C:\WPdata\TYPE3202\russell 32,2062 red.wpdC:\WPdata\TYPE3202\russell January 11, 2013 (7:57 pm) The BertrandRussellResearchCentre,McMasterU. russell: Brace, 1994), remain viablepositionsforcurre bloated ontology, etc.);thirdly,toar of company objection,theembarrassment continuetoplaguetheneo-logicistmovement(thebad that ety ofproblems Russell’s formof logicism o between Russell’slogicismandthemore tify moreclearly the primary meta in all at dismissed asnotreallylogicism isscarcelydiscussed in logicism form of Russell’s logicism. of presenting anupdatedform andFrege’s revised versionof of mathematicsidentifythemselvesas advocatesoftheso-called“neo Certain ing discovery, a discovery thate discovery ing discovery,a astonish- the GermanmathematicianandlogicianKurtGödelmadean (Peterborough, Ont.: Broadview P., 2003), p.85. P.,2003), (Peterborough, Ont.:Broadview ed., Sullivan, tury I W 2 1 nity, reducibility andsoon).In Forotherexamples,seeScottSoames, z SeeHerrick,Paul website for , Vol. 1: developed for a popular logic textbook developed forapopular now weseethesentimentrepeated. awhilefashionabletosaythatlogicismwasdead.Even for t was the JournalofBertrandRussellStudie The DawnofAnalysis http://www.manyworldsof Logicism andthePhilosophyofLangua RUSSELL’S LOGICISM NEO-LOGICISM AND klement Philosophy /U.Mass–Amherst T Kevin C.Klement Amherst, ers moreelegantand satisfactorysolutionsto a vari- i. introduction z (Princeton: Princeton U. P U. (Princeton:Princeton The ManyWorldsofLogic nt philosophersofmathematics. @ -ontological andmethodologicaldi thispaperIhave three aims: philos.umass.edu T {{ logic.com ma gue that neo-Russellian formsoflogicism that gue ectively endedthehopesoflogi- “neo-Fregeans” (e.g.,HaleandWright), (in lightofitsassumptionaxioms Philosophical Analysis -logicist” movementinthephilosophy , 01003, n.s.32(winter2012–13): 127–59 s thisliteratureand,whenitis,often recent forms;secondly,toarguethat richnessobjection,worriesabouta , © 2004; accessed January 2010. January accessed , ©2004; 1 As recently as2010,a website ge: SelectionsfromFregeandRussell usa 2 claimedthat“inthe1920’s, issn 0036-01631; ., 2003),p.157,andArthur (FortWorth:Harcourt in theTwentiethCen- W online1913-8032 rstly, toiden- T erences C:\WPdata\TYPE3202\russell 32,2062 red.wpdC:\WPdata\TYPE3202\russell January 11, 2013 (7:57 pm) most in The decades. recent to resuscitateaformoflogicismin thinkers certain by for thisdeclineareowedlargelytotheattempts thanks The decline. (!), Gödelprovedthat“logicismisfalse”,andindeed: not provable”,and“withtightdeductiveproofsthatconvinced everyone” arithmeticthatareinprinciple of truths cists”, namely,that“thereare kevin c.klement 128 Essays towardsaNeo-Fregea U.P.,1983),andBobHaleCrispinWright, Aberdeen tempted toprove)anysuchthing. evenhavingat- card, sinceI’mnotawareofGödelhavingproven(or Apparently, I need to turn in my logician card andtheory even my someone theory, it cannotbe derived from pure logicToday, logiciansagree that although alone. Thus was an exciting new tency of Frege’s Giventheinconsis- frame theirpositionasakindofneo-Fregeanism. topic foranotheroccasion. Fregeanism (indeed, partsofitaredecidedlyanti-Fregean), butthatisa formofneo- a view theyendupendorsingcanreallybeunderstoodas a source of inspiration. Ihavesomeseriousdoubtsastowhat extent the ples theyattributetoFrege,andFr Frege himselfbelaboured,theyendorseprinci- length particularpuzzles oftheirpositionstohis,theydiscussat details tics. Theycomparethe position asanattempttosalvage Frege’s overall philosophy of mathema- life—but theydescribetheir endorse Fregeanlogicismattheendofhis totheletter—evenFrege himselfdidnot down be thesameasFrege’s axiomatized set-theoreticfoundati than tosettheory,whereasofcourseGödel’ somehow thinksGödel’sresultsposeagreate of them,Ithink,actuallyen Boolos, RichardHeckandKitFinehaveco ofco kind me, thoughwithoutthesame Neil Tennant andStewart Shapiro have also with hiscollaborators,principallyBobHale. 4 3 However, thetendencyto make such bold pronouncements isonthe WhatisparticularlystrangeaboutHerri See especially Crispin Wright, Wright, Crispin Seeespecially W nally provenfalse. X uential attemptisfoundintheworkofCrispinWrightalong uential Grundgesetze n PhilosophyofMathematics dorse thepositioninend. on formathematicsiscomplete. , the kind of logicism theyadvocatecannot Frege’s ConceptionofNumbersasObjects mmitment totheproject.Moreover,George 3 ege’s workisconstantlyportrayedas done work insympathywiththeirprogram- r problem for reducing mathematics tologic r problemforreducing mathematics can be derived from set canbederivedfrom mathematics ntributed in important ways, though none ntributed inimportantways, s resultsequallyshowthatnorecursively ck’s way of putting the point is thathe is ck’s wayofputtingthepoint (Oxford: Clarendon P.,2001). (Oxford:Clarendon 4 Wright andHaleboth Wright The Reason’sProperStudy: (Aberdeen: C:\WPdata\TYPE3202\russell 32,2062 red.wpdC:\WPdata\TYPE3202\russell January 11, 2013 (7:57 pm) bad reasons. Axiom ofIn non-logical characterofitsaxiomatic ifestly Shapiro’s discussions of neo-logicist settheories,forexample. Shapiro’s discussionsofneo-logicist nitely extensibleconceptsarediscussedinsomedetailWrightand Russell’sviewsoninde to saythatWrightignoresRussellaltogether; other pillarofhistoricallogicism,BertrandRussell.Itwould not befair of interplaybetweenthe work done by Wright-styleneo-logicistsandthe axiomatizing logic itself. Indeed, second-orderlogicisknown tobe Indeed, axiomatizing logicitself. recursively of If not,alltheresultsshowisimpossibility omatization. vinced thatlogicitselfneededtobecapableofacomplete recursive axi- would onlyposeanimpedimentto truths arelogicaltruths,Gödel’sresults mathematical that be thethesis a theorem.Ifweunderstandlogicismto as truth number-theoretic every systemfornumbertheorythatiscomplete,i.e.,contains axiomatized resultsshowisthattherenorecursively unfamiliar, whattheGödel be may to besomethingIneedn’trehearse;butquicklyforthosewho as Icantell,irreproachable,andought far as is, logicism. Hisexplanation Wright goes on to explain why it is thatGödel’s results do not vitiate Russell’s ownattemptin that and Russe eral viewwouldseemtobethat grant thatalldependsonwhatwetake upinnumber-theoreticlogicism.Some,maybe,would sort oftruthwrapped two exceptions,virtuallyno With oneor of SymbolicLogic comparing BernardLinskyandEdward most extended statementWrightmakesaboutRussell’sformoflogicism: most understood. He fully at all,onceitis position seems to be thatRussell’sform oflogicismwasn’treally justifying.Awidespread needs mathematics theirdeviancefromwhich fortheirown,orasarivalapproachinthephilosophyof guidance and form of pletely ignoreRussell’s to saythatWright,andthoseofhisschool,almostcom- think itisfair Generality In this paper, I instead hope partlytoredressthenearlycompletelack 6 5 See, e.g., Shapiro and Wright, “All Things Inde “AllThings andWright, See,e.g.,Shapiro Wright, z , ed. A. Rayo and G. Uzquiano (NewYork: Oxford pp.255–304. U. P., 2007), G. Uzquiano and , ed.A.Rayo Frege’s Conception ofNumbersasObjects Frege’s W 6 nity) andbytheincompletenessth z 12(2006):60. Principia Mathematica Neo-Logicism andRussell’sLogicism logicism Zalta,“WhatIs ll’s ParadoxputpaidtoFrege’sattempt, to be comprised in “logic”. But the gen- But “logic”. to becomprisedin logicism if we were somehow con- logicism ifweweresomehow one nowadaysacceptsthatthereisany re iswhatis,asfarIknow,the basis (displayed most signally by the basis(displayedby mostsignally , eitherasasourceofinspiration z , pp. 130–1; on this point it is worth is it , pp.130–1;onthispoint eorems of Gödel. But these are of Gödel.Butthese eorems W nitely Extensible”,in Neo-Logicism?”, Neo-Logicism?”, z is vitiated bytheman- The Bulletin 5 Absolute ButI 129 W - C:\WPdata\TYPE3202\russell 32,2062 red.wpdC:\WPdata\TYPE3202\russell January 11, 2013 (7:57 pm) without a cost. Taking a Russellian approach also means abandoning the a cost.TakingRussellianapproachalsomeansabandoning without I cantell,isuntouchedbytheGödeltheorems. di have theresourcestoprovideeloquentandsatisfactorysolutions startingpoint.IbelieveRussell’sviews methodological ontological and Russell cametodi iswidelyknown,andindeed,that matters aremoresophisticatedthan current neo-logicistsface.I argue that Russell’s considered viewsonthese problemsthatthe most importantoutstandingpuzzlesandpotential the to to issues,ifnotpreciselythesameas,thenatleastverysimilar more unfortunate,preciselybecauseIthinkRussellgave serious thought the all neo-logicist movement,theneglectofRussell’sworkis temporary neo-logicists, andothersinterestedinassessingthecon- the cibility. For of in given theseeminglynon-logicalprinciples form oflogicismatall, a to understandhowRussell’sviewsconstitute because Russell’sviewsarenotwellunderstood.Inparticular,few seem part to takeRussell’slogicismasseriouslyitshouldbe.Thisisin not tomatic of a general trendamongworkingphilosophersofmathematics logical truth.Indeed,WrightdoesnotdiscussRussell’slogicismfurther. could not in principleholdthatitcouldn’tbeunderstood as any kind of the needforanaxiomof in for logicism,itisadi who doubt that second-orderlogicisreallylogic, second-order logical truths are logical truths. second-order logicaltruthsare falsityofthesuppositionthat the truths anymorethantheyestablish truths arelogical mathematical that thesis the don’t establishthefalsityof ofGödel’sresults.Hence,thoseresults incomplete duetoacorollary kevin c.klement 130 some of logicismthatholdonlytheweakerview however,Ra logical system.Disappointingly, canbe truths that holdmathematical primarily results Gödel there arguesthatthe Philosophy ofMathematicsandLogic “L Rayo, seeAgustín logicism, Harvard U. P., 1986), and John Burgess, John Harvard U.P.,1986),and U Wright does not discuss the otheralleged the Wright doesnotdiscuss In what follows, I want to argue that this isunfortunate,andsymp- In whatfollows,Iwanttoarguethat 8 7 See, e.g., W. V. Quine, See,e.g.,W.V.Quine, For a more sophisticated look at thei look Foramoresophisticated recursively axiomatized system culties faced by contemporary logicists, thoughtheydonotcome culties facedbycontemporarylogicists, T erent answers largelyasaresultofdi answers erent T erent problemfromGödel’sresults.) The PhilosophyofLogic ogicism Reconsidered”,in W nity, though given Wright’s own views, he nity, thoughgivenWright’sownviews, z , ed. Shapiro (Oxford: Oxford U. P., 2005). Rayo 2005). (Oxford:OxfordU.P., , ed.Shapiro or other Fixing Frege deduced inasingle recursively axiomatized that for every mathematical truth,thereis thatforeverymathematical pose a problemforproof-theoreticlogicisms pose in which it can be proven,which,asfar inwhichitcan mportance ofGödel’sresultsinassessing yo does not discuss proof-theoretic forms proof-theoretic yo doesnotdiscuss z (Princeton:PrincetonU. P., 2005). 7 z (Ofcourse,thereare those , 2nd edn. (Cambridge, Mass.: , 2ndedn.(Cambridge, X aw inRussell’slogicism, W 8 The Oxford Handbook ofthe Handbook The Oxford but if that isaproblem but nity, choiceandredu- T erent meta- C:\WPdata\TYPE3202\russell 32,2062 red.wpdC:\WPdata\TYPE3202\russell January 11, 2013 (7:57 pm) Landini, “Russell’s Intensional LogicofPr Landini, “Russell’sIntensional Logical Studies in EarlyAnalyticPhilosophy Russellan 1974); NinoCocchiarella,“Frege, truths or that arithmetic is simply a species of logic; he prefers to say that say to prefers he metaphysical statussimilartothata and temological logic; of species a simply is arithmetic canbeshowntoanalyticallytrue,andhencehasanepis- arithmetic that or truths occasionally called. or “Abstractionism”,asitis a primaryroletoabstractionprinciples, of neo-logicismthatgive forms those they deserve.Myfocuswillbeon Ishallnotbeabletogivetheserivalneo-logicismstheattention nately a few, Bostock, Cocchiarella,Landini,Sher,Hodes,LinskyandZalta,toname oranotherinrecentyears.Theseincludesuchthinkersas of logicism sometimes verydi currently dominantalternatives. neo-Russellian neo-logicisms are,ifanything,more promising than the ontology endorsedbyWright between languageand rather moresimplebutperhapsnaïveconnection Arithmetic”, Arithmetic”, terly Logic”, ofTarskian Conception “A Sher, Kluwer, 1998);Gila Thought, LanguageandOntology Logicism?”. Here, “ de analytical truthsorhavetheepistemologicalandmetaphysicalstatusof logic supplementedwithcertainadditionalprinciples that arethemselves bytheaxioms ofsecond-order Frege. Arithmeticaltruthsareentailed (AP) abstraction principles.Theseareprinciplestakingthefollowingform: chief di I beginbysketching Wright- and Hale-style neo-logicism, as wellthe W Wright usuallystopsshortofsayingthatarithmeticaltruthsarelogical 9 See David Bostock, SeeDavid z nitions 70 (1989): 341–68;HaroldHodes,“Logi 9 some of whom takebetteraccount ofRussell’swork,butunfortu- 2. k U w z ” and“ culties itfaces.Itshouldbenotedthattherehavebeenother, . Theseadditionalprinciplesparadigmaticallytake the formof contemporary abstractionistneo Journal of Philosophy Journal t zz T ” are two variables of the same logical type (e.g., two ” aretwovariablesofthesamelogicaltype(e.g., ; erent, attempts by other scholars to justifysomeform scholars other erent, attemptsby Logic andArithmetic: k ; t y ( S z , ed.F.OriliaandW.J.Rappaport(Amsterdam: z ( 81(1984):123–49,and k et al z ) Neo-Logicism andRussell’sLogicism 5 z (Columbus: Ohio State U.P.,1987);Gregory (Columbus: . Intheend,I think the prospects for

opositions: aResurrect S cism and theOntologicalCommitmentsof cism d Logicism:aLogicalReconstruction”,in ( NaturalNumbers t zz ) ø

E z ( k , Linsky andZalta,“Whatis t Paci zz )) z (Oxford:ClarendonP., - W logicism ion ofLogicism?”,in c PhilosophicalQuar- T orded toitby 131 C:\WPdata\TYPE3202\russell 32,2062 red.wpdC:\WPdata\TYPE3202\russell January 11, 2013 (7:57 pm) some equivalencerelationbetweenthingsofthetype where “ and itisworthpausingbrie neo-logicism, Abstractionist most importantcasefortheevaluationof variables, etc.),“ individual variables,twomonadic pr kevin c.klement 132 endon P.,1998),pp.386–7. Company”, in (HP) F de Boolos, “SavingFregefromContradiction”,in come tobecalledHume’sprinciple, Hale as abstraction principles,whenlegitimate,areunderstoodbyWrightand type of Wright, “AllThingsInde Set Theory”, Shapiro, Harvard U.P.,1998),pp.171–82; F one another. Peano arithmetic.Otherabstractionprinciplesare thought su and rulesforsecond-orderlogic,su with thestandardaxioms together are equinumerous.Hume’sprinciple, saying thatthenumberof Read togetherwiththisde terms oftheform“ stipulating theiridentityconditions,introducingtheentitiesdenotedby establish thebasictruthsofrealnumbersandevensettheory. z

and – W 11 10 is perhapsthe principle Obtaining thePeanopostulatesfromHume’s The most important such principle for number theory is what has what is The mostimportantsuchprinciplefornumbertheory See Hale, “Reals by Abstraction”, in in byAbstraction”, “Reals SeeHale, nable using second-order logical constants alone, thatholdsbetween constants logical nable usingsecond-order Forperhapsnotverygoodreasons;see G

5 G F k z df whenthe implicitly

and

– ' British JournalforthePhilosophyofScience G The PhilosophyofMathematicsToday R y z ( ” abbreviates theequivalencerelationofequinumerosity, ” abbreviates ; t z and forming anindividualterm,and“ and x S z [ de ” is a function sign operating on expressions ofthe expressions ” isafunctionsignoperatingon Fx ; F S z s andthe F W z

( W z

nitely Extensible”. ÷ k ; ning the function sign“ ning thefunction z )” intothedomainofdiscourse.

G ' ; F zz y (#( X x z zz s isthenumberof W (

y todiscusshowthe result is obtained. The ; Gy nition, Hume’s principle can be read as be nition, Hume’sprinciplecan F G y y

) z

; s canbe put in 1–1correspondencewith ¸ 5

z Rxy “Prolegomenon toAnyFutureNeo-Logicist z [( Reason’s ProperStudy #( edicate variables, twodyadicrelation U Michael Dummett, “Neo Michael Dummett, 10 Rxy z )] ces for the proof ofsecond-order proof ces forthe whichcanbestatedasfollows: G Logic, Logicand y ¸

) ¸

;

z , ed. Matthias Schirn (Oxford: Clar- Schirn , ed.Matthias ø Rxz y z [

Gy F z z ) G 54(2003):59–91;Shapiroand

– º z

s ifandonly ( z S ÷ G Ryx ” and,bymeansof y

) ' z , pp.399–420;George x

¸ zz (

(Cambridge,Mass: Rzx Fx k -Fregeans: inBad E and

zz ¸ z ” standsfor )

Rxy ÷ U F

t cient to y z )] z and z . Such 11 5 ¸ z z G ]) C:\WPdata\TYPE3202\russell 32,2062 red.wpdC:\WPdata\TYPE3202\russell January 11, 2013 (7:57 pm) using the#()functorintroducedbyHume’sprincipleasfollows: be de may relation, andthenotionofanaturalnumber.These usingthenotionofzero,successor Peano postulatesmaybestated necessary, since the Peano postulates cannotbesatis necessary, sincethePeanopostulates establish theexistenceofin ing, canbeusedwithHume’s principle to establish the existenceofone strapping”; one concept, guaranteed by logic tobeinstantiatedby noth- “boot- domain. Forlaterdiscussion,Ishallrefertothephenomenonas Grundgesetze derArithmetik principle, seeRich plus Hume’s The Peanopostulatesarethen: is instantiatedby instantiated bythreethings,andsoon. The concept of being a naturalnumberlessthanorequaltotwois then is instantiatedbytwothings. one to a naturalnumberlessthanorequal can usethistoestablishtheexistenc is onthetable,we one Once zero. Thisestablishestheexistenceofone. namely theconceptofbeingidenticalto more thanzero-manythings, thereisaconceptinstantiatedbyone tively, wehavetoestablishthat Induc- principle requiresmakinguseofthenumbersthemselves. Hume’s requires showing that for every natural number requires showingthatforevery ber. Theuniquenesspartisnothardtoestablish,theexistenceis.It that every natural numberhasauniquesuccessorthatis a natural num- The mostdi 12 This testi Formoreoftheproof Peano postul 3. 5. 0 Nx 4. Sxy 2. 1.

w w w w w 5 N ; ;

; ;

5 5 df W F x zz U x x ( #( es tothepowerofHume’sprinciple;fromit alone, one can

zz df z

0 df [ [  ; cult, byfar,of these to obtain is thesecond,whichstates Nx

F

) ; ' l y

y S (

F x F ; n 0 z z

( y z ( ÷

[ z ) z x many things, plus one more. The proofofthisfrom more. one manythings,plus ' x F z , ( ¸

y Syx z 0

' 5 ( ; / z 0 [ ) y ”, y y Fz

) x

; ; ¸ Journal ofSymbolicLogic z ¸ ))

z ard Heck, “The Development of Arithmetic inFrege’s z Szx ¸ ; z

z ( ( y W y Nz Fy

5 nitely many numbers, whichofcourseis numbers, nitely many ;

z ÷

Neo-Logicism andRussell’sLogicism z

¸ #( ¸ z

(

Syz

Fy Pxz y F z

5

y e of two, since theconceptofbeing two, of e

) ¸ z

÷

z ø Syz ¸

ates in the second-order logical calculus ates inthesecond-orderlogical x 12

z ) Fz

5

y z ÷ z

z ) 5 #(

÷ Fz z z 58(1993):579–600. l

z ; )] z n w ) , thereisaconceptthat x z z ÷ ( z ( Fw Nx

Fx W

¸ ed withina z z ] ÷

w

Fx 5 /

z )] z z ))] W W nite ned 133 C:\WPdata\TYPE3202\russell 32,2062 red.wpdC:\WPdata\TYPE3202\russell January 11, 2013 (7:57 pm) false instance) twonumbers.BorrowingametaphorfromFrege, ex a positiontobecomeawareofthe itand is in a position to recognize an instance of it as true or false isin the rightsideof on occurs that logic stands theformulaofsecond-order a number,andthatanyonewhoproperlyunder- mean whenwespeakof of thenotionnumber,thatit,ine consequence analytic pears. Nevertheless,WrightbelievesthatHume’sprincipleisan ap- it which in us toeliminatethesign“#()”fromallthosecontexts us ofade truth, ashaving,roughly,boththeepistemologicalandmetaphysical stat- tion fortheclaimthatHume’sprinciplecanbethought of as an analytic right side. quanti itum Rossberg (Oxford:OxfordU.P.,forthcoming). Arithmetic the leftcouldnotbe instances of the concepts lated so thattheentitiesnamedbyterms number. Oncewehaveit,canestablishanother,and so on, kevin c.klement 134 compatible withsuchapossibility. It ishighlyunlikelythatFrege’sow precise formofarticulation. the disparate decompositionsdependingon structure, but havinga as the thoughtsexpressedbycertainformulaecanbeanalyzed never foundthismetaphorparticularlyhelpful. It seems tosuggestthat have the correspondinginstanceonright.I of content the of ization describes aninstanceoftheleftsideasa“recarving”orreconceptual- ingful anddiscretepartsofappropriatekindstruesentences. syntactic andcombinatorialpropertiesofsingular terms to serve as mean- for languagetoreferobjectsis is constitutivelypriortoreference.Whatit think) toFrege,thattruth (problematically,I Wright endorsestheview,whichhealsoattributes tosayaboutit. more have views, butWright’s,andthankfullyhedoes (New York: Routledge, 2002),Ch.3,and“ (New York: tinction”, in All the more so givenitsdeductivepower,Wrightowesusanexplana- 14 13 GottlobFrege, As I have argued elsewhere;seeKlement, argued AsIhave z . This would notbepossibleifHume’sprinciplewereformu- would . This W ed individual variables thatoccurinthefullexpressionof ed individualvariables z (Evanston: NorthwesternU.P.,1974),§64. (Evanston: W nition. It is not a standardsortofde not nition. Itis A Companion to Frege’s Grundgesetze derArithmetik A CompaniontoFrege’sGrundgesetze X uidly, sothatthesamethoughtiscapableofradically Die GrundlagenderArithmetik for certainexpressions which havethe n viewsonthe nature ofthoughtsis 14 Grundgesetze ButtheissuehereisnotFrege’s istence ofoneor(inthecasea Frege andtheLogicofSenseReference z , translated as X anking theidentitysignon W F andtheSense/ReferenceDis- nition; itdoesnotallow z and T ect, explains what we ect, explainswhat G z , ed. P. Ebert and M. and , ed.P.Ebert , orvaluesofthe The Foundations of Foundations The 13 ad in Wright W n- C:\WPdata\TYPE3202\russell 32,2062 red.wpdC:\WPdata\TYPE3202\russell January 11, 2013 (7:57 pm) ciple Analytic?”, in the analyticity of Hume’s principle suggests that there wouldbeanything that suggests principle the analyticityofHume’s veri given classoftermsarefunctioningas a the so by … whenithasbeenestablished, and provided we are able to able and providedweare )”, We arefree,then,tointroducetermssuchasthoseformedwith“#( telligible questionwhethersuchterms genuine terms,theirreferencewillbetoo criteria, true,thenitfollowsthatth Objection. which theyappear,and 3.1 follow. of Russell’sworkto certain issues,thosemostgermanetomydiscussion my discussionto limit in parttechnical,meta-ontological.Ishall what othergrounditcouldbetrue.Theproblemsare on even or know it it sort ofstatus of precisely whatHume’sprincipledoeswithregardtothesortalconcept conditions, thisisenoughtoguaran independently theobtainingofthese stipulatively—and canrecognize really or otherkindofanalyticaltruth,itis di the statusofakindde have not does it if we speakofnumbers; principle doesseeminsomewayanexplanationofwhatwemean when It is di appear. ditions ofarithmeticalstatements already understoodlogicalvocabulary,andthereby w 16 15 Perhaps the most notorious problemisknownastheBadCompany Perhaps themostnotorious number W Wright, Wright, The BadCompanyObjection See Dummett, “Neo-Fregeans: in Bad Co Bad in SeeDummett,“Neo-Fregeans: ed that certain appropriate sentence ed thatcertainappropriate are suchobjects. 3. U w cult todenytheintuitiveappealofWright’sposition.Hume’s problems withabstractionistneo z ; it 16 Frege’s ConceptionofNumbersasObjects Nothing inwhatwe’veseensofarWright’sdefenceof Nothing W xes theidentityconditionsofnumbersintermsan xes Logic, Logicand could 15 have,onwhatothergroundwecouldcometo W x their identity conditions—we maydoso x theiridentity W Neo-Logicism andRussell’sLogicism x thetruthconditionsofstatementsin ose terms do genuinely refer. do And ose being terms z , pp.301–14. in which statements aboutnumbers in whichstatements really singular terms, and when it has been has singularterms,andwhenit tee that these terms refer.Thisis tee thatthese bjects. There is to beno further, in- s containingthemare,byordinary rt of syntactic criteria sketched, that rt ofsyntacticcriteriasketched, mpany”, andBoolos,“IsHume’sPrin- U haveareference,whetherthere cult tounderstandwhatother z , p.14. W - xes the truthcon- xes logicism W nition 135 C:\WPdata\TYPE3202\russell 32,2062 red.wpdC:\WPdata\TYPE3202\russell January 11, 2013 (7:57 pm) (BLV) second-order predicatecalculus,itistypicallystatedasfollows: instead, i.e., ciples orimplicitde equivalence relationatall?Theanswerisno. There are abstraction prin- we canabstractentities,oneforeachgroupofequivalents— relation, stipulative de anyabstractionprincipleofform(AP)asakind taking wrong with kevin c.klement 136 (Things) ofhis sistent system himself blamed for the derivabilityofRussell’sparadoxwithinincon- cally impossible.ThemostfamousisFrege’sBasicLawV,whichFrege principle which,farfrom being analytical truths, are demonstrablylogi- Here “ (Ord) ilarity to(HP)isthis: apparent sim- risome fortheAbstractionistneo-logicist,givenitsgreat ately toRussell’sparadoxofpredication. immedi- clear violationofCantor’stheorem,andleads a concept, every justamounts tothesuppositionthatauniqueobjectexistsfor But this tensional. Itwouldnotbedi Concepts tion involves whatyoumighttaketobetheequivalencerela- discussed case kind oflogicalmistake. Wright owes usanexplanationofthedi would beguiltyofsome person a such Yet “the extensionofaconcept”. is tobemeantby ulatively true,trueinvirtueofadecisionaboutwhat not onlyanalyticallytrue,butstip- as resulting contradictiontakingthis 17 Another “bad” abstraction principle which should beparticularlywor- which principle Another “bad”abstraction And Basic Law Visnottheonlyproblematicexample.Onerarely Law And Basic Thosewhodemurfromidentitybetweenco par excellence R

. F ; S M z and y ; ” abbreviatestheformula(againexpressibleusingpurely z W ( ; M F nition. But is it true that starting from nition. Butisittruethatstartingfrom F

z ( ;

G F ; ; , identity. G y ) z R G havethesameextensionifandonlytheyare coex- W ø z

(Ext( zz nitions of almost precisely the same form as Hume’s precisely thesameformas almost of nitions ; (Thing(

S M Grundgesetze z (Ord( z ( G F y y )), or more Fregeanly, )), ormoreFregeanly, ) 17 5 U F R y cult to imagine someoneignorantofthe Ext( ) y ) 5 5 Thing( Ord( G . Reconstructedwithinamodern y ) ø ncepts mayutilize indiscernibility here S

y G ; ) y ø x ) ; zz ( ø M Fx

R z (

M

z F

. ø b 5 S z ( F y Gx ) zz G ( z b any y z z )) )) ) ø equivalence

M T b z ( erence. G z ( z b any z ))). C:\WPdata\TYPE3202\russell 32,2062 red.wpdC:\WPdata\TYPE3202\russell January 11, 2013 (7:57 pm) only satis is Hume’s principle,inleadingtofullsecond-orderPeanoarithmetic, order-type ofrelation similar relations.Theabstractionsaystheorder-typeofrelation Where “ contradiction. Wright (HP),however,wegeta with is consistentonitsown.Inconjunction of theabstractionprincipleas tency under onebutnottheother. holds between ordinals ratherthancardinals. to benothingmorenorless than Hume’s Principle modi look attoomanyexamples;it’llsu principles thatareonlysatis logical vocabulary) claimingthat logical vocabulary) ciple yields a contradiction from the Burali-Forti paradox. ciple yieldsacontradictionfromtheBurali-Forti which isconsistent,but incompatible with oneanother. each of lead tocontradiction,therearepairsofabstractionprinciples, 3.2 abstraction principles,butthisleadsintothenextproblem. statusfor to thinkthatconsistencyalonecannotsalvageanyprivileged reason given thebadcompanyitkeeps.Indeed,wehaveverygood play, explain howitcanservethesortoffoundationalrole Wright wants it to is, ifsecond-orderPeanoarithmeticis).Butthisdoesnot (and indeed (HP)doesseemtobeconsistent between thisand(HP),mostnotably of FormalLogic 202–19; AlanWeir,“Neo-Fregeanis why itis(PP),not(HP),thatwe (PP) Boolos’s With these sorts of examples,theneo-logicistcannotcite inconsis- 20 19 18 Not only are there abstraction principles,which,takeninisolation, Not onlyarethereabstraction w SeeHodes,“Logicism and the Ontologica Fortheconnectionbetweenor See Boolos, “The Standard See“The Equalityof Boolos, Numbers”, in The EmbarrassmentofRichesObjection s parity principle W able in ” representsthe(equivalence)relationof z 44 (2003): 13–48. 44(2003): ; F F and

; W G nite domains.Thereareotherconsistentabstraction z (Parity S G et al z ifandonly , statedasfollows: i T . mustgivesomenon- there is an even numberofobjectsfalling even thereisan z ( m: anEmbarrassment of Riches”, W F der-types and ordinals, see der-types andordinals, able in 19 y Neo-Logicism andRussell’sLogicism ) Ofcourse,therearesomedi 5 R should reject, and why one must be must should reject,andwhyone U Parity and grounds for rejecting it, since (PP) groundsforrejectingit,since R ce in this context to examine one. ce inthiscontexttoexamine z and l CommitmentsofArithmetic”,p.138n. W nite domains,oneofwhichis S z ( areisomorphicorordinally G S z y areisomorphic.Thisprin- ) ø ad hoc

Logic, Logicand F

di s IMP V G ering evenly z Notre DameJournal explanationfor W y z 20 , Ch.7. ) ed to dealwith 18 Weneednot Yet,it seems T R erences z isthe z that z , pp. 137 C:\WPdata\TYPE3202\russell 32,2062 red.wpdC:\WPdata\TYPE3202\russell January 11, 2013 (7:57 pm) a e the abstractionto Concerning ciples. 3.3 help butfeelthestrainwearingoncurrentAbstractionistdoctrine. ones, includingconservativeness, permutation-invariance, non-in separate outtheacceptableabstractionprinciples from theunacceptable objections, anumberofcriteriahave privileged epistemologicalandfoundationalrole. impossibilitywhereastheotherenjoysa an relegated tothestatusof kevin c.klement 138 In theproposition: priori formed inorderforittobejusti knowthatherbeliefwasreliably someone that not typicallyrequire do who knowledge, might bemadetoreliabilistsaboutempirical parison be anacceptableone),onlythat it is to it what principle inquestionisoneoftheacceptableones(oreven it to,theneo-logicistwouldnot insist that someone must justi foundational epistemologicaland the play site features.Presumablyinorderforsuchanabstractionto requi- the that agivenabstractionprinciplehas establish to meta-theory the quite abitofsophisticatedlogicalandmathematicalapparatusin do to the neo-logicistrequiresthem yet hopeless,buteventhoseproposals that come closetodoingthework they want.Thesearchcontinues,anditisnot incompatible withwhat but thoseneededfor real analysis, set theoryandsoon)nonethatare stractions theneo-logicistwants(includingnotjustHume’sprinciple, ad hoc it tosaythatisgenerallyagreednoindependentlymotivated(non- review thetechnicaldetailsoftheseproposalshere.Su not We need irenicity, andothers. modesty, non-paradox-exploitativeness,stability, Fregeanism: an Embarrassment ofRiches”. Embarrassment an Fregeanism: “Is Hume’s Principle Analytic?”, in Frege writes: isidenticalwiththedirectionofline 21 prin- abstraction This objectionstemsfromFrege’sowndiscussionof Riches of Embarrassment response toboththeBadCompanyand In w See, e.g., Kit Fine, Fine, See,e.g.,Kit The JuliusCaesarProblem z realm, however, this seems a little more puzzling,andonecannot little a seems this realm,however, z ) criterion yet suggested unproblematically validates all those ab- those ) criterionyetsuggestedunproblematicallyvalidatesall The LimitsofAbstraction Reason’s ProperStudy W ed;it is enough that it was. In the in fact be an acceptable one. A com- A one. in factbeanacceptable are complicated enoughthatittakes are been considered for attempting to for beenconsidered W b z catory roletheneo-logicistneeds (Oxford:ClarendonP.,2002);Wright, z ifandonly they are parallel, T ect thatthedirectionofline z , pp. 307–32; Weir,“Neo- 307–32; , pp. know thatthe X ation, U ce 21 a C:\WPdata\TYPE3202\russell 32,2062 red.wpdC:\WPdata\TYPE3202\russell January 11, 2013 (7:57 pm) crop upinsomeotherguise,say, as the direction of line means ofrecognizingthisobjectasthe direction oftheEarth’saxis;butthatisnothankstoourde which looksnonsensical.Naturallynoon Eart the of is thesameasdirection not provideforallcases.Itwillnot,fo direction. Wright, “To Bury Caesar …”, in in …”, Caesar Bury “To Wright, the directionof not really does what expressionsfornumbersmayornotreferto,andhence another way,sayEngland,orJuliusCaesar.Hence,itdoesnotreally number ofsomeconceptwouldbeidenticaltoanobjectintroduced conditionsthe what cept. Itdoesnotallowustodetermineunder bers bothexplicitlydescribingthenumberas the number of some con- truth conditionsforanidentityclaimformedwithtwonames of num- forallidentitystatementsnumbers.Itonlygivesusthe conditions vocabulary,doesnotevengosofarasprovidingthetruth understood whic conditions ofanysentencesin ustounderstandthetruth allow pressions fornumbersthatwould providing thesortofde from far Hume’s principle,so the other, and hence if the identity conditions of the one are a special a the other,andhenceifidentityconditionsofoneare can onlyeverbetrueifthesortalofoneisasub-concept of thesortal identity statementformedbetweenentities falling underdi toWrightandHale,an one personisidenticaltoanother.According Caesar is legitimateonlyiftherearedeterminateconditions under which whether ornotsomethingisidenticaltotheperson of that thequestion assume only exactly whatpersonalidentityamountsto,butwecan out under othersortalconcepts.Itisofcoursenotoriouslydi equinumerous. Countries, like England,andpeople,suchas Caesar, fall aregovernedbywhetherornotconcepts whose identityconditions numbers, it tells us what sortofobjectsnumbers must be, namely, those have aresponse. 23 22 Wright andHaleareofcoursewellawarethisdi See Wright, SeeWright, Frege, 22 The FoundationsofArithmetic W “the directionof x theirmeaningatall,orsoonemightargue. a Frege’s ConceptionofNumbersasObjects z plays thepartofanobject,andourde 23 To simplify: (HP) introduces the sortal conceptof sortal Tosimplify:(HP)introducesthe a z Reason’s ProperStudy isidenticalwiththedirectionof Neo-Logicism andRussell’sLogicism r instance, decide for us whether England whether for us decide r instance, h’s axis—if Imaybeforgivenanexample h’s z , §66. same again,in case it should happen to h they appearintermsofalready h e isgoingtoconfuseEnglandwiththe z , pp.335–96. z , pp. 116–17, and Haleand and 116–17, , pp. b z . But thismeansdoes W nition a U W U culty, anddo T b nition ofex- z erent sortals cult to spell ” W T nition of nition ords usa 139 W x C:\WPdata\TYPE3202\russell 32,2062 red.wpdC:\WPdata\TYPE3202\russell January 11, 2013 (7:57 pm) Analytic?”; andRayo, “Logicism Reconsidered”. concept sortal the of could onlybeasub-concept Thesortalconcept case oftheidentityconditionsother. kevinc.klement 140 (HP) cannot be true for modelswith (HP) cannotbetruefor to be—manywouldsay,is objectionleavesitatthat:alogicaltruthissupposed the crude formof the existenceofoneobject,muchlessin must bedevoidofontologicalcommitment,andshouldnotevenentail Logic logic. of part existential claimsofmathematicscanbea substantive numbers tofunctionssetswhat-have-you,nothing that entails the postulating theexistenceofvariouskindsthings,from by does ematics math- and assumptions, logic itselfmustnotmakeanysubstantive since basically, that species ofamoregenerallinecriticismwhichargues, is a might argue,(HP)cannothavethestatusofalogicaltruth.This neo-logicism, butitissomethin principle suchas(HP)whichisonlysatis neo-logicists’allowingthemselvestoadopta simply objectingtothe 3.4 many remainsceptical. response still requiresquiteabitofexplanationtobe perfectly clear, and how personalidentityistobecons beingequinumerous.Sincethisisevidentlynot terms oftwoconcepts being identicaltoagivenpersoncouldbeexplicatedin something’s factual contentorsigni lytical truths, asnotempiricallyveri by-product ofthelogicalpositivistinsistencethatana- than alingering attitudeseemslittlemore the theory-neutral argumentotherwise,and scarcelyheardofa I’ve tology? Otherthanbyappealtomodeltheory, nothaveitsownon- those objectsarelogicalobjects?Whymustlogic ging. Whycan’titbealogicaltruththatcertainobjectsexist,solongas trines. remain despitethenearly universal rejection of mostotherpositivistdoc- 25 24 Objections of thissortcomeinvariousstripes.The Objections The crude form of theobjectionseemstobeentirelyquestion-beg- of form The crude The model-theoreticargument,asstated,isnotmuchbetter.A w I abstain from Iabstain attributingthis crude ob See Dummett, “Neo-Fregeans:See inBad Worries aboutanoverfullontology W cance g I’veheardinconversation. de W z — ned z one that has inexplicably managedto inexplicably has one that trued, numbers are neverpeople.This asbeing—truemodels. inall W able, must be completelydevoidof must able, Company”; Boolos,“IsHume’sPrinciple jection toanyparticularcommentatoron W W W nite domains,andhence,one able in nitely many. number W nite domains. W rst derivesfrom z if,somehow, 24 25 person But The C:\WPdata\TYPE3202\russell 32,2062 red.wpdC:\WPdata\TYPE3202\russell January 11, 2013 (7:57 pm) U. P.,1990). see John Etchemendy, Etchemendy, see John the logicalconsequencerelationitde “ to takingthe knowledge, nooneobjects in of thecontentanother recarving allows ustoinfertheirexistence,bea objects. Howcanonestatement,whichreferstoobjectsand two distinct instance theexistenceof ognizing theexistenceofoneobject,andafalse rec- of means a the lefthalf.Atrueinstanceofhalfprovides cannotbesaidabout same and isnotaboutanyindividualobjects,the loaded instance ofthelefthalf.Whilerighthalfisnotexistentially instance ofnumbers orsomeothersort of abstra the right half whethertheybe argument fortheexistenceofvariouslogicalobjects, of somebe putthisway.Theneo-logicist of objects.Theinitialworrymight abstractionscrutinizing the means wherebytheneo-logicist argues fortheexistence principle (AP) into an not. should should countasalogicalpossibilityandwhat what andthatcannotbeestablishedpriortoconsidering legitimate model, what shouldcountasa established already isn’t alogicaltruthonceitis model-theoretic argumentcanonlybeusedtoestablish that something dismiss their project before it starts, merelyinvirtueofwhereitends. it before dismiss theirproject for anin described earlier.Demandingthatth in domains asrelevant. objects,thereisnoneedtoconsiderso-calledmodelswith many pletely analytical arguments can convince usoftheexistencein ranging overalltheobjectsthereare.Ifheisrightinthinkingthatcom- takes theindividualquanti tion. AsWrightinsists,he double-arrow inthemiddleofitwere (HP) thatitwouldcomeoutasfalseifthebiconditional to objection no is models aresimplyexcludedfromconsideration.Similarly,it called which the identity relationwouldhaveadi which theidentity ; W Reason’s ProperStudy 27 26 from derives A moresophisticatedversionofthiscriticism,however, Here one must further rememberthattheneo-logicistproofofan further Here onemust x nity ofobjectsstemsfrom(HP),bymeansthebootstrapping Wright’s explicitreplytosuchworriesis Andthisisevenifwegrantthatthemo 5 W x nity of objects before they are allowed to posit (HP) is just to just of objectsbeforetheyareallowedtoposit(HP)is nity z ” as a logical truthbecauseitisnotsatis logical ” asa The ConceptofLogical Consequence z , pp.256–71. W Neo-Logicism andRussell’sLogicism nes getthingsright,whic del-theoretic conceptionof logical truth and ction, requires reconceptualizing an ction, requiresreconceptualizing similar; see,e.g., similar; ey givesomeindependentgrounds reinterpreted as exclusivedisjunc- W rst-order lawofself-identity T erent extension.Thoseso- z (Cambridge,Mass:Harvard “Response toDummett”, W h is itself questionable; ers asunrestricted, W ed by models in 26 Tomy W nitely W nite 141 27 C:\WPdata\TYPE3202\russell 32,2062 red.wpdC:\WPdata\TYPE3202\russell January 11, 2013 (7:57 pm) to which there canbenoseriousdoubtsaboutthetruth-preservation of postulat entities the of logical status a viewoftheonto- to committed are they ism. However,theyadmitthat committed toanythingasin are they refers toobjects found on thelefthalfof(HP)isafeatureitssyntacticarticulation; it Thecommitmentandreferencetoobjects uses. of theexpressionsit committed toitsexistence,isnotindependentof the syntactic character to whichwhetherornota certain claimreferstoanobject,andhenceis doctrine suchasthepriorityoftruthtoreferenceaccording ontological stipulation or intoexistenceby objects new that doesnot?Onecannotintroduce kevinc.klement 142 restatement of“( saying “Thedisjunctionof what Imean.Objectslikedisjunctions and conjunctions must exist, since ifIexplain standards “my incar”canbeevaluatedastruebyordinary in referringto ual termstoreferinstancesofsomesortalconcept be evaluatedastrue “by ordinarystandards”thatintelligiblyusesindivid- about existence.Anytheorythatcan in particularakindofmaximalism Chalmers (Oxford: Oxford U. P., 2009), pp.178–212. (Oxford:OxfordU.P.,2009), Chalmers logicists commit themselvestoa“radically promiscuous ontology”, of prioritythesis,neo- kind this on argued thatinbasingtheirposition theprioritythesisentails.Inarecentpaper,MattiEklundhas ments probing a bit furtherintotheontologicalandmeta-ontologicalcommit- semantics—you nameit—andwecanno allthebigissuesinontology,meta-ontology,andphilosophical almost and objects almostlikecarsexceptfor the di carwise in mygaragearranged molecules the than “the carinmygarage”rather Cars must exist,notjustmolecules, or insofarastheyareinsidegarages” 30 29 28 In theirresponseto Eklund and others, Full evaluation of this response would moreorlessrequiresettling would Full evaluationofthisresponse MattiEklund,“Neo-FregeanOntology”, Hale andWright,Metaon “The

Ibid. q z ”, is a perfectlyintelligibleand,byordinarystandards,truthful a is ”, z . To use one of Eklund’s examples, incars(thatis,“would-be examples, . TouseoneofEklund’s W F at. In response, the neo-logicist must appeal toameta- appeal at. Inresponse,theneo-logicistmust z because s; if we can intelligibly talkintermsof“ intelligibly s; ifwecan p z

¸

q ) | = z itemployssingularexpressionsforthem. ( p z

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q tology ofAbstraction”,in )”. q ed byabstractionprinciples according is entailed bytheconjunction of 29 since itisintelligibletospeakabout X ) mustexist,sincesentencesabout ationary as this kind of maximal- of ationary asthiskind Philosophical Perspectives T erence thattheyonlyexistwhen t do this here.Butitisworth do t 30 HaleandWrightdenythat Metametaphysics F z willbesuccessful F z 20(2006):102. z s”, F z s exist. z , ed.D. 28 and p C:\WPdata\TYPE3202\russell 32,2062 red.wpdC:\WPdata\TYPE3202\russell January 11, 2013 (7:57 pm) form ofsentencesthatexpressthem. a patible with what they call a “Tractarian”accountofcontentsasstates di conceptualized. HaleandWrightalsoadmitthattheircountenancing of or of merely onthebasisofwhatcanbemeaningfullyspoken ontology, large ifthisdoesnotleadtomaximalism,itstillleadsafairly Even have a“sense”inFregeanterminology, reference is therebyguaranteed. conceptual a is there as long so allowsfortalkaboutsuchentitiestobemeaningful,or apparatus that that suggest to less or more is This them. them to exist whenever exempli whenever them toexist ofagenerousmetaphysicsthatpostulates like theabundantproperties properties inaconservativemetaphysics,butrathermore “sparse” even this, theseentitiesmustbeunderstoodaslesslikeconcreteobjects,or (AP) totheleft.Tosecure of instance an of the movefromrightside ment about whether Russellhimself something somewhat controversial, as thereseemstobesomedisagree- the logicist projectistobeconceived.Puttingitthat way is already to say neo-logicists, areatleasthighlyrelated. ber ofissues which, if not identical tothosediscussedbycontemporary num- a with confrontation are readytoturnRussell,andRussell’s We conceived his project. If anything, Russell thought that mathematics was mathematics that conceived hisproject.Ifanything,Russellthought epistemological statusofde the of logicalongwithadditionalprincipleshavingroughly truths by tailed beliefs inarithmeticaltruthsbyshowingthemtobeen- our footing of one:tosecuretheepistemological their projectasanepistemological is notwithoutitsmerits,butIbelieveitmistaken. such principlesthemselves be logically oranalyticallytrue.Thisreading rest ofmathematicscanbederived, are fromwhichthe principles fundamental the identifyingwhat and (2) of pure(unapplied)mathematicscanbereducedtologicalvocabulary, pure logic,butonlytworelatedprojects:(1)showingthatthevocabulary to betruthsof mathematics of truths the so muchinterestedinshowing ject aswenowconceiveof it. I have heard itclaimedthatRussellwasnot T T airs witha The I believe Wright and others of like mind fundamentally conceiveof I believeWrightandothersoflikemindfundamentally erent “recarvings”ofthe same content make their viewlikelyincom- W rst point of comparison I want to explore has to do withhow to rst pointofcomparisonIwanttoexplorehas W xed form or structure uniquely mirrored bythesyntactic xed formorstructureuniquely 4. w russell W Neo-Logicism andRussell’sLogicism nitions. This iscertainlynothowRussell W ’ s methodology cation conditionscanbespeci withoutnecessarilyrequiringthat was z atallengagedinalogicistpro- W ed for 143 C:\WPdata\TYPE3202\russell 32,2062 red.wpdC:\WPdata\TYPE3202\russell January 11, 2013 (7:57 pm) problems withintheoriginaltheoryarefoundtobewhollyunnecessary paradoxes, conundrumsorother to rise giving concepts orassumptions are eliminated.Whensuccessful,the terminology aspects oftheoriginal between variousthesesofthetheoryareexplicit,andvagueorunclear concepts aremadeclear,thelogicalinterrelations various between tions was desirableabouttheoriginal,but taking anewforminwhichconnec- bridge CompaniontoBertrandRussell pp. 272–3. Fordiscussions Essays inAnalysis Regressive 11: 22–3);and“The 163–8, 176–7; OKEW better known. mystery whythisaspe a somewhat of explicit aboutinmorethanadozenplaces.Indeed,itis was fairly he application ofageneralphilosophicalmethodologythat an as instead Russellunderstoodhisproject epistemologically. ground, solid on already kevinc.klement 144 a along” by a given collectionofstatements.Thegoalis rather to provide “whatwemeantall discover to is notaprocesswherebyweattempt second the “synthetic” stage. 2003), pp.310–31. Logic”, in More speci terms oftheresults in original bodyofknowledge knowledge. Thesecondphaseconsists principles whichmightbethoughttounderlietheoriginalbodyof unde of “data”,toacertainminimalstock Theaimistoworkbackwardsfromthesebeliefs,takenasakind zling. vague, imprecise,disuni is thoughttobemoreorlesscorrect, one beginswithacertain theory, doctrine orcollectionofbeliefs,which ti deemed worthpreserving in terms ofthe“minimumvocabulary”iden- original theoryfromthebasicprinciplesorgeneraltruthssoidenti W replacement 32 31 of The methodologyconsists Russell callsthisprocedure“analysis”, ed inthe For some ofRussell’sexpli For Thoughsometimeshereservesthattermforthe 4 , pp. 144–5; , pp.144–5; Papers W MPD cally, onede W z z (hereafter (hereafter fortheoriginaldoctrine,somethingthatpreserveswhat z 6:33; 31 rst phase, and derives or deduces the main tenets of the of rst phase,andderivesordeducesthemaintenets z , pp.98–9,162–3;“ReplytoCrit IMP PLA z , pp. 1–2;“ThePhilosophica , ee PaulHager, “Russell’s Me EA z , W Papers z Method of Discovering thePremisesofMathematics”,in of Method ), ed.Douglas Lackey (New Yo ed, overly complexorinsomeotherway puz- cit see,e.g., cit statements, W nes thoseelementsoftheoriginaldoctrine z 8: 157–63, 234–35; “Logical Atomism”, in in Atomism”, 8:157–63,234–35;“Logical z , ed. N. Gri a two-phaseprocess.Inthe but istakento be in certain regards in rebuilding orreconstructingthe ct ofRussell’sphilosophyisnot 32 U but it should be clear that it n (Cambridge:CambridgeU.P., icisms”, in Schilpp, p. 687 ( icisms”, inSchilpp,p.687 W PoM thod ofAnalysis”,in l ImportanceofMathematical ned conceptsandgeneral W rst phaseonly, dubbingthe rst z , pp.1–2,129–30; rk: GeorgeBraziller,1973), W W rst phase. rst phase, The Cam- PM Papers Papers z W 1:59; ed. z 9: C:\WPdata\TYPE3202\russell 32,2062 red.wpdC:\WPdata\TYPE3202\russell January 11, 2013 (7:57 pm) sellian. and facilitatesthediscoveryofnewresults. and henceinvites system, deductive the procedurearrangesitsresultsasa by somethinglessproblematic.Finally, or capableofbeingsupplanted it obliquely relates.Russell’smetaphysicsis“Tractarian”, it which to disconnect betweentheformittookandoffacts the principlesorconceptsitassume original bodyofknowledgewasvagu physical? Ibelieveso.Thepresuppositionseemstobethat insofar asthe ( more orlessindubitablefortheprocesstogetunderway. suchas“2+2 claims uptheepistemologicalstatusofarithmetical his goalisnottoshore mathematics, is explicitthatinprovidingalogicalfoundationfor Russell it. canbededucedfrom probablyfalse which is by which these propositionscouldbetrue th and dubitable canbededucedfromit, largely inductive,namelythat always as foracceptinganyotherproposition,is The reasonforacceptinganaxiom, Wittgenstein, ite logicalstructuresindependentofourconceptualschemes.Unlike like Hale and Wright’s, in thesensethathethoughtfactshadde new theory.AsRussellputthingsin toprovideinductiveevidenceforthe thought knowledge, andtheyare body of original resulting theoryaretheirreproachableaspectsof bythe esis testingintheempiricalsciences.Thedatatobeexplained and revealingaboutmathematicaltheorieswasretained. toRussell,providedthatwhichisuseful been completelyunimportant wouldhave more sophisticatedvocabularyofworkingmathematicians the either ordinarystatementsaboutnumbersoreven of structure tactic nor evenwhatmathematicians mean. Preserving the vocabularyorsyn- tell uswhatpeoplemeanwhentheyspeakofnumbers, aim wasnotto 34 33 of terms in In providingan“analysis”ofmathematics If Russell’s goalwasnotepistemological,canitbeconsidered meta- If Russell’s As Russell described it, the process isakintotheprocedureofhypoth- As Russelldescribedit,the Or,moreexactly,Wittgenst See Ludwig Wittgenstein, Wittgenstein, SeeLudwig 34 however,Russelldidnotthinkthatordinarylanguage’s 5 4”.Indeed,thosehavetobetakenasalready Tractatus Logico-Philosophicus ein’s metaphysicsinthe Neo-Logicism andRussell’sLogicism many propositionswhicharenearlyin- s disguised,therewasboundtobea e, and logical connectionsbetween e, and Principia Mathematica at no equallyplausibleway isknown at iftheaxiomwerefalse,andnothing Tractatus (London: Kegan Paul, (London:Kegan z was (broadly)Rus- z logic, Russell’s 33 andthusun- z : PM z 1: 59) 1: W 145 n- C:\WPdata\TYPE3202\russell 32,2062 red.wpdC:\WPdata\TYPE3202\russell January 11, 2013 (7:57 pm) to theformer. by meansofrelatingthelatter concepts and vocabulary of mathematical principle, togainanunderstanding second-order logicisinaposition, of thelanguage understanding an does thinkthatanyonearmedwith vocabulary canbede logicists. Ofcourse,whileWrightdoesnotthinkthatmathematical a pointofagreementwiththecontemporaryneo- is this whether ornot may bemanydi case such that be manysuch (within its istransitive,symmetricaland which that “whenthereisanyrelation statedit,thisprincipleasserted rather than“neo-Fregeans”.)AsRussell ists HaleandWrightwouldless misleadingly be labeled “neo-Peanists” pp. 229–55. di wasthatoflogic.Itis mathematics apparatus neededtoreconstruct to theconclusion that theminimumvocabularyandconceptual came tomathematics,Russell In applyingthemethodologyjustdescribed called the “principle of abstraction” liberally. called the“principleofabstraction” they what used who in theworkofPeanoandhisassociates, abstraction” guages closertothatofreality. ofanalysiswasinparttobringthestructureourscienti goal structure alreadyre kevinc.klement 146 Trench, Trubner,1922),§5.5563. e the principle to postulate a relation the principletopostulate functions asderivedfromrelations,andsotook above. Russellregarded §210). The 1895), p.45. de T U W ect, thisistorealizeaversionoftheJuliusCaesarproblem:satisfying 36 35 earlyontothenotionofa“de exposed Russell hadbeen ne anewentity See Wright, “On theHarmlessImpredicativityofN “On SeeWright, See, e.g., Giuseppe Peano, Peano, See,e.g.,Giuseppe cult tosay,giventheirratherdi u and 5. w mathematical termsasincompletesymbols u W v wouldbear f arerelatedby eld) re hereisine 35 S z s T z — erent interpretationsof X X z exive, then, if thisrelationholdsbetween which is just the same insight asthatin(AP),there which isjustthesameinsight f ected the structure ofobjectivereality;indeed,the ected thestructure W z ( ned outrightvocabulary, intermslogical of he u S T z ), whichistobeidenticalwith tothesameobjectwhich ect afunction;itisthesame as the E Formulaire demathématiques z . Russell realized, however,thattherecould Russell . T erent conceptionsoftheirprojects, S foreveryequivalencerelation S thatwouldmakeit true. In 36 (Indeed,theAbstraction- = ”, in z , Vol.1(Turin:Bocca, Reason’s ProperStudy v z bears f z ( u W v z S and nition by z )” ( S in(AP) just in just W c lan- PoM v , we E z , , C:\WPdata\TYPE3202\russell 32,2062 red.wpdC:\WPdata\TYPE3202\russell January 11, 2013 (7:57 pm) class” theory,termsforclassesmaybeeliminatedcontextually.Inthe Russell’s “no to course, thisisnottheendofanalysis,asaccording The numberofclass plifying) theresultsofanumberintermediatede not anythatderive from the word“two”. (representin constants will containnon-logical it S (AP) foragivenequivalencerelation term writes “Nc”.Inappearance,thiswasusedasafunctorappliedtoclass Mathematica f holding between soughttoovercomethisbytaking Russell 320), Order-Types”, in giving aspeci into asentencecontainingnonon-logicalconstants. any sentence inwhichthattermappearscan be translated, whole-scale, replaceit.Rather, thereisamethodwhereby cal vocabularythatcould for whichonecannot lation, afterRussell’sanalysis, becomes for Russell thekindofexpression mathematical term,foranumber, function, for amathematicalre- be de Russell toobelievedthatmathematicalvocabularycouldnot the mature they appear inamorecomplicatedway.Thereis sense, then, in which ties, butneverthelesscontribute to themeaningofexpressionsinwhich “incomplete symbols”,orsymbolswhichdo not stand directly for enti- of classes,eventuallycomingtotheconclusionthatclasstermsare reality from thebasicprinciplesofclasstheory. ofabstractionas“capable ofproof regarded theprinciple makes tothe sentence. The analysis of amixed sentence such as “I have twobrothers,” z ( ( k 38 37 Let ustakeasourexampletheclosestnotationfoundin In hismaturework,ofcourse,Russellcametohave doubts about the u Or, more precisely, containing nonon-l precisely, Or,more See also Russell, “On SomeDi Russell,“On Seealso z z ) and ) takenastheclassofall a W ned outrightin logical vocabulary. Almostwithoutexception,any . Modernizing Russell’s notation somewhat,andapplying(orsim- Russell’s Modernizing . S ( t to the neo-logicist #( ) involvedin(HP),whichRussell ) totheneo-logicist#( zz W ) mustbe. EA c referenceto“ u , pp.149–51. andtheclassofobjectstowhichitbears a W z istheclassofall classes equinumerous with nd asingleuni 37 Nc( In his earliest work in formallogic( Inhisearliestwork a U Neo-Logicism andRussell’sLogicism x ) culties inthe Theory ofTrans f towhich 5 z ( u df z )” (or“

ogical constantsaspartofthecontribution b E ˆ W g me,thebrotherhoodrelation,etc.),but z isnotenough to pin downwhat ( ed expressionformedusinglogi- b

– u a S bears ) ( k z )” ifyouprefer).With E S W tobetherelation , Russell, earlyon, Russell, , 38 nitions, wehave: W nite Numbersand z ” ( PoM E Principia , thereby Papers , §157) a . Of 147 3: C:\WPdata\TYPE3202\russell 32,2062 red.wpdC:\WPdata\TYPE3202\russell January 11, 2013 (7:57 pm) expression “Nc( potential scope ambiguities, should inprincipleallowustoeliminate the These contextualde case of a term for a class of individuals, i.e., one of the form case ofatermforclassindividuals,i.e.,onetheform kevinc.klement 148 class relatedtoit. a givenmemberofthemultisetde of plicity” sets, e.g.,setsofordinals,andthewould-be may beanalyzedasinvolvingquanti tisets”, but multisets can be de tisets”, butmultisetscanbe have, by classes, sothat: quanti inated intermsof Quanti becomes a statement involving higher-order quanti higher-order becomes astatementinvolving (Ext the classtermincompletesymbolstheirusualextensionality,i.e.: higher-order de no constantsbesideslogicalconstants. ables, andviceversa( class vari- for genuine individualvariables,thereisacorrespondingone with, quanti quanti Such classes justde propositional functionvariablealongwith the kind of quanti 39 The contextualde The In the case of classes ofclasses,suchasNc(

R PM * ) W zz ’s contextualde cation overclasses,usingdummyGreek variables, is to beelim- PM F z [ W W zz f b ’s ˆ W cation over individuals, so that for each quanti each for that so individuals, over cation cation overclassesisnotidenticalto,butisomorphic z z ( [ ned. By * W cb z z ˆ a 20.01: nition ofidentitygivesstatementsformed using z ( b )” inanycontextwhichitappears,sothattheresult ˆ c z z )] ( W z cb z W nitions, alongwithcertainconventionsfor resolving )] 5 W PM nition of classes doesnotdirec ofclasses nition ; nitions forclasstermsalongwiththestandard zz 5 PM df ) a W

5 z df ' W ned using 1:196). y ca

cation overpropositionalfunctionsde zz M '

b ’s ˆ F z ( zz

* [ 5 z zz xb ; [ 20.08 z ; df a z

x PM ) z ; ( z ( ca ø F members of themultiset,with“multi- of members F W 39 zz W y ’s class logicbymean ’s ned as the number oftermstheoriginal z z c !

: cation usinganevenhigher-type z z ;

x z ø [ z b

z z ˆ ø z z ( (

M F cb

a z c z ! z z ! z ), astatement“about”one z x z

a ø z tly allow for a theory of “mul- of allow foratheory tly z )] ) z ) ¸ xb ¸

f

F W z ( zz F ) z cation, containing ( s ofrelationsbetween M z z ! z z ˆ z z z ! )] z

b ˆ W zz W )] cation over z ˆ er rulefor z ( c z W z ), we ning C:\WPdata\TYPE3202\russell 32,2062 red.wpdC:\WPdata\TYPE3202\russell January 11, 2013 (7:57 pm) requires a context of use so that it can be resolved intoastatementof requires acontextofusesothatitcanberesolved entity onitsown;it meaningful bynamingorrepresentingasingular tributes inaregularwaytoanysentence which itappears,butisnot meaning thatitcon- ber isinRussell’sparlancean“incompletesymbol”, through thequanti connection withentitiescomesinindirectly All own referentialpowers. that these apparent terms are terms, properlyspeaking,ordohavetheir This isnotenoughtosecuretheresult numbers. for terms apparent with de “reconceived” as(orreallymade be can alone that certainstatementsexpressibleusinglogicalvocabulary admits Russellrejectsthepriorityprinciple.Inasense, power. unlike Wright, does not takesurfacegrammarasindicativeofreferential And so, (HP can berewritten: And indeed,given how quanti of writing statementsofpurelogicsothatthey ratherbyinventinganewway but ditions aretobegovernedby(HP), aretermsreferringtoobjectswhoseidentitycon- stipulating thatthere cal statements in order to gain an understandingofthetruth-conditionsmathemati- logic hasallthatheorsheneeds standing andknowledgeofhigher-order thing thatappearsalmostidenticalto(HP)above. (HP equivalence relation.Wehave: resultthatequinumerosityisan logical therein, alongwiththepurely classtheoryandthede the reconstructed as consequencesof principle, notasbasicassumptionsofthesystem,but If we were to abbreviate “Nc( form “Nc( conditions issimplyaby-productofthede when thecontextualde butinwhichitcanbeseen, objects havingsuchidentityconditions, In theend,Russelltoobelievesthatsomeonepossessinganunder- In One ofthechiefdi R2 R1 ) ) W ; nally, in the end, we get the following versions ofHume’s nally, intheend,wegetfollowingversions a F )” is not understood by Russell asagenuineterm.Russell, )” isnotunderstoodbyRussell

; as he analyzes them as heanalyzes G z (Nc( ; W a cational apparatusofthesystem.Atermfor a num- z ; z z ˆ T b W z ( erences, of course, isthatanexpressionofthe course, erences, of F nitions are unpacked, thathavingtheseidentity z (Nc( z z ! z z z )) a W z z ˆ Neo-Logicism andRussell’sLogicism 5 z ) cation over classes is tobeconstrued,this ( F 5 Nc( . However, this is not done by simply z z ! W Nc( z z nitionally equivalentto)statements z ))” assimply#( ))” z ˆ z ( G b z z ) z ! z W z ø W z )) nition ofnumbersutilized nitions.

a appear ø

–

F

b z z ! F z z x ˆ ) z toutilizetermsfor

y ), we’dgetsome- – G z z ! z x ˆ z ) 149 C:\WPdata\TYPE3202\russell 32,2062 red.wpdC:\WPdata\TYPE3202\russell January 11, 2013 (7:57 pm) course, theprecisemeansofeliminatingthemputsconstraints onwhat eliminating numbertermsinanycontext which they may appear. Of de The contextual Hume’s principle. ( to havesuchpropertiesisconstructedarti appears whatever comprises] hassmoothlogicalproperties… it entities raw materialoftheworld[i.e.,genuine sell putitlater,“noneofthe tions guarantees,purelylogically,thattheyareso individuated. AsRus- de damental principle”northeresultofade tobeassumedasa“newfun- needs numbers isneithersomethingthat ing ornon-obtainingofanequinume governed bytheobtain- are conditions truth troduced asentitieswhose the philosophyof mathematics in particular. Those advantagesarethese: when itcomestothelogicistprojectin advantages of number possess a metaphysics. However,theydo questions inphilosophicalsemanticsand in thepresent context; it leads straightaway to manyofthefundamental position isnotpossible neo-logicist ontology rivallingtheAbstractionist a relativeassessmentofthemasmeta- and logicalconstructions, assessmentofRussell’sviewsonthenatureincompletesymbols Full entities thatitseemstopointto. nor arethe“apparent” basic, as taken 165–6). Thisprincipleitselfisnot greater complexity. kevin c.klement 150 principle whichdispenseswithabstraction”( Hence, the mature Russell claims thatitmightjustas well be called “the work ofHaleandWright,butwithoutpostulatinganynewentities. the Peanists,orin in found principles abstraction of work ofthekind do the to us version ofthe“principleabstraction”,asheseesit,allows de seeminglyforthemwithinsentenceswithwell- are plete symbolsthat certain discourse classes andnumbers,which as such “things”, apparent as labelsforthose PoM W W (2) It provides an immediate solution totheJuliusCaesarproblem. Itprovidesanimmediatesolution (2) (1) Itprovidesdirectinsightinto Russell usesthephrases “logical constructions” and“logical nition. The way in which numbers are introduced as logical construc- as nition. Thewayinwhichnumbersareintroduced ned truth conditions inthesensejustdescribed.Russell’smature ned truthconditions , 2nd edn., p. xi). Numbers are constructed in order toobey order in edn.,p.xi).Numbersareconstructed , 2nd 6. w advantages ofincompletesymbols appears W nitions o to be aboutonlybecauseoftheuseincom- to T ered byRussell how rosity relation.Thisfactabout it isthatnumberscanbein- it W W cially in order to have them” have cially inorderto nition which is notreallya which nition OKEW do provideameansfor 4 , p.40; Papers W ctions” 9: C:\WPdata\TYPE3202\russell 32,2062 red.wpdC:\WPdata\TYPE3202\russell January 11, 2013 (7:57 pm) One needonlyde (Ext version, isalsoobtainable: cated inworksbothearlyandlate. of this method issomethingRussellrepeatedlyandself-consciouslyadvo- discourse aboutwould-beentitieswith relation, a result of the shape aboveisobtainable.Indeed,reconstructing that provided the appropriatetype,andthen, that mightbeo similarcanbedonefor numbers. Something of derlying logicallawsaswelltheprecisenatureofconstruction a new fundamental assumption, butasaderivedtheoremfrom the un- pany andEmbarrassmentofRichesproblems. more haphazardlyorganizedconceptualsystemwithwhichwebegan. createdbythe poses ofthemethodologyistoeliminateconundrums one ofthepur- everyday lifeandevenordinarymathematics.Indeed, mathematics iso tactically similar,andhence,canposethequestion.Russell’sanalysisof Russell’s nosethatordinarylanguagetreatsthesetwoexpressionsassyn- or notJulius Caesar is whether individual maygo.Hence,thequestionof an of name a where may notbemeaningfullyplacedinaposition that numberexpressions ti areeliminatedinfavorofquan- classes ofclasses,andsuchclassterms samelogicaltype.Numbertermsareconstruedas the of expressions above in(Ext areonlymeaningfulwhenthetwosides tity statements de higher-order logic,andbecauseofthe Russell’s in sign contexts they maymeaningfullyappearin.Theidentitysignisa de W We have seem how it is that a version of (HP) can be obtained, not as not obtained, be can (HP) of version a that is it how seem have We (3) Rathermoreindirectly,itprovidessolutions to both the Bad Com- 40 Indeed, wehavealreadyseenahigher-typeversionofBasicLawV cation over higher-orderpropositionalfunctions.Theconsequenceis cation Seee.g., R z ) ; PLA F R

* ; z T , Lect. ), a version for classes of individuals, closer to Frege’s to ), aversionforclassesofindividuals,closer T G ered bytheneo-logicist.Thegeneralformisagain: W ered as a z ( ; ne z x ˆ k z ( viii z F S ; 0 z t z ( ! ; “LogicalAtomism”, is not a meaningful question. It is no skin o isnotameaningfulquestion.Itnoskin z k zz x ( z z S ) ) as replacement for 5 ( k

z m ) z ˆ ˆ Neo-Logicism andRussell’sLogicism z ( 5 E G z (

z z 40 ! k S z z , ( z ) t m such identity conditions by means by conditions identity such z ø ) z ), where ø z

themathematicaldiscourseof ; Papers

E x z ( z E ( F k any z isindeedanequivalence z z z , ! 9:164–5. m z x t abstractionprinciple

is anothervariableof zz ø )) G z z ! z x W z )) nition, iden- X anked by W ned 151 T C:\WPdata\TYPE3202\russell 32,2062 red.wpdC:\WPdata\TYPE3202\russell January 11, 2013 (7:57 pm) Principia away incontext. are incompletesymbolsthatcanonlybede these noted, as wehave Instead, individuals. terms, andhencedonotname genuine as regarded X versions donotleadtocontradiction.The reason is that the expressions ing aresultoftheshape(Ord)above. numbers are aspecies,wereobtainedin Order-types, orwhatRussellcalled“relationnumbers”,of which ordinal kevin c.klement 152 which may or may not be included in the very extensions that(Ext which mayornotbeincludedintheveryextensions of theproofanin inconsistency betweenthemisaresult The Principle. version ofHume’s notincompatiblewiththeRussellian is (PP). Thisversion,however, from ofconceptsdi extensions are the classofallclassesthat wishes, introduce an incompletesymbolfor“theparityof arenolongerembarrassing.Onemay,ifone riches The the sameway. company here. is nobad prevents itfromleadingtotheBurali-Fortiparadox.Sothere withby(Ord)itself.This dealt are the relationswhoserelationtypes can fallinthe that entities new version of(Ord)doesnotgiveus to Russell’sparadox.TheRussellian deals with,andhencedoesnotlead of theknowndomainquanti selves inturntobefedbackintotheright side, the unending expansion de jects. Without(HP)deliveringnewobjects which can inturnbeusedto X numberexpressions the bootstrapping describedearlier—isblockedif case of (PP).Thewayinwhich(HP)leadsto an in result leading totheconsequencethat domain must be principles play nicely with one another.Indeed,theirmutualconsistency tions, the principlessuper theidentitystatementsonleftinvolvelogicalconstruc- When way. ivation ofa The Russellian principle(Ext Russellian The bols of thetypeof“ of bols anking the identity sign on the lefthalfoftheprinciplearenolonger the anking theidentitysignon anking the identity on the left are not taken asgenuinetermsforob- taken not anking theidentityonleftare W Given the nature of the construction of class terms, the type theory of theory type the terms, class the natureofconstruction Given theirAbstractionistcounterparts,theseRussellian unlike However, The EmbarrassmentofRichesproblemissolvedinalmostprecisely ne ever more expansive concepts applicable to such objects, them- ne evermoreexpansiveconceptsapplicabletosuchobjects, F , orsomethingsimilar,therebyarriveatof the formof requires“ W nite domain fromtheparityprincipleproceeds in a similar m ˆ k W E z nite domain of objects from (HP),andasimilar objects nite domainof ” orintermsofwhich“ z ( k , W m cially similar to theneo-logicist’sabstraction similar cially z )” to haveadistinctlogicaltypefromsym- )” R W z ) doesnotintroducenewindividuals cation does notgetunderway. The der- Principia k z z bythis method, yield- ” maybeconstructed. W nity ofentities—the F T y ering evenly ”, de W nite inthe W W ned as eld of W ned R z ) C:\WPdata\TYPE3202\russell 32,2062 red.wpdC:\WPdata\TYPE3202\russell January 11, 2013 (7:57 pm) is alsoacorollaryofthegene ism: anEmbarrassment of Riches”. Of course, icative abstraction principles are consistent ative toset theories as weak as Z. higher-order quanti inthebasicaxiomsofhigher-orderlogicand beyond thoseinvolved as,indeed,itentailsnogenuineontologicalcommitments ontology, leadstoa“radicallypromiscuous” worries thattheRussellianapproach can beproven. di structurewitha the or (perhapsbetter)encodeencrypt,aspectsof though itiswithinourmeanstoin structure, al- univocal a singletruthwhich,whenfullyarticulated,hasan spondence holdingbetweentwo concepts ontheother.Instead,wehave hand,andanexistentialclaimabouta1–1corre- one the numbers on a plasticene nature thatcanbemoldedtoseen as an identity between our solarsystemas there are members of theBradyfamilyisnotonewith ized”. The reality behind thetruththatthereareequally many planets in of a state a intuitive pictureofhowitisthat abstraction principlesasimplicitde controversially besaidaboutunderstanding the neo-logicistversionsof properly understandingthecontextualde structure ofthede surface the ing de onemayhavethatis of higher-orderlogicalone.Anyappearance are not“newthings”inadditiontowhatoneis committed to by means introduce a new way of talking;the logicalconstructionstheyintroduce orfunctionsrelationsclassesreally numbers emulate talkabout for theadvantages.TheRussellianversionofHume’sprinciplecannot tions, iswithoutitscosts.Indeed,thecostsare aslogicalconstruc- objects complete symbols,andhencemathematical mathematicaltermsasin- of None ofthisistosaythattheconception T W (4) Andforreasonsthatshouldal 41 Indeed, it seems that the Russellian approach provides a much more Indeed, itseemsthattheRussellianapproachprovidesamuch erently appearingsurfaceform. ning into existence anynewentitiesisanillusionengenderedbytak- Theproofwouldberemarkablysimilarto 41 W cation. The incomplete symbolsthatcanbeusedto The cation. 7. ral consistency ofthelogic of w the lossofinfinity Neo-Logicism andRussell’sLogicism W ned symbolsuncriticallyandwithout vent modesofsymbolismthathide, W with one another; see Weir, “Neo-Fregean- see Weir, another; one with their mutualconsistencywithoneanother nitions. ready be clear, one need have no have need ready beclear,one the proof given by Weir that all pred- all that theproofgivenbyWeir W nitions. The samecannotun- nitions. T airs canbe“reconceptual- PM z , whichcanbeshownrel- X ip sideofthecoins 153 C:\WPdata\TYPE3202\russell 32,2062 red.wpdC:\WPdata\TYPE3202\russell January 11, 2013 (7:57 pm) classes ofobjectsis2 not thefoundationaltruths. vocabulary of ting thatatbestthe Indeed, Russell is often read here as simply giving up the game, admit- in an powerless toconvinceusoftheexistence Philosophy Introduction toMathematical a naturalnumberasitscardinality. arith- Peano the formofassumingthat universal class ofindividualsdoesnothave standard attain to order of quanti In principle. metic outrightinRussell’ssystem,oneneedsto Hume’s of version lian numbers, andhence,cannotbe obtained, unmodi entailstheexistenceofin ard Peanoarithmetic bootstrapping isout.Stand- approach, are notthemselvesobjectsonthis classes ofclasses,andtherefore,the of objectsis be usedtoestablishtheexistenceofanin kevin c.klement 154 for thePhilosophyofScience Oxford1962), p. U. 699; P., Alan Musgrave, “Logicism Revisited”, obviously-logical axiom. Itdoesraiseanumberofissuesregarding the axiom. obviously-logical asimilarnot- that theirapproachwouldfareanybetterwithout beguntoexplore,anditisnotclear only neo-logicists havethemselves would be needed, e.g., thearithmeticofin Theareasinwhichit analysis. real truths ofnaturalnumbertheoryor parts ofRussell’sreconstructionmathematics,i.e.,its treatment of the neededforthelower not axiom, weneedonlymentionherethatitis For the multiplicative content myselfwithsomerathersnarkyreplies. axioms wouldrequireamorein-depthstudythanispossible here. Ishall andreducibility neo-logicists. Fullexaminationofthemultiplicative di raise alsodon’tgetattheheartof asareconstructiveone.Theissuesthey project the natureofRussell’s in establishinglogicismastrongsense. tion thatdisquali of anin assumption the the axiomofchoice)andreducibility, of (Russell’s version Russell was explicit about this in many places and, in works suchas works in and, Russell wasexplicitaboutthisinmanyplaces 42 These criticisms seem too quick to me,anddon’tdoproperjustice These criticismsseemtooquick See, e.g., William Kneale and Martha Kneale, Kneale, Kneale Martha and See,e.g.,William W cation for theindividualvariablesisin n , the numberofclasses objects is 2 W es Russell’sfoundationalproject as any kind ofsuccess z 28(1977):112. W nity is seen as an evidently non-logical assump- non-logical evidently nity isseenasan 42 2 Together withthemultiplicativeaxiom n . Henceif mathematics reducestothatoflogic, number of numbers.Since numbers T erences betweenRussellandthe W , admittedthatlogicaloneis n nity of objects. If the number of objects.Ifthe nity W The DevelopmentofLogic z is nite numbers,areareasthat W nite, so is thenumberof so nite, assume W W W nite, usuallytakenin n nitely manydistinct W , andthenumberof ed, from the Russel- ed, fromthe nity ofindividuals. thatthedomain The BritishJournal z (Oxford: C:\WPdata\TYPE3202\russell 32,2062 red.wpdC:\WPdata\TYPE3202\russell January 11, 2013 (7:57 pm) ilar to how the multiplicative axiom is treated, butdi treated, is ilar tohowthemultiplicativeaxiom in if thereare distinct naturalnumbershavethesamesuccessor,they instead prove that ticular, ratherthanprovingthe fourth Peano postulate, i.e., thatnotwo asconsequent.Inpar- taken astheantecedentandstandardresult in ditional wheretheassumptionofan quire theassumptionofanin re- rather thanprovingthoseresultsofstandardPeanoarithmeticthat the postulationof“choicesets”forin interpretationofRussell’shigher-orderlogic proper fact: more suitableplacetoframethe debate. However, here is a littleknown than havingaproofofitfromone’sbasiclogicalassumptionsseems axiom ofreducibilitycompletelymoot. of the issue the render account ofmathematicaldiscourse,whichwould of second-orderlogic,alongwiththeRussellianlogicalconstructionist be freetoadopttheneo-logicistconception would logicist day current be ally, andwhiletherearesubstantiali standing ofthesemantics and justi again hastodowiththeproperunder- once think, I here, disagreement theycouchtheirtheories.The for thesecond-orderlogicinwhich closer to the latter in adoptinganimpredicativecomprehensionprinciple fault, since he himself describes describes hehimself since fault, believe that even their inductive or “regressivemethod”providessu or believe thateventheirinductive like reducibility,forexample,Whitehead and Russellapparentlydonot other “primitivepropositions”are,includingthe reducibility axiom.Un- forward simpletheoryoftypes.Wright,Hale adoption ofarami sell’s stractionist approachestologicism.Theneedforitatall stems from Rus- assessing therelativemeritsofRussellianversusAb- in for help logicism andthemorerecentforms. fromthecomparisonofRussell’sform issues besttreatedseparately in there tobe in termsofquanti 43 The issues raised by the need to assume aprincipleofin assume The issuesraisedbytheneedto The axiomofreducibilityisalso,Ithink,not a fruitfulplacetolook Although to befair,themiscon Principia Mathematica W nitely manyindividuals,thenthisisthe case. This is sim- W nitely complexpropositionalfunctions;buttheseare W cation over“propositionalfunctions”seems to require W ed theory of types, rather than a more straight- more ed theoryoftypes,ratherthana it thatwayinvariousplaces( z does not have an axiom of in of axiom an have not does W ception that itis anaxiomof ception Neo-Logicism andRussell’sLogicism nite domain,theyproveinsteadacon- nite W ssues theretobediscussed,awould- cation for higher-order logic gener- cation forhigher-orderlogic W nite collectionstobeanalyzable W nite domainofindividualsis et al. z — LK assumesomething z z especially astaking , p.97; T PM W erent fromhow nity. z is Russell’s own is IMP W nity rather 43 Instead, z , p.131). U 155 - C:\WPdata\TYPE3202\russell 32,2062 red.wpdC:\WPdata\TYPE3202\russell January 11, 2013 (7:57 pm) issues relatetothedebateover those oftheAb attitudes with various mathematicalnotionsbothpre-andpo 273–303; Musgrave,“Logicism Philosopher oftheCentury theorem inthesimple form cannot provethePythagorean one that grounds the object tologicismon notbeatallcontroversial. Noonewould ways madeexplicit,should hidden assumptionsthatarenotal- mathematicians infactprovehave thatworking Russell’s, thinkingthatthetruemathematicalprinciples “if-thenism”, cient justi kevin c.klement 156 enough to Russell that when Wh when enough toRussellthat as anundischargedantecedentor“hypothesis”. Logicism sequents oflogicaltruths.This,however,istrivial.Anytruth truths arelogicaltruths;itonlyshowsthatmathematicalcon- 46 (1981):247–63. Princeton U.P.,2000),pp.385,400. see I.Grattan-Guinness, results in the submitteddraftofVolume gard Russell’sadoptionofanin quences of thoseprincipleswouldbelogicaltruths. Should we then re- the conse- to principles basic conditional fromtheconjunctionoftheir empirical sciencescanoften be arrangedasdeductivesystemssothatthe ofsomelogicaltruths(e.g., is theconsequent to bepreferred. doctrine, itis original sents theformofactualfacthintedatby mathematical so-calledentities.Ifadi preserve allaspectsoftheoriginal one;hisaimwasnotto ber thatRussell’sprojectwasareconstructive ante conceived withoutanysuppressed truth isnecessarilyPeanopostulate4,astraditionally pure mathematical ambitions? 46 45 44 approachlike Indeed, evenwithoutbuyingintoareconstructivist opens upRussell’slogicismtothechargeof arguably This maneuver It depends on whether,attheendofday,youinsistthat on depends It See Hilary Putnam, “The Thesis Thesis “The Putnam, SeeHilary Treatingin Foralongerdiscussionof the complex in z (NewYork:Palgrave W cation fortakingthein 45 W thatis,itfailsactuallytoshowmathematical nity inthisway,di 46 , ed. R. Schoenman (London: Allen & Unwin, 1967), pp. 1967), Unwin, , ed.R.Schoenman(London:Allen& The SearchforMathematicalRoots,1870–1940 Macmillan,2012). Revisited”; AlbertoCo Revisited”; stractionist school, and even a discussion ofhowthese discussion evena school, and stractionist if-thenism,seeSébastienGandon, itehead mistakenly left it o leftit mistakenly itehead a T z 2 W erent from reducibility, + nity antecedent as trivializinghislogicist that MathematicsIsLogic”,in W ii b ways of conceiving and talking about talking ways ofconceivingand z nity axiomtobetrue. Hence it is left , Russellpaidtohavethe mistake corrected; 2

5 terplay between Russell’ terplay between st-analysis, aswellcomparisonofthese T cedents. Again, wemustremem- cedents.

c erent logicalformbetterrepre- erent z 2 outright.Obviously,onecan T a, “KantandRussell”, 44 if T asarestrictiontovarious was apparentlyimportant z p then Russell’s Unknown s attitudetowards Bertrand Russell: Bertrand p), andeven p z (Princeton: z whatever Synthese real C:\WPdata\TYPE3202\russell 32,2062 red.wpdC:\WPdata\TYPE3202\russell January 11, 2013 (7:57 pm) prove. apartofwhatthemathematicianisevenattemptingto not is triangles there areanyapplicationsofthetheoremtoactual matician. That a backgroundassumptiontotheworkingmathe- to betakensimplyas triangle existinginaEuclideanspaceisapt the about especially thepart thinks ofthetheorem,and one when mind to what immediatelycomes twenty- likeafarlessdauntingroutefor seems neo-logicisms, thisactually son tothelitany of problems thatcontinuetoplagueAbstractionist-style Russellian orneo-Russellianlogicism.Bycompari- pects foranykindof very prematuresimplytoconcludethatthisissuealoneruinsthepros- issue ofin sophical issues thatgetraisedbyRussell’sformoflogicism,oreventhe Obviously here, we haveonlybeguntoscratchthesurface of the philo- suitably large are then not.Ifallthatisneededfortheapplicationofmathematics not, if dividuals, thenPeanoarithmeticwillhaveanapplicationtoit; describe space. If thetical systemstotheconcreteworld which of these sets of axiomsactually worldwe actually livehold ofspace.Itisamattertheapplication of thesevariousmathema- in has in actually the axiomsofanygivenEuclideanornon-Euclideangeometry even attempttoprovethat or prove not did it isappropriatethatRussell involve the application of mathematics totheconcreteworld.Similarly, arein there tension, whetherornot on whetherornotthereare in only provethat 2011), Ch.2. 2011), at results arealreadyforthcoming,evenwithoutanynewexistentialposits, Euclidean space, and some It might beargued It And indeed, there are some alternative avenuesthatmightlook And indeed,therearesomealternative 47 Andindeed, has GregoryLandini, beensoargued—see b are the lengths of the other sides, and thetriangleexistsina and arethelengthsofothersides, 8. stageofthehierarchytypes( W w rst centurylogicisttoexplore. conclusion: prospectsforfuturelogicisms W nity in particular. However, it seems clear that it wouldbe it that nity inparticular.However,itseemsclear W nite numbers, then,asRussellpointsout,therelevant numbers, nite if

then c z is the length of the hypotenuse ofarighttriangle, the of length isthe 47

a that puremathematicsoughtnottotakeastand z 2 + b z 2

W 5 Neo-Logicism andRussell’sLogicism nitely many individuals, or even, by ex- by nitely manyindividuals,oreven,

c z 2 . Theseassumptionsarenotpartof W nitely manynumbers.Suchissues IMP , p.133). Russell z (London:Routledge, W nitely manyin- 157 a C:\WPdata\TYPE3202\russell 32,2062 red.wpdC:\WPdata\TYPE3202\russell January 11, 2013 (7:57 pm) ( 2parts, Theorem”, “Logic andExistentialCommitment”, pli Other Essays means ofabstractionprinciples. additions tothelistofthingsthat might perhaps be bestintroducedby of in principle a adding for One suchwouldbetoobviatetheneed tempting. kevin c.klement 158 Verlag, 2005),pp.132–3. simply oninductivegroundsalone than amongthe“rawmaterial”ofworld. logical properties”asconstructionsrather ing thingsthathave“smooth would seemtoinvolvemoving away from aRussellianapproachoftreat- sceptical aboutthisoption,however,sincethemostplausibleroutes Iamsomewhat more cannot haveitsownontologicalcommitments. attitudes towardsclassesandnumbers. objectivebeingsareofapiecewithhis propositions asself-subsistent Russell’sreasonsforeventuallyabandoning think, complaint, and,I ditions. Myownresearches into the subject haveonlycon “creatures ofdarkness”, worriedthatthesewere notoriously Fregean senses,orwhatnot.Quine entities, betheypropositionsorsomethingmorelikerei be? TheonlyotherkindofthingIcanthinkwouldbeintensional classes, orsets,numbers, concepts-as-things, then whataretheyto arenot must beabletoindividuatethem.Iftheobjectsinquestion order toensurethatthenumberofthesethingsisindeedin an in plausible way of explaining theirtruthisknown—thenourargumentfor of,andnoequally us todeducemanytruthswearealreadyconvinced think that this road is not just a road to bad company iftheneo-Russel- bad to think thatthisroadisnotjusta that Ihaveheardnogoodargumenttothee be regardedasalogicalor can objects http://onlinelibrary.wiley.com W 51 50 49 48 meanings haverecentlybeenproposedas Indeed, itisstrikingthat If taking an in ed”, in in ed”, For discussion, see, e.g., Peter Simons, “Gray’s “Gray’s see,e.g.,PeterSimons, discussion, For Formoreonthispoint,seeKlement, its have may view AlthoughIthinkthat Quine, “Quanti Quine, W W nity by seeking some new argument to the e nity byseekingsomenewargumenttothe nity must establish the existence ofobjectsacertainkind.In existence nity mustestablishthe On Denoting1905–2005 z , rev. edn. (Cambridge, Mass: Harvard U.P.,1975),p.188. Harvard Mass: , rev.edn.(Cambridge, Philosophy Compass W W nity of objects asalogicaltruthisnottobeurged objects of nity ers andPropositionalAttitudes”,in 49 bewildering us withtheirelusiveidentitycon- ), accessedJanuary2013. z , ed. G. Imaguire and B. Li B. and , ed.G.Imaguire 51 Logique etanalyse 5(2010):16–28and29–41;anonlinejournal One would need very goodreasonto Onewouldneed 48 —since asRussellputsit,itallows defenders too; see, e.g., Matthew McKeon, defenders too;see,e.g.,Matthew “Russell, His Paradoxes, and Cantor’s and Paradoxes, His “Russell, analyticaltruth.Iarguedearlier 50 Elegy T z ect that 47(2004):409–23. z withoutTears:RussellSim- T nsky (Munich: Philosophia The WaysofParadoxand ect thatanin in principle W W rmed Quine’s ed meanings, ed W nite, we W nity of logic C:\WPdata\TYPE3202\russell 32,2062 red.wpdC:\WPdata\TYPE3202\russell January 11, 2013 (7:57 pm) given anyknock-downarguments to the e should alsosaythatIdonotthinkhave I either. down thisavenue defending aneo-Russellianlogicismmightnotbefruitfullyexplored tionist. any advantageovertheneo-FregeanAbstrac- maintain to is lian logicist pronounce its death, or even the death of one of itsforms,prematurely. one of pronounce itsdeath,oreventhedeath to logicismasawhole: regard with made have same mistakethatsomany stractionist forms of neo-logicism. I ambynomeansreadytomakethe of themorerobustdi that theyhavecertainattractions,andavoidmany argue only aimedto forms oflogicismhaveanyprospectsinthetwenty- However, this is certainlynotconclusiveevidencethatprospectsfor is However, this U culties continuing to plague thedominantAb- plague culties continuingto Neo-Logicism andRussell’sLogicism T ect that W only rst century.Ihave z neo-Russellian 159