EssaysonFrege’s Basic LawsofArithmetic
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PHILIPA.EBERTANDMARCUSROSSBERG
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MichaelHallett
13MathematicalCreationinFrege’s Grundgesetze
PhilipA.EbertandMarcusRossberg
EricSnyderandStewartShapiro
15Frege’sLittleTheoremandFrege’sWayOut
RoyT.Cook
16“HowdidtheserpentofinconsistencyenterFrege’sparadise?”411
CrispinWright
17Second-OrderAbstractionBeforeandAfterRussell’sParadox437
MatthiasSchirn
RichardKimberlyHeck
MichaelKremer
20ABriefHistoryofEnglishTranslationsofFrege’sWritings
MichaelBeaney
21Translating‘Bedeutung’inFrege’sWritings:ACaseStudyand CautionaryTaleintheHistoryandPhilosophyofTranslation588 MichaelBeaney
Foreword
GottlobFregepublishedhis GrundgesetzederArithmetik intwovolumes;the firstappearedin1893,thesecondin1903. Grundgesetze wastofulfillFrege’s ambitiontodemonstratethatarithmeticandanalysisarereducibletologic, andthusestablishaviewthatwenowcall‘logicism’.Inpreviouswork,in particularin Begriffsschrift (1879)and DieGrundlagenderArithmetik (1884), Fregeprovidessomeoftheformalandphilosophicalfoundationsforhislogicism.Morespecifically,hisfirstbookcontainstheinitialformulationofthelogicalsystem—theeponymous Begriffsschrift—whilehissecondbookeschews formulaealtogetherandoffersaphilosophicalfoundationforhislogicistpositioninthephilosophyofmathematics.Bothpublicationsare,intheirown right,groundbreaking.
Begriffsschrift constitutes“perhapsthegreatestsinglecontributiontologicevermadeanditwas,inanyevent,themostimportantadvancesince Aristotle.”1 Ontheotherhand,thephilosophicalmethodologyof Grundlagen,inparticularitsanalyticwritingstyle,ledmanytothinkofFregeasone ofthefoundingfathersofanalyticphilosophy.MichaelDummettconsiders Grundlagen “themostbrilliantpieceofphilosophicalwritingofitslength everpenned”.2 Inthisworkscholarslocatetheoriginsofthelinguisticturn inphilosophy,whichshapedmuchoftwentieth-centuryphilosophy.
However,Frege’sstandinginthehistoryofanalyticphilosophydoesnot simplyderivefromthesetwogroundbreakingbooks,butitisalsobasedon aseriesoflecturesandarticlesthatFregewrotejustbeforethepublication ofthefirstvolumeof Grundgesetze. FunctionundBegriff (1891),‘ÜberSinn undBedeutung’(1892),and‘UeberBegriffundGegenstand’(1892)make foratrilogyofGodfather-likeproportionswithpartIIbeing,ofcourse,one ofthemostwidelyreadarticlesinphilosophy.Itsinfluenceinthephilosophy oflanguageandinlinguisticswouldbehardtoexaggerate.
Howthendoes Grundgesetze,aworkthatFregewithoutdoubtintended tobehis magnumopus,fitintothelistofFrege’sphilosophicalandlogical achievement?Mostscholarshave,untilrecently,viewedFrege’s Grundgesetze
1AlexanderGeorgeandRichardKimberlyHeck(2000),‘Frege,Gottlob’,inEdwardCraig (ed.), ConciseRoutledgeEncyclopediaofPhilosophy,LondonandNewYork:Routledge,page296. (Orig.publ.underthename“RichardG.Heck,Jr”.)
2MichaelDummett(2007),‘IntellectualAutobiography’,inRandallE.AuxierandLewis EdwinHahn,eds. ThePhilosophyofMichaelDummett,vol.XXXIof TheLibraryofLivingPhilosophers,ChicagoandLaSalle,Ill.:OpenCourt,page9.
ascontainingforthemostparts,merelytheformaldetailsofhislogicistprojectwhichwenowknowtohavefailed.Frege’sderivationsoffundamental arithmeticalprinciples,whichcoverabouthalfofeithervolume,arebased onaninconsistentaxiom:theinfamousBasicLawV.Itisthusperhapsno surprisethatlittleattentionwaspaidtoFrege’spredominatelyformalwork.
Interestingly,however,itis Grundgesetze thatcontainsFrege’smostforcefulandmostfamousrejectionofpsychologisminlogic,andtherelevantpassagesfromtheForewordofthefirstvolumewerethefirstpiecesofFrege’scorpustobetranslatedintoEnglish.Moreover,Wittgenstein,Russell,Jourdain, Peano,andotherphilosophersandmathematiciansofthetimewhoseriously engagedwithFrege’sworkpaidcloseattentionto Grundgesetze.Nonetheless, throughouttherestofthetwentiethcenturymoreandmoreemphasiswas placedonFrege’sotherwritings—writingsthatareindependentofthefailureofhislogicalfoundations,orindeedindependentofhislogicismmore generally.
In1983,CrispinWrightpublished Frege’sConceptionofNumbersasObjects,inwhichhereconstructedFrege’sderivationoftheaxiomsofarithmetic fromwhatwouldcometobeknownas“Hume’sPrinciple”,or“HP”.Wright alsoconjecturedtherethatHP,unlikeBasicLawV,isconsistent.3 ThisinspiredGeorgeBoolostolookmorecarefullyatFrege’sinformalderivationsin Grundlagen,andalsoattheformalworkin Begriffsschrift. 4 Boolos’sstudent RichardKimberlyHeckthenbeganinvestigatingFrege’stechnicalworkon arithmeticin Grundgesetze. 5 Aroundthesametime,PeterSimonsandMichael DummettalsopublishedtheirdiscussionsofFrege’sformalworkonthereal numbersinPartIIIof Grundgesetze. 6 TheincreasinginterestinFrege’sproofs wasaccompaniedbyarenewedinterestinFrege’sphilosophyofmathematics, givenalsotheriseofaviewlabelled‘neo-logicism’,‘neo-Fregeanism’,orsimply ‘abstractionism’,aschampionedbyCrispinWrightandBobHale,7 andthe
3ForthefirstpublishedconsistencyproofsofHPseetworeviewsof Frege’sConceptionof NumbersasObjects:onebyJohnP.Burgess(1984, PhilosophicalReview 93:638–40),andtheother byAllenHazen(1985, AustralasianJournalofPhilosophy 63:251–4);andseeBoolos(1987),as citedinnote4below.
4GeorgeBoolos(1987),‘TheConsistencyofFrege’sFoundationsofArithmetic’,inJudith JarvisThomson(ed.), OnBeingandSaying:EssaysforRichardCartwright,Cambridge,Mass.: MITPress,pages3–20;GeorgeBoolos(1985),‘Readingthe Begriffsschrift’, Mind 94:331–44.
5RichardKimberlyHeck(1993),‘TheDevelopmentofArithmeticinFrege’s Grundgesetzeder Arithmetik’, JournalofSymbolicLogic 58:579–601(originallypublishedunderthename“Richard G.Heck,Jr”).Seealsotheirmorerecentbooks Frege’sTheorem (2011)and ReadingFrege’s Grundgesetze(2012),bothOxford:ClarendonPress.
6PeterSimons(1987),‘Frege’sTheoryoftheRealNumbers’, HistoryandPhilosophyofLogic 8:25–44;MichaelDummett(1991), Frege:PhilosophyofMathematics.London:Duckworth.
7Comparehere,e.g.,CrispinWright(1983), Frege’sConceptionofNumbersasObjects,Aberdeen:AberdeenUniversityPress;BobHale(1987), AbstractObjects,Oxford:BasilBlackwell; BobHaleandCrispinWright(2001), TheReasons’ProperStudy:EssaystowardsaNeo-Fregean PhilosophyofMathematics,Oxford:ClarendonPress;PhilipA.EbertandMarcusRossberg,eds. (2016) Abstractionism:EssaysinPhilosophyofMathematics,Oxford:OxfordUniversityPress.
workbyMichaelDummett.YetabroaderengagementwithFrege’s magnum opus facedtheobstaclethatonlypartsofthetwovolumeshadbeentranslated. TogetherwithCrispinWright,westartedworkonanewtranslationof Grundgesetze in2003,and,withthehelpofmanydistinguishedFregescholars,the firstcompleteEnglishtranslation, BasicLawsofArithmetic,volumesIandII, appearedin2013.8
ThepresentvolumeisthefirstcollectionofessaysthatfocusesonFrege’s Grundgesetze andaimstohighlightthetechnicalaswellasphilosophicalrichnessofFrege’smajorwork.Thecompanionbringstogethertwenty-twoFrege scholars,whosecontributionsdiscussawiderangeoftopicsarisingfromboth volumesof Grundgesetze.Andsomaythiscollection,evenifbelatedly,contributetoarenaissanceof Grundgesetze andhelptoestablishthisworkasone ofFrege’smanymasterpieces.
Wewouldliketothankoureditor,PeterMomtchiloffatOxfordUniversityPress,forhissupportandhispatience.ThankstoDeniseBannerman formeticulousproof-reading.Typesettinganearly700-pagevolumeinLATEX provedtobeyetanotherbigchallenge,andwewouldliketothankColin McCullough-BennerandAndrewParisiwhoprovidedinvaluableassistance withthisarduoustask.WewouldalsoliketothankColinforhisworkonthe Indexandhisadditionalhelpwithcopyeditingandproof-reading.
AndthankstoCrispin,forgettingitallstarted,andforeverything.
PhilipEbert,Stirling,Scotland MarcusRossberg,Storrs,Conn.,USA
8Formoredetailsonearliertranslationsaswellasthefirstcompletetranslationof Grundgesetze, andhowthelattercameabout,consultCrispinWright’sForewordandourTranslators’IntroductioninGottlobFrege(2013), BasicLawsofArithmetic.DerivedUsingConcept-Script,volumesI andII,trans.anded.PhilipA.EbertandMarcusRossbergwithCrispinWright.Oxford:Oxford UniversityPress.
Contributors
MichaelBeaney isProfessorofHistoryofAnalyticPhilosophyattheHumboldt-UniversitätzuBerlinandProfessorofPhilosophyatKing’sCollegeLondon.
PatriciaA.Blanchette isGlynnFamilyHonorsChairofPhilosophyatthe UniversityofNotreDame.
RoyT.Cook isCLAScholaroftheCollegeandJohnM.DolanProfessorof PhilosophyattheUniversityofMinnesota,TwinCities.
PhilipA.Ebert isSeniorLecturerinPhilosophyattheUniversityofStirling.
MichaelHallett isJohnFrothinghamProfessorofLogicandMetaphysicsat McGillUniversity,Montreal.
RichardKimberlyHeck isProfessorofPhilosophyatBrownUniversity,in Providence,RhodeIsland.
KevinC.Klement isProfessorofPhilosophyattheUniversityofMassachusettsAmherst.
MichaelKremer istheMaryR.MortonProfessorofPhilosophyandinthe CollegeattheUniversityofChicago.
ØysteinLinnebo isProfessorofPhilosophyattheUniversityofOslo.
RobertC.May isDistinguishedProfessorofPhilosophyandLinguisticsat theUniversityofCalifornia,Davis.
WalterB.Pedriali isAssociateLecturerinPhilosophyattheUniversityofSt Andrews.
ErichH.Reck isProfessorofPhilosophyattheUniversityofCaliforniaat Riverside.
MarcusRossberg isAssociateProfessorofPhilosophyattheUniversityof Connecticut.
MatthiasSchirn isProfessorEmeritusofPhilosophyattheUniversityofMunichandamemberoftheMunichCenterforMathematicalPhilosophy.
Contributors xi
StewartShapiro isO’DonnellProfessorofPhilosophyatTheOhioStateUniversity,DistinguishedVisitingProfessorattheUniversityofConnecticut,andDistinguishedPresidentialFellowattheHebrewUniversityof Jerusalem
PeterSimons, FBA,MRIA,isProfessorEmeritusofPhilosophyatTrinity CollegeDublin.
EricSnyder isPostdoctoralFellowattheMunichCenterforMathematical Philosophy,LMUMunich,andAssociateProfessorofPhilosophyat AshokaUniversity.
WilliamStirton worksasanadministrativeassistantforEdinburghLeisure Ltd.
JamieTappenden isProfessorofPhilosophyattheUniversityofMichigan, AnnArbor.
KaiF.Wehmeier isDean’sProfessorofLogicandPhilosophyofScienceand ofLanguageScienceattheUniversityofCalifornia,Irvine.
JoanWeiner isProfessorofPhilosophyatIndianaUniversity.
CrispinWright, FBA,FRSE,FAAAS,isProfessorofPhilosophyatNewYork University,ProfessorofPhilosophicalResearchattheUniversityofStirling,andRegiusProfessorofLogicEmeritusatAberdeenUniversity.
TheBasicLawsofCardinalNumber
RichardKimberlyHeck
Fregebeginshis GrundgesetzederArithmetik asfollows:
Inmy GrundlagenderArithmetik,Iaimedtomakeitplausiblethatarithmeticisa branchoflogicandneedstorelyneitheronexperiencenorintuitionasabasisforits proofs.Inthepresentbookthisisnowtobeestablishedbydeductionofthesimplest lawsofcardinalnumberbylogicalmeansalone. (Grundgesetze I,1)
Theplausibilityofwhatisnowcalled‘logicism’wassupposedtohavebeen establishednotonlybythephilosophicalargumentsin DieGrundlagen but, moreimportantly,bytheproofsofbasicarithmeticalprinciplesthatFrege sketchesin§§70–83.Butthecharacterofthoseargumentsleftalargelacuna: Idonotclaimtohavemadetheanalyticcharacterofarithmeticalpropositionsmore thanplausible,1 becauseitcanalwaysstillbedoubtedwhethertheyarededuciblesolely frompurelylogicallaws,orwhethersomeothertypeofpremissisnotinvolvedatsome pointintheirproofwithoutournoticingit.Thismisgivingwillnotbecompletely allayedevenbytheindicationsIhavegivenoftheproofofsomeofthepropositions; itcanonlyberemovedbyproducingachainofdeductionswithnolinkmissing,such thatnostepinitistakenwhichdoesnotconformtosomeoneofasmallnumberof principlesofinferencerecognizedaspurelylogical. (Frege,1884,§90)
Buthowcanwebesurethatnolinkismissing?Thatproblemwastheone thathadledtoFrege’sinterestinlogic,ashemakesexplicitin Begriffsschrift: [W]edividealltruthsthatrequirejustificationintotwokinds,thoseforwhichthe proofcanbecarriedoutpurelybymeansoflogicandthoseforwhichitmustbesupportedbyfactsofexperience.…[W]henIcametoconsiderthequestiontowhichof thesetwokindsthejudgmentsofarithmeticbelong,Ifirsthadtoascertainhowfar onecouldproceedinarithmeticbymeansofinferencesalone,withthesolesupportof thoselawsofthoughtthattranscendallparticulars.…Topreventanythingintuitive frompenetratinghereunnoticed,Ihadtobendeveryefforttokeepthechainofinferencesfreeofgaps.Inattemptingtocomplywiththisrequirementinthestrictestway possibleIfoundtheinadequacyoflanguagetobeanobstacle;nomatterhowunwieldy
1Austintranslatesthisas‘probable’,butIhavealteredthetranslation,sinceFregeusesthe samewordhereasin Grundgesetze
RichardKimberlyHeck
theexpressionsIwasreadytoaccept,Iwaslessandlessable,astherelationsbecame moreandmorecomplex,toattaintheprecisionthatmypurposerequired.
(Frege,1879,5–6)
WhatFregeneededtodo,then,tofillthelacuna,wastoprovideformalproofs ofthevariouspropositionshehadonlyproveninformallyin DieGrundlagen. Fregeseemsalreadytohaveachievedsomethingalongtheselineseven beforehewrote DieGrundlagen.InaletterwritteninAugust1882,hesays: IhavenownearlycompletedabookinwhichItreattheconceptofnumberand demonstratethatthefirstprinciplesofcomputation,whichuptonowhavegenerally beenregardedasunprovableaxioms,canbeprovedfromdefinitionsbymeansoflogical lawsalone,sothattheymayhavetoberegardedasanalyticjudgementsinKant’ssense. ItwillnotsurprisemeandIevenexpectthatyouwillraisesomedoubtsaboutthis andimaginethatthereisamistakeinthedefinitions,inthat,tobepossible,they presupposejudgementswhichIhavefailedtonotice,orinthatsomeotheressential contentfromanothersourceofknowledgehascreptinunawares.Myconfidencethat thishasnothappenedisbasedontheapplicationofmyconcept-script,whichwillnot letthroughanythingthatwasnotexpresslypresupposed…(Frege,1980,99–100) Nonetheless,itwouldbemorethanadecadeafterFregewrotethosewords, andnineyearsafterthepublicationof DieGrundlagen,beforeheactually wouldprovidethegap-freeproofshehadpromised.2
ThoseproofsarecontainedinPartIIof Grundgesetze.PartIofthebookis devotedtothe‘ExpositionoftheConcept-Script’,thatis,totheexplanation oftheformallanguageinwhichFrege’sproofswillbestatedandoftheformal theoryinwhichtheywillbedeveloped,thatis,ofthebasiclawsandrulesof inferenceofhissystem(seeHeck,2012,PartI).PartIIcontainsthe‘Proofs oftheBasicLawsofCardinalNumber’.AlloftheDedekind–Peanoaxioms forarithmeticareproventhere,includingthestatementthateverynatural numberhasasuccessor,whoseproofFregehadsketchedin§§82–3of Die Grundlagen.
ItisthereforeclearthatPartIIof Grundgesetze playsanimportantrole inFrege’sphilosophyofmathematics.Thatmakesitreallyquiteastonishing thatithasonlyrecentlybeenpublishedinEnglishtranslation(Frege,2013).3 Thereare,ofcourse,severalreasonswhythatis.Frege’sformalsystemis,asis well-known,inconsistent,sinceRussell’sParadoxisderivableintheconceptscriptfromFrege’sBasicLawV.OnemightthereforesupposethatFrege’s proofscanbeoflittleinterest,sinceanythingcanbeproveninaninconsistent system.ThereisalsotheproblemofFrege’snotation,whichisutterlyunlike thatusedbyanyotherauthorandwhichhasareputationforbeingdifficultto
2Idiscusssomeofthereasonsforthedelayinmyothercontributiontothisvolume(Heck, 2019).
3TherewerenotranslationsatallavailablewhenIstartedworkingon Grundgesetze inthe early1990s,untilJasonStanleyandIdidavery(very)roughoneinthesummerof1992.That wasusedinaseminarGeorgeBoolosandItaughttogetherin1993.Itwasoneofthefirstthings IputonmywebsitewhenIgotone,around1996,andatleastafewotherpeopleuseditin seminarsoftheirown.
read.4 Infact,however,thoseofuswhohavelearnedtoreaditknowthatitis notdifficulttoread.Rather,itsunfamiliaritymakesitsomethingofachallenge to learn toread.Andwehaveknownsincethemid-1980sthatFrege’ssystem, thoughinconsistent,isnotirremediablyinconsistent.Aswasfirstobserved byPeterGeach(1955,570),andemphasizedshortlythereafterbyCharles Parsons(1995,198),Frege’sownargumentsin DieGrundlagen makevery limitedappealtoBasicLawV,whichisthesourceoftheinconsistency.Law Visusedonlyintheproofofwhatisnowknownas“Hume’sPrinciple”,or HP:Thenumberof F sisthesameasthenumberof Gsif,andonlyif,the F s areinone–onecorrespondencewiththe Gs.Theremainderoftheargument appealsonlytoHP.And,asCrispinWright(1983,154–8)conjecturedand severalpeoplethenproved(Burgess,1984;Hazen,1985;Boolos,1998a),HP isconsistent.So,asWright(1983,§xix)showedindetail,Frege’sproofsin Die Grundlagen canbereconstructedinaconsistentsub-theoryoftheinconsistent theoryheimplicitlyassumes.
Theobviousquestion,whichGeorgeBoolosdirectedtomeinthesummer of1991,iswhethersomethingsimilarbutstrongeristrueof Grundgesetze.In Frege:PhilosophyofMathematics,whichwaspublishedthatyear,SirMichael Dummettseemstoassertthatthereis:
CrispinWrightdevotesawholesectionofhisbook…todemonstratingthat,ifwe weretotake[HP]asanimplicitorcontextualdefinitionofthecardinalityoperator, wecouldstillderiveallthesametheoremsasFregedoes.Hecouldhaveachievedthe sameresultwithlesstroublebyobservingthatFregehimselfgivesjustsuchaderivation ofthosetheorems.Hederivesthemfrom[HP],withnofurtherappealtohisexplicit definition. (Dummett,1991,123)
WhatBoolosaskedmewassimplywhetherthisistrue.Isettoreading Grundgesetze andsoondiscoveredthat,ifitwas,itwasgoingtotakeworktoshow it.Itiseasyenoughtoverifythat,afterprovingHP,Fregemakes“nofurther appealtohisexplicitdefinition”.Butthatisnotenough.ThecrucialquestioniswhetherFregemakesnofurtherappeal toBasicLawV,andhemost certainlydoes.HardlyapageofPartIIlackstermsforvalue-ranges,ofwhich extensionsofconceptsareaspecialcase,andthelogicallawgoverningvaluerangenamesis,ofcourse,BasicLawV.Moreprecisely,duetothedetailsof howFregeformalizesvariousnotions,almosteveryresultheprovesdepends uponhisTheorem1,whichisageneralizationoftheprincipleknownasnaïve comprehension: a ∈{x : Fx}≡ Fa
AndthatprincipleleadsdirectlytoRussell’sParadox,oncewetake Fξ tobe: ξ/ ∈ ξ and a tobe: {x : x/ ∈ x}. 5
4Nottomentiontypeset.IscannedtheformulasforthetranslationJasonandIdid.How theywerehandledinthenewtranslationiswell-relatedbyEbertandRossberg(Frege,2013, xxx–xxxii).
5Theorem1itselfisprovenfromBasicLawVandFrege’sdefinitionoftheanalogue,for value-ranges,ofmembership(Heck,2012,§1.2).
RichardKimberlyHeck
ButitalsoquicklybecamecleartomethatmanyoftheusesFregemakes ofvalue-rangescaneasilybeeliminated.Forexample,Fregealmostalways quantifiesovertheextensionsofconceptsinsteadofoverconcepts,sothatwe findthingslike:
But,asjustillustrated,thisiseasilyremedied.AndFrege’sotherusesofvaluerangesprovedtobeeliminableaswell.SoDummettturnedouttoberight,in spiritifnotindetail:Modulousesofvalue-rangesthatareessentiallyjustfor convenience,PartIIof Grundgesetze reallydoescontainaformalderivation ofaxiomsforarithmeticfromHP.
AndthereismuchmoreinPartII.Frege’sproofofaxiomsforarithmetic comprisesonlyaboutathirdofit.Intheremainder,Fregeprovesanumberof resultsconcerningfinitude,infinity,andtherelationshipbetweenthesetwo notions.WhenIexaminedthoseproofsclosely,itturnedoutthatFregeused LawVinthem,too,onlyforconvenience.AndtherewasmuchofphilosophicalinterestbothinFrege’sformalargumentsandintheinformaldiscussion ofthemcontainedinthe“Analysis”sections.
Mygoalinthischapter,then,istoprovideabriefoverviewofwhatFrege accomplishesinPartIIandtogivesomeindicationofthephilosophicaland historicalinterestthismaterialhas.Furtherdetails,andactualargumentsfor theinterpretiveclaimstobemadebelow,canbefoundinPartIIofmybook ReadingFrege’s Grundgesetze,ofwhichthischapterisakindofprécis.
AndsincethischapterismeanttoprovideanintroductiontoFrege’s formalworkonarithmetic,Iwillpresenthisresultsusingmodernnotation, soastomakethediscussionmoreaccessible.Iwillalsosilentlytranslateaway Frege’srelianceuponvalue-ranges,sincethatservesonlytoobscurehisaccomplishments.6
1.1THEPROOFOFHP
Frege’sfirsttaskin Grundgesetze istoproveHP,whichmaybestated,inmodernnotation,as:
Here,‘Nx : Fx’istoberead:thenumberof F s.
6IshallalsosilentlyaltersomeofthetranslationsfromwhichIquote,tomakethemuniform intheirterminology.
Frege’sformulationofHPmightinitiallyseemverydifferent,andnotjust becausehisnotationissodifferent.7 TranslatingFrege’snotationintoours,of course,hewouldwriteHPas:
Nx : Fx = Nx : Gx ≡∃R[Map(R)(F,G) ∧ Map(Conv(R))(G,F )]
Here,‘Map(R)(F,G)’,whichFregewouldwriteas‘f SgS⟩r’,8 mayberead: R mapsthe F sintothe Gs.Conv(R),whichFregewouldwriteas‘Ur’,isthe converseof R,definedtheobviousway:
Conv(R)(a,b) df ≡ Rba
SoHPitself,asFregewouldformulateit,saysthatthenumberof F sisthe sameasthenumberof Gsif,andonlyif,thereisarelationthatmapsthe F s intothe Gsandwhoseconversemapsthe Gsintothe F s. Themappingrelationitselfisdefinedasfollows(Grundgesetze I,§38):
Map(R)(F,G) df ≡ Func(R) ∧∀x(Fx →∃y(Rxy ∧ Gy))
Here,‘Func(R)’,whichFregewouldwriteas‘Ir’,meansthat R is“singlevalued”or“functional”.Ittooisdefinedtheobviousway(Grundgesetze I, §37):
Func(R) df ≡∀x∀y(Rxy →∀z(Rxz → y = z))
So R mapsthe F sintothe Gsjustincase R issingle-valuedandeach F is relatedby Rξη tosome G.Notecarefully: into,not onto.That R mapsthe F s intothe Gssays,ofitself,nothingwhatsoeverabouttherelativecardinalities ofthe F sandthe Gs:Aslongasthereisatleastone G,therewillalwaysbe arelationwhichmapsthe F sintothe Gs,inFrege’ssense,whateverconcept Fξ maybe.
ToseetherelationofFrege’sformulationofHPtotheusualone,unpack theright-handsideofhisversionHPusingthedefinitions:
∃R[Map(R)(F,G) ∧
Map(Conv(R))(G,F )]
7Theorem32,whichistheright-to-leftdirection,reads:
”u = ”v uS(vS⟩q) vS(uS⟩Uq)
Talkaboutdifferent!Green,Rossberg,andEbert(2015)discussFrege’snotationindetail.
8Hereandbelow,Ishalluseuppercaselettersforconceptsandrelationsandthecorresponding lowercaselettersfortheextensionsofthoseconceptsandrelations.
∃R[Func(R) ∧
∀x(Fx →∃y(Rxy ∧ Gy)) ∧ Func(Conv(R)) ∧
∀x(Gx →∃y(Conv(R)(x,y) ∧ Fy))]
∃R[∀x∀y(Rxy →∀z(Rxz → y = z)) ∧
∀x(Fx →∃y(Rxy ∧ Gy)) ∧ ∀x∀y(Conv(R)(x,y) →∀z(Conv(R)(x,z) → y = z)) ∧
∀x(Gx →∃y(Conv(R)(x,y) ∧ Fy))]
∃R[∀x∀y(Rxy →∀z(Rxz → y = z)) ∧
∀x(Fx →∃y(Rxy ∧ Gy)) ∧ ∀x∀y(Ryx →∀z(Rzx → y = z)) ∧ ∀x(Gx →∃y(Ryx ∧ Fy))]
WhatFregehasdoneisgroup‘∀x∀y(Rxy →∀z(Rxz → y = z))’and ‘∀x(Fx →∃y(Gy∧Rxy)’inthefirstconjunct,‘Map(R)(F,G)’,andtogroup ‘∀x∀y(Rxz →∀z(Ryz → x = y))’and‘∀y(Gy →∃x(Fx ∧ Rxy)’inthe secondconjunct,‘Map(Conv(R))(G,F )’.Wearemoreinclinednowadaysto grouptheconjuncts‘∀x∀y(Rxy →∀z(Rxz → y = z))’and‘∀x∀y(Rxz → ∀z(Ryz → x = y))’(R isaone–onefunction…)and‘∀x(Fx →∃y(Gy ∧ Rxy)’and‘∀y(Gy →∃x(Fx ∧ Rxy)’(…fromthe F sontothe Gs).So,in theend,thedifferencebetweenFrege’sformulationandoursismostlyoneof emphasis,thoughFrege’sformulationhassometechnicaladvantagesoverthe usualone(Heck,2012,§6.3).
Inmodernpresentations,‘Nx : Fx’istreatedasaprimitivenotiongovernedbyHP,whichisitselftreatedasanaxiom.Frege,byconstrast,means toproveHPandsodefines‘Nx : Fx’intermsofextensions.9 Now,in Grundgesetze,Fregetreatsextensionsasakindofvalue-range,buthisdefinitionof numberin Grundgesetze isotherwisethesameastheonegivenin§68of Die Grundlagen:10
Thenumberof F sistheextensionoftheconcept:is[theextensionof] aconceptthatisequinumerouswith F . Wecanformalizethisas:
Nx : Fx df ≡ ˆ x{∃G[(x =ˆ y(Gy) ∧ Eq(F,G)]}
9Whatmostobviouslycorrespondstoour‘Nx : Fx’isFrege’s‘”f’.Infact,however,as GregoryLandinipointedouttome,‘Nx : Fx’isdefinableinFrege’ssystemas‘” –εFε’.
10Frege’sdefinitiondoesnotcontainthebracketedoccurrenceofthephrase‘theextensionof’. Iargueelsewhere(Heck,2019,§18.1)thatitisnonethelesswhathemeans.
where‘Eq(F,G)’abbreviates: ∃R[Map(R)(F,G) ∧ Map(Conv(R))(G,F )]. Here‘ˆ x(Fx)’means:theextensionoftheconcept F ,andthenotionofextensionistobegovernedbyaversionofBasicLawV:11
TheproofofHPthenneedslittlemorethantheobservationthatEqisan equivalencerelation.
Infact,however,asMayandWehmeier(2019)pointoutintheircontributiontothisvolume,FregeneveractuallyprovesHPasabiconditional:He provesitstwodirections,butneverbotherstoputthemtogether.Therightto-leftdirectionisTheorem32,whichisthegoalofthefirstchapterofPartII, ChapterAlpha.12 Theleft-to-rightdirectionisTheorem49,whichisprovenin ChapterBeta.Theproofoftheformerisquitestraightforward,anditfollows theoutlinein§73of DieGrundlagen closely.Theproofneedsonlythetransitivityandsymmetryofequinumerosity.Theproofof(49)thatFregegivesis somewhatpeculiar,becauseitusesthedefinitionofnumberinamoreessentialwaythanitreallyshould.Thereis,however,asimplerproof,whichFrege musthaveknown,thatneedsonlythereflexivityofequinumerosity(Heck, 2012,§6.8).
Anotherpointworthnotingabouttheseproofsisthattheproofof(32) needsonlytheright-to-leftdirectionofLawV,whichFregecallsLawVaand whichisthe“safe”direction,whereas(49)needstheleft-to-rightdirection, LawVb,whichisthe“unsafe”direction,theonethatgivesrisetoRussell’s Paradox.Onreflection,thisshouldnotbesurprising,since(32)isthe“safe” directionofHP,whichbyitselfhasnosignificantontologicalconsequences, sinceitiscompatiblewiththerebeingonlyonenumber,sharedbyalltheconceptsthereare.Theorem49,ontheotherhand,istheontologicallyprofligate directionofHP,whichentailstheexistenceofinfinitelymanynumbers.
1.2THEAXIOMSOFARITHMETIC
AfterhavingprovenHP,Fregeturnshisattentiontotheproofsofvariousfundamentalprinciplesconcerningcardinalnumbers,includingwhatwenowcall theDedekind–Peanoaxioms,forwhichseeTable1.1.Here‘Nξ’isapredicate toberead‘ξ isanaturalnumber’,and‘Pξη’isapredicatetobereadas‘ξ immediatelyprecedes η inthenumber-series’.Toprovetheseaxioms,Frege mustofcoursedefinethearithmeticalnotionsthatoccurinthem.
11Assaid,Fregeactuallyworkswiththemoregeneralnotionofavalue-range,but,surprisingly,henevermakesuseofthemoregeneralnotion.Allthevalue-rangesinwhichheisactually interestedin Grundgesetze areextensionsofconcepts.
12Fregedoesnotcallthesedivisionschapters,butitseemstheobviousnameforthem.
1. N0
Table1.1.OneVersionoftheDedekind–PeanoAxioms
2. ∀x∀y(Nx ∧ Pxy → Ny)
3. ∀x(Nx →∃y(Pxy))
4. ¬∃x(Nx ∧ Px0)
5. ∀x(Nx →∀y∀z(Pxy ∧ Pxz → y = z))
6. ∀x∀y∀z(Nx ∧ Ny ∧ Pxz ∧ Pyz → x = y)
7. ∀F [F 0 ∧∀x(Nx ∧ Fx →∀y(Pxy → Fy)) →∀x(Nx → Fx)]
ThedefinitionsFregegivesin Grundgesetze arethesameastheonesgiven in DieGrundlagen.Fregedefineszero,whichhewritesas‘0’,asthenumberof objectsthatarenotself-identical(Grundgesetze I,§41;seeFrege,1884,§74): 0 df ≡ Nx : x = x
Hisdefinitionofpredecession,whichhewritesas‘mS(nSs)’,isasfollows (Grundgesetze I,§43):
mn df ≡∃F ∃x[Fx ∧ n = Nz : Fz ∧ m = Nz : (Fz ∧
Thatis, m precedes n if,asFregeputsitin DieGrundlagen,‘thereexistsa concept F ,andanobjectfallingunderit x,suchthattheNumberwhich belongstotheconcept F is n andtheNumberwhichbelongstotheconcept “fallingunder F butnotidenticalwith x”is m’(Frege,1884,§76).Weshall returntothedefinitionof‘Nξ’.
FregeprovesAxiom5inChapterBetaasTheorem71;Axiom6inChapter Γ asTheorem89;andAxiom4inChapterEpsilonasTheorem108.The proofsarestraightforward,butthereisaphilosophicaldiscussionthatoccurs duringFrege’sinformalexpositionoftheproofofTheorem71thatisofsubstantialinterest.ItconcernstheproofofTheorem66:
Fc ∧ Gb ∧ Nz : (Gz ∧ z = b)= Nz : (Fz ∧ z = c) → Nz : Fz = Nz : Gz (66) whichisthekeylemmaintheproofof(71).13 Toprove(66),whatwewant toshowisthat,ifthereisaone–onecorrelationbetweenthe Gsotherthan
13SupposethatPxy andPxw.Then,bythedefinitionof‘P’,thereare F and c suchthat:
Fc ∧ Nz : Fz = y ∧ Nz : (Fz ∧ z = c)= x andthereare G and b suchthat:
Gb ∧ Nz : Gz = w ∧ Nz : (Gz ∧ z = b)= x
SoNz : (Gz ∧ z = b)= x = Nz : (Fz ∧ z = c),and(66)nowimpliesthatNz : Gz = Nz : Fz, so w = y,andwearedone.
b andthe F sotherthan c,andif b isa G and c isan F ,thenthereisalsoa one–onecorrelationbetweenthe F sandthe Gs.Fregewrites:14
Ifoneweretofollowtheusualpracticeofmathematicians,onemightsaysomething likethis:wecorrelatetheobjects,otherthan b,fallingundertheconcept G,withthe objects,otherthan c,fallingundertheconcept F bymeansoftheknownrelation, andwecorrelate b with c.Inthisway,wehavemappedtheconcept G intotheconcept F and,conversely,thelatterintotheformer.So…thecardinalnumbersthatbelong tothemareequal.Thisisindeedmuchbrieferthantheprooftofollowwhichsome, misunderstandingmyproject,willdeploreonaccountofitslength.Whatisitthatwe aredoingwhenwecorrelateobjectsforthepurposeofaproof?(Grundgesetze I,§66) Thereis,ofcourse,nothingunusualaboutthesortofreasoningFregerehearses,buthehasaquestiontoraiseaboutit.Itisnot,ofcourse,thathe thinkssuchreasoningmightbeinvalid.Buthewantstoknowwhatjustifies it.
Fregefirstemphasizesthat,whenweestablishacorrelationinthissense, wedonotcreateanythingbut“merelybringtoattention,apprehend,whatis alreadythere”(compareFrege,1884,§26).Hethenseizestheopportunityto takeaswipeatpsychologism.Havingslainthatfamiliarfoe,Fregeconsiders thequestionhowwemightformulate“apostulate,inthestyleofEuclid”,that permitssuchcorrelations,answering:
[It]wouldhavetobeunderstoodthisway:‘Anyobjectiscorrelatedwithanyobject’ or‘Thereisacorrelationbetweenanyobjectandanyobject’.Whatthenissucha correlationifitisnothingsubjective,createdonlybyourmaking?However,aparticular correlationofanobjecttoanobjectisnotwhatcanbeatissuehere…;ratherwerequire agenusofcorrelations,sotospeak,whatwehavesofarcalled,andwillcontinueto call,arelation. (Grundgesetze I,§66) Sowhattheactofcorrelatingbringstoattentionisarelation,andFregegoes ontodiscusshowtheconcept-scriptallowsustospecifysuchrelationspreciselyandtoshowthattheyhavethevariouspropertiesweneedthemtohave. Infact,inthiscase,Fregegoesontosuggest,theusualreasoningencourages ustooverlookcertainsubtleties,whichemergeinhiscareful,rigorouspresentationoftheproof.
WhatisatissuehereisaquestionfundamentaltoFrege’slogicism.The informalargumenthementionsinvolvesjustthesortoftoxicmixofreason andintuitionthattendstoobscuretheepistemologicalstatusoftheresult proved.Onecrucialquestion,inparticular,ishowweknowthatthe“correlations”weneedactuallyexist.And,solongasonethinksofcorrelatingas somethingwedo,ratherthanofcorrelationsassomethingwediscover,this willtendtomakeonesupposethattheexistenceofcorrelationsdependssomehowuponmentalactivity.Frege’sviewis,ofcourse,different.Inhissystem, theexistenceofconcepts,relations,andthelikeisguaranteedbyRule9on
14Inhisexposition,Fregespeaksofsuchthingsas“the u-concept”,bywhichhemeansthe conceptwhosevalue-range u is.Ihavesilentlyreplacedsuchtalkwithdirectreferencestoconcepts.
RichardKimberlyHeck
thelistgivenin§48.Thisruleallowsfortheuniformreplacementofafree variable,ofarbitrarytype,byanywell-formedexpressionoftheappropriate type.Suchasubstitutionprincipleisequivalenttothecomprehensionscheme ofsecond-orderlogic
R∀x∀y[Rxy ≡ A(x,y)] whichassertstheexistenceofarelationco-extensivewithanygivenformulaof thelanguage(solongasitdoesnotcontainthevariable R free).SoFrege’sview isthat logicitself commitsustotheexistenceofanyrelationwecandescribe. Since“correlations”arejustrelations,itisthuslogic,andnotpsychology,that affirmstheirexistence.
Toreturntotheproofsoftheaxioms,then,whatremaintobeprovedatthe endofChapterEpsilonaretheaxiomsconcerningthenotionofanatural(or finite)number.SowenowneedtoconsiderFrege’sdefinitionofthatnotion. Itis,again,essentiallythesameasthatgivenin DieGrundlagen,andituses Frege’sdefinitionoftheso-called ancestral,whichheintroducesin§23of Begriffsschrift.Givenarelation Q,wesaythataconcept F is hereditaryinthe Q-series justincase,whenever x is F ,eachobjecttowhich Q relatesitis F :
x∀y(Fx ∧ Qxy
Wenowsaythatanobject b follows anobject a inthe Q-seriesjustincase b fallsundereveryconceptthatishereditaryinthe Q-seriesandunderwhich eachobjecttowhich Q relates a falls.Formally,writing‘Q∗ab’for‘b follows a inthe Q-series’,Frege’sdefinitionofthe strong ancestral,15 whichhewrites as‘aS(bSMq)’,is(Grundgesetze I,§45):
∗ab df ≡∀F [∀x(Qax → Fx) ∧∀x∀y(Fx ∧ Qxy → Fy) → Fb]
Fregethendefinesthe weak ancestral,whichhewritesas‘aS(bSRq)’,thus (Grundgesetze I,§46):
Theconcept Nξ isthendefinableasP∗=0ξ.Soanobjectisanaturalnumber justincaseitbelongstotheP-seriesbeginningwith 0.Fregehasnospecial symbolforthis.Hedoes,however,regularlyread‘P∗=0ξ’as‘ξ isafinitenumber’(e.g.,in Grundgesetze I,§108).
Axioms1and2thenfollowfromgeneralfactsabouttheancestral,Theorems140and133,respectively.Famously,Axiom7,theinductionaxiom, alsofollowsfromFrege’sdefinition.Mattersaremorecomplicatedthanoften seemstobesupposed,however.Wecanquiteeasilyprove:
15Thestrongancestralisso-calledbecauseweneednothave Q∗aa,whereaswealwayshave Q∗= aa
fromwhich
followsbysubstitution.Sowehave
bysimplelogicalmanipulationsandthedefinitionof‘N’.Butthisisweaker thanAxiom7.Thehypothesisofinductionisnotthat, whenever x is F ,its successoris F ;itisonlythat,whenever x isanaturalnumber thatis F ,its successoris F .Thatis,inductionis:
Thisiseasyenoughtoprove—itfollowsbysubstitutionfromFrege’sTheorem 152—butthedifferencebetweenitandwhatthedefinitionoftheancestral deliversimmediatelyturnsouttobehistoricallysignificant,asweshallsee shortly.16
Theonlyremainingaxiom,then,isAxiom3,whichassertsthateverynumberhasasuccessor.LyingbehindFrege’sproofofAxiom3isapictureofhow thenaturalnumbersaregenerated.Thegenerativeprocessbegins,ofcourse, withzero.Fregeinsiststhatzeroexists,evenifnothingelsedoes,becausezero isthenumberofthingsthatarenon-self-identical,andthenon-self-identical thingsexistevenifnothingatalldoes.Butifzeroexists,thenthereisanumberthatisthenumberofthingsthatarelessthanorequaltozero,andthat numberwemaycall‘one’.ByHP,oneisnotzero:Therecanbenoone–one mapbetweenthethingsthatarenon-self-identicalandthethingsthatareless thanorequaltozero,sincethereisatleastonethinglessthanorequaltozero, namely,zero.Butnowbothzeroandoneexist,andsothereisanumberthat isthenumberofthingsthatarelessthanorequaltoone.Callthatnumber ‘two’.ByHP,twocanbeneitherzeronorone.Sozero,one,andtwoexist, andthereisanumberthatisthenumberofthingslessthanorequaltotwo… Formally,whatwewanttoproveis:
whichisthecentralresultofChapterH(Eta)andwhichsays,roughly,that everynaturalnumberissucceededbythenumberofnumbersintheP-series endingwithit(roughly,thenumberofnumberslessthanorequaltoit).The proofproceedsbyinduction,wherewesubstitute:
(ξ, Nx :P∗=xξ) for‘Fξ’.Soweneedtoprovethatzerofallsunderthisconcept:
16Thedifferenceisalsoofsometechnicalimport,sincetheweakerprincipleiseasilyprovable eveninpredicativesystemsthatdonotallowustoproveinduction(Heck,2011).
(154) andthatitishereditaryintheP-seriesbeginningwith0:
(150)
Theproofof(154)iseasy.
Theinterestliesintheproofof(150).Itfollowsbygeneralizationfrom:
Toprovethis,supposethat d isanaturalnumber,that d precedesthenumber ofmembersoftheP-seriesendingwith d,andthat d precedes a.Wemust showthat a precedesthenumberofmembersoftheP-seriesendingwith a. Todoso,wemust,bythedefinitionofP,findsomeconcept F andsome object x fallingunder F suchthat a isthenumberof F sotherthan x andthe numberof F sisthesameasthenumberofmembersoftheP-seriesending in a.Thatis,wemustshowthat:
TheconceptinquestionistobeP∗=ξa;theobjectinquestionistobe a itself. Hence,wemustshowthat:
Thelasttwoconjunctsaretrivial.Thefirstwemayderivefrom:
bythetransitivityofidentity.TheformerfollowsfromthefactthatPissinglevalued,since,byhypothesis,wehavebothPda andP(d, Nx :P∗=xd).The latter,inturn,istheconsequentof:
whichFregederivesfrom:
andtheextensionalityofthecardinalityoperator:
HederivesTheorem149α fromthefollowingtworesults:
whichareTheorems148α and148ζ.Forthelatter,weneedthecentralresult ofChapterZeta:
Thissaysthatthereareno“loops”inthenaturalseriesofnumbers,andittoo isprovedbyinduction.
TheargumentherewilllookfamiliartoanyonewhohasstudiedFrege’s informalproof,in§§82–3of DieGrundlagen,thateverynaturalnumberhas asuccessor.Theproofin Grundgesetze hasmuchthesamestructure,andmany ofthestepsintheproofarealsomentionedin DieGrundlagen.Forexample, (145)isthelastpropositionmentionedbyFregein§83.Butthereareimportantdifferencesbetweenthetwoproofs,aswell.AclosereadingofFrege’s discussionoftheproofin§114of Grundgesetze shows,infact,thatFregewas awarethattheproofsketchedin DieGrundlagen isactuallyincorrect(Boolos andHeck,2011;Heck,2012,§6.7).Theearlierproofpurportstorelyonly uponwhat,asmentionedbefore,triviallyfollowsfromthedefinitionofnaturalnumber:
ratherthanuponmathematicalinductionproper:
Inparticular,theproofin DieGrundlagen wassupposedtogovia:
whichisadirectformalizationofthepropositionmarked‘1.’in§82of Die Grundlagen andwhichisalso(150ε)minusitsfirstconjunct,P
=0d.Frege mentionsthisformulaexplicitlyin§114—itistheformulalabeled(α)—only thentosay,inafootnote,thatit“is,itseems,unprovable…”.Itishardto seewhyFregewouldsomuchashavediscussedthispropositionifitdidnot figurecruciallyinhisearlierargument.Andthat,tome,isthemostimpressive evidencethatFregeknewhisearlierproofwasflawed.
Itfollows,presumably,thatFregecannotactuallyhavehadafullyworked out,formalproofoftheexistenceofsuccessorswhenhewrote DieGrundlagen. Hesimplycouldnothavemadesuchamistakeotherwise.17 Moreinterestingly,itmakesitplainthatFregewasawarethattheremightbepropositions ofhisformallanguagethathewouldregardastruebutthatwerenonetheless unprovablefromtheBasicLawshewasthenpreparedtoaccept.Forthere isgoodreasontothinkFregeregardedthe“unprovable”formula(α)asbeingtrue.ThereisaverysimpleargumentforitthatusesDedekind’sresult thateveryinfinitesetisDedekindinfinite,aclaimthatFregeacceptedastrue, though,likemanymathematiciansofhistime,heregardedDedekind’sproof asinsufficientlyrigorous(Frege,1892,271).I’llreturntothispointbelow.
Finally,carefulanalysisofFrege’sproofin Grundgesetze revealsthatthe onlyfactsaboutnumberstowhichitessentiallyappealsaretheextensionality
17Andthat,inturn,suggeststhatthemanusciptmentionedintheletterfrom1882cannot havecontainedsuchaproof,whichmakesitaninterestingquestionwhatitdidcontain.Frege saysintheletterthathehassetouttoprove“thefirstprinciplesofcomputation”,andtheexistence ofsuccessorsisnotnaturallysodescribed.
