeliminatetheconflict,itisimplausibleasaninterpretationoftheclaimsofnumbertheorymadebymathematiciansbeforethe19thcentury,astheydidnotyethavetheconceptofanomegasequence.Otherresponsesarepossible,butthoseIknowaboutaredifficultandnolesscontentious.Timetolookatotherviews.
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Logicism:NumbersasSpecialSetsOntheneo-classicalviewthatnumbersarejustsets,therearemanytwos,manythrees,andsoon,insteadofjustonenumbertwo,onenumberthree,etc.Intheprevioussection,wefoundthatthisviewrunsintotroublewithnumbertheory.Analternativethatavoidsthatproblemistotakeacardinalnumberktobesomethinglikeaspeciestowhichallthek-memberedsetsbelong.
ViewsofthiskindwereputforwardbyGottlobFrege(1980,pp.7980)andBertrandRussell(1919),aspartoftheirphilosophythatthemathematicsofnumberispartofpurelogic.Withwrinklesironedout,theclaimis:
Thenumberkisthesetofk-memberedsets.
Thisrunsintotwoproblemsonemathematical,theothermetaphysical.Themathematicalproblemisthatthereisnosetofsetswithexactly1member;hencetherewouldbenocardinalnumber1.Fromtheassumptionthatthereissuchaset,twouncontroversialprinciplesaboutsets(unionandseparation)leadstraighttoRussellsparadox.Russell(1908)evadestheparadoxbymeansofhistheoryoftypes.Thedetailsofhistheoryoftypesneednotdetainus,butanessentialelementisthatthingsareregardedasfallingintoexclusivelayersortypesordinaryindividualitemsareoftype0,setsofindividualsareoftype1,setsofsetsofindividualsareoftype2,andingeneralsetsofthingsoftypenareoftypen+1.Theneachnumberksplitsintomany,thesetofallk-memberedsetsofthingsoftype0,thesetofallk-memberedsetsofthingsoftype1,andsoon.
Russellsmany-typesviewfacesseveralproblems.First,ittakesusbacktothedisadvantageoftheneo-classicalview,havingmanyones,manytwos,etc.Secondly,itconflictswithmathematicalpractice,whichallowsthatsomesetsofdifferenttypehavethesamecardinalnumber.Finally,toestablishthecorrectnessoftheprinciplesofnumbertheory,Russellhadtoassumethatthereareinfinitelymanyindividuals,butthisisclearlyawry(andcontrarytohislogicistoutlook),becauseweknowthattheprinciplesofnumbertheoryaretruewithoutknowingthatthereareinfinitelymanyindividuals.
Bothversionsoflogicismfacethemetaphysicalobjection.TheargumenthereisforFregesversion;thesameargumentappliestoRussellsversionfornumbersofindividuals(thingsoftype0.)Callatwo-memberedsetapair,forshort.Thepropositiontobechallengedentailsthatthenumber2isthesetofallpairs.CalltheactualsetofpairsP.ThesetofCharlesWindsorssons,{William,Harry},isamemberofP.NowconsiderthepossiblecircumstancethatHarryhadneverbeenconceived:theset{William,Harry}wouldnothaveexisted;soPwouldnothaveexisted;sothesetthatwouldhavebeenthesetofpairsisnotP,butsomeotherset.Ingeneral,whichsetisthesetofpairsdependsoncontingentevents,justastheidentityofthe43rdUSpresidentitwouldhavebeenGorenotBush,hadtheSupremeCourtorderedarerunoftheFloridaballot.Butdoestheidentityofthenumber2dependoncontingentevents,suchastheresultsofroyalmating?Surelynot.Thenumberofprotonsinthenucleusofaheliumatomisthesameinallpossiblecircumstances;andthatnumberis2inallpossiblecircumstances.Sotheidentityofthenumber2isnotdependentoncircumstances,whereastheidentityofthesetofpairsisdependentoncircumstances.So2isnotthesetofpairs.Aparallelargumentworksforanyotherpositivecardinalnumber.
Althoughthelogicistproposaliswrong,therearemanymathematicallyadequatewaysofrepresentingthefinitecardinalnumbersassets.Settheoryhassettledononeofthesesystemsofsetrepresentationsasthemostconvenient,butweshouldnottakethesetrepresentationsofcardinalnumberstobethethingsrepresented.ThepointiscogentlyarguedforinBenacerraf(1965).Weshouldbenomoretemptedtothinkthatcardinalnumbersreallyaresetsofacertainkindthanthatspatialpointsreallyareorderedtriplesofrealnumbers.
Itisclear,however,thatthereisanintimaterelationbetweenacardinalnumbernandsetswithexactlynmembers.Anysatisfactoryanswertoourquestionmustmakethisrelationclear.Asatisfactoryanswer,however,mustalsomakeitpossibletoaccountforourcognitivegraspofsomecardinalnumbers,whichinallofusantedatesknowledgeofevenmoderatelysophisticatedsettheory.Soweneedtoturnawayfromsettheory(and,forthesamereason,fromanymathematicaltheory)andlookinanotherdirection.
NumbersasSet-sizesCardinalnumbersareanswerstoquestionsoftheformHowmanyFsarethere?.Thisgivesusabigclue.Answers
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toquestionsbeginningHowmuch,Howfar,Howlong,Howheavy,Howloudandothersofthisformaresizes,quantitiesormagnitudes.SoareanswerstoquestionsbeginningHowmany:cardinalnumbersaresizesofsets.Thisistheset-sizeview.Thereisdisagreementaboutwhethereverydefinitepluralityofthingsconstitutesaset;theset-sizeviewallowsthatadefinitepluralityhasanumericalsize,thatis,acardinalnumber,evenifthepluralityisnotaset.
Letmesayup-frontthattheset-sizeviewofcardinalnumbersistheviewIjudgetobecorrect.Animmediateadvantageoftheset-sizeviewisthatitisconsonantwiththewayweordinarilythinkandtalk.Whenwetalkoffamilysizesorclasssizes,werefertothenumberoffamilymembersorthenumberofpupilsinaclass.Theset-sizeviewalsorevealstheconnectionbetweenthenumbernandsetswithexactlynmembers:thenumberniswhatallandonlysetswithexactlynmembersareboundtohaveincommon,namely,theirsize.ThisviewwasexpressedbythephilosopherJohnLocke(1975)andmorerecentlysupportedin(Giaquinto2001).
Acardinalnumber,ontheset-sizeview,isnotanobject,butapropertyofsets.Thishaspromptedtwomajorphilosophicalobjections:
(1)Propertiesarenotreal;talkofpropertiesisamerefaondeparler.(2)Ifnumberswereproperties,theywouldbecausallyinert;sowecouldnothaveknowledgeofthem.
Iwillrespondtothesequestionsinturn.Inrespondingtothesecond,Iwillarguethatempiricalworkonnumericalcognitionrevealshowknowledgeofcardinalsizeispossible.
Debateabouttherealityofproperties(oruniversals)stretchesfromatleastmediaevaltimestothelate20thcentury(Mellor&Oliver,1997).However,wecancutthroughthesescholasticthicketsbynotingthatempiricalsciencequiteoftendeliversproperlysubstantiatedjudgmentsabouttherealityorunrealityofproperties.Forexample,JosephPriestleythoughtthatallcombustiblematerialcontainedphlogiston,asubstancethatisliberatedincombustionfromthematerial,withthedephlogisticatedsubstanceleftasanashorresidue.Onthistheory,acandleflameinanenclosedlanternwillgooutbecausethecontainedairwillbecomesaturatedwithphlogiston.AntoineLavoisierheldthatthereisnosuchsubstanceasphlogistonandnosuchpropertyasphlogistonsaturation.Combustioninvolvesabsorptionofoxygen,ratherthanreleaseofphlogiston,andacandleflameinanenclosedlanternwillgooutbecausethecontainedairwillbecomedepletedofoxygen.Eventually,thejudgmentsofLavoisierweresubstantiatednothingcouldbephlogistonsaturatedbecausethereisnosuchpropertyasphlogistonsaturation,butoxygendepletionisreal.Medicineandpsychiatrymakesimilarjudgments:possessionbydemonsisnotarealcondition;butmultiplepersonalitydisordermayberealandbipolardisorderdefinitelyis.Theconclusionmustbethatsomeputativepropertiesarerealandsomeareunreal.
Aresetsizesreal?Locke(1975,II.VIII.17)includednumberinhislistofrealproperties,incontrasttosensoryqualitiessuchasflavours,whichhetooktobeinus,ratherthaninthesubstancestowhichweattributethem.AboutnumberLockewasright.Scientistsappealtothenumberofprotonsinthenucleusofanatomtoexplainpropertiesoftheatom;theydonotexplainthenumberasamerelysubjectivephenomenonlikearainbow.Thisisreasontobelievethatthecardinalnumberofthesetofprotonsinaheliumnucleus,forexample,isarealpropertyofthatset,apropertythatdoesnotdependonusorourmentallife.Moreover,thefactthatthenumberofelectronsinanatomandthenumberofprotonsinitsnucleusarethesameisasignificantobjectivefact.Thenumberoflegsonanormalspider,thenumberofmajorbranchesofasnowflakethesesurelyarerealpropertiesoftherelevantsets,nottobeexplainedawayasillusoryphenomenaormerewaysoftalking.
Thesecondobjectiontotheset-sizeviewisthatifnumbersaresetsizeswecouldnothaveknowledgeofthem,forthefollowingreason.Setsizes,beingproperties,cannothaveanycausaleffectonus:theyemitnosignals,leavenotraces,andhavenoinfluenceonperceptiblethings;thereforetheycannotbeknown.
Thislastinferentialstepisthemainerror.Itmayarisefromusingasageneralmodelofknowingthingsamodelthatisappropriateforphysicalobjects,especiallySpelke-objects;butitisnotappropriateformoreabstractkindsofthing,suchaspropertiesandrelations.Aspropertiesdonotcausallyinteractwithotherthings,wecannothaveknowledgeoftheminthewaythatwehaveknowledgeofplanetsandprotons.YetweoftenknowsuchthingsasBeethovenspastoralsymphony,thelettersoftheGreekalphabet,orotherthingsthatdonotthemselveshavecausaleffectsonotherthings.However,theirinstances,thesoundsofanactualperformanceortheactualinscriptionsofGreekletters,dohavecausaleffectsonus:weperceivethem.Wecancometoknowamusical
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compositionoralettertypethroughrepeatedexposuretotheirphysicalinstances.
ToknowBeethovenspastoralsymphonyitisenoughthanwecanrecogniseperformancesasperformancesofBeethovenspastoralsymphonyandtotellthemapartfromperformancesofothermusic.ToknowtheGreeklowercasealphaitisenoughthanwecanrecogniseinscriptionsofitasinscriptionsofthelowercasealphaandtellthemapartfrominscriptionsofotherletters.Theparallelholdsforcardinalnumbers.Toknowthenumbernitisenoughthatonecanrecognisesetsofthingsasn-memberedanddiscriminaten-memberedsetsfromsetswithfewerormorethannmembers.
Howisitpossibletoacquirethiscapacityfornumberrecognitionanddiscrimination?Onthismatter,philosophersmustattendtothefindingsofcognitivescience.First,thedataprovideevidencethatwehaveaninnatelygivennumbersense,thatis,asystemofmentalmagnituderepresentationsofroughcardinalsize,withaneuralbasisintheintraparietalsulcus(Butterworth&Walsh,2011).Theevidencecomesfromavarietyofsources:experimentsonhealthyadultsandchildren,clinicaltestsonbraindamagedpatients,brainimaging,andstudiesonanimalsfromparrotstoprimates(Butterworth,1999,chapters36;Dehaene,1997,Chapters13,7,8;andseveralbookchaptersinthishandbookthatdescribethemostrecentlineofresearch).
Therearenodedicatedexteroreceptorsandthereisnospecialisedorganfornumberdetection.Sowhyanumbersense?Onereasonisthatourcapacityfordetectingnumberdoesnotinvolveapplyingaprocedure(suchascounting);itissubjectivelyimmediate.Anotherreasonisthatnumberdetectionhasthesignaturefeaturesofotherquantitysenses,suchassenseofduration(CohenKadosh,Lammertyn,&Izard,2008;Walsh,2003).Oneofthesefeaturesisthedistanceeffect:thesmallerthedistancebetweentwolevelsofaquantity(forfixedmean),theharderitistodistinguishthem.Ittakeslongertodistinguish7from9than4from12,whateverthestimulusformat(e.g.randomdotarrays,arabicnumerals,sequencesofknocks).Theotherfeatureisthemagnitudeeffect:thegreaterthemeanoftwolevelsofaquantity(forfixeddistance),theharderitistodistinguishthem.Ittakeslongertodistinguish8from10than2from4.Theseeffectsfollowfromthekindofformula(Welford,1960)towhichthereactiontimedataforsingleanddoubledigitnumbercomparisonconform:
whereListhelargernumber,Sisthesmaller,andaandkdenoteconstants.(Butterworth,Zorzi,Girelli,&Jonckheere,2001;Dehaene,1989;Hinrichs,Yurko,&,Hu,1981;Moyer&Landauer,1967,1973).Thisistypicalofresponsedataforcomparisonofotherphysicalmagnitudes,suchasline-length,loudness,andduration.Finally,experimentswithpre-linguisticchildrenandwithanimalslackinglanguageorsymbolsystemshowthattheytoohaveacapacityfornumberdiscrimination(seeBeranetal.,Agrillo,andMcCrink,thisvolume).Soitisreasonabletopositaninnatenumbersense.
Thisnumbersensedecreasesinprecisionasnumbersincrease,enablingustogaugeapproximatesizeforlargernumbers,butfornumbers1,2,and3itisprecise.Thisispredictedbyaneuralnetworkmodelforthenumbersense(Dehaene&Changeux,1993),althoughitmaybeduetoasecondsystemofmentalrepresentation(Feigenson,Dehaene,&Spelke,2004).
Thenumbersenseisjustoneoftheresourcesofnumericalcognition.Thereisalsotheculturallysuppliedinstrumentofverbalcounting,whichenablesustodeterminecardinalsizepreciselyforsetstoogreattobegaugedwithprecisionbythenumbersense.Countingalsohelpsusappreciateafeaturethatseemstodistinguishset-sizefromothermagnitudes,namelydiscreteness.Betweenanytwolengthsthereis(orseemstobe)anintermediatelength,buteachset-sizehasanimmediatesuccessor,withnoset-sizeinbetween.
Practicewithverbalcountingmayproduceinusanassociationofnumbersenserepresentationswithourrepresentationsofnumberwords,thencewithourrepresentationsofnumerals(seeSarnecka,thisvolume);anditmayhelpsharpenournumbersenserepresentations,sothattherangeofnumbersrepresentedwithprecisionextendsbeyond3(thoughperhapsnotveryfar).Familiaritywithcountingalsosuppliesuswithuniquelyidentifyingpositionalinformationaboutnumberswithinourcountingrange.Thus1isthefirstnumberand2isthenext,3thenextafter2,andsoon.
Allthisissurelyenoughforpossessionofconceptsforthefirstfewpositivecardinals.Withtheseresourcesitispossiblenotmerelytodiscriminatebetween3-memberedsetsandsetswithmoreorfewerthan3members,but
RT = a + k. log[L/(LS)],
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alsotorecognisesuchasetas3-membered.Thesameforotherverysmallpositivecardinalnumbers.Sowehavetheframeworkatleastforanaccountofhowwecanhavecognitivegraspofthesenumbers,withoutappealingtomodesofcognitionnotrecognisedbycognitivescientists.
Thenumber0isaspecialcase(seeTzelgovetal.,thisvolume).Weprobablydonothaveanynumbersenserepresentationforzero(Wynn&Chang,1998).Itisthecardinalsizeoftheemptyset.Theemptysetcanseemtobeanartificialposit,butitdoesnotseemsoartificialwhenoneconsidersallthesetsofpossiblewinningsinatwo-personzero-sumgameplayedforvaluableitems;moreover,theexistenceofauniqueemptysetisprovablefromtheestablishedaxiomsofsettheory.Sowecanknowthecardinalnumber0bydescriptionasthecardinalsizeoftheemptyset.Wedonotgrasp0aswegraspthesmallpositivenumbers;wecannotliterallyrecognisethatasethaszeromembers,thoughwemaydeduceit.
Whataboutlargernumbers?Whenweknowanidentifyingdescriptionofanumberintermsofsmallernumbers,wecanknowthenumberdescriptively(assumingwealreadyknowthesmallernumbers).Oftenwehavemorethanoneidentifyingdescriptionofanumber,givingusabettergraspofit.Forexample,weknowfiveidentifyingdescriptionsof10asthesumoftwopositivenumbersandoneidentifyingdescriptionofitastheproductoftwosmallernumbers(treatingtheorderofoperandsasirrelevant.)ByknowinganidentifyingdescriptionofanumberImeanthatwecanretrievetherelevantnumberfactfrommemory.Wecan,ofcourse,figureoutmanymorethanfiveidentifyingdescriptionsof53asthesumoftwonumbers,butbeforehavinglearntthoseadditionfactstheycannotmakeusmorefamiliarwiththenumber.Contrast53with60,ofwhichnumerateadultsknowfiveidentifyingdescriptionsasaproductoftwosmallernumbers(230,320,etc.)andthreeassumsoftwodecades(10+50,20+40,etc.)
Thesethenarewaysinwhichwecanproperlybesaidtoknowanumber:bymeansofournumbersense,byavarietyofidentifyingdescriptionsintermsofsmallernumbers,andbyacombinationofthesetwo.Beyondnumbersknowableinthosewaysarenumbersstillsmallenoughtorefertobymeansoftheirdecimalnotation:viewingdecimalnumeralsinblocksofthreedigits(fromtheright)givesussomerelativeawarenessofsize.Stillfurtheroutarenumberswhichwecandesignate,butnottranscodeintotheirdecimalnumeralsandwhichutterlydefeatournumbersense,suchas9^(9^(9^9)),wheren^pdenotesntothepowerofp.Ofcourse,mostnumberswilllietotallybeyondourabilitytorefertothem,usingwhateverisourcurrentlymostcompactnotation;evenifweallowforeverimprovingmeansofreference(usingsymbolsforfastergrowingfunctions),mostwillremainoutsidethelightconeofhumanintellect.
Myclaims,insummary,arethese.Cardinalnumbersaresizepropertiesofsets(orofdefinitecollectionsordefinitepluralities).Somecardinalnumberscanbeknown.Verysmallnumberscanbeknownbymeansofthenumbersenseandthepracticeofcounting.Thisknowledgeisnotaquasi-perceptionofthenumbernitself,butacapacityforrecognisingn-memberedsetsasn-memberedandfordiscriminatingsetsofnitemsfromsetswithfewerormoreitems.Somelargernumbersareknowableinadifferentway,asthecardinalnumberdesignatedbyoneormoreidentifyingdescriptionsintermsofsmallernumbers,whenthesedescriptionsarestoredinmemory.
Thisaccountofthenatureofcardinalnumbershasthesecrucialadvantages.Itmakesclearthenecessaryrelationofacardinalnumberntosetswithexactlynmembers,itavoidsanyconflictwithnumbertheory,anditpermitsanaccountofourknowledgeofcardinalnumberswithintheframeworkofcognitivescience.
TheFiniteCardinalNumbersandtheNaturalNumbersShouldwetakethenaturalnumbersofmodernnumbertheorytobethefinitecardinalnumbers?Numbertheorybecameahighlyabstractsubjectinthe19thcentury,withtheworkofthemathematiciansKummer,Kronecker,andespeciallyDedekind.Therewasafocusonstructuralpropertiesandrelationsthathasremainedeversince.Thesubjectmatterofmodernnumbertheorycanbedescribedasthestructureofnaturalnumbersanditsextensionstoothernumberstructures.Thesequenceoffinitecardinalnumbershasthestructureofnaturalnumbers,butindefinitelymanyothermathematicalsequencesalsohavethatstructure.Soweshouldnottakethenaturalnumberstobethefinitecardinalnumbers.
Tomakethepointabitclearer,letusgointoalittledetail.Thefinitecardinalnumbershaveanaturalordering:startingwithzero,eachnumbernisimmediatelysucceededbyn+1(thesizeofsetswithn+1members),and
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eachnumberisreachedbyfinitelymanyapplicationsofthissuccessoroperationstartingwithzero.ThisorderingissharedwithanysequenceofthingssatisfyingthefollowingconditionsgivenbyDedekind(1996):
(1)Thereisasingledistinguishedelementcallitthezero.(2)Thereisaunaryoperationscallitthesuccessoroperationsuchthateveryelementxhasauniqueimmediatesuccessorelements(x).(3)Thezeroisnotthesuccessorofanyelement.(4)Notwoelementshavethesamesuccessor.(5)Anysetthatcontainsthezeroandthatcontainss(x)ifitcontainsxisasetthatcontainsalltheelements.
SomeexamplesaregiveninTable1.
Table1.Instancesofthestructureofthenaturalnumbers.
Startingelement Successoroperation
Cardinalnumbers Zero
Hilbertnumerals I I
VonNeumannordinals (theemptyset)
Zermeloordinals
Anytwoofthesesequences,knowncollectivelyasomegasequences,areisomorphic:theirelementscanbepairedoffone-to-oneinawaythatpreservesorder,i.e.withoutanycrossovers.Inotherwords,theseorderedsetshavethesamestructure,thestructureofthenaturalnumbers.
Whatthenarethenaturalnumbers?Itisamistaketolookforsomemathematicalentitiesspecifiableindependentlyofthestructure.Thisleavestwooptions.Oneistodenythatthequestionhasanabsoluteanswer;theelementsofanyomegasequencecanserveasnaturalnumbers,butnoneisprivilegedastherealsequenceofnaturalnumbers,asBenacerraf(1965)argues.Onthisview,thefullcontentofatheoremofDedekindPeanonumbertheoryisapropositiontacitlyaboutallomegasequences.Theotheroptionistotakethenaturalnumberstobethepositionsinthenaturalnumberstructure:0istheinitialposition,andforanypositionx,s(x)isthenextpositionalong(Shapiro,1997).
Myaimhereismerelytorelatethefinitecardinalnumberstothenaturalnumbersand,forthatpurpose,itisnotnecessarytoadjudicatebetweenthetwoviewsofthenaturalnumbersjustpresented.Onthefirstview,asthefinitecardinalsintheirnaturalorderingconstituteoneofthemanysequenceswhichinstantiatethenaturalnumberstructure,itispermissibleinasuitablecontexttothinkandtalkasthoughtheyarethenaturalnumbers.Thiswouldbeamannerofspeakingorthinking,notanexpressionofmetaphysicalfact.Onthealternativeview,therelationshipbetweenthefinitecardinalsandthenaturalnumbersis(again)notidentity;itisoccupation.Thefinitecardinalsintheirnaturalsequenceoccupythepositionsofthestructureofnaturalnumbers.
Eitherway,thetheoryofnaturalnumbershasgreatergeneralityandabstractnessthanthetheoryoffinitecardinalnumbers,beingaboutfeaturesofthestructurecommontoallomegasequences.Here,then,isanareaforfuturecognitiveresearch.Whatcognitiveresourcesareinvolved,andhowaretheyinvolved,inthedevelopmentofonesgraspofnumbertheory?
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MarcusGiaquintoMarcusGiaquinto,UniversityCollegeLondon
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Cognitive Foundations of Human Number Representations and Mental Arithmetic
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Subject: Psychology,CognitivePsychology,CognitiveNeuroscience
OnlinePublicationDate: Nov2014
DOI: 10.1093/oxfordhb/9780199642342.013.61
CognitiveFoundationsofHumanNumberRepresentationsandMentalArithmetic OliverLindemannandMartinH.FischerTheOxfordHandbookofNumericalCognition(Forthcoming)EditedbyRoiCohenKadoshandAnnDowker
OxfordHandbooksOnline
AbstractandKeywords
Thechaptersinthissectionofthevolumerevealthestrikingvarietyofhumannumericalcognition.Thesectioncomprisesfourchaptersthatfocusondifferentaspectsoftherepresentationofnumericalknowledge,aswellasthreechaptersthatexaminetheseveralcognitiveprocessesinvolvedinthemanipulationofnumbersduringsimplementalarithmetic.Theyshowhowchronometricanalyses,incombinationwithcleverexperimentaldesigns,canrevealthecognitiveprocessesandrepresentationsunderlyingthisimpressivecollectionofcognitiveskills.Ourgoalinthisoverviewchapteristohighlightcommonthemesthatconnectthesecontributions.Inparticular,wesuggestlinksbetweenthepresentcontributions,allofwhicharefirmlygroundedinthetraditionalinformation-processingapproachtothehumanmind,andthemorerecentlyemergedembodiedcognitionperspective,accordingtowhichallknowledgerepresentationsremainassociatedwiththosesensoryandmotorfeaturesthatwereactivatedduringacquisitionofthatknowledge.Keywords:humannumberrepresentation,mentalarithmetic,embodiedcognition,knowledgerepresentation,embodiednumberprocessing,spatial-numericalassociation,intuitivereasoning
RepresentationofNumericalKnowledgeThephilosopherHenriPoincar(18541912)statedthatintuitions,andnotformallogic,arethefoundationuponwhichhumansbasetheirunderstandingofmathematics(McLarty,1997).Interestingly,modernpsychologicalresearchprovidesempiricalsupportforPoincarsnotionandshowsthatasenseofnumbersispartofahumanscoreknowledgethatisalreadypresentearlyonininfancy.Theoriginandtheunderlyingcognitivecodesonwhichthisnumbersenseisgroundedhave,however,sofarnotbeenfullyunderstood.Thisiswherethechapterscollectedinthisvolumedeliversignificantadvancesinourunderstandingofthecomponentprocessesandrepresentationsinvolvedinnumericalcognitionandarithmetic.
PrimitivesofNumberRepresentationThechapterbyTzelgovetal.(thisvolume)discussesseveralbasiccognitivemechanismsunderlyingtheprocessingofArabicdigits.Whileweknowthathumansandnon-humananimalssharetheabilitytoprocessapproximatemagnitudesandnumerosityinformation(seealsothechaptersbyAgrilloandbyBeranPerdue&Evans,thisvolume),onlyhumanspossesstheabilitytogeneratenumericalnotationsystemsthatallowforasymbolicrepresentationofexactquantitiesofnaturalnumbers.Throughoutcivilisation,thesenotationalsystemsbecamemoreandmoresophisticated(e.g.,Ifrah,1981):theprogressiveintroductionofsyntacticfeatures,suchastheplace-valueprincipletocodemagnitudeswithmulti-digitnumbers,thepolaritysigntodenotenegativevalues,orfractionsymbolstodenotenon-naturalnumbers,madeitpossibletogeneratecompoundexpressionsto
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Cognitive Foundations of Human Number Representations and Mental Arithmetic
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representmagnitudesthatdonotcorrespondtosimplesingle-digitnumbers(seealsoNuerketal.,thisvolume).
Intheirchapterinthisvolume,Tzelgovandcolleaguesreporttheirlong-runningresearchprogramaimedatidentifyingelementaryentitiescalledprimitivesforcognitivenumericalrepresentations.Whilemathematiciansoftenconsidertheprimenumberstobesuchelementaryunits(sincetheymakeupallothernaturalnumbers),theauthorstakeapsychologicalviewanddefineprimitivesasnumberswhosemeaningsareholisticallyretrievedfrommemorywithoutfurtherprocessing.Incontrast,thesemanticsofnon-primitivenumbersaregeneratedon-linefromprimitivesinordertoperformaspecifictask.Inotherwords,thedirectandautomaticmeaningretrievalfrommemoryis,accordingtoTzelgovandcolleagues,thecentralprocessingcriterionbywhichanumericalprimitivecanbeidentified.Oneapproachtoinvestigatesuchautomaticityofnumberprocessingistodeterminethesizecongruityeffect,thatis,theinteractionbetweennumericalmagnitudemeaningandphysicalsizeofthenumbersymbolbeingprocessed:moreefficientprocessingincongruentconditions(e.g.,1printedinsmallfontor9printedinlargefont)establishessuchautomaticity(Henik&Tzelgov,1982).Itisnoteworthythatthisinteractionpointstoaninescapablelinkbetweensensoryexperienceandconceptualrepresentationofmagnitudes,whichisacoreaspectoftheembodiedcognitionapproach(cf.Barsalou,2008).ThechapterbyTzelgovandcolleaguesprovidesadetailedreviewofstudiesonnumericalprimitivesandalsoencompassesworkonmulti-digitnumbers(cf.Nuerketal.,thisvolume),fractions,negativenumbers,andthenumberzero.Theauthorsconcludethatnotonlynaturalsingle-digitnumbersbutalsosomedouble-digitnumbersandcertaintypesoffractionsseemtobeholisticallyrepresented.Togetherwiththebasicconceptofplace-value,theyshould,therefore,beconceivedasprimitivesofnumberrepresentation.
Inthemodernnumericalcognitionliterature,theconceptofaholisticrepresentationofnumbermeaningisoftenlinkedtothenotionofamentalnumberline,thatis,thehypothesisthatnumbersaresystematicallyassociatedtospatialcodes,asifmagnitudeswererepresen