Michael Beaney

8/13/2019 Michael Beaney

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Instructions for use

TitleFrege, his Logic and his Philosophy Interview w ith M ichaelB eaney

A uthor(s) B eaney, M ichael; B o, C hen; N akatogaw a, K oji

C itation Journal of the G raduate School of Letters, 5: 1-25

Issue D ate 2010-03

D oc U R L http://hdl.handle.net/2115/42862

R ight

Type bulletin (article)

A dditionalInform ation

H okkaido U niversity C ollection of S cholarly and A cadem ic P apers : H U S C A P
http://eprints.lib.hokudai.ac.jp/dspace/about.en.jsphttp://eprints.lib.hokudai.ac.jp/dspace/about.en.jsp


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Frege,hisLogicandhisPhilosophy

InterviewwithMichaelBeaney

MichaelBEANEY

ChenBO

KojiNAKATOGAWA

Abstract:Theinterview beginswithanoutlineofGottlobFregeslife,academiccareerandthe

receptionofhisideasbylaterphilosopherssuchasBertrandRussellandLudwigWittgenstein.

Fregesmaincontributionstologicandphilosophyaresummarized,andthekeyideasofhisthreemainbooksBegriffsschrift,DieGrundlagenderArithmetikandGrundgesetzederArithmetik

areexplained. ParticularattentionispaidtoFregesfundamentalclaim thatastatementof

numbercontainsanassertionaboutaconceptandtotheCantor-HumePrinciple,whichplay

acentralroleinhislogicistprojecttheattempttoshow thatarithmeticcanbereducedto

logic. AlsodiscussedarethreeofFregesimportantessays,whichelucidatehisview ofconcepts

asfunctionsandhisdistinctionsbetweenconceptandobjectandbetweenSinnandBedeutung.

Hereparticularattentionispaidtoproblemsconcerninghisnotionofanextensionofaconcept

(ormoregenerally,hisnotionofavalue-rangeofafunction),thetranslationofBedeutung,and

theapplicationofthedistinctionbetweenSinnandBedeutungtodifferentkindsoflinguisticexpressions. RecentdevelopmentsofFregesideasby,forexample,neo-logicistsarementioned,

andthereisalsodiscussionofFregesconceptionofthoughts. Theinterview endswithsome

remarksoncertaininfluencesonFrege,suchasfrom neo-Kantianism,andsomesuggestionsfor

furtherreading.

(ReceivedonDecember24,2009)

1.Introduction:Fregeslife,academiccareerandinfluence

CB

1):Prof. MichaelBeaney,Im verygladtohavetheopportunitytointerview you. You

areawell-knownFregescholar. YoueditedTheFregeReader(1997)and co-editedGottlob

Frege:CriticalAssessmentsofLeadingPhilosophers(2005),whichareveryhelpfulforteaching

1

Thisinterview issupportedbynationalsocialsciencefund07AXZ003,P.R.China.

JournaloftheGraduateSchoolofLetters,HokkaidoUniversity

Vol.5;pp.1-25,March2010

C2010bytheGraduateSchoolofLetters,HokkaidoUniversity

MichaelBEANEY:mab505york.ac.uk

ChenBO:chenbophil.pku.edu.cnKojiNAKATOGAWA:kojinakatogawa.jp


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andresearch onFrege. IonceusedTheFregeReaderasatextbook formygraduateclass.

Sincetherearefew biographicalaccountsofFrege,ChinesereadersarenotsoclearaboutFrege

asapersonandalsoasalogicianandphilosopher. Fregemaybeslightlymorefamiliarto

Japanesereaders,butin Japan,too,heisstillrelatively unknown. Isthatnotright,Prof.

Nakatogawa?

KN2):Yes,thatistrue. Fregeswritingshaveindeed been translated into Japanese,but

discussionofFregesphilosophyisstillrelativelyrare.

CB:Inthelightofthis,then,Prof. Beaney,couldyoufirstsketchthecontoursofFreges

life,especiallyhischaracterandhisacademiccareer?

MB3):Prof. Chen,Im gladtoknow thatthebooksIhaveeditedhavebeenuseful,andIm

veryhappytotalktobothyouandProf. NakatogawaaboutFrege,whosework,inmyopinion,liesatthebasisofwhatwenow callanalyticphilosophy,thedominanttraditioninphilosophy

intheEnglish-speakingworld.

GottlobFregewasborninWismar,atownontheBalticcoastinnorthernGermany,on8

November1848. Hisparentswereteachers,andhewasthefirstoftwosons. Hewasbaptised

intotheLutheranchurch,andremainedaLutheranfortherestofhislife,althoughIam notsure

how deeplyreligioushewas,andIdonotthink,inanycase,thatithadanysignificantinfluence

onhislogicalwork. Littleisknownabouthisyoungerbrother(bornin1852),butweknow a

bitmoreabouthisparents. Hisfather(bornin1809)wasprincipalofaprivategirlsschool,and

hismother(bornin1815)wasateacherthere. Hisfatherdiedfrom typhusin1866,however,andhismotherthentookoverasprincipal. ShewasclearlyveryclosetoFrege. Whensheretired

from herjobin1876,sheleftWismartwoyearslatertolivewithFrege,andremainedwithhim

untilshediedin1898. ItwaswithhermoneythatFregewasabletobuyahousein1887,ashe

neverearned enough asa professor. Fregeattended thegrammarschool(Gymnasium)in

Wismarfrom 1864to1869,whereheseemstohavehadanaverageeducation.

In1869heenteredtheUniversityofJena,ineastGermany,andtookcoursesinmathematics,

physics,chemistry and philosophy (with Kuno Fischeron Kant),beforetransferring to the

UniversityofGottingen,atthattimeoneoftheleadingcentresofmathematics,wherehetook

furthercoursesin mathematics,physicsand philosophy (thistimewith Hermann Lotze onphilosophyofreligion). In1873hewasawardedhisdoctoratewithadissertationentitledOn

aGeometricalRepresentation ofImaginaryFormsin thePlane. Heimmediatelywrotehis

Habilitationsschrift,on MethodsofCalculation based on an Extension oftheConceptof

Magnitude. Thiswasrequiredtogainauniversitylecturingpost,andFregecompleteditinjust

afew monthsanunusuallyshorttime. OntherecommendationofErnstAbbe,hismentor

atJena,hewasthenappointedtoteachanalyticgeometryand thetheoryoffunctionsin the

DepartmentofMathematicsatJena,coveringforCarlSnell,whohadbecomeill. Snellsillness

wasthereasonwhyFregehadbeenencouragedtodohisHabilitationsschriftasquicklyashe

could. FregewastoremainatJenauntilheretiredin1917. Hewaspromotedtoausserordentli-cherProfessor(roughlyequivalenttoAssociateProfessor,althoughwithonlyasmallsalary)in

MichaelBEANEY,ChenBO,KojiNAKATOGAWA2


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1879,andtoordentlicherHonorarprofessor(FullProfessor,althoughonlyanhonorarypost)in

1896. Fregeneverattained aproperly salaried university professorship,and wasrelianton

moneygiventohim bytheCarlZeissFoundation,whichAbbehadsetupin1889. Abbehad

workedcloselywithZeissintheopticsindustryinJena,anditwasthesuccessofthisindustry

thatenabledFregetopursuehisresearch.

ThethreebooksthatFrege published in hislifetime wereBegriffsschrift(Conceptual

Notation)in1879,DieGrundlagenderArithmetik(TheFoundationsofArithmetic)in1884,and

GrundgesetzederArithmetik(BasicLawsofArithmetic),thefirstvolumeofwhichappearedin

1893andthesecondvolumein1903. Fregesmainaim inthesebookswastodemonstratethe

logicistthesisthatarithmeticisreducibletologic. InBegriffsschrifthegavehisfirstexposition

ofthelogicalsystem bymeansofwhicharithmeticwastobereduced. IntheGrundlagenhe

offeredaninformalaccountofhislogicistproject,criticizingotherviewsaboutarithmetic,such

asthoseofKantandMill. IntheGrundgesetzeherefinedhislogicalsystem andattemptedto

demonstrateformallyhislogicistthesis. In1902,however,asthesecondvolumewasgoingtopress,hereceivedaletterfrom BertrandRussellinforminghim ofacontradictioninhissystem

thecontradiction weknow now asRussellsparadox. Although Fregehastilywrotean

appendixattemptingtorespondtotheparadox,hesoonrealizedthattheresponsedidntwork,

andwasledtoabandonhislogicistproject. Hecontinuedtodevelophisphilosophicalideas,

however,and to correspond with othermathematiciansand philosophers,and published a

numberofinfluentialpapers.

ThereisnodoubtthatRussellsparadoxdealtaterribleblow toFrege,andthiscameata

difficulttimein hispersonallife. Fregehad married MargareteLieseberg(born 1856)from

Wismarin1887,whentheymovedintoanewlybuilthouseat29ForstweginJenawithFregesmother. ButFregeswifeclearlyhadsomeillness,aboutwhichweknow nothing,andthismade

itdifficulttocareforFregesmotherinthelastfew yearsofherlife. Shemovedtoanursing

homein1896,anddiedtwoyearslater. Butthen,in1904,Margaretedied,too,andFregewas

leftwithjusthishousekeeperMetaArndt(bornin1879)forcompany. In1908,however,he

adoptedason,Alfred(bornin1903),whobecamehisheirin1921. FregeandMargaretehadhad

nochildrenthemselves,butitseemsthatFregelikedchildren(anddogs),andwasapparentlya

goodfathertoAlfred.

Thereisnodoubt,too,thatFregesideaswerenotunderstood,letaloneaccepted,whenthey

werefirstformulated,andthismusthavebeenveryfrustrating,especiallyasFregewriteswithaclaritysecondtononeinGerman-languagephilosophy,inmyopinion. From the1890sonwards

hiswritingsincreasinglyshow signsofbitternessashesubjectedtheviewsofhiscontemporaries,

includingthoseofsomeofhiscolleaguesatJena,todevastatingcriticism. RudolfCarnap,who

attendedsomeofFregeslecturesbetween1910and1914,reportsthatFregelookedoldbeyond

hisyears,wasextremelyshyandintroverted,andrarelyturnedtotheaudiencewhenhelectured.

HisideasneverthelessmadeadeepimpressiononCarnap,astheydidonLudwigWittgenstein,

whometFregeonthreeoccasionstodiscussphilosophybetween1911and1913. ItwasFrege

whorecommendedthatWittgensteinstudywithRussell,anddespitehiscriticismsofFregesideas,

Wittgenstein held Fregein thehighestregard throughouthislife. Fregemayhavehadfewstudents,buttwoofthem becametwoofthegreatestphilosophersofthetwentiethcentury. If

Frege,hisLogicandhisPhilosophyInterview withMichaelBeaney 3


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hehadlivedlonger,Im surehewouldhavediedhappier.

Fregesprofessionaldisillusionmentandthesadnessinhispersonallifegraduallytookatoll

onhishealth,andinthelastfew yearsofhiscareerhewasabletodolessandlessteaching.

Whenhefinallyretiredin1917,hemovedbacktohishomelandontheBalticCoast,abletobuy

ahouseatatimeofeconomiccollapsewithagiftofmoneyfrom Wittgenstein. Hepublished

thethreeessaysthatcomposehisLogicalInvestigations,buthealsokeptadiaryinwhichhe

expressessomeunpleasantright-wingandanti-Semiticviews. Suchviewswerenotuncommon

inGermanyintheaftermathoftheFirstWorldWar,butIneverthelessfinditverysad that

someoneofFregesextraordinaryintellectualcalibreshouldhavesunktothisinthetwilightof

hislife. Hediedon26Julyattheageof77.

CB:Asyousay,Fregesideaswerenotappreciatedatthetimehewrote,evenbyhisclosest

colleagues,andhewasarelativelyunknownfigureoutsideJena. Butthingschangeddramati-

callyinthetwentiethcentury. Fregebecomesregardedasthefounderofmodernlogic,thatis,mathematicallogic,andalsoasthefatherofanalyticphilosophy. Prof. Beaney,Id liketo

know,towhatortowhom doyouattributethischange? Husserl? Russell? Wittgenstein?

Carnap? Dummett? Orsomeoneorsomethingelse?

MB:Yes,yourerightthatthetransitioninourappreciationofFregeinthetwentiethcentury

hasbeendramatic. EdmundHusserlknew Fregeswork,whicharguablyconvincedHusserlof

theerrorofhisearlypsychologism,andtheycorrespondedin1891andagainin1906. ButIdont

thinkthatmanypeopleweredrawn to Fregethrough HusserlalthoughMartin Heidegger

discussesFregeinsomeofhiswritings. Russellclaimedthathewasresponsiblefordrawingattention to Fregeswork in 1903. ItistruethatAppendix A ofRussellsPrinciplesof

MathematicscontainsthefirstsubstantialdiscussionofFregesideasinEnglish. Butitwasthe

ItalianmathematicianGiuseppePeanowhohaddirectedRusselltoFrege,andtherewereother

GermanlogiciansandphilosopherswhohadalsodiscussedFregesideasearlier,suchasErnst

SchroderandBennoKerry. CarnapwasprofoundlyinfluencedbyFrege,butheadmitshimself

thatthisonlyhappenedwhenhereadhisworksmorecarefullyaftertheFirstWorldWarrather

thanatthetimeheattendedFregeslectures. IhavealreadyspokenaboutWittgensteinsregard

forFrege,whom hethanksforhisgreatworksin thepreface to theTractatusLogico-

Philosophicus,andthereisnodoubtthataspeoplesoughttounderstandWittgensteinsphiloso-phy,especiallytheideasoftheTractatus,theywereinevitablydrawntoFrege. Indeed,Icansay

thatthiswasexactlyhow IbecameinterestedinFregeintryingtounderstandWittgensteinas

astudentinOxfordinthe1980s. AsfarasMichaelDummettisconcerned,hispioneeringbook,

Frege:PhilosophyofLanguage,publishedin1973,washugelysignificantintheblossomingof

interestinFregethatoccurredfrom the1970s. Butoneshouldmention,too,thetranslationof

FregesGrundlagenthatJ.L.Austinpublishedin1950,andtheselectionofFregeswritingsthat

wastranslated byPeterGeach and MaxBlack and published in 1951. Wittgenstein advised

GeachandBlackontheirselectionandevenlentthem hisowncopiesofsomeofFregesworks.

ThereisalsothepublicationofFregesNachgelasseneSchriftenin1969(translatedintoEnglishasPosthumousWritingsin1979)andhisWissenschaftlicherBriefwechselin1976(translatedinto

4 MichaelBEANEY,ChenBO,KojiNAKATOGAWA


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EnglishasPhilosophicalandMathematicalCorrespondencein1980)tonote,indicatingrecogni-

tion ofFregesimportancein GermanybeforeDummettsinfluencein theEnglish-speaking

world. So Ithinkthereisacomplexstorytotellaboutthegradualappreciation ofFreges

philosophy,astorythatitselfshedsinterestinglightonFregesideas.

KN:WouldyousaythattherewasanythinginFregescharacter,behaviourorideasthem-

selvesthatcontributedtothelackofrecognitioninhislifetimeandtohisgrowinginfluenceafter

hisdeath?

MB:Well,thatsagoodquestion. Asfarashischaracterisconcerned,wemightcompare

FregewithRussell,whoinmanywayswashisexactopposite,despitethefactthattheywereboth,

formuchoftheircareers,concernedtodemonstratelogicism. Bornanaristocrat,Russellstrode

theworldstagethroughouthisworkinglifewithsupremeconfidence. Russellhadextraordinary

energyandwroteonahugerangeoftopics,from mathematicallogicandmetaphysicstomarriageandsex. Im surethatifFregehadbeenmoreoutgoingandwell-connected,hewouldhavehad

greaterinfluencein hislifetime. Herarelyattended conferences,forexample,orgavetalks

outsideJena. WhileRussellpursuedloveaffairs,Fregewalkedhisdog. Thereisalsothenature

ofFregeslogicalnotationitselfhisBegriffsschrift. Thisnevercaughtonamonglogicians,

andishardertolearnandwritethanmodernnotation. Indeed,itmayamuseyoutohearthat

Fregewascriticizedbyoneofhisfirstreviewers(Schroder)forindulgingintheJapanesepractice

ofwritingvertically. Thetwo-dimensionalnatureofhisnotation certainlyhindered people

from appreciatingFregesquantificationallogic,anditwasonlywhenthislogicwasdeveloped

byPeanoandRussellthatittookontheform withwhichwearefamiliartoday. ThefailureofFregesownlogicism mayalsohaveledphilosopherstothinkthattherewaslittleofvaluein

Fregeswork-nothingthathadntbeenimprovedonbyRussellandWittgenstein. Itwasonly

whenphilosophersstartedinvestigatingRussellsandWittgensteinsphilosophiesinproperdetail

thattheyrealizedtheimportanceofFregesideasandwereledtoexplorethoseideasintheirown

right,whichopenedup alternativepossibilitiesofdevelopmentand understandingthan those

pursuedbyRussellandWittgenstein.

2.Fregesmaincontributionstologicandphilosophy

KN:How wouldyousummarizeFregesmaincontributionstologicandphilosophy?

MB:Fregeisrightlyregardedasthefounderofmodernlogic,givingthefirstexpositionof

quantificationallogicin hisBegriffsschriftof1879. Healso soughtto justifythatlogicby

clarifyingitsuseoffunction-argumentanalysisbasedonanontologicaldistinctionbetween

functionandobject,foundeddeepinthenatureofthings,asheputitinhisessayFunction

andConcept. Quantificationallogicwasfarmorepowerfulthantraditionallogic,andwiththis

new logicathisdisposal,Fregewasabletoformalizearithmeticalpropositionsinawaythat

hadntbeenpossiblebefore. Forthefirsttimeinhistory,logicism thusbecameafeasibleprojectandinpursuingthisproject,Fregealsomadeanimportantcontributiontothephilosophyof

5Frege,hisLogicandhisPhilosophyInterview withMichaelBeaney


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mathematics. Furthermore,inthinkingthroughthephilosophicalimplicationsofbothhislogic

andhislogicism,Fregewasledtodraw anumberoffurtherdistinctions,suchasthatbetween

SinnandBedeutung,whichhavebeenenormouslyinfluentialinthedevelopmentofmodern

philosophyoflanguageandmind.

CB:Asyouhavesaid,Fregeisalogicist:hetriestoreducemathematicstologic. Ithink

manyreaderswouldliketoknow exactlywhathislogicism is. AndwhatdoesFregemeanby

logic? WhatsimilaritiesanddifferencesaretherebetweenFreges,Kantsand contemporary

conceptionsoflogic? Prof. Beaney,couldyouclarifythesequestionsforus?

MB:Well,thefirstthingtonoteisthatFregeisalogicistaboutarithmetic,notgeometry.

Thatis,hethinksthatarithmeticcanbereducedtologic,buthedoesnotunlikeRussell

thinkthatgeometrycanalsobereducedtologic(forexample,viaanalyticgeometry). Hethinks

geometricaltruthsarespatialtruths,whichwerecognizethrough intuition(Anschauung,asKantcalledit),whereasarithmeticaltruthsareconceptualtruths,whichwerecognizebyreason

(Vernunft). InKantsterminology,Fregeheldthatgeometricaltruthsaresyntheticapriori,

whilearithmetical(andlogical)truthsareanalyticapriori. UnlikeKant,however,Fregedid

notthinkthatanalyticityimpliedtriviality. AccordingtoKant,logicaltruthsareanalyticand

thustrivial,butinclaimingthatarithmeticcouldbereducedtologic,Fregedidnotconcludethat

arithmeticaltruthsarethereforetrivial. Onthecontrary,hewasconcernedtoshow how logical

truthscanadvanceourknowledge.

Fregeisoftenregardedashavinghadauniversalistconceptionoflogic,accordingtowhich

logicallawsareuniversaltruthsthatareapplicabletoeverything,inotherwords,towhatevercanbeconceived. Hedidnotholdaschematicconception,accordingtowhichlogicisthestudy

oflogicalformsorschemata. Nordidhedistinguishbetweenlogicandmetalogic,thoughthe

extentto which he nevertheless(implicitly)pursued metatheoreticalinvestigationsishotly

debatedamongFregescholars. OneaspectofFregesuniversalistconceptionishisdoctrinethat

everyconceptmustbedefinedforallobjects. Heregardedvagueconceptsasoutsidethedomain

oflogicandthereforedeficient. Logicianstodaytendtodisagree:theymightthink,forexample,

thatvagueconceptsandtheproblemstheycausesuchasthesoritesparadox justshowthe

needtodevelopadifferentlogic,suchasintuitionisticlogicorfuzzylogic,todealwiththem.

3.FregesBegriffsschrift

CB:Letusnow talkaboutFregesfirstbookBegriffsschrift(1879). WhatdidFregeachieve

inthisbook? Couldyousummarizethisachievementforus?

MB:In hisBegriffsschriftFregegavethe firstexposition in history ofpredicatelogic,

introducinganotation forquantification,andalsoofferedanaxiomatization ofpropositional

logic. Heshowedhow mathematicalpropositionscouldbeformalizedandsucceededingiving

a purely logicalanalysisofmathematicalinduction. Helaterdeveloped hislogicalsystemfurther,butwhatheachievedinthisshortbookwastrulyremarkable,althoughittookalong



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timeforlogicianstoappreciateit.

KN:WhatdifferencesaretherebetweenFregeslogicalsystem and traditionalAristotelian

logic?

MB:Aristotelianlogichadhadgreatdifficultyinanalysingstatementsofmultiplegenerality.

Ininventingquantificationallogic,Fregeshowedhow thesecouldbeeasilyformalized. Healso

showedhow thetwotraditionalpartsoflogic,syllogistictheoryandpropositionallogic,could

beintegratedintoonecomprehensivesystem,asystem thatisfarmorepowerfulthananything

remotelydreamtofinAristotelianphilosophy.

KN:Whatcontribution did theBegriffsschriftmaketo Fregeslogicistproject? What

remainedtobedone?

MB:TheBegriffsschriftprovidedthebasiclogicalsystem bymeansofwhicharithmetical

propositionscouldbeformalizedandproved,andhislogicalanalysisofmathematicalinduction

throughhislogicaldefinitionofaone-onerelationwasalsoimportant. Whatremained

tobedonewastoprovidelogicaldefinitionsofallarithmeticalconcepts,includingnumberterms

themselves,which isthetask thatFregeundertook in hissecond book,DieGrundlagender

Arithmetik(1884).

4.FregesGrundlagen

CB:Verywell,so letusnow turn to thatbook,which manypeopleregard asFreges

masterpiece. DummettclaimsthattheGrundlagenmay justly becalled thefirstwork of

analyticalphilosophy. Prof. Beaney,doyouagreewiththis,andcouldyououtlinethemain

achievementsofthisbook?

MB:IdoagreethattheGrundlagenisamasterpieceandcanindeedberegardedasthefirst

work ofanalyticphilosophy. In thefirsthalfofthebook Fregeprovidessomepowerful

objectionstoearlieraccountsofarithmetic,suchasKantstheoryofarithmeticassynthetica

prioriandMillsempiricistview. In thesecond halfhesketcheshisown account,definingnumbersasextensionsofconcepts,and drawingon hisBegriffsschrift,defining thesuccessor

relation. Ineffect,heshowshow toderivetheDedekind-Peanoaxioms,whichwenow takeas

definingthenaturalnumbersequence.

CB:FregelaysdownthreeprinciplestoadheretointheintroductiontotheGrundlagen.

Thefirstistheanti-psychologism principle:Theremustbeasharpseparationofthepsychologi-

calfrom thelogical,thesubjectivefrom theobjective. Myquestionsarethefollowing. What

exactlyisthepsychologism to which Fregeobjects? Whatarehismain argumentsagainst

psychologism? How doyouevaluatehisanti-psychologism anditsinfluence?



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MB:Thepsychologism towhichFregeobjectsisessentiallytheview thatlogicallawsare

psychologicallawsofthought,understood asdescriptionsofthewayweactuallythink. For

Frege,thisdoesnotdojusticetothenormativecharacteroflogic. Logictellsushow weought

tothink,nothow weactuallythink. Fregesaysmuchmoreaboutallthisintheprefacetothe

Grundgesetze. Buthismainargumentisstraightforward. Justaswedontsaythatsomethingismorallyrightjustbecausethemajorityofpeoplebelieveitisright,oractasifitisright,sowe

dontsaythatanargumentislogicallyvalidjustbecausethemajorityofpeoplethinkitisvalid,

orreasonasifitisvalid. Onthismatter,inmyview,Fregeisabsolutelyright. Ofcourse,ifhe

is,then Ican feelhappy saying thiseven ifeveryoneelsedisagrees! Asithappens,many

philosophersdoagree,althoughtheissueofwhethernormativitycanbeexplainednaturalistically

isfiercelycontestedatthemoment.

CB:ThesecondofFregesthreeprinciplesisthecontextprinciple:Themeaningofaword

mustbeaskedforinthecontextofaproposition,notinisolation. Whatroledoesthisprincipleplay in Fregesphilosophy ofmathematics,and whatisitsrelation (ifany)to the anti-

psychologism principle?

MB:InthemainargumentoftheGrundlagen,Fregeappealsto thecontextprinciplein

explaininghow itispossibletoapprehendnumbers,giventhattheyareabstractobjects,thatis,

objectsthatarenotlocated in spaceand time. Fregesanswerisingenious:weapprehend

numbersbygraspingthemeaningsofnumberterms,whichwedobyunderstandingthesensesof

sentencesinwhichnumbertermsappear. Thisgivesusaclueastotherelationbetween the

contextprincipleand theanti-psychologism principle. Forifwethoughtthatgrasping themeaningofaterm wereamatterofhavinganappropriateidea,understoodpsychologically,then

wemighthavedifficultyinfindingideascorrespondingtonumbers. ButaccordingtoFrege,to

graspthemeaningofanumberterm,itisenoughtounderstandthesenseofasentenceinwhich

thenumberterm appears,andthissenseistobeexplainedlogically,notpsychologically.

CB:Prof. Beaney,wewillcometoFregesthirdprincipleinaminute. Butcouldyoufirst

clarifyFregesclaim thatastatementofnumbercontainsanassertionaboutaconcept? This

claim isveryimportantinFregesphilosophy,isitnot? Couldyouexplainitssignificance?

MB:Yes,youarequiterightabouttheimportanceofthisclaim:itliesattheveryheartof

Fregesaccount. LetustakeoneofFregesownexamples,thepropositionthatJupiterhasfour

moons. WemightbetemptedtointerpretthispropositionaspredicatingofJupitertheproperty

ofhavingfourmoons,butwemightthenfindithardto analysethepropertyofhavingfour

moons. AccordingtoFrege,however,thepropositionistobeunderstoodassayingsomething

notaboutanobjectbutaboutaconcept:itdoesnotpredicateoftheobjectJupitertheproperty

ofhavingfourmoons;instead,itpredicatesoftheconceptmoonofJupiterthepropertyhasfour

instances. Thepropertyhasfourinstancesisasecond-levelproperty,thatis,apropertythat

holdsofafirst-levelproperty(inthiscasethepropertyofbeingamoonofJupiter),andthekeythingaboutthepropertyhasfourinstancesisthatitcanbedefinedpurelylogically.



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InteachingandwritingaboutFrege,Ioftenexplainthesignificanceofthisanalysisbytaking

theexampleofanegativeexistentialstatement,suchasUnicornsdonotexist. Hereagainwe

mightbetemptedtoconstruethisassayingsomethingaboutobjectsattributingtheproperty

ofnon-existencetounicorns. Butwhat,then,aretheseunicorns? Musttheynotsomehow exist

orsubsistinsomenon-actualworldinorderfortheretobeasubjectofourproposition?OntheFregeanaccount,however,todenythatsomethingexistsistosaythattherelevantconcept

hasnoinstances:thereisnoneedtopositanymysteriousobject. Tosaythatunicornsdonot

exististo saythattheconceptunicornisnotinstantiated,whichcanbeeasilyformalized in

predicatelogicas

(x)Fx,whereFxrepresentsxisaunicorn.

Similarly,tosaythatGodexistsistosaythattheconceptGodisinstantiated,i.e.,todeny

thattheconcepthasno(i.e.zero)instances. (Ifwewantedtosaythatthereisoneandonlyone

God,wewouldhavetodenyaswellthattheconcepthastwoormoreinstances.)Onthisview,

existenceisnolongerseenasafirst-levelproperty,butinstead,existentialstatementsareanalysed

in termsofthesecond-levelpropertyisinstantiated,represented by meansoftheexistentialquantifier. AsFregenotesinsection53oftheGrundlagen,thisoffersaneatdiagnosisofwhat

iswrongwiththetraditionalontologicalargument. Fregesanalysisofnumberandexistential

statementsisthusawonderfulexampleofthepoweroflogicalanalysis.

5.TheCantor-HumePrincipleandtheJuliusCaesarproblem

CB:Dummetthaswrittenthatsection62oftheGrundlagenisarguablythemostpregnant

philosophicalparagrapheverwritten. ItintroduceswhathasbeencalledHumesPrinciple,

aboutwhichtherehasbeenalotofdebateinrecentyears. Itconnects,too,withtheso-calledJuliusCaesarproblem. Somyquestionsarethese. WhatexactlyisHumesPrinciple? What

roledoesitplayinFregeslogicistproject? Whyisitcontroversial? How isitrelatedtothe

JuliusCaesarproblem,and whatwasFregessolution to thisproblem? Whatisthelatest

thinking aboutthese matters? Is Humes Principle seen as analytic,for example? Prof.

Beaney,couldyoushedsomelightonthesequestionsforus?

MB:Well,Prof. Chen,weareindeednow attheveryheartofFregesphilosophy,andyou

haveaskedsomekeyquestionsaboutwhichtherehasindeedbeenmuchdebate. Again,letme

agreewithDummettabouttheimportanceofsection62oftheGrundlagen,althoughDummetthasalsosuggestedthatitisinsection62thatthelinguisticturnfirsttookplaceinphilosophy.

Ithinkthisisanexaggeration,forreasonsthatmightalreadybeclearfrom whatIhavejustsaid.

ForFregesanalysisofnumberstatementsalsoexemplifieswhatpeoplehaveinmindintalking

ofthelinguisticturntheidea,thatis,thatphilosophicalproblemscanbesolvedbytransform-

ingonepropositionintoanother,byshowingwhatamisleadingpropositionreallymeansby

paraphrasingit. Numberstatementsarereallyaboutconcepts,notobjects,accordingtoFrege.

Butasexaggerationsgo,Dummettssuggestion servesausefulpurpose:ithighlightsanother

importantexampleofwhatImyselfhavecalledinterpretiveortransformativeanalysis. Before

answeringyourquestions,IshouldalsonotethatIprefertotalkoftheCantor-HumePrinciple,sincethisdoesbetterjusticetoCantorsroleinthestory. Humewasonlythinkingofthefinite



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case,anditwasCantorwhofirstuseditasanexplicitprinciple.

TheCantor-HumePrincipleassertstheequivalencebetweenthefollowingtwopropositions,

whichIshalllabel(Na)and(Nb):

(Na)TheconceptFisequinumeroustotheconceptG.

(Nb)ThenumberofFsisidenticalwiththenumberofGs.Letusstartwith(Na). TosaythattwoconceptsFandGareequinumerous(gleichzahlig)isto

saythattheobjectsthatfallundertheconceptFcanbeone-onecorrelatedwiththeobjectsthat

fallundertheconceptG,inotherwords,thatthereareasmanyFsasGs. Andthisisjustto

saythenumberofFsisthesameasthenumberofGs,whichiswhat(Nb)says. So(Na)and

(Nb)areindeedequivalent.

TheroleoftheCantor-HumePrinciplecannow beexplainedasfollows. Recallthat,on

Fregesview,weapprehendnumbersbygraspingthemeaningofnumbertermstermssuchas

thenumberofFs. Wedothisaccordingtothecontextprinciple-bygraspingthesenseof

sentencesinwhichthenumbertermsappearsentencessuchas(Nb). Butwhatisthesenseof(Nb)? BytheCantor-HumePrinciple,(Nb)isequivalentto(Na). Sowegraspthesenseof

(Nb)bygraspingthesenseof(Na)andacceptingtheprinciple. Furthermore,thecrucialpoint

about(Na)isthatitcanbedefinedpurelylogically,sinceone-onecorrelation canbedefined

logically. SoifweputthecontextprincipleandtheCantor-HumePrincipletogether,thenit

looksasifourknowledgeoflogicisenoughtoexplainourapprehensionofnumbers.

Thisseemstoogoodtobetrue,andintheGrundlagen,Fregehimselfimmediatelygoeson

toraiseanobjection. ThisistheJuliusCaesarproblem,which,inessence,canbestated as

follows. WhiletheCantor-HumePrincipleallowsustotellwhentwonumbersarethesameor

notaslongastheyaregiventousasnumbers(theyarethesameiftherelevantconceptsareequinumerous),itdoesnottelluswhetheranobjectgiventousasanumberisthesameasan

objectnotgiventousasanumber,forexample,JuliusCaesar. Forallweknoworatany

rate,ifallweknow istheCantor-HumePrinciplethenumber7mightactuallybeJulius

CaesarorMao Tse-tung. ThePrincipledoesnot,in otherwords,laydown asufficient

criterionofidentityfornumbers.

Itisforthisreason thatFregegoeson to definenumbers,notimplicitly,viathecontext

principle,butexplicitly,byidentifyingthem withappropriateextensionsofconceptsexten-

sionsofconcepts,thatis,thatcanbedefinedpurelylogically. Thenumber0,forexample,is

definedintermsoftheconceptnotidenticalwithitself,whichcanberepresentedlogicallyasx x. Sincenothingisnotidenticalwithitself,thenumberofthingsthatfallunderthis

conceptisthenumber0. Infact,thenumber0isthenumberofthingsthatfallunderany

conceptthatisequinumerous(touseFregesterm)totheconceptnotidenticalwithitself;and

thisleadstotheexplicitdefinitionofthenumber0astheextensionoftheconceptequinumerous

totheconceptnotidenticalwithitself. (IexplainthedetailsofFregesaccountinTheFrege

Reader,pp.116-20,towhichIcanreferanyonewhowouldliketoknow more.)Havingprovided

hisexplicitdefinitions,FregecanthenderivetheCantor-HumePrinciple,whichcaninturnbe

usedtoderivetheDedekind-Peanoaxioms.

TheCantor-HumePrincipleisan exampleofwhatisnow widelycalled an abstractionprinciple,whichseekstodefineabstractobjectsofacertainkind(suchasnumbers)intermsof



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someequivalencerelationholdingbetweenobjectsofsomeother(morebasic)kind. Another

examplewhichFregehimselfgivesisdefiningdirectionsintermsofparallelism. Comparewhat

Istatedas(Na)and(Nb)withthefollowing:

(Da)Lineaisparalleltolineb.

(Db)Thedirectionoflineaisidenticalwiththedirectionoflineb.Iftwolinesareparallel,thentheirdirectionsarethesame,andviceversa. So(Da)and(Db)are

equivalent,andwecandefinedirectionsinananalogouswaytodefiningnumbers.

Therehasbeenagreatdealofdebateaboutabstractionprinciplesoverthelastfew years.

Whichonesaregoodandwhichonesarebad? TheCantor-HumePrincipleseemstobeagood

one. Ifso,thenwhycannotwesimplytakeitasanaxiom? WewouldhavetoanswertheJulius

Caesarproblem,butwhatifwejusttook thePrincipleasconstitutiveofourknowledgeof

numbers? Wouldthisbetreatingitasineffectanalytic? Canweregarditaslogical? Ifso,

thenshouldweagreethatFregewasessentiallyrightinhislogicism? Thesearesomeofthe

questionsthatphilosophersofmathematicstodayarevigorouslydebating.4)

CB:Prof. Beaney,allthisisvery interesting. Iam nota specialistin philosophy of

mathematicsmyself,butitseemstomethatyouaresayingthatFregesphilosophyofmathematics

isstillverymuchalive. AttheendoftheGrundlagen,Fregeconcludesthatfrom allthathas

gonebefore,theanalyticandapriorinatureofarithmeticaltruthshasthusemergedashighly

probable. DoyouthinkFregewasrighthere,bearinginmindtheattackthatQuinelatermade

onanalyticity? Isthisconnectedwithwhatisnow calledFregestheorem? Couldyoubriefly

explainthattheorem tous?

MB:Certainly,IthinkthatthereismuchofvalueinFregesphilosophy. Butletmesayone

thingaboutanalyticity. Interestingly,aftertheGrundlagen,Fregehimselfnevertalksagainof

theanalyticityofarithmetic;whatheclaimsisthatarithmeticcanbereducedto logic. My

suspicionisthathesooncametohavedoubtsaboutthenotionofanalyticity,longbeforeQuine.

Sothekeyissue,inmyview,iswhetherarithmeticcanbereducedtologic,andthatraisesthe

questionaboutthenatureoflogic,aboutwhichIhavealreadysaidtherehasbeenmuchrecent

debate. Fregeslogicisasystem ofsecond-orderlogic,thatis,asallowingquantificationover

functionsaswellasobjects. ButQuine,forexample,notoriouslydeniedthatsecond-orderlogic

wasproperlylogic. Imyselfthinkitis,buttheimportantquestionsconcernwhatyoucandowithwhatkindoflogic,notwhatisthecorrectlogicinanyabsolutesense. Sothisdoesindeed

relatetowhatisnow calledFregestheorem,whichstatesthattheDedekind-Peanoaxiomscan

bededuced,withinsecond-orderlogic,from theCantor-HumePrinciple. ThisiswhatFrege

showed in theGrundlagen. So ifweagreethatsecond-orderlogicislogic,and also (more

controversially)thattheCantor-HumePrincipleisalogicalprinciple,thenwedohaveaversion

oflogicism. Sologicism isindeedaliveissue.

6.Functions,conceptsandobjects

CB:Now letusdiscussFregesimportantpaperFunctionandConcept(1891). Youhave



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usintosayingwhatisnot,strictlyspeaking,correct. Butwecandobetterthanthis-following

asuggestionthatDummettfirstmade. Recallthataconcept,forFrege,mustbedefinedforall

objects. Sowecanthinkofconceptsasdividingallobjectsintothosethatfallunderitandthose

thatdonot. SowhatwearetryingtosayinsayingTheconcepthorseisaconcept(falsely,

accordingtoFrege)isjustthateverythingiseitherahorseornotahorse. Andthiscan be

readilyformalizedinpredicatelogicas(x)(Hx

Hx),whereHxrepresentsxisahorse.

Fregewouldbeperfectlyhappywiththis. JustaswesawinthecaseofUnicornsdonotexist,

whichcanmisleadusintothinkingthatunicornsmustexistinsomesense,wecanrephrasethe

problematicpropositiontomakeclearwhatisreallybeingsaid. Thisisanotherexampleof

whatIcallinterpretiveortransformativeanalysisresolving philosophicalproblemsby

rephrasingthepropositionsthathavegeneratedpuzzlement. AlthoughFregedidnotresolvethe

paradoxoftheconcepthorsehimself,heprovidedthetoolsandthegeneralstrategybymeansof

whichtodoso.

7.SinnandBedeutung

CB:Now letusturntoFregespaperOnSinnandBedeutung(1892),whichishismost

famousandinfluentialwork. InyourintroductiontoTheFregeReader,inwhichthispaperis

reprinted,youwroteasectiononthetranslationofBedeutung,whichhasbeenacontroversial

issueintheliteratureonFrege. Ifoundthisveryinformativeandaskedoneofmygraduate

studentstotranslateitintoChinese. Asyouknow,thishasnow beenpublishedinthejournal

WorldPhilosophy(March2008). Socouldyoujustbrieflytelluswhatyourowndecisionwas

astohow totranslateBedeutung?

MB:Yes,Ican certainlytellyou whatmydecision waswhich wasactuallyto leave

Bedeutunguntranslated! ButbeforeIsaysomethingtodefendthisdecision,letmetakethis

opportunity ofthanking you foryourwork in having whatIwrotepublished in Chinese

translation. IexplainthehistoryofthetranslationintoEnglishofBedeutung,asitisusedin

Fregeswork;soIreferanyChinesereaderinterestedinthishistorytotherelevantissueofWorld

Philosophy. Bedeutunghasbeen variouslytranslated asreference,meaning,denotation,

significance,indicationandnominatum;andthereareargumentssomegood,somebad

forandagainsteachofthese. Preciselybecauseofthis,andallthecontroversytherehasbeen,Ithoughtitbestin an edition ofFregeswritingsintended forwideuse,to leaveit

untranslated. Thatwayreaderswouldbeabletomakeuptheirownmindastohow itshould

betranslatedineachcontext,alltheoccurrencesofthekeywordBedeutungbeinghighlighted

byitsnon-translation. IfIhadtochooseasingleEnglishwordmyselftotranslateBedeutung,

Iwouldpickreference. Butthatworksbetterinsomeplacesthaninothers,soinwritingabout

FregeItendtousetheGermanterm untranslated,explainingwhatitmeansinanygivencontext

whereverappropriate.

Perhapsatthispoint,Prof. Chen,Icould ask you aquestion,sinceyou recognizethe

philosophicalimportanceofissuesoftranslation. How hasBedeutungbeentranslatedintoChinese? Are there also a range ofpossible wordsthatmightbeusedcorresponding,



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perhaps,toEnglishwordssuchasreferenceandmeaning? Iwouldbeinterestedtoknow if

ChinesetranslatorshavehadsimilarproblemsintranslatingBedeutungorindeed,anyother

keyterm inFregeswork.

CB:InChina,therehasnotbeenmuchworkonFregesofar,althoughinteresthasbeen

growingrapidlyrecently. OneChinesescholar,Prof. WangLu,edited andtranslated into

ChineseTheSelectedPhilosophicalPapersofGottlobFrege(1994). Healso translatedDie

GrundlagenderArithmetik(TheFoundationsofArithmetic). HansSlugasbook,Gottlob

Frege(1980),hasalsobeentranslatedintoChinesebyanotherscholar. Chinesescholarsusually

acceptEnglishtranslations,thatis,withSinntranslatedassense,translatedastheChineseword

,andBedeutungtranslatedasreference,translatedastheChineseword. Sometimes

Sinnisalsotranslatedas (meaning),andBedeutungas (reference). Inrecent

years,influencedbythedebateaboutEnglishtranslationsofBedeutung,someChinesescholars

haveinsistedthatSinnshouldbetranslatedas (meaning)andBedeutungshould betranslatedas (significance). ButIdisagree. Ifound yourcommentsand suggestions

aboutthetranslationofBedeutungquitereasonable.

MB:Prof. Nakatogawa, perhaps I could ask you the same questions. How has

Bedeutungbeen translated into Japanese? Aretherewordsin Japanesethatcorrespond to

Englishwordssuchasreferenceandmeaning?

KN:Atfirst,JapanesescholarstranslatedSinnas(inChinesecharacters)or(in

hiragana,Japanesephoneticsymbols),transliteratedasimi. Bedeutungwasatfirsttranslated

asorshiji. TheEnglishtranslation ofimiisusuallysenseormeaning,whiletheEnglishtranslation ofshijiisreference. Researcherswho cameto Fregestudies

later,however,tend to useimiforBedeutung,andorigiforSinn.

Therearesomescholarswhosuggesttheuseofanotherphoneticsystem,katakana,employing

i mifor sense. Each translation emphasizes different aspects ofSinn and

Bedeutung,andreflectsdifferentinterpretationsofFregestexts. MostJapaneseFregescholars

thinkthatBedeutungisacommonwordinGerman,usedforjustmeaning. Ifimi(mean-

ing)isusedforBedeutung,thereferringfunctionofthewordseemstobeweakened.

MB:Well,itsinterestingthatthereareindeedsimilarissuesintranslatingFregestermsintobothChineseandJapanese. Itiseasytoforgetthattranslatinginvolvesinterpreting,andweneed

tobeverycarefulinreadingphilosophicalworksintranslation. Translatingaphilosophical

workitselfrequiresphilosophicalskills,andIm afraidthattherearealotofphilosopherswho

donotappreciatethis. Ithinkreadingandtranslatingphilosophywritteninadifferentlanguage

from onesownisanessentialskillforaphilosophertoacquire:onelearnstobeverysensitive

totheexactwordsbeingusedtoexplainanideaorformulateanargument.

CB:With thisin mind,then,how would you explain Fregesfundamentaldistinction

betweenSinnandBedeutung? Whatmotivated thatdistinction? Whathasitto do withFregeslogicistproject? Andwhatproblemsdoesitraise?



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MB:Thereareanumberofmotivationsforthedistinction. Fregecametothinkthatall

threetypesoflinguisticexpression-names,conceptwords(orfunctionterms,moregenerally)and

sentencesallhadbothaSinnandaBedeutung. Butthereasonsarenotthesameineachcase

(andsomereasonsarebetterthanothers). Thesimplestwaytoexplainthedistinction,though,

asFregehimselfdoesatthebeginningofhisfamousessay,isbyconsideringtheinformativeness

ofidentitystatements. A statementoftheform a a(e.g.7 7)istriviallytruewhilea

statementoftheform a b(e.g.4 3 5 2),assumingitistrue,cantellussomething.

Inhisearlywork,FregehadusedthetermsInhalt(content)SinnandBedeutungmoreor

elsesynonymously. Butifthecontentofaandbisthesame,thenhow canastatementof

the form a bbe informative? Freges answer is to distinguish between Sinnand

Bedeutung(introducingadivisionwithinhisearliernotionofcontent). TheBedeutungofa

nameistheobjectreferredto,whiletheSinnindicatesthewayinwhichthisobjectisreferred

to. (YouwillseethatIhaveusedtheEnglishverbrefertoexplainFregesideahere. Wecan

thenspeakoftheobjectreferredtoasthereferenceor(perhapsbetter)referentofthename.)SoconsiderFregesownfamousexampleofaninformativeidentitystatement,Themorning

staristheeveningstar. Thetwonamesthemorningstarandtheeveningstarrefertothesame

object,namely,Venus(which,ofcourse,isactuallyaplanet,notastar):theyhavethesame

Bedeutung. Thisiswhytheidentitystatementiscorrect. Buttheyhavedifferentsensesand

Iwillnow usethewordsenseasthetranslationofSinn,asIthinkthatisfairlyunproblematic.

ThemorningstarreferstoVenusasitappearsinthemorningandtheeveningstarrefersto

Venusasitappearsintheevening. WecansenseVenusindifferentways,andthisiswhywe

havethetwoexpressionstoreflectthesedifferentways. AsFregenotes,itwasanempirical

discoverythatthebrightstarweseeinthemorningisthesameasthebrightstarweseeintheevening.

Fregedrew thesamedistinctionforconceptwords. Theconceptwordsequilateraltriangle

andequiangulartriangle,forexample,havethesameBedeutung,accordingtoFrege,whichis

theconceptitself. Theyalsohavethesameextension,whichisthesetofobjectsthatfallunder

thisconcept(roughlyspeaking,sincewewouldneedtotalkofpairings,onFregesview,aswe

haveseen). ButFregeneverthelessdistinguishesbetween theconceptitself(an unsaturated

entity)and theextension oftheconcept(which isan object). Whiletheyreferto thesame

concept,however,theydosoindifferentways,onereflectingtheideaofequilateralityandthe

otherreflecting the ideaofequiangularity. Thus,even though allequilateraltrianglesareequiangulartriangles,and allequiangulartrianglesareequilateraltriangles,thetwo concept

wordshavedifferentsenses. ThedistinctionbetweenSinnandBedeutungislessintuitivehere,

Ithink,sincewemightbeinclinedtosaythattherearedifferentconceptshere,understanding

conceptsintensionallyratherthanextensionally. Fregecanbedefendedonthis,buthereisone

issueaboutwhichtherehasbeendisagreement.

Healsodrew thesamedistinctionforsentences,andthishasprovedparticularlycontrover-

sial. Accordingto Frege,theBedeutungofasentenceisatruth-value,and itssenseisthe

thoughtitexpresses. SotheBedeutungofThemorningstaristheeveningstaristheTrue,and

itssenseisthethought(roughly)thatthebrightstarweseeinthemorningisthesameasthebrightstarweseeintheevening. WehavealreadytalkedofFregesdoctrinethatconceptsarefunctions



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thatmap objectsonto oneofthetwo truth-values. Wehave also seen thattheresultof

completingaconceptwordwithanameisasentence. SoifanobjectistheBedeutungofa

name,andaconceptistheBedeutungofaconceptword,thentheBedeutungofthesentencethat

resultsfrom completingaconceptwordwithanameisthevalueoftherelevantconceptforthe

relevantobject,inotherwords,oneofthetwotruth-values. Wecanthinkofthesetruth-values

inmanydifferentways,eachofthesewaysbeingreflectedinacorrespondingsentencewhosesense

istherelevantthought.

Itseemsoddtotalkofthemeaningofasentenceasatruth-value,whichisoneofthereasons

whyIdonotfavourmeaningasthetranslationofBedeutung(thoughithastobesaidthatit

alsosoundsoddinGerman). Butreferenceseemsnottobethatmuchbetter,whichiswhy

peoplehavesuggested significanceinstead. Butwhateveritsmeritin thecaseofsentences,

significanceishighlyinappropriateinthecaseofnames,sothisterm shouldberejected,too.

WethusseewhyBedeutunghasprovedsohardtotranslate:nothingwechooseworksproperly

andequallywellforallthreecases-names,functiontermsandsentences.ThefactthatFregedrawsthedistinction forallthreecasessuggeststhatthecasesare

analogous. Butmanypeoplehavearguedthatthisismisleading. Giventhattruth-valuesare

objects,accordingto Frege,sentencesareakind ofname. Thisview hasbeen particularly

criticized. ThereareimportantdifferencesbetweenourapprehensionofobjectssuchasVenus,

asreflected in ouruseofnamessuch asthemorning starand theevening star,and our

apprehensionoftruth-values,whichwesomehow graspinthinkingthoughts,expressedinouruse

ofsentences.

ThedistinctionbetweenSinnandBedeutungseemsbestmotivatedinthecaseofnames. But

thereareproblemshere,too. Fregeallowsthattherecanbesenseswithoutreferents,andindeed,thismightbetakenasanothermotivation forthedistinction. Fregetalksinafew placesof

senseasamodeofpresentationofareferent;butifthereisnoreferent,thenhow cantherebe

amodeofpresentationofit? Morefrequently,Fregetalksofsenseasamodeofdetermination

ofareferent,whichisbetter,sinceitmayturnoutthatthereisnothingdetermined. Butifaname

hasnoreferent,thenanysentenceinwhichthenameappearswilllackatruth-value,sincethe

BedeutungofasentenceisdeterminedbytheBedeutungofitsparts. Soitlooksasifwehave

ruledouttherebeinganyfictionaltruths,suchasHarryPotterisaboywizard. Atanyrate,for

aFregeanphilosophyoflanguagetobesustained,weneedtogivesomeaccountoffiction.

Philosophersalsonow distinguishbetweennames,suchasVenus,anddefinitedescriptions,suchasthemorningstar. Fregetreatedthem asthesame,butthereareverygoodreasonsfor

holdingthattheirlinguisticrolesarequitedifferent. Onewaytoexpressadifferenceistosay

thatnamesmusthavereference(iftheyaretomeananythingatall)butnot,perhaps,sense,while

definitedescriptionsmusthavesensebutnotnecessarilyreference. Thisisonlyoneofanumber

ofpossibleviewsonthematter. AlltheseissuesandmanyothersthatFregesideashave

raisedarebeingvigorouslydebatedtoday.

You alsoaskedabouttheconnection betweenFregesdistinction andhislogicistproject.

Thereisalotthatcanbesaidaboutthisaswell. Butletmejustnotehereonecentralconnection.

RecallthatFregewantedtoshow how arithmeticaltruthscanbeinformative,i.e.non-trivial,even iftheycanbereduced to logicaltruths. Thedistinction betweenSinnandBedeutung



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allowsthis. Taketheexampleof7 5 12. AccordingtoFrege,thisisalogicaltruth,but

itisnottrivial,unlike7 7,because7 5and12havedifferentsenses,justlikethemorning

starandtheeveningstar.

8.FregesGrundgesetze,Russellsparadoxandextensionsofconcepts

CB:ThisbringsusbacknicelytoFregeslogicistproject. Soletusnow talkaboutFreges

thirdbook,GrundgesetzederArithmetik(vol.1,1893;vol.2,1903). Myquestionsaresimilarto

thoseIasked abouttheBegriffsschrift. WhatwasFregesaim in thisbook? Whatdid he

actuallyachieve?

MB:Fregesaim wastodemonstrateformallywhathehadonlysketchedinformallyinthe

Grundlagenthatarithmeticcouldbereducedtologic. Hegaveamoresophisticatedexposi-

tionofhislogicalsystem,andinthesecond volume,healsoprovidedpowerfulobjectionstoexistingviewsoftherealnumbers,justasintheGrundlagenhehadcriticizedpreviousviewsof

thenaturalnumbers. Heattacked psychologism andformalismveryconvincingly,in my

opinionaswellastheideasofhiscontemporariessuch asCantorand Dedekind. He

plannedathirdvolume,tocompletehislogicistproject,butasyouknow,aletterhereceivedfrom

RussellinJune1902,asthesecondvolumewasinpress,dealtadevastatinglyblow tohiswork.

CB:Yes,indeed,RussellslettertoFregeisveryfamous,andIwantedtoaskyouaboutthis

aswell. WhatistheparadoxthatRusselldiscoveredinFregessystem? Canyouexplainitto

us,andhow itarosein Fregessystem? WhatwasFregesreaction? Whatsolutionsto theparadoxarepossible?

MB:Theparadoxarosefrom Fregesappealtoextensionsofconcepts. RecallthatFrege

endedupintheGrundlagendefiningnumbersintermsofextensionsofconcepts. Butwhatare

extensionsofconcepts? Fregesunderstandingofthesewas,in effect,encapsulated in anew

axiom thathelaiddown inGrundgesetzederArithmetik. ThiswashisinfamousAxiom V,

assertingtheequivalencebetweenthefollowingtwopropositions:

(Va)ThefunctionFhasthesamevalueforeachargumentasthefunctionG.

(Vb)Thevalue-range(Wertverlauf)ofthefunctionFisidenticalwiththevalue-rangeofthefunctionG.

Aswehaveseen,Fregesnotionofavalue-rangeofafunctionisageneralizationoftheideaof

anextensionofaconcept. (Va)and(Vb)thusyieldthefollowingasaspecialcase:

(Ca)TheconceptF appliestothesameobjectsastheconceptG (i.e.,whateverfallsunder

conceptF fallsunderconceptG,andviceversa).

(Cb)TheextensionoftheconceptF isidenticalwiththeextensionoftheconceptG.

Letusnotestraightawaytheanalogybetween(Ca)and(Cb)ormoregenerally,(Va)and(Vb)

and(Na)and(Nb),theequivalencebetweenwhichisassertedbytheCantor-HumePrinciple.

Axiom V,inotherwords,hasexactlythesameform astheCantor-HumePrinciple. AsFregesaw it,Axiom V ensuresthatevery(legitimate)concepthasanextension(ormoregenerally,that



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everyfunctionhasavalue-range)injustthesamewayastheCantor-HumePrincipleguarantees

thatnumbertermshaveaBedeutung.

Russellsparadox,asitarisesin Fregessystem,cannow bestated asfollows. Ifevery

conceptisdefinedforallobjects(asFregeheld,aswehaveseen),theneveryconceptcanbe

thoughtofasdividingallobjectsintothosethatdo,andthosethatdo not,fallunderit. If

extensionsofconceptsareobjects(asFregehadassumed theywere,justlikenumbers),then

extensionsthemselvescanbedividedintothosethatfallundertheconceptwhoseextensionthey

are(forexample,theextensionoftheconceptisanextension)andthosethatdonot(forexample,

theextensionoftheconceptisahorse). Butnow considertheconceptistheextensionofa

conceptunderwhichitdoesnotfall. Doestheextensionofthisconceptfallundertheconcept

ornot? Ifitdoes,then itdoesnot,andifitdoesnot,then itdoes. Wehavearrived ata

contradiction:thisisRussellsparadox.

Considernow thecaseinwhichtheconceptFandtheconceptG areoneandthesame.

Thentheyhavethesameextension,sothat(Cb)istrue. Butifthisconceptistheconceptistheextensionofaconceptunderwhichitdoesnotfall,thenitisnotthecasethatanythingthatfalls

underthisconcept(theconceptF)fallsunderthisconcept(theconceptG),asthecounterexample

ofitsownextensionshows,sothat(Ca)isfalse. Axiom V,whichassertstheequivalencebetween

(Ca)and(Cb),isthereforefalse. Axiom V,whichFregehadwantedtotreatasalogicallaw,

farfrom beingalogicaltruth,isntevenatruthatall!

Whathasgonewrong? Inmyview,whatisresponsiblefortheparadoxistheassumption

thatextensionsofconceptsareobjectsofthesamekindoronthesamelevelastheobjects

thatfallunderthoseconcepts. AsIsaidearlier,principlessuchastheCantor-HumePrinciple

arenow calledabstractionprinciples,andAxiom V isalsoanabstractionprinciple. WhiletheCantor-HumePrinciplemightbethoughtagoodabstractionprinciple,however,Axiom Vwould

seem tobeabadone. WhatisespeciallyproblematicinAxiom V,atleastasFregeunderstood

it,istheassumption thattheextensionsofconceptsorvalue-rangestherebyimplicitly

definedarealreadyinthedomainofobjectsoverwhichtheequivalencerelationstatedin(Ca)

or(Va)istakentohold.

Talkofabstractionprinciplessuggestsanobviousanswer:extensionsofconceptsshouldbe

seenasabstractedoutoftherelevantequivalencerelation:whetheroneseesthem asgenuine

objectsornot,theyarecertainlynotobjectsalreadyintheoriginaldomain. Thiswasessentially

theresponsethatRussellgaveindevelopinghistheoryoftypesasananswertotheparadox.Therearebase-levelobjects,first-levelextensionsofconcepts,second-levelextensionsofconcepts,

andsoon. Recognizingsuchahierarchyofobjectsenablesonetoavoidtheparadox,although

thedifficultquestionthenbecomeshow todevelopatheoryoftypesthatstillmakeslogicism a

viableposition. Russellendeduphavingtointroducefurtheraxiomswhoselogicalstatuswas

bynomeansclear. Butthatisanotherlongstoryinthehistoryoflogicandanalyticphilosophy.

AstoFrege,hisinitialresponse,whichhegaveinanappendixhastilywrittenasthesecond

volumeofGrundgesetzewasgoingtopress,wassimplytodisallow conceptsfrom applyingto

theirownextensions,restrictingAxiom V accordingly. Buthesoonsaw thatthisresponsewas

inadequate:amongotherthings,itseemedadhocratherthanphilosophicallymotivated. Herecognized thepossibilityofRussellsresponse-treating extensionsofconceptsasimproper



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objects,asFregeputit. ButforFrege,thecomplexityoftheresultingtheoryconflictedwithhis

universalistconceptionoflogic,thatis,withhisview thatlogicalprinciplesoughttoapplyto

allkindsofobjectsunrestrictedly;andheeventuallycametoabandonhislogicism.

9.Neo-logicismandthedevelopmentofFregesideas

KN:Inrecentyearstherehasbeensomethingofarevivaloflogicism underthenameof

neo-logicismorneo-Fregeanism. How doesthisdifferfrom Fregesand Russellslogicism,

andwhoareitsmainadvocates? Whatisyourownview onthis?

MB:Wetalkedearlieraboutwhatisnow called Fregestheorem,which statesthatthe

Dedekind-Peanoaxioms,whichdefinearithmetic,canbededucedwithinsecond-orderlogicfrom

theCantor-HumePrinciple. Itisthisresultthatguidesneo-logicism. NeitherFregenorRussell

waspreparedtoclaim thattheCantor-HumePrinciplewasafundamentallogicalprinciple:theythoughtthatit,too,had to bederived from logicallawsand definitions. Butifwecould

somehow arguethatitisindeedafundamentallogicalprinciple,thenanew form oflogicism is

possible. Twoofthemostimportantneo-logicistsareCrispinWrightandBobHale. Theyhave

also tried to identify abstraction principlesby meansofwhich to define therealnumbers,

understoodasmeasuringnumbersratherthancountingnumbers. Fregethoughtthatthereal

numbersand thenaturalnumberswerequitedifferententities,and offered differentlogicist

accounts. Wrightand Halehavefollowed Fregein this. My own view oflogicism,asI

suggestedbefore,isthatitalldependsonwhatyoutakeaslogic. ThequestionIsarithmetic

reducibletologic?,itseemstome,doesnotpermitasimpleYesorNoanswer.

CB:Neo-logicism isonewayinwhichphilosophershavedevelopedFregesideas. Inwhat

otherwayshavehisideasbeendevelopedinrecentyears?

MB:Fregeisnow widelyrecognizedasoneofthefoundersofanalyticphilosophy,andin

mostareasofanalyticphilosophy,hisideasarereferredto,discussed,criticizedanddeveloped.

Soafullanswerwouldinvolvetellingthewholehistoryofanalyticphilosophy. Havingsaid

this,though,thereislittledoubtthatFregesideashavebeen particularlyinfluentialin the

philosophy oflanguageand thephilosophy ofmind,with thedistinction betweenSinnandBedeutunglyingatthecoreofdebate. Totakejustoneexample,therehasbeenmuchrecent

discussion ofindexicals-terms,such asI,you,hereand now,whosereferencedepends

systematicallyonthecontextofuse. Butwhilethereferencemightbeclearonanygivenoccasion

ofuse,whatisthesenseofsuchterms? Fregehimselfmakesonlyafewremarksaboutindexicals,

and itiscontroversialjusthow histheory ofsense should bedeveloped to accommodate

indexicals.

10.Fregesconceptionofthoughts(Gedanken)

CB:Inhislateryears,Fregetalkedoftherebeingathirdrealminadditiontotheouter



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issueandithasgeneratedalotofdebate5)butthebestcriterionthatcanbeoffered(inmy

view)issomethingonthefollowinglines:

TwosentencesAandB(asusedinagivencontext)expressthesamethoughtifandonlyif

anyonewhounderstandsbothsentences(asusedinthatcontext)canimmediatelyrecognize

thatAistrue(orfalse)iftheyrecognizethatBistrue(orfalse),andviceversa.Therearestillproblemswithsuchacriterion,butthedetailsarenotrelevanttothepointInow

wanttomake. Letusspeakoftwosentencesbeingcognitivelyequivalentwhentheystandin

therelationship captured in thecriterion. Wecanthen formulateacontextualdefinition of

thought(asthesenseofasentence)inthesamewayascontextualdefinitionscanbeofferedof

termsforabstractobjectssuchasnumbers,directions,value-rangesand extensionsofconcepts.

Whatwecansay,inotherwords,isthatthefollowingtwopropositionsareequivalent:

(Sa)SentenceAiscognitivelyequivalenttosentenceB.

(Sb)ThesenseofA(thethoughtexpressedbyA)isidenticalwiththesenseofB(thethought

expressedbyB).Youwillseethat(Sa)and(Sb)haveexactlythesameform as(Na)and(Nb),(Da)and(Db),(Va)

and(Vb),and(Ca)and(Cb),whichwediscussedearlier. A propositionassertinganidentity

betweenobjectsofacertainkindistakenasdefinablebymeansofapropositionassertingan

equivalencerelationbetweenobjectsofsomeotherkind. Thisoffersausefulwaytounderstand

Fregesconceptionofthought. Foryoucanseenow whyFregeregardedsenses(thoughts)as

objectsinjustthesamewayasheregardednumbers,directions,value-rangesandextensions

ofconceptsasobjects. They can allbedefined bymeansofwhatwewould now callan

abstractionprinciple.

Recognizingthatwhatwehavehereisanabstractionprinciple(thoughthisisnothowFregewouldhaveputit)alsoholdsthekeytohow IwouldreviseFregesconceptionofthoughts.

Wecanindeedtalkofthoughtsasobjects,inthesamewayaswecantalkofnumbersasobjects.

Butsuch talk isto beexplained in termsoftherelevantequivalencerelations. Justaswe

apprehend numbersbyunderstandingthesensesofsentencesin whichnumberstermsappear,

explainedbyourunderstandingofone-onecorrelation,sotoowegraspthoughtsbyunderstand-

ingthesensesofsentencesinwhichthoughttermsappear,explainedbyourunderstandingof

cognitiveequivalence. IgraspthoughtstotheextentthatIcanuseandunderstandsentencesthat

expressthosethoughtsandrecognizewhensentencesarecognitivelyequivalent,asexhibitedin

ourpracticesofinference,rephrasalandsubstitutionoftermsforoneanother. AsIwouldputit,then,thoughtsarenotobjectsexistingindependentlyofourpracticesofusingsentences. Talk

ofathirdrealm,inmyview,ismisleading. Fregewasrighttoemphasizethatthoughtsare

objective,andwecanagreethattheyareobjects. Buttheyareobjectsabstractedfrom our

practicesofusingsentences,nottimelessentities. Sotoconclude,Prof. Chen,youarequite

righttobeconfusedbywhatFregesaysaboutthoughts,butIthinkthereisawayofsiftingout

whatisrightinhisaccountfrom whatismisleading. Indeed,philosopherseversincehavebeen

tryingtodojustthis,thoughIwouldntsaythatthereisyetagreementonthematter.



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11.Neo-Kantianism

KN:Prof. ChenaskedyouaboutFregesinfluenceon laterphilosophers,and whatyou

havejustsaidhighlightsafurtherissuethathasinspiredsubsequentworkinthephilosophyof

languageandmind. IwouldliketoaskyouabouttheinfluenceofearlierphilosophersonFrege

himself. Whatarethelatestthoughtsanddebatesaboutthis? InJapan,thereisagreatdealof

interestinNeo-Kantianism,whichinfluencedJapanesethoughtafter1868. Whatinfluencedid

Neo-Kantianism,inparticular,haveonFrege?

MB:IntheearlydaysofFregescholarshiptherewaslittleconcernwiththeinfluenceson

Fregehimself. Dummettnotoriouslyclaimedintheintroductiontohis1973bookthatFreges

logicwasbornfrom Fregesbrainunfertilizedbyexternalinfluences. Wehavecomealong

wayinourunderstandingofFregesincethen. TherehasbeenresearchdonebyMarkWilson

andJamieTappenden,amongothersonthemathematicalbackgroundtoFregeswork;andtherehasbeenmuchdiscussionofFregesphilosophicalpredecessorsandcontemporaries. Hans

Sluga,inthebookonFregehepublishedin1980,wasoneofthefirsttoemphasizetheinfluence

ofsuchphilosophersasTrendelenburgandLotze;andinrecentyears,GottfriedGabriel,whois

ProfessorofLogicandTheoryofScienceatFregesownuniversity,theUniversityofJena,has

beenwritingaseriesofpapersontheinfluenceofkeyGermanthinkersonFrege. Wetalked

earlieraboutthecentralclaim ofFregesGrundlagen,thatanumberstatementcontainsan

assertion aboutaconcept. Thebasicideaherecamefrom Herbart. TheinfluenceofNeo-

Kantianism isevenmorepervasive,especiallytheNeo-Kantianism oftheso-calledSouth-West(or

Baden)school. Fregesanti-psychologism,hisview ofthenormativenatureoflogicallawsandhistheoryofjudgementcanallbebroadlylocatedintheNeo-Kantiantradition. Thevalue

theory developed byWindelband and Rickertmightalso besingled outashaving had an

influenceonFrege,reflectedinhisconceptionoftheBedeutungofasentenceasitstruth-value

whichhasledGabriel,inparticular,tosuggestthatBedeutungisbesttranslatedassignifi-

cance.

12.Suggestionsforfurtherreading

CB:ThereisclearlyagreatdealofworkbeingdoneonallaspectsofFregesphilosophy,bothhistoricalandsystematic. Sofinally,Prof. Beaney,canweaskyoutorecommendsome

secondaryliteratureonFregetoChineseandJapanesereaderswhoareinterestedinunderstand-

ingFregesphilosophyfurther?

MB:A usefulintroductorybookisFregeExplainedbyJoanWeiner(OpenCourt,Chicago,

2004),arevisedandexpandedversionofherearlierFrege(OxfordUniversityPress,1999). My

ownbook,Frege:MakingSense(Duckworth,London,1996)isratherlonger,butIwouldhope

itprovidesagoodintroduction toallthemain ideasofFregesphilosophyin particular,

showinghow hisphilosophyoflanguageemergesfrom hislogicistproject. AlthoughIwroteitmorethantenyearsago,andfeelcleareraboutsomethingsnow,Istillagreewithmostofit! The



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bestbook onFregesphilosophyofmathematicsisMichaelDummettsFrege:Philosophyof

Mathematics(Duckworth,London,1991). Thiswasthelong-awaited sequelto hisFrege:

PhilosophyofLanguage(Duckworth,London,1973),thepioneeringbookImentionedatthe

beginningofourinterview. Thisfirstbooksaw Fregetoomuch,Ithink,asamodernphiloso-

pheroflanguage. Mostscholarsrecognizenow how muchFregesphilosophyoflanguagewas

subservienttohisphilosophyofmathematics,andDummettgivesanexcellentaccountofthe

latterinhissequel.

Finally,letmealsomentionafour-volumecollectionofpapersonFregethatIeditedwith

ErichReckoftheUniversityofCaliforniaatRiverside,GottlobFrege:CriticalAssessmentsof

LeadingPhilosophers(Routledge,London,2005). ThefourvolumesareonFregesphilosophy

incontext,hisphilosophyoflogic,hisphilosophyofmathematics,andhisphilosophyofthought

andlanguage. Eachvolumehasitsownintroduction,andthe67papersweselectedwereall

written orpublished in thetwenty-yearperiod from 1986 to 2005. Thework Ihavejust

mentionedbyMarkWilson,JamieTappendenandGottfriedGabriel,forexample,canbefoundinthesevolumes. Thecollectionwouldcertainlygiveanyreaderagoodsenseoftheworkthat

isnow beingdoneonFrege.

CB:Prof. Beaney,thatisveryhelpful,andthankyouverymuchforagreeingtotalktous.

Iam sureChineseandJapanesereaderswillhavelearntalotaboutFrege,hislogicand his

philosophy,andaboutsomeofthedebatesthathisideashavegenerated. Wewishyouallthe

bestinyourcontinuedworkonFregeandthehistoryofanalyticphilosophy,andinallyourother

philosophicalactivities.

MB:Well,thankyoubothverymuchforyourquestionsandforyourowninterest. Ihope

ourinterview willresultinafarhighernumberbeingassignedtotheconceptreaderofFrege.

Notes

1)CBstandsforChenBo.

2)KN standsforKojiNakatogawa.

3)MBstandsforMichaelBeaney.

4)Thesecond volumeofBeaney and Reck (2005)containsarticlesrelevantto thedebatein philosophyof

mathematics.5)Readersinterestedinthedebateinphilosophyoflanguagecanfindrelevantarticlesinthefourthvolumeof

BeaneyandReck(2005).

References

FortheoriginalworkofGottlobFrege,seeBeaney(1997).

Beaney,Michael.1996.Frege:MakingSense.London:Duckworth.

Beaney,Michael(editor).1997.TheFregeReader.Oxford:Blackwell.

Beaney,Michael.2005.ImaginationandCreativity.MiltonKeynes:OpenUniversity.

Beaney,Michael(editor).2007.TheAnalyticTurn:AnalysisinEarlyAnalyticPhilosophyandPhenomenology.

London:Routledge.Beaney,MichaelandReck,ErichH.(editors).2005.GottlobFrege:CriticalAssessmentsofLeadingPhilosophers,



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fourvolumes.London:Routledge.

Dummett,Michael.1973.Frege:PhilosophyofLanguage,London:Duckworth.Secondedition1981.

Dummett,Michael.1991.Frege:PhilosophyofMathematics,London:Duckworth.

Sluga,Hans.1980.GottlobFrege.London:Routledge.Secondedition,1990.

Weiner,Joan.2004.FregeExplained.Chicago:OpenCourt.

Editorialnote

ThisarticleisbasedondiscussionsbetweentheauthorswhichtookplacebothactuallyintheUK andvirtually

ontheInternetbetweenthesummerof2008andJanuary2009. AlthoughmostofitwaswrittenbyProfessor

Beaney,somepartsreflectthediscussionswithProfessorNakatogawa,andthemainstructureoftheinterview was

setbyProfessorChenBo.

Abouttheauthors

MICHAEL BEANEYabbreviated to MB,BA,BPhil,DPhil(Oxford),ProfessorofPhilosophyatthe

UniversityofYork,isawell-knownFregescholar. Hismainresearchinterestsareinthephilosophyoflanguage,

logic,mathematicsandmind,andthehistoryofphilosophy,especiallyanalyticphilosophy. Heistheauthorof

Frege:MakingSense(Duckworth,1996),ImaginationandCreativity(OpenUniversity,2005),andover50papers

andarticles;heisalsotheeditorofTheFregeReader(Blackwell,1997)andTheAnalyticTurn:AnalysisinEarly

AnalyticPhilosophyandPhenomenology(Routledge,2007),andtheco-editor(withErichReck)ofGottlobFrege:

CriticalAssessmentsofLeadingPhilosophers(4vols.,Routledge,2005).

CHEN BOabbreviatedtoCB,PhD,ProfessorofPhilosophyatPekingUniversity,China;academicvisitor

atOxfordbetweenAugust2007andAugust2008.

KOJINAKATOGAW AabbreviatedtoKN,PhD,ProfessorofPhilosophyattheUniversityofHokkaido,

Japan;academicvisitoratOxfordbetweenMarch2008andSeptember2008.


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Michael Beaney