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ThinObjects ThinObjects AnAbstractionistAccount ØysteinLinnebo
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FormydaughtersAlmaandFrida 2.ThinObjectsviaCriteriaofIdentity21
PartII.Comparisons 4.AbstractionandtheQuestionofSymmetry77
7.6.1Anultra-thinconceptionofreference
7.6.2Semanticallyconstrainedcontentrecarving
8.ReferencebyAbstraction135
8.4WhytheNon-reductionistInterpretationisPreferable
8.4.1Theprincipleofcharity
8.4.2Theprincipleofcompositionality
8.4.3Cognitiveconstraintsonaninterpretation
8.5WhytheNon-reductionistInterpretationisAvailable
Preface Thisbookisaboutapromisingbutelusiveidea.Arethereobjectsthatare“thin”inthe sensethattheirexistencedoesnotmakeasubstantialdemandontheworld?Frege famouslythoughtso.Heclaimedthattheequinumerosityoftheknivesandtheforks onaproperlysettablesufficesfortheretobeobjectssuchasthenumberofknives andthenumberofforks,andfortheseobjectstobeidentical.Versionsoftheideaof thinobjectshavebeendefendedbycontemporaryphilosophersaswell.Forexample, BobHaleandCrispinWrightassertthat whatittakesfor“thenumberof F s=thenumberof Gs”tobetrueisexactlywhatittakesfor the F stobeequinumerouswiththe Gs,nomore,noless.[…]Thereisnogapformetaphysics toplug.1
Thetruthoftheequinumerosityclaimissaidtobe“conceptuallysufficient”forthe truthofthenumberidentity(ibid.).Or,asAgustínRayocolorfullyputsit,onceGod hadseentoitthatthe F sareequinumerouswiththe Gs,“therewasnothing extra she hadtodo”toensuretheexistenceofthenumberof F andthenumberof G,andtheir identity(Rayo,2013,p.4;emphasisinoriginal).
Theideaofthinobjectsholdsgreatphilosophicalpromise.Iftheexistenceofcertain objectsdoesnotmakeasubstantialdemandontheworld,thenknowledgeofsuch objectswillbecomparativelyeasytoattain.OntheFregeanview,forexample,it sufficesforknowledgeoftheexistenceandidentityoftwonumbersthatanunproblematicfactaboutknivesandforksbeknown.Indeed,theideaofthinobjectsmay wellbetheonlywaytoreconciletheneedforanontologyofmathematicalobjects withtheneedforaplausibleepistemology.Anotherattractionoftheideaofthin objectsconcernsontology.Iflittleornothingisrequiredfortheexistenceofobjects ofsomesort,thennowonderthereisanabundanceofsuchobjects.Thelessthat isrequiredfortheexistenceofcertainobjects,themoresuchobjectstherewillbe. Thus,ifmathematicalobjectsarethin,thiswillexplainthestrikingfactthatmathematicsoperateswithanontologythatisfarmoreabundantthanthatofanyother science.
Theideaofthinobjectsiselusive,however.Thecharacterizationjustofferedis impreciseandpartlymetaphorical.Whatdoesitreallymeantosaythattheexistence ofcertainobjects“makesnosubstantialdemandontheworld”?Indeed,ifthetruth of“thenumberof F s=thenumberof Gs”requiresnomorethanthatof“the F sare
1 (HaleandWright,2009b,pp.187and193).Bothofthepassagesquotedinthisparagraphhavebeen adaptedslightlytofitourpresentexample.
equinumerouswiththe Gs”,perhapstheformersentenceisjusta façondeparler for thelatter.Tobeconvincing,theideaofthinobjectshastobeproperlyexplained.
Thisbookattemptstodeveloptheneededexplanationsbydrawingonsome Fregeanideas.Ishouldsaystraightaway,though,thatmyambitionsarenotprimarily exegetical.IusesomeFregeanideasthatIfindinterestinginanattempttoanswer someimportantphilosophicalquestions.Byandlarge,Idonotclaimthatthe argumentsandviewsdevelopedinthisbookcoincidewithFrege’s.Someoftheviews Idefendarepatentlyun-Fregean.
Mystrategyformakingsenseofthinobjectshasasimplestructure.Ibeginwith theFregeanideathatanobject,inthemostgeneralsenseoftheword,isapossible referentofasingularterm.Thequestionofwhatobjectsthereareisthustransformed intothequestionofwhatformsofsingularreferencearepossible.Thismeansthat anyaccountthatmakessingularreferenceeasytoachievemakesitcorrespondingly easyforobjectstoexist.AsecondFregeanideaisnowinvokedtoarguethatsingular referencecanindeedbeeasytoachieve.Accordingtothissecondidea,thereisa closelinkbetweenreferenceandcriteriaofidentity.Roughlyspeaking,itsuffices forasingulartermtoreferthatthetermhasbeenassociatedwithaspecification ofthewould-bereferent,whichfiguresinanappropriatecriterionofidentity.For instance,itsufficesforadirectiontermtoreferthatithasbeenassociatedwitha lineandissubjecttoacriterionofidentitythattakestwolinestospecifythesame directionjustincasetheyareparallel.2 Inthisway,thesecondFregeanideamakeseasy referenceavailable.AndbymeansofthefirstFregeanidea,easyreferenceensureseasy being.Mystrategyformakingsenseofthinobjectscanthusbedepictedbytheupper twoarrows(representingexplanatorymoves)inthefollowingtriangleofinterrelated concepts:
Myconcernwithcriteriaofidentityleadstoaninterestin abstractionprinciples, whichareprinciplesoftheform:
2 Admittedly,wewouldobtainabetterfitwithourordinaryconceptofdirectionbyconsideringinstead directed linesorlinesegmentsandtheequivalencerelationof“co-orientation”,definedasparallelism plus samenessoforientation.Weshallkeepthisfamousexampleunchanged,however,asthementionedwrinkle doesnotaffectanythingofphilosophicalimportance.
where α and β arevariablesofsometype,§isanoperatorthatappliestosuch variablestoformsingularterms,and ∼ standsforanequivalencerelationonthe kindsofitemsoverwhichthevariablesrange.AnexamplemadefamousbyFregeis theaforementionedprinciplethatthedirectionsoftwolinesareidenticaljustincase thelinesareparallel.Mypreferredwayofunderstandinganabstractionprincipleis simplyasaspecialtypeofcriterionofidentity.
Howdoesmyproposedroutetothinobjectscomparewithothersexploredinthe literature?MydebttoFregeisobvious.Ihavealsoprofitedenormouslyfromthe writingsofMichaelDummettandtheneo-FregeansBobHaleandCrispinWright. Assoonasonezoomsinontheconceptualterrain,however,itbecomesclearthatthe routetobetraveledinthisbookdivergesinimportantrespectsfromthepathsalready explored.Unliketheneo-Fregeans,Ihavenoneedfortheso-called“syntacticpriority thesis”,whichascribestosyntacticcategoriesacertainpriorityoverontologicalones. AndIamcriticaloftheideaof“contentrecarving”,whichiscentraltoFrege’sproject inthe Grundlagen (butnot,Iargue,inthe Grundgesetze)andtotheprojectsofthe neo-FregeansaswellasRayo.
MyviewisinsomerespectsclosertoDummett’sthantothatoftheneo-Fregeans. IshareDummett’spreferenceforaparticularlyunproblematicformofabstraction, whichIcall predicative.Onthisformofabstraction,anyquestionaboutthe“new” abstractacanbereducedtoaquestionaboutthe“old”entitiesonwhichweabstract. Aparadigmexampleisthecaseofdirections,whereweabstractonlinestoobtain theirdirections.Thisabstractionispredicativebecauseanyquestionabouttheresultingdirectionscanbeansweredonthebasissolelyofthelinesintermsofwhich thedirectionsarespecified.Iarguethatpredicativeabstractionprinciplescanbe laiddownwith nopresuppositionswhatsoever.Butmyargumentdoesnotextend toimpredicativeprinciples.Thismakespredicativeabstractionprinciplesuniquely wellsuitedtoserveinanaccountofthinobjects.Myapproachextendseventothe predicativeversionofFrege’sinfamousBasicLawV.This“law”servesasthemain engineofanabstractionistaccountofsetsthatIdevelopandshowtojustifythestrong butwidelyacceptedsettheoryZF.
Therestrictiontopredicativeabstractionresultsinanentirelynaturalclassof abstractionprinciples,whichhasnounacceptablemembers(orso-called“bad companions”).Myaccountthereforeavoidsthe“badcompanyproblem”.Instead, Ifaceacomplementarychallenge.Althoughpredicativeabstractionprinciplesare uniquelyunproblematicandfreeofpresuppositions,theyaremathematicallyweak. Myresponsetothischallengeconsistsofanovelaccountof“dynamicabstraction”, whichisoneofthedistinctivefeaturesoftheapproachdevelopedinthisbook.Since abstractionoftenresultsinalargerdomain,wecanusethisextendeddomainto providecriteriaofidentityforyetfurtherobjects,whichcanthusbeobtainedby furtherstepsofabstraction.(Thisobservationisrepresentedbythelowerarrowin theabovediagram.)Thesuccessive“formation”ofsetsdescribedbytheinfluential iterativeconceptionofsetsisjustoneinstanceofthemoregeneralphenomenonof
dynamicabstraction.Legitimateabstractionstepsareiteratedindefinitelytobuild upeverlargerdomainsofabstractobjects.Dynamicabstractioncanbeseenasa developmentandextensionofthefamousiterativeconceptionofsets.
Aseconddistinctivefeatureofmyapproachisthedevelopmentoftheideaof thinobjects.Supposewespeakabasiclanguageconcernedwithacertainrangeof entities(say,lines).Suppose ∼ isanequivalencerelationonsomeoftheseentities(say, parallelism).Thenitislegitimatetoadoptanextendedlanguageinwhichwespeak preciselyasifwehavesuccessfullyabstractedon ∼ (say,byspeakingalsoaboutthe directionsofthelineswithwhichwebegan).Iarguewehavereasontoascribetothis extendedlanguageagenuineformofreferencetoabstractobjects.Sincetheseobjects neednotbeinthedomainoftheoriginallanguage,wecanintroduceyetanother languageextension,wherewetalkaboutyetmoreobjects.Infact,thereisnoendto thisprocessofformingevermoreexpressivelanguages.
Somewordsaboutmethodologyareinorder.Imakefairlyextensiveuseoflogical andmathematicaltools.Formaldefinitionsareprovided,andtheoremsproved.Iam undernoillusionsaboutwhatthismethodologyachieves.AsKripkeobserves,“There isnomathematicalsubstituteforphilosophy”(Kripke,1976,p.416).Definitionsand theoremsdonotbythemselvessolveanyphilosophicalproblems,atleastnotofthe sortthatwilloccupyushere.Thevalueoftheformalmethodstobeemployedlies intheprecisionandrigorthattheymakepossible,notinreplacingmoretraditional philosophicaltheorizing.Butexperienceshowsthatprecisionmattersinthediscussionsthatwillconcernus.Itisthereforescientificallyinexcusablenottoaspiretoa highlevelofprecision.Infact,muchofthematerialtobediscussedlendsitselfto amathematicallypreciseinvestigation.Whiletheuseofformalmethodsdoesnotby itselfsolveanyphilosophicalproblems,itimposesanintellectualdisciplinethatmakes itmorelikelythatourphilosophicalargumentswillbearfruit.3
Aquickoverviewofthebookmaybehelpful.PartIisintendedasaself-contained introductiontothemainideasdevelopedinthebookasawhole.Chapter1setsthe stagebyintroducingtheideaofthinobjects,explainingitsattractionsaswellassome difficulties.Thisdiscussionculminatesinadetailed“jobdescription”fortheideaof thinobjects.Thisjobdescriptionisformulatedintermsofanotionofoneclaim sufficing foranother—althoughtheontologicalcommitmentsofthelatterexceed thoseoftheformer.Byformulatingsomeconstraintsonthenotionofsufficiency, Iprovideaprecisecharacterizationofwhatitwouldtaketosubstantiatetheidea ofthinobjects.Chapter2introducesmyowncandidateforthejob.Iexplainthe Fregeanconceptionofobjecthoodandtheideathatanappropriateuseofcriteriaof identitycansufficetoconstituterelationsofreference.Chapter3introducestheideaof dynamicabstraction.TheformofabstractionexplainedinChapter2canbeiterated,
3 Compare(Williamson,2007).
resultingineverlargerdomains.Iarguethatthisdynamicapproachissuperiortothe dominant“static”approach,bothphilosophicallyandtechnically.
PartIIcomparesmyownapproachwithsomeotherattemptstodeveloptheidea ofthinobjects.Ibegin,inChapter4,bydescribingandcriticizingsomesymmetricconceptionsofabstractionaccordingtowhichthetwosidesofanacceptable abstractionprincipleprovidedifferent“recarvings”ofoneandthesamecontent. InChapter5,Iexplainandrejectsome“ultra-thin”conceptionsofreferenceand objecthood,whichgomuchfurtherthanmyownthinconception.OnetargetisHale andWright’s“syntacticprioritythesis”,whichholdsthatitsufficesforanexpressionto referthatitbehavessyntacticallyandinferentiallyjustlikeasingulartermandfigures inatrue(atomic)sentence.Theultra-thinconceptionsmakethenotionofreference semanticallyidle,Iargue,andgiverisetoinexplicablerelationsofreference.The importantdistinctionbetweenpredicativeandimpredicativeabstractionisexplained inChapter6.Iarguethattheformertypeofabstractionissuperiortothelatter,atleast forthepurposesofdevelopingtheideaofthinobjects.Onlypredicativeabstraction allowsustomakesenseoftheattractiveideaoftherebeingno“metaphysicalgap” betweenthetwosidesofanabstractionprinciple.Finally,inChapter7,Idiscussa venerablesourceofmotivationfortheapproachpursuedinthisbook,namelyFrege’s contextprinciple,whichurgesusnevertoaskforthemeaningofanexpressionin isolationbutonlyinthecontextofacompletesentence.Variousinterpretationsof thisinfluentialbutsomewhatobscureprinciplearediscussed,anditsroleinFrege’s philosophicalprojectisanalyzed.
PartIIIspellsouttheideasintroducedinPartI.Ibegin,inChapter8,bydeveloping indetailanexampleofhowanappropriateuseofcriteriaofidentitycanensure easyreference.Chapter9addressestheJuliusCaesarproblem,whichconcernscrosscategoryidentitiessuchas“Caesar=3”.Althoughlogicleavesusfreetoresolve suchidentitiesinanywaywewish,Iobservethatourlinguisticpracticesoften embodyanimplicitchoicetoregardsuchidentitiesasfalse.Chapter10examines theimportantexampleofthenaturalnumbers.Idefendanordinalconceptionofthe naturalnumbers,ratherthanthecardinalconceptionthatisgenerallyfavoredamong thinkersinfluencedbyFrege.Thepenultimatechapterreturnstothequestionofhow thinobjectsshouldbeunderstood.Whilemyviewisobviouslyaformofontological realismaboutabstractobjects,thisrealismisdistinguishedfrommorerobustforms ofmathematicalPlatonism.IusethisslightretreatfromPlatonismtoexplainhow thinobjectsareepistemologicallytractable.Thefinalchapterappliesthedynamic approachtoabstractiontotheimportantexampleofsets.Thisresultsinanaccount ofordinaryZFCsettheory.
Themajordependenciesamongthechaptersaredepictedbythefollowingdiagram. The viabrevissima providedbyPartIisindicatedinbold.
Manyoftheideasdevelopedinthisbookhavehadalongperiodofgestation. ThecentralideaofthinobjectsfiguredprominentlyalreadyinmyPhDdissertation (Linnebo,2002b)andanarticle(laterabandoned)fromthesameperiod(Linnebo, 2002a).Atfirst,thisideawasdevelopedinastructuralistmanner.Later,anabstractionistdevelopmentoftheideawasexploredin(Linnebo,2005)andcontinuedin (Linnebo,2008)and(Linnebo,2009b).Thesethreearticlescontainthegermsoflarge partsofthisbook,butarenowentirelysupersededbyit.Theideaofinvokingthin objectstodevelopaplausibleepistemologyofmathematicshasitsrootsinthefinal sectionof(Linnebo,2006a).Theseconddistinctivefeatureofthisbook—namelythat ofdynamicabstraction—hasitsoriginsin(Linnebo,2006b)and(Linnebo,2009a) (whichwascompletedin2007).
Someofthechaptersdrawonpreviouslypublishedmaterial.InPartI,theopening foursectionsofChapter1arebasedon(Linnebo,2012a),whichisnowsuperseded bythischapter.Section2.3derivesfromSection4of(Linnebo,2005),which(as mentioned)issupersededbythisbook.Theremainingmaterialismostlynew.In PartII,Sections4.2and4.3arebasedon(Linnebo,2014),andSection6.2on(Linnebo, 2016a).ThesetwoarticlesexpandonthethemesofChapters4and6,respectively. Chapter7closelyfollows(Linnebo,forthcoming).InPartIII,Chapters8,10,and12 arebasedon(Linnebo,2012b),(Linnebo,2009c),and(Linnebo,2013),respectively, butwithoccasionalimprovements.Chapter9andSection11.5makesomelimited useof(Linnebo,2005)and(Linnebo,2008),respectively,bothofwhichare(as mentioned)supersededbythisbook.
Therearemanypeopletobethanked.SpecialthankstoBobHaleandAgustín Rayoforourcountlessdiscussionsandtheirsterlingcontributionasrefereesfor OxfordUniversityPress,aswellastoPeterMomtchiloffforhispatienceandsound advice.Ihavebenefitedenormouslyfromwrittencommentsanddiscussionsofideas
developedinthismanuscript;thankstoSolveigAasen,BahramAssadian,NeilBarton, RobBassett,ChristianBeyer,SusanneBobzien,FrancescaBoccuni,EinarDuenger Bøhn,RoyCook,PhilipEbert,MattiEklund,AnthonyEverett,JensErikFenstad, SalvatoreFlorio,DagfinnFøllesdal,PeterFritz,OlavGjelsvik,VolkerHalbach,Mirja Hartimo,RichardHeck,SimonHewitt,LeonHorsten,KeithHossack,Torfinn Huvenes,NickJones,FrodeKjosavik,JönneKriener,JamesLadyman,HannesLeitgeb, JonLitland,MicheleLubrano,JonnyMcIntosh,DavidNicolas,CharlesParsons,Alex Paseau,JonathanPayne,RichardPettigrew,MichaelRescorla,SamRoberts,Marcus Rossberg,IanRumfitt,AndreaSereni,StewartShapiro,JamesStudd,TolgahanToy, RafalUrbaniak,GabrielUzquiano,AlbertVisser,SeanWalsh,TimothyWilliamson, CrispinWright,aswellastheparticipantsatalargenumberofconferencesand workshopswherethismaterialwaspresented.ThankstoHansRobinSolbergfor preparingtheindex.ThisprojectwasinitiatedwiththehelpofanAHRC-funded researchleave(grantAH/E003753/1)andfinallybroughttoitscompletionduring twotermsasaVisitingFellowatAllSoulsCollege,Oxford.Igratefullyacknowledge theirsupport.
PARTI Essentials InSearchofThinObjects 1.1Introduction Kantfamouslyarguedthatallexistenceclaimsaresynthetic.1 Anexistenceclaim canneverbeestablishedbyconceptualanalysisalonebutalwaysrequiresanappeal tointuitionorperception,thusmakingtheclaimsynthetic.Thisviewisboldly rejectedinFrege’s FoundationsofArithmetic (Frege,1953),whereFregedefendsan accountofarithmeticthatcombinesaformofontologicalrealismwithlogicism.His realismconsistsintakingarithmetictobeaboutrealobjectsexistingindependently ofallhumanorothercognizers.Andhislogicismconsistsintakingthetruths ofpurearithmetictorestonjustlogicanddefinitionsandthusbeanalytic.Most philosophersnowprobablyagreewithKantinthisdebateanddenythattheexistence ofmathematicalobjectscanbeestablishedonthebasisoflogicandconceptual analysisalone.ThisiswhyGeorgeBoolos,onlyslightlytongue-in-cheek,canoffera one-linerefutationofFregeanlogicism:“Arithmeticimpliesthattherearetwodistinct numbers”(Boolos,1997,p.302),whereaslogicandconceptualanalysis—Boolostakes usalltoknow—cannotunderwriteanyexistenceclaims(otherthanperhapsofone object,soastostreamlinelogicaltheory).2
However,thedisagreementbetweenKantandFregelivesoninadifferentform. Evenifweconcedethattherearenoanalyticexistenceclaims,wemayaskwhether thereareobjectswhoseexistencedoesnot(looselyspeaking)makeasubstantial demandontheworld.Thatis,arethereobjectsthatare“thin”inthesensethattheir existencedoesnot(againlooselyspeaking)amounttoverymuch?Presumably,an analytictruthdoesnotmakeasubstantialdemandontheworld.3 Butperhapsbeing analyticisnottheonlywaytoavoidimposingasubstantialdemand.Insteadofasking Frege’squestionofwhetherthereareexistenceclaimsthatareanalytic,wecanaskthe broaderquestionofwhetherthereareexistenceclaimsthatare“non-demanding”—in somesenseyettobeclarified.
Anumberofphilosophershavebeenattractedtothisidea.Twoclassicexamples arefoundinthephilosophyofmathematics.First,thereistheviewthattheexistence
1 See(Kant,1997,B622–3). 2 Seealso(Boolos,1997,pp.199and214).
3 Analyticitymustherebeunderstoodinametaphysicalratherthanepistemologicalsense(Boghossian, 1996).IcannotdiscussherewhetherFrege’srationalismledhimtodepartfromatraditionalconceptionof (metaphysical)analyticity.See(MacFarlane,2002)forsomerelevantdiscussion.
insearchofthinobjects
oftheobjectsdescribedbyatheoryofpuremathematicsamountstonothingmore thantheconsistencyorcoherenceofthistheory.Thisviewhasbeenheldbymany leadingmathematiciansandcontinuestoexertastronginfluenceoncontemporary philosophersofmathematics.4 Then,thereistheviewassociatedwithFregethatthe equinumerosityoftwoconceptssufficesfortheexistenceofanumberrepresenting thecardinalityofbothconcepts.Forinstance,thefactthattheknivesandtheforks onatablecanbeone-to-onecorrelatedissaidtosufficefortheexistenceofanumber thatrepresentsthecardinalityofboththeknivesandtheforks.5 AgustínRayonicely capturestheideawhenhewritesthata“subtlePlatonist”suchasFrege
believesthatforthenumberofthe F stobeeight justis fortheretobeeightplanets.Sowhen Godcreatedeightplanetsshe thereby madeitthecasethatthenumberoftheplanetswaseight. (Rayo,2016,p.203;emphasisinoriginal)
Iamnotclaimingthatthereisasingle,sharplyarticulatedviewunderlyingallthese views,onlythattheyareallattemptstodeveloptheas-yetfuzzyideathatthereare objectswhoseexistencedoesnotmakeasubstantialdemandontheworld.
Wehavetalkedaboutobjectsbeingthininan absolute sense,namelythattheir existencedoesnotmakeasubstantialdemandontheworld.Anobjectcanalsobethin relativetosomeotherobjects if,giventheexistenceoftheseotherobjects,theexistence oftheobjectinquestionmakesnosubstantial further demand.Someoneattractedto theviewthatpuresetsarethinintheabsolutesenseislikelyalsotobeattractedtothe viewthatanimpuresetisthinrelativetotheurelements(i.e.non-sets)thatfigurein itstransitiveclosure.Theexistenceofasetofallthebooksinmyoffice,forexample, requireslittleornothingbeyondtheexistenceofthebooks.Moreover,amereological summaybethinrelativetoitsparts.Forexample,theexistenceofamereologicalsum ofallmybooksrequireslittleornothingbeyondtheexistenceofthesebooks.6
Ishallrefertoanyviewaccordingtowhichthereareobjectsthatarethinineither theabsoluteortherelativesenseasaformof metaontologicalminimalism,orjust minimalism forshort.Thelabelrequiressomeexplanation.Whileontologyisthe studyofwhatthereis,metaontologyisthestudyofthekeyconceptsofontology,such asexistenceandobjecthood.7 Aviewisthereforeaformofmetaontologicalminimalisminsofarasitholdsthatexistenceandobjecthoodhaveaminimalcharacter. Minimalistsneednotholdthat all objectsarethin.Theirclaimisthatourconceptofan object permits thinobjects.Additional“thickness”canofcoursederivefromthekind ofobjectinquestion.Elementaryparticles,forexample,arethickinthesensethat theirexistencemakesasubstantialdemandontheworld.Buttheirthicknessderives fromwhatitistobeanelementaryparticle,notfromwhatitistobeanobject.
4 Seeforinstance(Parsons,1990),(Resnik,1997),and(Shapiro,1997).
5 Seeforinstance(Wright,1983)andtheessayscollectedin(HaleandWright,2001a).
6 Philosophersattractedtothisviewinclude(Lewis,1991,Section3.6)and(Sider,2007).
7 Seeforinstance(Eklund,2006a).
Metaontologicalminimalismhasconsequencesconcerningontologyproper. The thinnertheconceptofanobject,themoreobjectstheretendtobe. Metaontological minimalismthustendstosupportagenerousontology.8 Bycontrast,agenerous ontologydoesnotbyitselfsupportmetaontologicalminimalism.Theuniversemight justhappentocontainanabundanceofobjectswhoseexistencemakessubstantial demandsontheworld.
JustasmetaontologicalminimalistsareheirstotheFregeanviewthatthereare analyticexistenceclaims,therearealsoheirstothecontrastingKantianview.Hartry Fieldhasattackedtheideathatmathematicalobjectsarethin,sometimesmentioning theKantianoriginofhiscriticism.9 Andvariousmetaphysiciansrejecttheidea thatmereologicalsumsarethinrelativetotheirparts.10 Justaswiththeoriginal Kantianrejectionofanalyticexistenceclaims,thiscontemporaryrejectionofthin objectsstrikesmanyphilosophersasplausible.Metaontologicalminimalismcan comeacrossasapieceofphilosophicalmagicthataspirestoconjureupsomething outofnothing—or,intherelativecase,toconjureupmoreoutofless.
Thechapterisorganizedasfollows.Inthenexttwosections,Ioutlinetwoinfluentialapproachestotheideaofthinobjectsthatarefoundinthephilosophyof mathematicsandthatwerementionedabove.Then,Iexaminetheappealoftheidea. Basedonthisexamination,Iformulatesomelogicalandphilosophicalconstraints thatanyviableformofmetaontologicalminimalismmustsatisfy.Wethusobtain a“jobdescription”,andthetaskofthebookistofindasuitablecandidateforthe job.Thechapterendswithanattempttodramaticallyreducethefieldofacceptable candidatesbyrejectingthecustomarysymmetricconceptionofabstractioninfavor ofanasymmetricconception.Theleft-handsideofanabstractionprinciplemakes demandsontheworldthatgobeyondthoseoftheright-handside.Thinobjectsare neverthelesssecuredbecausetheformerdemandsdonot substantially exceedthe latter.Forthetruthsontheleftaregroundedinthetruthsontheright.
1.2CoherentistMinimalism Oneclassicexampleofmetaontologicalminimalismistheviewthatthecoherence ofamathematicaltheorysufficesfortheexistenceoftheobjectsthatthetheory purportstodescribe.Sinceitiscoherenttosupplementtheordinaryrealnumber line R withtwoinfinitenumbers −∞ and +∞,forexample,the extendedreal numberline R = R ∪{−∞, +∞} exists.Andsinceitiscoherenttosupplement R withtheimaginaryunit i = √ 1andalltheothercomplexnumbers,thecomplex field C exists.Allthattheexistenceofthesenewmathematicalobjectsinvolves, accordingtotheviewinquestion,isthecoherenceofthetheoriesthatdescribethe relevantstructures.Letusrefertothisasa coherentist approachtothinobjects.
8 See(Eklund,2006b)foradiscussionofsomeextremelyabundantontologiesthatmayariseinthisway.
9 See(Field,1989,pp.5and79–80). 10 Seeforinstance(RosenandDorr,2002).
Thisapproachenjoyswidespreadsupportwithinmathematicsitselfandisdefended byseveralprominentmathematicians.InhiscorrespondencewithFrege,forexample, DavidHilbertwrote:
AslongasIhavebeenthinking,writingandlecturingonthesethings,Ihavebeensaying theexactreverse:ifthearbitrarilygivenaxiomsdonotcontradicteachotherwithalltheir consequences,thentheyaretrueandthethingsdefinedbythemexist.Thisisformethe criterionoftruthandexistence.11
Asiswellknown,theword‘criterion’isambiguousbetweenametaphysicalmeaning (adefiningcharacteristic)andanepistemologicalone(amarkbywhichsomething canberecognized).Sincethecontextfavorsthemetaphysicalreading,thepassage isnaturallyreadasanendorsementofmetaontologicalminimalism,notjustofan extravagantontology.
AsimilarviewisendorsedbyGeorgCantor:
Mathematicsisinitsdevelopmententirelyfreeandonlyboundintheself-evidentrespectthat itsconceptsmustbothbeconsistentwitheachotherandalsostandinexactrelationships, orderedbydefinitions,tothoseconceptswhichhavepreviouslybeenintroducedandare alreadyathandandestablished.12
Itmaybeobjectedthat,whilethispassagedefendsanextremelygenerousontology, itisnotadefenseofmetaontologicalminimalism.Inresponse,weobservethatthe passageisconcernedwithwhatCantorcalls“immanentreality”,whichisamatterof occupying“anentirelydeterminateplaceinourunderstanding”.Cantorcontraststhis with“transientreality”,whichrequiresthatamathematicalobjectbe“anexpression orcopyoftheeventsandrelationshipsintheexternalworldwhichconfrontsthe intellect”(p.895).Hefeelscompelledtoprovideanargumentthattheformerkindof existenceensuresthelatter.Themostplausibleinterpretation,Ithink,isthatCantor seeksaformofmetaontologicalminimalismwithrespecttoimmanentexistencebut merelyagenerousontologyconcerningtransientexistence.
Thecoherentistapproachtothinobjectshasenjoyedwidespreadsupportamong philosophersaswell.Astructuralistversionoftheapproachhasinrecentdecades beendefendedbycentralphilosophersofmathematicssuchasCharlesParsons, MichaelResnik,andStewartShapiro.13 Forinstance,Shapiroincludesthefollowing “coherenceprinciple”inhistheoryofmathematicalstructures:
Coherence:If ϕ isacoherentformulainasecond-orderlanguage,thenthereisastructurethat satisfies ϕ .(Shapiro,1997,p.95)
11 LettertoFregeofDecember29,1899,in(Frege,1980).See(Ewald,1996,p.1105)foranotherexample fromHilbert.
12 See(Cantor,1883),translatedin(Ewald,1996,p.896).
13 Seetheworkscitedinfootnote4.Alsorelevantisthe“equivalencethesis”of(Putnam,1967).
ItisinstructivetocomparethisprinciplewithTarski’ssemanticaccountoflogical consistencyandconsequence.OnTarski’sanalysis,atheory T issaidtobesemanticallyconsistent(orcoherent)justincasethereisamathematicalmodelof T .The coherenceprinciplecanberegardedasareversalofthisanalysis:wenowattempt toaccountforwhatmodelsorstructuresthereareintermsofwhattheoriesare coherent.14
Shapironotonlyendorsesthecoherenceprinciplebutmakessomestrikingclaims aboutitsphilosophicalstatus.Hecomparestheontologicallycommittedclaimthat thereisacertainmathematicalstructurewiththe(apparently)ontologicallyinnocent claimthatitispossiblefortheretobeinstancesofthisstructure.Theseclaimsare “equivalent”(p.96),hecontends,and“[i]nasense[…]saythesamething,using differentprimitives”(p.97).Shapiro’sviewisthusaversionofcoherentistminimalism, centeredontheclaimthat
thereisamodelof T ⇔ T iscoherent where ϕ ⇔ ψ meansthat ϕ and ψ “saythesamething”.
Thecoherentistapproachcanbeextendedtoobjectsthatarethinonlyina relativesense.Coherencedoesnotsufficefortheexistenceof“thick”objectssuchas electrons.Butgiventheexistenceofcertainthickconstituents,coherencemaysuffice fortheexistenceoffurtherobjectsthatarethinrelativetotheseconstituents.Given theexistenceoftwoelectrons,forexample,theirsetandmereologicalsummayexist simplybecausetheexistenceofsuchobjectsiscoherent.
Iscoherentistminimalismtenable?Iremainneutralonthequestion.Mypresent aimistodevelopanddefendanalternativeformofminimalismbasedonFregean abstraction.Mypursuitofthisaimisunaffectedbythesuccessorfailureofthe coherentistalternative.
1.3AbstractionistMinimalism AnotherclassicexampleofmetaontologicalminimalismderivesfromFregeandhas beendevelopedbytheneo-FregeansHaleandWright.Fregefirstargues(alonglines thatwillbeoutlinedinSection1.4)thatthereareabstractmathematicalobjects.He thenpausestoconsiderachallenge:
How,then,arethenumberstobegiventous,ifwecannothaveanyideasorintuitionsofthem? (Frege,1953,§62)
Thatis,howcanwehaveepistemicorsemantic“access”tonumbers,giventhattheir abstractnessprecludesanykindofperceptionofthemorexperimentaldetection?
14 Thisisnottosaythatwepossessanotionofcoherencethatisindependentofmathematics.Ourview onquestionsofcoherencewillbeinformedbyandbesensitivetosettheory.Hereweusesomemathematics toexplicateaphilosophicalnotion,whichinturnisusedtoprovideaphilosophicalinterpretationof mathematics.See(Shapiro,1997,pp.135–6)fordiscussion.
