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Promoting Maths to the General Public Page 1 of 12 Subject: Psychology, Cognitive Psychology, Educational Psychology Online Publication Date: Feb 2015 DOI: 10.1093/oxfordhb/9780199642342.013.47 Promoting Maths to the General Public Chris J. Budd The Oxford Handbook of Numerical Cognition (Forthcoming) Edited by Roi Cohen Kadosh and Ann Dowker Oxford Handbooks Online Abstract and Keywords In this chapter I will address the issue that whilst mathematics is vital to all of our lives, playing a vital role in modern technology and even helping us to understand the brain, it is often perceived to be a dry, boring, and useless subject. The chapter will explore various ways that mathematics can be presented to the general public in a way that makes it seem to be exciting and relevant, and captures its essence without dumbing it down. In particular it will look at strategies that have been shown to work well with the public including the use of careful real life examples and relating maths to people, hands on maths at science fairs, and maths in the media and on the Internet. The chapter includes some case studies of what does and does not work in the field of maths communication. Keywords: Mathematics, popularization, formulae, science fairs, masterclasses What’s it all about? Mathematics is all around us, it plays a vital role in much of modern technology from Google to the Internet and from space travel to the mobile phone. It is central to every school student’s education, and anyone needing to get a mortgage, buy a car, sort out their household bills, or just understand the vast amount of information now thrown at them, needs to know some maths. Maths is even used to help us understand, and image, the complex networks and patterns in the brain and many of the processes of perception. However, like the air around us, the importance of mathematics is often invisible and poorly understood, and as a result many people are left unaware of the vital role that it could, and does, play in their lives. In an increasingly technology and information driven world this is potentially a major problem. However, we have to be honest, mathematics and its relevance, is a difficult subject to communicate to the general public. It certainly doesn’t have the instant appeal of sex and violence that we find in other areas (although it does have applications to these) and there is a proud cultural tradition in the UK that it is good to be bad at maths. For example when I appeared once on the One Show, both presenters were very keen to tell me that they were rubbish at maths and that it didn’t seemto have done themany harm! (I do wonder whether they would have said the same to a famous author, artist, or actor.) Maths is also perceived as a dry subject without any applications (this is also very untrue and I will discuss this later) and this perception does put a lot of school students (and indeed their teachers) off. Finally, and (perhaps this is what makes it especially hard to communicate), maths is a linear subject, and a lot of background knowledge, and indeed investment of time, is required of any audience to whom you might want to communicate its beauty and effectiveness. For example, one of the most important way that maths affects all of our lives is through the application of the methods of calculus. But very few people have heard of calculus, and those that have are generally scared by the very name. It also takes time and energy to communicate maths well and (to be honest), most mathematicians are not born communicators (in fact rather the opposite). However, it

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  • Promoting Maths to the General Public

    Page 1 of 12

    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

  • Promoting Maths to the General Public

    Page 2 of 12

    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

    2

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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

    2 2

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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.

    0

    0

    0

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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

  • Philosophy of Number

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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?

    ReferencesAgrillo,C.(Thisvolume)NumericalandArithmeticabilitiesinnon-primatespecies.InRoiCohenKadosh&AnnDowker(Eds.)TheOxfordHandbookofMathematicalCognition.Oxford:OxfordUniversityPress.

    +1

    {} {}

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    Balaguer,M.(2011).FictionalisminthePhilosophyofmathematics.InE.N.Zalta(Ed.),TheStanfordEncyclopediaofPhilosophy(Autumn2011edn).Availableathttp://plato.stanford.edu/archives/fall2011/entries/fictionalism-mathematics/

    Benacerraf,P.(1965).Whatnumberscouldnotbe.PhilosophicalReview74,4773.

    Beran,M.J.,Perdue,B.M.,&Evans,T.A.(Thisvolume).MonkeyMathematicalAbilities.InRoiCohenKadosh&AnnDowker(Eds.)TheOxfordHandbookofMathematicalCognition.Oxford:OxfordUniversityPress.

    Berkeley,G.(1956).LettertoMolyneux[1709].InA.Luce&T.Jessop(Eds),TheWorksofGeorgeBerkeley,BishopofCloyne,Volume8.London:Nelson.

    Berkeley,G.(1989).PhilosophicalCommentaries,G.Thomas(ed.).London:Garland.

    Brouwer,L.(1983).IntuitionismandFormalism[1913].InP.Benacerraf&H.Putnam(Eds).PhilosophyofMathematics,2ndedn.Cambridge:CambridgeUniversityPress.

    Butterworth,B.(1999).TheMathematicalBrain.London:Macmillan.

    Butterworth,B.,&Walsh,V.(2011).Neuralbasisofmathematicalcognition.CurrentBiology,21,13371420.

    Butterworth,B.,Zorzi,M.,Girelli,L.,&Jonckheere,A.(2001).Storageandretrievalofadditionfacts:theroleofnumbercomparison.QuarterlyJournalofExperimentalPsychology,54A,10051029.

    Cantor,G.(1955).ContributionstotheFoundingoftheTheoryofTransfiniteNumbers[1895],transl.P.Jourdain.NewYork:DoverPublications.

    CohenKadosh,R.,Lammertyn,J.,&Izard,V.(2008).Arenumbersspecial?Anoverviewofchronometric,neuroimaging,developmentalandcomparativestudiesofmagnituderepresentation.ProgressinNeurobiology,84,132147.

    Dedekind,R.(1996).WassindundwassollendieZahlen?Vieweg,Braunschweig.InW.Ewald(Ed.),FromKanttoHilbert.ASourceBookintheFoundationsofMathematics,Vol.2.Oxford:OxfordUniversityPress(originalpublished1888).

    Dehaene,S.(1989).Thepsychophysicsofnumericalcomparison:are-examinationofapparentlyincompatibledata.PerceptionandPsychophysics,45,557566.

    Dehaene,S.(1997).TheNumberSense.Oxford:OxfordUniversityPress.

    Dehaene,S.,&Changeux,J.(1993)Developmentofelementarynumericalabilities:aneuronalmodel.JournalofCognitiveNeuroscience,5,390407.

    Dummett,M.(1977).ElementsofIntuitionism.Oxford:OxfordUniversityPress.

    Euclid(2002).EuclidsElements:AllThirteenBooksCompleteinOneVolume,transl.T.Heath.SantaFe,NM:GreenLionPress.

    Feigenson,L.,Dehaene,S.,&Spelke,E.(2004).Coresystemsofnumber.TrendsinCognitiveSciences,8,307314.

    Field,H.(1980).ScienceWithoutNumbers.Princeton,NJ:PrincetonUniversityPress.

    Frege,G.(1980).TheFoundationsofArithmetic[1884],transl.J.Austin.Evanston,IL:NorthwesternUniversityPress.

    Giaquinto,M.(2001).Knowingnumbers.JournalofPhilosophy,98,518.

    Hilbert,D.(1967).OntheInfinite.InJ.vanHeijenoort(Ed.),FromFregetoGdel:asourcebookinmathematicallogic,transl.S.Bauer-Mengelberg.Cambridge,MA.:HarvardUniversityPress,367392.

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    Hinrichs,J.,Yurko,D.,&Hu,J.(1981).Two-digitnumbercomparison:useofplaceinformation.JournalofExperimentalPsychology:HumanPerceptionandPerformance,7,890901.

    Kripke,S.(1992).Logicism,Wittgenstein,andDeReBeliefsAboutNumbers.Unpublishedlectures.

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    McCrink,K.,&Birdsall,W.(Thisvolume).NumericalAbilitiesandArithmeticinInfancy.InRoiCohenKadosh&AnnDowker(Eds.)TheOxfordHandbookofMathematicalCognition.Oxford:OxfordUniversityPress.

    Mellor,D.,&Oliver,A.(1997).Properties.Oxford:OxfordUniversityPress.

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  • Philosophy of Number

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    MarcusGiaquintoMarcusGiaquinto,UniversityCollegeLondon

  • Cognitive Foundations of Human Number Representations and Mental Arithmetic

    Page 1 of 9

    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

  • Cognitive Foundations of Human Number Representations and Mental Arithmetic

    Page 2 of 9

    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


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