Foundations of Math_Metaphysics, Epistemology, Structure

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Scots Philosophical AssociationUniversity of St Andrews

Foundations of Mathematics: Metaphysics, Epistemology, StructureAuthor(s): Stewart Shapiro

Source:The Philosophical Quarterly,

Vol. 54, No. 214 (Jan., 2004), pp. 16-37Published by: Oxford University Presson behalf of the Scots Philosophical Associationand theUniversity of St. AndrewsStable URL: http://www.jstor.org/stable/3543074.

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The hilosophicaluarterly,ol.4,No.214 January004ISSN oo3I-8o94

FOUNDATIONS OF MATHEMATICS:METAPHYSICS, EPISTEMOLOGY, STRUCTUREBY STEWARTHAPIRO

Since irtuallyveryathematicalheoryanbe nterpretedn et heory,heattersafoundationformathematics.Whetheret heory,s opposedo ny f tsrivals,s the ightoundationormathematicsependsn what foundationsfor.One urposesphilosophical,oprovidehemetaphysicalasisormathematics.nothersepistemic,o rovidehe asis f llmathematicalknowledge.nothersto erve athematics,y endingnsightnto he ariousields.notherstoproviden arenaor xploringelationsnd nteractionsetweenathematicalields,heirelativestrengths,tc.Givenhe ifferentoals,heres ittleointodeterminingsingleoundationor llofmathematics.A numberofmathematical nd philosophical rameworksre touted s afoundationfmathematics,ometimess the one and onlyfoundation.hemostprominent,fcourse, s settheory. he received odificationfthis stheaxiomsystemFC, but there re other ettheories,s well as extensionsof ZF withnew axioms ike V = L, determinacy,nd largecardinalprin-ciples.Otherproposedfoundations,ach with corpsofdedicated dvoc-ates,are higher-orderogic,structuralism,raditionalogicism, eo-logicistabstraction,roof heory,amifiedype heorynd categoryheory. re thevariousfoundationalistlaims ncompatiblewith one another, r can weagree thatmathematics an have more thanone foundation,r perhapsdifferentoundationserving ifferenturposes?Some people argue thatthose whowork n foundationsre deluding hemselvesntothinkingheyare doing somethingmportant.Mathematicsneitherhas nor needs afoundation.Whatarewe to make ofthose laims?To makeprogressn these uestions,we mustgetclearer bout whatweare asking.To labour the obvious,the answersto the questionsdependlargelynwhat 'foundation'sandwhat foundationsfor.Quiteoften nphilosophy he most important art of a questionis to figure ut themeaning f thewords n thequestion.The crucial temshere, fcourse, re'foundation'nd mathematics'.hopetohavesomethingosayabout bothofthese, sing hefirstuestion o lluminate he econd.? The Editors of The PhilosophicalQuarterly,oo004.ublished by Blackwell Publishing,9600oo arsington Road, Oxford ox4 2D0, UK,and350MainStreet, alden,MA2148,USA.

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FOUNDATIONS OF MATHEMATICS 17To underminenydrama thispapermayhave, note herethat shallkeepcomingback to argumentsn favour fstructuralism.he paper is aspin-offrom nd extension fa paper on set-theoreticsoundationsub-lished n2000.1It is clear that he word foundation' asmanydifferenteanings.As aresult,much of the discussion n thevarious ssues s at crosspurposes, rstarts ut nthatwayat least.Forexample, omeonemight rgue,ormightjust take it as obvious, hattheconsequencerelationunderlying found-ationmust be complete, r effective,nd then dismiss econd-orderogic(with tandard emantics)n thesegrounds.But advocatesof second-orderlogicare aware that tsconsequencerelation s not effective.o they we)

musthavesomethinglse nmindby foundation' nd even logic'.To be sure, hediscussion eednotremain t crosspurposes.Once thedifferentenses of the termsare made clear, one side may argue thattheother'snotionof foundation's notvery nterestingr worthwhile.proposeto delimit,n verybroad terms, omedifferentensesof founda-tion'. This willallow me to separateoutat leasta fewof thedisputes romoneanother.

I. ONTOLOGYOne sortoffoundations metaphysical.n this ense,a foundation rovidestheultimate ntologyormathematics,tatingwhatmathematicss about. neitherphilosophicalor mathematicalterms,the proposed foundationspecifieshereferencefmathematicalerms nd therangeofmathematicalquantifiers.o a set-theoreticoundationalist ouldargue,or perhaps ustclaim, hat llmathematicalbjects numbers, unctions,eometric ointsand lines, opological paces,groups, tc.- areactuallyets, nd thatZFC isan accurate nd sufficientlyichdescriptionftheseultimatemathematicalobjects.The pure set-theoreticierarchyV is the real subject-matterfmathematics all mathematics.imilarly,ny traditional regean ogicistwho hasadoptedthis ntological erspective ouldclaimthatmathematicalobjects re logicalobjects certain xtensionsr courses fvalues,perhaps.Fromthisperspective,heneo-logicist ould claimthatall mathematicalobjects,or at leastall themathematicalbjectswhich are ofinterest,reI Shapiro, 'Set-TheoreticFoundations',Analytichilosophynd Logic:Proceedingsf theTwentieth orld ongressfPhilosophy(PhilosophyDocumentationCenter,BowlingGreenStateUniversity,000),pp. 183-96.That projectbenefitted rom discussionn thefounda-tions of mathematics mail list a fewyears ago, participantsn whichincludedPenelopeMaddy,HarveyFriedman, ohnMayberry, obertTragesser ndNeilTennant;apologiestothose havenotmentioned. he list sarchived twww.cs.nyu.edu/pipermail/fom.C The Editors fThe hilosophicalujarterly,004



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18 STEWART SHAPIROgenerated y egitimatebstractionrinciples. he categoryheorist laimsthat all mathematicalbjectsare arrows or objects) n categories; nd atleast one kindofstructuraliste.g.,myself)laimsthatmathematicalbjectsare places in structures.notherkind of structuralistGeoffrey ellman)and anotherkind oflogicistRussell)take the foundational orkto showthatmathematics asnodistinctiventology.Argumentsnfavour f second-orderogic typicallyo not nvolve nto-logicalclaims of thisnature, ne wayor another.Pure second-orderogicdoes not have an ontology, xceptfor omerelativelynnocuous tems iketheempty ropertyor set)and the universal roperty,nd these are notsufficientlyobust oground non-trivial athematicalheory.

Since Plato,philosophers ave detected differenceetween heir wnperspectivend that of the mathematicians. he metaphysical ature ofmathematical bjects s a distinctivelyhilosophical oncern.Most mathe-maticians re not interestedn such ultimate ntological uestions.Theycare, qua mathematicians,bout the ultimate nature of mathematicalobjects only to the extentthat this nature bears on theirprofessionalconcerns the mathematicalropertiesf numbers nd points, or xample.Ofcourse, hisdistinctionependson what hemathematical,s opposedtometaphysical, roperties re, and thisin turndependson what mathe-matics s.I do notknow how one wouldgo about arguing hatthere s a uniqueontological oundation or ll ofmathematics, uch essarguing hatV,ortheuniverse fante emtructures,r of theobjectsdelivered yabstractionprinciples,tc., s thisuniqueontological oundation. o be sure,one caninterpretvery xistingmathematicalheoryn settheory,n some sense of'interpret'. nd one can axiomatizestructureheory nd theninterpretexistingmathematicalheoriesn it.2And one can interpret athematicaltheories n the category f categories.A similarresultdoes not seem tobe available for traditionalFregean logicismor ramified ype theory(withouthe axiomofreducibility).he standard ttemptt the formersnotconsistentandis thus oostrong),nd the atterppearsto be tooweak.At present, t is an open questionwhether here are relatedresults orneo-logicism.3But whatdo these nterpretationesultshow, ven whenwe have them?The most one can conclude is thatthe favoured ntological oundationcannot be ruled out on purely ogical grounds.For example,we mustconcede that et-theoreticoundationalistso not contradicthemselvesn

2See myPhilosophyfMathematics:tructurendOntologyOxfordUP, 1997),PP. 93-7.3 See, forexample, my Prolegomenon o AnyFutureNeo-Logicist et Theory',BritishJournalforhe hilosophyf cience,4 (2003),PP. 59-91.C The Editors f The hilosophicaluarterly,oo4



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FOUNDATIONS OF MATHEMATICS 19claiming hat V is the unique foundation.Neitherdo the ante em truc-turalists,ith nalogous laims, tc. But howdoes a set-theoreticntologicalfoundationalist,or xample,go on to establish,r evenarguefor, heclaimthat Vis theuniquefoundation? ince we can interpretverymathematicaltheoryn more thanone system, ow do we knowwhich s theright ne?Presumablyhere an be onlyone 'being'or 'intrinsic ature'ofanygiventype fmathematicalbject.The problem s compoundedwhen we notethatthere s typicallymorethan one wayto interpret givenmathematicalheory,ikearithmetic,neach ftheproposedontological oundations.or example, ven ifwe endup settlingn V as the ultimate oundation,heontological oundationalistmust stillresolve the Benacerraf roblemof showingust which set thenumber is,or which et s identical o eachpointofEuclideanspace,etc.4Againthere an presumablye onlyone being'or intrinsic ature' feachindividualmathematicalbject ike henumber .The Benacerraf ilemma s one ofthe bestsupports or tructuralism.na sense,the intrinsic atureof the number2 is what all of its variousinterpretationsave in common,namely, eingthe indicatedplace in anc-series.

Supposethat omeoneclaims hat realmA istheuniquefoundation orall ofmathematics,nd someone else claimsthat nother ealmB is. Sup-pose also thateach managesto interpretxtantmathematicsnto his ownpreferredntology. etus assume lsothat headvocateofA agrees hat hetheoryfB isa legitimateranch fmathematics;e ust nsists hatB is notthe realfoundation. he advocateofB makestheanalogousclaimaboutA,concedingthat t is a legitimate,ut non-foundational,ealm ofmathe-matics.Giventhe mutual nterpretation,here s thus n interpretationfBinA, and there s an interpretationfA in B. The problemfor neutraloutsider s tofigureut which if ither) fthesegives herealontology,ndwhich sa merereinterpretation.One reasonforthisstandoffs thatmathematicstself oes not decidebetweenthe alternativentological oundations. s far as mathematicssconcerned, ny of themwill do, or we can just eschew an ontologicalfoundationltogether. orwhat t is worth, hestructuralistof ust aboutanystripe) asa neatexplanationfthese pparentlynresolvabletandoffs.The reasonwhyour advocates ofthefoundations and B can interpretevery xtantmathematicalheory,ncludinghe other's oundation,s thatthe twofoundationalheorieshave a commonstructure. or example, t4 P. Benacerraf,WhatNumbersCould Not Be', Philosophicaleview,4 (1965),PP. 47-73,repr. n P. Benacerraf nd H. Putnam eds),PhilosophyfMathematics,ndedn (CambridgeUP,1983), p. 272-94.

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20 STEWART HAPIROleastformally,he set-theoreticierarchynd the realm ofstructuresrelittlemorethan notational ariants f each other.The set-theoreticier-archy s designedto be a maximaldomain,in thatit can model everyisomorphismype. ince isomorphic ystemsharea structure,hecompre-hensiveness f V amounts o itsability o exemplifyvery tructure. ndonce again,formathematics,tructures all thatmatters. he samegoesfortheuniverse f nte emtructures,hecategoryfcategories,tc.Thereis a modestvariation n thetheme fontological oundations,nethatdoes notpresuppose nythingbout prior)metaphysicalatures rthelike, nd so does notfallprey o thesepriority isputes. he pragmaticallymindedphilosopherW.V. Quine supports set-theoreticoundation ngroundsof economy,or ontologicalparsimony.t is not a questionofdiscoveringhe ntrinsicature four familiarmathematicalbjects.RatherQuine makesa proposal,n general cientificrounds, orcleaningup andrefurbishingur web of beliefs the ship of Neurath.Set theorys anestablished ranchofmathematics ithnumerouspplicationsnempiricalscience.So, givenQuineandoctrines, e are committedotheexistence fsets, ike t or not.Moreover,we can interpretvery xtantmathematicaltheory used in science)in the set-theoreticierarchy.Given that,whyshould we have numbers, unctions,oints, ines and sets n our ontology,when sets alone will do? In otherwords,since we can get away withinterpretingathematicalheoriesnsettheory, e shoulddo so,regardingV(or,forQuine, L) as the olemathematicalntology.Quinean doctrinesiketherelativityfontologynd the nscrutabilityfreference llowforthepossibilityf alternativencompatible oundations,bothequally cceptableon generalpragmatic/scientificrounds. o we arenotexactly ommittedo sets.n accepting heweb ofbelief,we are com-mitted o a structures richas thatofsome settheory r other. et theoryand an alternativeoundation hat s ustas strong eednot be competitors.In choosingone or theother, t is nota matter fgetting hings ight rwrong.In that ase,wemightwonderwhy t s that here an be morethanonefoundation fmathematics.t cannotmerelybe a matter f the under-determinationftheory ydata, sincepuremathematics as no 'data' inQuine's sense beyond omputationsnd the ike).The answer, gain, sthatmathematicss thescienceofstructure.ach ofthenon-competingounda-tionshas an instance feverymathematicaltructuremployedn science,and as far s mathematicsoes, tructuresallthatmatters.

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FOUNDATIONS OF MATHEMATICS 21

II. EPISTEMOLOGYA secondsenseof foundation's epistemic. ere thephilosopherrgues, rsimply laims, hat heproposedfoundationrovides heultimatejustjficationfor achfounded ranch fmathematics.hisnotionmight ave itsnaturalhomewith ogicism. regeexplicitlylaimedthathis definitionsrovide heproperepistemic asis ofarithmeticnd analysiswithinogic.The goal ofhis ogicismwas to show that rithmeticruths re analytic,nd he definedthatnotionnexplicitlypistemicerms:

... thesedistinctions etween a priorind a posteriori,yntheticnd analytic,concern,as I see it, not the content of the judgement but the justificationfor makingthe judgement. Where there is no such justification, he possibility f drawingthedistinctions anishes.... When a proposition s called a posteriorir analytic nmysense,this s not a judgement ... about theway in whichsome otherman has come, perhapserroneously, o believe it true; rather, t is a judgement about the ultimategroundupon which rests he ustification orholding ttrue.... The problembecomes ... thatoffinding heproofoftheproposition, nd offollowingtup rightback to theprimi-tive truths.f, n carrying ut thisprocess,we come onlyon general ogical laws andon definitions,henthe truth s an analytic ne.5Today,we clean up Frege'sdevelopmentyreplacing logic'withlogicplussettheory',nd thenchangethedefinitionsokeepthe numbers rombeingproper lasses.6 he correspondingpistemic oundationalistlaim, fanyonewants omake t, s that heultimate easonfor he nductionxiomforarithmetic,r fortheparallel postulate n Euclideangeometry,tc., sthatonce theproper dentificationsremade,thoseprinciplesreprovablein ZFC. Similarly,heanalogousepistemic laimfor hecategoryheoristwould be that heproper pistemic asis for hetruthsfmathematicsies ntheir ollowingromheaxiomsofcategory heory lusdefinitions.Whether heclaim nvokes heset-theoreticierarchy,ogicist efinitions,categories,nte emtructures,rwhatever,urepistemic oundationalistassome workto do in order to clarify he position.One possible goal, Isuppose, stoprovide ecurityobranches fmathematicsike rithmeticndanalysis. n thecase oflogicism, orexample,we might lleviatepotentialdoubts boutthe naturalnumbers yshowing heaxioms and theorems f5Frege, Die GrundlagenerArithmetikBreslau: Koebner, 1884), tr.J.L. Austin as TheFoundationsfArithmetic,ndedn (Oxford:Blackwell, 959), 3. In a footnote,regenoted thathe did not ntend oassign newsensetothe termsanalytic',synthetic',a priori',a posteriori',but only o state ccurately hatearlierwriters, ant inparticular, ave meantbythem'.6 See, for xample,A. GeorgeandDJ. Velleman,PhilosophiesfMathematicsOxford:Black-well, 002), ch.3.

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22 STEWART SHAPIROarithmetico be logicaltruths.Thiswas notFrege'sorientation. s far s Iknow,he did notseriouslyonsider hepossibilityfdoubting rithmetic.)This orientationoepistemicoundationalisms a non-starter hen tcomesto a set-theoreticoundation. n anyreasonable cale,ZFC is lesssecurethan Peano arithmetic. ow can doubts about the natural numbersbeaddressedby showinghow arithmetican be derived n set theory? hesame would hold of any othercomprehensiveccount,such as categorytheoryrstructureheory.Another trong ersion fepistemic oundationalismouldbe the viewthat no one knowsthetruths f anybranch ofmathematicsntilhe hasprovedthemfrom he true foundation. ccordingly,o one beforeFregeand Cantor had strict nowledge bout the naturalnumbers, uclideanspace, etc. This strikesme as absurd.SurelyEuclid,Archimedes, auchyand Gauss knewsomethingbout the naturalnumbers,fanyonedoes oreverdid.I thinkwe can safelyet aside these trong ersions fepistemic ounda-tionalism. moremodestpositionwouldbe thattheproposedfoundationprovides he ultimateustificationor xioms and theorems hatwe alreadyknow. We prove the axioms fromthe foundation, ot because of anyinsecurityn theaxioms,norbecausewe do notalreadyknowtheaxioms,but to shedlight n their pistemic asis. To paraphraseFrege, t is goodmathematicalndphilosophical ractice oprovewhatone can. In order oprove propositions aken to be axioms,we providedefinitions f theprimitivesfthefounded heorynterms fthefoundation. sFrege Grund-lagen,2) put t:

The aim ofproofs, nfact, otmerelyoplacethetruthf propositioneyond lldoubt, utalsotoaffords insightnto hedependence ftruthsponone another.After e haveconvincedurselveshat boulder s immovable.. there emainshefurtheruestion, hat s tthat upportst osecurely?One possible esult f this uest, he result rege imedfor,s a demonstra-tion hat hepropositionsnquestion reanalyticnd/orknowable priori.Perhapsthis notion offoundations as metaphysicals it is epistemic,despite heuse ofnotionsike proof'and 'justification'.t is not a questionofwhethereknow, or xample, hat + 5 = 12, to takeFrege's and Kant's)ownexample.There sreally oquestion ut thatwe do know hat.Nor is ta questionof howwe know that7 + 5 = 12. We knewthatpropositionlongbefore hefoundationalworkbegan.Moreover,our ownknowledgedid not need to go, and in fact did not go, via the proposedfoundingdefinitions. e ustdidthe um.Fregewasinterestednobjective roundingrelationsmongpropositions,erhaps omethinglongthe inesofBernard? The Editors of ThePhilosophical uarterly,00oo4



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FOUNDATIONS OF MATHEMATICS 23Bolzano's ground-consequenceelation.7 his seemsto drive wedgebe-tween he tate fbeingustifiedndtheultimate round r ustificationfaproposition.

Putting his terminologicaland exegetical)matteraside, the modestepistemic oundationalisms still roblematic.t is one thing o interprettheoryA in a theory , and another o claim thatB provides heultimatejustificationorA (via the interpretation).rege,forexample,was surelyaware that Euclidean geometry an be interpretedn R3, via ordinaryanalyticgeometry. et he did not hold that real analysisprovidestheground rproof r ultimateustificationorgeometry.8fhe had,hewouldhave heldthatgeometrys analytic. ut nfactFregeacceptedthe Kantianthesis hatEuclideangeometryssyntheticpriori.Even ifFregewaswrong boutthestatus fgeometryorreal analysis),this hicken nd egg ssue eems ntractable.olzano noted hat heground-consequencerelation s asymmetric.fp is infact heground fq,thenq isnotthegroundofp. As above,Euclideangeometryan be interpretednrealanalysis. ut realanalysis an be interpretednEuclideangeometry.odo wehold that ealanalysis rovides heultimateustificationor uclideangeometry,r thatgeometryrovides he ultimateustificationor nalysis?Do we provefacts boutreal numbers y invoking efinitionsn terms fpoints, r doweprovefacts boutpoints nd linesby nvokingefinitionsnterms frealnumbers?If therewere any epistemic et-theoreticoundationalists,heywouldclaimthat heultimatepistemic asis for he basicprinciplesfarithmeticandfor thefirst-orderersion f)Hume'sprinciplere theaxiomsof ZFC.Of course, hesuccessfulnterpretationfevery xtantmathematicalheoryin ZFC does not establish pistemic oundationalism.s withontologicalfoundationalism,hemost hegrand nterpretationhows s thatwe cannotrule out epistemic et-theoreticoundationalismn logicalgrounds.Thepersonwhoclaims hatZFC provides ltimateustificationoesnot contra-dicthimself. do not knowwhatfurtheronsiderationsould be broughtobearon therelevantpistemiclaims.The literaturen neo-logicismuggests n even moremodesttypeofepistemic oundationalism,ne thatdoes notrely n objectivemetaphysicaland asymmetricroundingelations etween ropositions.9he idea is thatthefoundationrovides neway nwhich hemathematicalropositionsnquestioncould have become known. t does not matterwhether nyone

7See B. Bolzano, TheoryfScience1837), r.R. George Univ.ofCalifornia ress, 972).8 See R. Heck, 'Finitude and Hume's Principle', ournalfPhilosophicalogic, 6 (1997),PP. 589-617.9Here I am indebted oconversations ithCrispinWright.C The Editors fThe hilosophicaluarterly,004



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24 STEWART SHAPIROactually ame to know thepropositionsia theproposedfoundation. negoal of theenterprises to show thatthe mathematical ropositionsre apriorinowable. his is only) o showthat hey dmit fbecoming nownnan a priorimanner but this s what t means to be a priorinowable.Theprogrammes a reconstructivepistemology.Frege'stheorem s that the Peano-Dedekindpostulates an be derivedfromHume's principle. o the Peano axiomscan enjoywhateverpistemicstatusHume'sprinciple njoys. fthe attersanalytic,r allbutanalytic,ran implicit efinition,r otherwise nown or knowable a priori,hen theaxiomsand thus the theorems f arithmetic an have thatsame specialepistemic edigree assuminghat he derivationreserveshe relevant pi-stemic tatus). he neo-logicisteed notthink hat hisprogrammerovidesthetrueustificationor he successor xiomand the otherPeano-Dedekindpostulates.Maybe there s no such thing.But theprogramme till hedslight n the tatus ftheaxioms ndconsequences. rispinWright rote:

Frege's heorem.. [ensures]hat hefundamentalaws ofarithmetican be derivedwithin systemf second-orderogic augmented y a principlewhose role is toexplain,fnot xactlyodefine,hegeneral otion f dentityfcardinal umber,ndthat his xplanation roceedsnterms f a notionwhich anbe definednterms fsecond-orderogic. f such an explanatoryrinciple.. can be regarded s analytic,then hat hould uffice.. todemonstrateheanalyticityf arithmetic.ven fthatterm s found roubling.. it willremain hatHume'sprinciple like nyprincipleservingmplicitlyo define certain oncept willbe availablewithoutignificantepistemologicalresupposition....o oneclear priorioute nto recognitionfthetruthof ... the fundamental aws of arithmetic .. will have been made out.... So,always rovidedhat oncept-formationyabstractionsaccepted,herewillbe an apriorioute rom masteryf econd-orderogic o a full nderstandingndgrasp fthe ruth f he undamentalawsof rithmetic.'0

Similarly, neo-logicist oundation or real analysiswould provide anabstraction rinciple hatgivesa possiblebasis forknowledge f the realnumbers,tc.Despitethecategoricityfsecond-orderrithmetic,heneo-logicistoesnot claim to have provideda possible epistemicfoundationfor everyarithmeticruth. et s be any truesentence n the languageof first- rsecond-order rithmetic. hen s is in facta (semantic) onsequenceofHume's principleplus the usual definitions.his alone does not tellusanythingbout theepistemic edigree fthe entence (orthepropositiontexpresses). o ascertain heepistemictatus f swe needtodetermine owweknow ormight now) hat is a consequence f thePeano axiomsor of

10C.J.G.Wright,On thePhilosophicalignificancefFrege's heorem',nR. Heck ed.),Language,hought,ndLogicOxfordUP, 1997), p. 201-44, tpp. 21O-II.? The Editors fThe hilosophicaluarterly,004



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FOUNDATIONS F MATHEMATICS 25Hume's principle lusthe definitionsif nfactwe do knowormight nowthis). fwe invoke ettheoryo showthat isa consequence fsecond-orderPeano arithmeticor,amountingothe amething,o showthat istrue fthenaturalnumbers thenwe have shownno morethanthat heepistemicstatus fthearithmeticentence maybe boundup with hatof settheory.As is suggested y thepassage fromWright bove, the neo-logicist nlyclaims that f n arithmeticroposition can be derivedromHume's prin-ciple plusdefinitions)na standard eductiveystemor econd-orderogic,thens is knownwithout ignificantpistemological resuppositions'.f s istrue, utcannotbe soderived,hen ts pistemictatus s left pen.It is thus onsistent ithmodest pistemic oundationalismhatdifferentsentencesnthe ame anguage andifferntheir pistemicedigrees. ame-nessof ubject-matteroesnotguarantee ameness fepistemictatus.11 eknowfrom he ncompletenessheorem hat here s no consistenteductivesystemhathasamong ts onsequences very rithmeticruth. he categor-icityof second-order rithmetic oncernsthe structure escribedby thetheory,nd isprettymuchorthogonalo theepistemicssueshere.The neo-logicistaces serious ersion f theforegoinghicken nd eggissue.Frege'stheorems that hePeano-Dedekind axioms of second-orderarithmetican be derivedfromHume's principle lus definitions. ut toprovidetherequisite pistemic asisforthe basic principles farithmetic,we need some assurance hat hetheory evelopedfromHume's principle,sometimes alled Fregearithmetic',s in fact rithmetic.n otherwords, omakegood theepistemic laimsofneo-logicism, e need some assurancethat the items definedfromHume's principleare in fact the naturalnumbers hatwe all know and love - the same naturalnumbers s werestudiedby Euclid,Archimedes, auchy and Gauss, not to mention henaturalnumbersused by ordinary eople in ordinary ank statements.Otherwise,he mostthattheneo-logicistas shown s that t s possible oobtain priorinowledgef a system hichhappens o be isomorphicothenaturalnumbers.Alongsimilarines, here re at least two differentroposedneo-logicisttreatmentsfrealanalysis.'2t ispossible, suppose, hat ne of themmightbe disqualified,ince it invokes n illegitimatebstraction rinciple.Butthis s unlikely,incethe abstraction rinciplesn the twotreatmentsrestructurallyimilar, nd fareequallyon proposedcriteria or abstractionprinciples.So each of the theoriesprovides a legitimateepistemic

11See my Induction nd Indefinitextensibility',ind, 07 1998),pp. 597-624-12 See B. Hale, 'Reals byAbstraction',hilosophiaathematica,(2000), pp. 100-23; Shapiro,'FregeMeetsDedekind',NotreameJournalf ormalogic, 1 2000), pp. 335-64; see also NeilTennant's ontributiono this ssue,pp. 105-33below.C The Editors fThe hilosophicaluarterly,004



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26 STEWART HAPIROfoundation or omemathematicalheory,f either f themdoes. Butwhatreason s there o hold thatone ofthem, r bothofthem, an providean apriorioutefrom masteryf second-orderogicto a full nderstandingndgraspof the truth f the fundamentalaws' of the real numbers?Whatreason s there othinkhat he abstractsroducedbythevariousprinciplesare infact hereal numbers?As with heontologicalssues, he structuralistofanystripe)s bemusedby these issues.All of the various accountsof arithmetic,orexample,deliver he samestructure,nd so all areas correct s an account an be. Ofcourse,some accountsare more perspicuous, nd some have importantmathematicalamificationssee the nextsection), utphilosophicallyhere snothing o choose between them.They are all correct, n thattheyalldeliver heproper tructure. o be sure, hestructuralistaynot be inter-ested n thepresent uestion oncerning ossible pistemic edigree.Evenso,there san importantpistemicole to be playedbythevarious ccountsof the naturalnumbers.They help to provideassurancethatthePeano-Dedekind descriptions coherent.That is, they help to show that thestructure xists or is possible).But there s no questionofgettingn iso-morphic mpostornstead ftherealthing.

In contrast, eo-logicistso take theissueseriously. heymustprovideassurance hattheabstractionrinciples eliver herequisite nowledge fthepromisedmathematicalbjects.One promising pproach, heonlyoneattemptedo far, sto invoke rege's laim that hetypical pplicationsf abranchofmathematicshouldflowdirectlyrom he correct ccountof thenatureoftheobjects. shall call thisFrege'sonstraint.ccordingly,ume'sprinciple rovides satisfyingprioriustificationor he Peano-Dedekindaxiomsbecause Hume'sprinciple ecapitulatescentral eature ftheuse ofthenaturalnumbersnmeasuringardinality.he other ccounts f arith-metic, ncluding he von Neumann finite rdinals, heZermelonumerals,the ante emtructure,nd even Frege'sdefinitionn terms f extensions,delivern isomorphicmpostor.UnlikeHale's, the accountofrealanalysisnmy FregeMeetsDedekind'eschewsFrege'sconstraint,nd goes straightor hestructure. full reat-mentofFrege'sconstraint ould takeme too far fieldhere; besides, donot have much to sayabout t.13 In anycase, it is notclear how the ana-logous epistemicssue shouldbe resolved orbranches fmathematicsuchas complex nalysis, unctionalnalysis nd homotopyheory, here hereseems to be no standardapplicationto latch onto. In most cases, theapplicationsame after hetheories erewelldeveloped.13For a lucid account of the issues,see Wright,Neo-FregeanFoundations for RealAnalysis', otreameJournalfFormalogic,1 2000),pp. 317-34.C The Editors fThe hilosophicaluarterly,004



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FOUNDATIONS F MATHEMATICS 27

III. MATHEMATICSA third ense of foundation' indsucid articulationyPenelopeMaddy.14Althoughhefocuses n settheory,imilar laimsmight e made on behalfofany comprehensiveoundationwith erhapsdifferentegreesofplaus-ibility).ollowingMaddy, beginwith passagefrom iannisMoschovakis'chapter ntitledAreSetsAllThere s?':

[Consider] the 'identification' f the ... geometricline ... with the set ... of realnumbers, ia the correspondence hich identifies' ach point ... with ts co-ordinate....What s theprecisemeaning f thisidentification'?ertainlyot hatointsare eal umbers.en havealwayshad direct eometricntuitionsboutpointswhichhavenothingo do with heir o-ordinates.... hatwe meanbythe identification'...is that hecorrespondence.. gives faithfulepresentationf[the ine] n [therealnumbers] hich llowsus to givearithmeticefinitionsor ll the useful eometricnotions nd to studyhemathematicalropertiesf[the ine]as ifpointswererealnumbers ... we ... discoverwithinthe universeof setsfaithfiulepresentationsf all themathematicalbjectsweneed, nd wewill tudyettheory.. as if llmathematicalobjectswere ets.The delicate roblemnspecificases s to formulatereciselyhecorrect efinitionf faithfulepresentation'ndtoprove hat ne such xists.15On Maddy'sgloss p. 26), the ob of set-theoreticoundationss to isolatethemathematicallyelevant eaturesfa mathematicalbjectand to findset-theoreticurrogatewith thosefeatures'. his spirit f as ifall objectsweresets' s differentrom hemetaphysicalnd epistemic oundationalismarticulated bove. The mathematical oundationalistoes notsay, nd per-haps does not care,whethernumbers, oints, tc.,are sets.Rather,onemakes mathematical se ofthe factthat some setsfaithfullyepresenthenumbers.What is thepurposeof thismathematicaloundationalism?addyputs twell p. 26;my talics):The answer .. lies nmathematicalather hanphilosophicalenefits. he force fset-theoreticoundationss to bring surrogatesor) ll mathematicalbjects nd(instantiationsf)allmathematicaltructuresntoone arena- theuniverse f setswhich llows herelationsndinteractionsetween hem o be clearly isplayedndinvestigated.urthermore,he et-theoreticxioms..haveonsequencesor xistingfields....Finally, erhapsmostfundamentally,hissingle,unified rena formathematicsprovides court ffinal ppealfor uestionsfmathematicalxistencendproof:fyou want to know f there s a mathematicalbjectof a certain ort,you ask(ultimately)fthere s a set-theoreticurrogatefthat ort; fyouwant o know f14P. Maddy,NaturalismnMathematicsOxfordUP, I997),pp. 25-35-15 .Moschovakis,otesn et heoryNewYork: pringer,994),PP.33-4.

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28 STEWART SHAPIROgiven tatementsprovable rdisprovable,oumean ultimately),rom he xioms fthe heoryf ets.

The mathematicalayoffs considerablepp.34-5):... vaguestructuresre made moreprecise, ld theoremsregivennewproofsndunified ith ther heoremshatpreviouslyeemeddistinct,imilar ypothesesretraced t thebasis ofdisparatemathematicalields,xistence uestions re givenexplicitmeaning,nprovableonjecturesan be identified,ewhypothesesansettleoldopenquestions,nd soon.

Maddy concludes that these mathematicalbenefits f foundations resufficient:No metaphysics,ntology,r epistemologys needed tosweetenthispot '. Nevertheless,he success of mathematical oundationalismassomephilosophical amifications.ot to sweeten hepot,but tohelpus tosee what hepot ooks ike.On the defusedmatter fexistence,Maddywrites hattheset-theoretichierarchyprovides court offinal ppeal forquestions f mathematicalexistence.. ifyouwant oknow f heres a mathematicalbjectof certainsort,you ask (ultimately)fthere s a set-theoreticurrogate f that sort'.Despite its reputation orclarity nd exactitude,mathematics as seenconsiderable ontroversy.eyondthenaturalnessfthe naturalnumbers,there are negative, rrational, ranscendental,maginary nd complexnumbers.n geometry,here re also idealpoints t infinitynd imaginaryelements. s is indicated ythenamesofthese ntities,heir xistencewasonce controversial. ithhindsight,here re essentiallyhreeways nwhich'new' entities ave been incorporatedntomathematical heories.16ne issimply opostulatehe existence fmathematical bjectsthatobeycertainlaws.Complexnumbersre likerealnumbers utclosedunder hetaking froots, nd idealpoints re likerealpoints ut not ocated nthe sameplaces.Of course, fsomeonedoubtsthe existence f theentities,ostulation egsthe question.The secondmethod s implicitefinition.he mathematiciangivesa descriptionf the system fentities, suallyby specifyingts aws,and then sserts hat hedescriptionppliesto any ollection hatobeysthestipulatedaws. The thirdmethod s construction,here themathematiciandefines he new entities s combinations f alreadyestablished bjects.Hamilton'sdefinitionfcomplexnumbers s pairsofreals fits hismould.This last s clearly hesafest nd mosteffective ethod.No extra ssump-tions remade;no questionsrebegged.In this last case, what is the relationship etween the controversialentities,ay,the complexnumbers, nd the constructedntities, rdered

16 ee E. Nagel, ImpossibleNumbers', n TeleologyevisitedndOtherssays n the hilosophyandHistoryf cienceColumbiaUP, 1979), p. 166-94.C The Editors f The hilosophicaluarterly,004



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FOUNDATIONS OF MATHEMATICS 29pairsof reals?One can think ftheproposedconstructionss giving ixeddenotations othenew terms. he imaginaryumber justis thepair.This is unnatural,for much the same reason as metaphysical ounda-tionalism s unnatural.To echo Benacerraf nd Moschovakis, he pair has propertiesne would not attribute o thecomplexnumber . Afruitfulutlookwouldbe to takeconstructionntandemwith ither ostula-tion or implicit efinition. constructionfa systemfobjects stablishesthat here re ystemsfobjects husdefined;nd so the mplicit efinitionsnotemptynd thepostulationsatleastcoherent. o adapttheterminologyofMaddyandMoschovakis,he constructionrovidessafe) urrogates,ndfaithfulepresentations,ftheerstwhileuestionablentities.

This is thecontextnwhich foundationrovides n arena for ettlingquestions fexistence.n thenineteenthentury,heconstructionsypicallytookplace inordinaryuclideangeometry. arkWilson llustrateshehist-oricaldevelopmentnd acceptance fa space-time ith n 'affine'tructureon thetemporal lices:... theacceptancef .. non-traditionaltructuresoses delicate roblemorphilosophyfmathematics,izhow can the novel tructurese brought nder heumbrella fsafemathematics?ertainly, e rightlyeel, fterufficientoodleshavebeendepositedn coffeehopnapkins,hatweunderstandhe ntendedtructure....But t shard o find fullyatisfactoryay hat ermits smoothntegrationfnon-standard tructuresntomathematics.... ewouldhopethatanycoherenttructurewecandream pisworthyfmathematicaltudy.. . The rub omeswhenwetryodetermine hether proposed tructures coherent' rnot.Raw 'intuition'annotalways e trusted;ven hegreatRiemann ccepted tructuress coherenthat aterturned utto be impossible.xistencerincipleseyondit eems kay o me'are neededtodecidewhetherproposed ovel tructuresgenuinelyoherent.. latenineteenth-centurymathematiciansrecognizedthat .. existenceprinciples .. need to piggybackeventuallypon someacceptedrangeofmoretraditional athematicaltructure,such s theontologicalrames f rithmeticrEuclidean eometry.n ...ourcentury,set theory as becomethe canonicalbackdrop o whichquestions f structuralexistencerereferred.l7There is no a priori eason to expecta unifiedmathematical oundation.Experiencewithparadoxesshouldmake us waryof the possibilityf atheoryfeverythingorevena theoryfsurrogatesor verything.utthisis ustwhatwe seemtohave with FC.Similarly,we have no advance reasonto expecta singlemathematicalfoundation.fwe have one foundation, hynot twoor more? fwe loseinterestn thethesis hatthere s a metaphysicalubstance hatunderliesall mathematical bjects,we need not view alternative oundations s17M.Wilson,There's Holeanda Bucket,earLeibniz',nP.Frencht l. eds),MidwestStudiesnPhilosophy8 MinnesotaUP, 1993), p. 202-41, atpp. 208-9.

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30 STEWART SHAPIROcompetitors. e alreadyknow hat hevery ame structurean havemanysurrogatesn the set-theoreticierarchy.Whycannot here e surrogatesnotherdomains as well? Let manyflowersry o bloom,even ifnot all ofthemdo.Of course, ftwo domainsare both egitimateoundations,n the fore-going ense, henpresumablyach is a legitimateranch fmathematics.oeach domain will have a surrogate or he other.This may spark debateover which is the surrogate nd which is the real item,but from themathematicalerspective,hat s a wrong-headed ispute.Wilson remarksthat anynotion hat he reals shouldn't e identified ith etsrepresentssgreata misunderstandingf mathematicalntology s the claim thattheyshould'. n these ontexts,alk f dentitysmisplaced.'8Moschovakisp. 34)wrote hat he delicateproblemnspecificases stoformulate recisely he correctdefinition f "faithfulepresentation"ndto provethat one such exists'.Can the faithfulness'f the surrogates eshown at all? Is the delicateproblem'even a mathematicaluestion? f theoriginal heory as been formalized,hentheanswerto bothquestions s'Yes'. A formalizations itself mathematicalbject, nd we can establishthe faithfulnessf thesurrogateso theformalizedheory. orexample,wecan show nmodeltheoryhat heformalizations categoricalnd that hesurrogatesre a model of theformalization. his establishesdequacy,ifanythingoes.However,no first-orderormalizationf theory ith n infinite odel scategorical. o inorder o establishdequacy,theformalizationeedstogobeyondfirst-orderogic. t seems o me that econd-orderogic,or at leastanon-compact ogic beyond first-order,s a crucial component,or pre-requisite, fthe mathematical oundationalistrogram.We need to knowwhat the structures beforewe can lookfor tssurrogatesn V or anywhereelse. am aware,ofcourse, hat here relingeringuestions oncerninghedeterminacyf thesecond-orderonsequence elation. or what t sworth,myownview is that thedeterminacyfsecond-orderogicstandsor fallswith the determinacyf mathematicstself.But this is not the place toengage nthatdispute.'19Withformalized heories, ne might tillponderthe adequacy of theformalization. ndwhat ftheoriginal heory as notbeen formalized? tthis evel,can one 'formulate recisely' mathematically 'the correctdefinitionf "faithfulepresentation"nd ... prove hat ne suchexists'?

18 ee my Structure nd Identity',n C. Wright nd F. MacBride (eds),BeingCommitted(OxfordUP, forthcoming).19 ee myFoundationsithoutoundationalism:Case or econd-Orderogic OxfordUP, i991),chs5, 9-? TheEditorsfThe hilosophicalUarterly,004



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FOUNDATIONS OF MATHEMATICS 31Church's thesisprovidesa case in point.The informalmathematicalnotion s that fcomputability,nd theproposed urrogatesthenotion farecursivefunctionor, to be precise,the set-theoreticurrogate f the

arithmeticotionofa recursive unction). therexamples re not hardtofind.What is therelationshipf thepre-theoreticotions fmeasure, rea,polyhedronnd continuityo theirofficial' et-theoreticefinitions?o wehave mathematical ertainty f faithfulness?f not, then why followMoschovakis nd speakofprovinghat surrogatexists?Here opinions ary, utthis s anothermatter hatwould takemetoofarafield. ufficetto notethatmathematicians,uamathematicians,ngage nthe activity f providing urrogates,nd sometimeswe have compellingreasonsto think surrogates correct.This is a prime exampleof whatKreiselcalls informaligour'.20n anycase,neither he delicate'problemofformulatinghe correct efinitionortheproblem festablishingt to becorrect re partof settheory.hat is, one cannotshowin ZFC that thedefinedurrogatesre indeed faithfulepresentations'ftheoriginaldeas,for hesimplereason thattheoriginaldeas are not n the anguageof settheory. he foundation annot stablish tsownadequacy.So at least somecentralactivity f mathematicianss not captured in the set-theoreticfoundation.nstead, t s an essential reliminaryoeach instance ftheset-theoreticrogramme.The Maddy-Moschovakisccountofset-theoreticoundationsrovidesmore gristfor the structuralist ill.21Maddy (p. 34) makes an analogybetweenmathematical et-theoreticoundationalismnd thenow commonthesis hat everythingtudiednnatural cience sphysical'. shallcall thislast physicalism'.Maddy pointsout that itdoesn't follow rom physical-ism]thatbotanists,eologists,nd astronomershould ll becomephysicists,should ll restrictheirmethods othose haracteristicfphysics'. imilarly,the set-theoretic athematical oundationalisteed not claim thateverybranchof mathematicshouldbe studied ythe methods fsettheory. orneed thecategoryheoristrguethateverybranchofmathematicshouldbe studied ythemethods fcategoryheory,tc.Quite correct.However,one crucialdisanalogy etweenmathematicalfoundationalismnd physicalismoncerns he talk of surrogates'.Maddyand Moschovakis othcorrectlynsist hatwe do not identifrealnumberswith ertain ets, ikeDedekindcuts ntherationalnumbers. llwe wantorneed are 'faithful epresentations'f the real numbers.In contrast,physicalism roclaims hat the resources f physics an providethe true

20See G. Kreisel, Informal igour ndCompleteness roofs',n . Lakatos ed.),Problemsnthe hilosophyfMathematicsAmsterdam: orthHolland,1967), p. 138-86.21Some ofthematerial hat ollowss drawnfrommy rticleSet-Theoretic oundations'.C The Editors fThe hilosophicaluarterlv,004



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32 STEWART SHAPIROidenti yfany legitimate hysical onceptor object,be it from hemistry,biology, sychology,tc. puttingsidedelicate ssues, uch as thedistinctionbetween ype dentitynd token dentity,n articulatinghysicalism).neimportant iscovery,or xample,was thatheat is meankinetic nergy.twill notdo fora cautiousphysicalisto demur from alk of identity'ndclaimonlythat mean kinetic nergys a 'faithfulepresentation'f heat,whateverhatmightmean.Maddy points ut that the set-theoreticxioms .. haveconsequences orexisting ields'.For example,set-theoreticheorems bout the finite onNeumannordinalscorrespond o truths bout the naturalnumbers, ndsome ofthese et-theoreticheorems re not deductive onsequences f theoriginal ormalizationsfarithmetic. he samegoesforvirtuallyvery ichfield fmathematics.y studyinghesurrogates, e can learn moreabouttheoriginals sometimes lotmore.In other contexts, ne would not expect this from faithful epre-sentations'.uppose havebeforeme a faithfulepresentationfGlasgowa very good map, for example, or perhaps a large set of differentialequations in quantum mechanics. s there something bout Glasgow,formulatednordinaryanguage, hat can learnfrom tudyinghe faithfulrepresentation,ut that could not learn,even in principle, y studyingGlasgow tself?How does the neat trickwork n mathematics?Whyis it thatfaithfulrepresentations surrogates are all that we want or need,and that nofurtheruestion f dentitys pertinent? hycan thesurrogateseplaceheoriginals, t least in principle, nd why is it thatwe can extend ourknowledgeftheoriginals ystudyinghe urrogates?The answers o thesequestionsie in thesloganthatmathematicss thescienceof structure. ll thatmatters bout thenaturalnumberss their e-lationshipsoone another.n a sense,we do notstudy henaturalnumbersinarithmetic,ut rather he naturalnumber tructure,heform ommon oanycountablynfiniteystemfobjectswith successor elation atisfyinginductionnd the otherPeano postulates. structurean be characterizedbyan axiomatization,hich s an implicitefinition.he characterizationssuccessfulf t is categorical ifall itsmodels are isomorphic.Here againwe see the rolefor logicthatgoes beyondfirstrder.)Since isomorphicmodels are equivalent,the relevantpropertiesof any model of theaxiomatization re the same,and so, in a sense, nymodel is as good asanyother.We can learnabout thestructureystudyingn exemplificationof t.Russellnotedthatthemathematicianan adopta version fstructural-ism, ven f hephilosopherannot:? The Editors fThe hilosophicaluarterly,oo4



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FOUNDATIONSOF MATHEMATICS 33... the mathematicianeednotconcern imself ith heparticular eing r ntrinsicnatureof his points,ines, nd planes,evenwhen he is speculatings an appliedmathematician....A] point'..has obesomethinghat snearlyspossibleatisfiesouraxioms, ut t does nothaveto be verymall' r withoutarts'.Whether rnotit s those hingssa matterf ndifference,olong s it atisfieshe xioms.fwecan... construct logicalstructure.. which[satisfies]ur geometricalxioms, hatstructuremay legitimately e called a 'point' ... we must .. say 'This objectwe haveconstructeds sufficientor hegeometer;tmaybe one ofmany bjects, nyofwhichwouldbe sufficient,ut hat s no concernfours ..'. This sonly n illustrationf hegeneral rinciplehatwhatmattersnmathematics.. isnot he ntrinsicature fourterms, ut thelogicalnatureof their nterrelations.e may say,of twosimilarrelations,hat hey avethe ame structure'.ormathematicalurposesthough otfor hose fpurephilosophy)he nly hingf mportancebout relations the asesin whichtholds, ot ts ntrinsicature.22

From theparentheticalemark t the end of thispassage,Russell eemstohaveheld thatunlikemathematicians,hilosopherso concern hemselveswith the beingor intrinsic ature' of mathematicalbjects.The ante emstructuralist,t least, takes the 'being' and 'intrinsic ature' of naturalnumbers, or xample, obe their elations oone another. he eliminativestructuralistenies hatmathematicalbjectshave a nature since na sensetheydo notexist s independent bjects). venthatdispute iesbeyond hepurviewofmathematics.n contemporaryargon,numbershave a func-tional definition.o there s not so largea gap between heperspectivefthemathematicianndthestructuralisthilosopherfmathematics.This iswhytheabove questions f dentityo not matter o the mathe-matician even ftheydo to thephilosopher);nd in a sense suchquestionsare misplaced. f one capturesthe structure,ne captures verythingfmathematical elevance.For purposesoffixing ruth-values,ny instanceof the structure illdo. Moreover,when we find n instance f a structurein a richcontext,uchas theset-theoreticierarchy,henwe can rely n apowerful heoryfthatcontext', FC, toshed ight nthe tructure.Advocates of set-theoreticoundations,n thismathematical ense,cancitesomedetailsnaddition othegeneric laimofcomprehensivenessthebare idea that n Vevery tructureindsn isomorphicopy.Maddy'spointthatdifferentranches fmathematics avetheir wndistinctive ethodo-logies sbeyondreproach.Neverthelesst sarguable hat heprimitivesndsomeofthedefinitionsn settheory ermeatemathematics.or example,anymathematician ho considers family fsets, ndexedby a set (e.g.,{SI i e I}), is nvokinghereplacementrinciple. ny lgebraist hoformsquotients invoking owerset. he set-theoreticefinitionsf orderedpair,

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34 STEWART SHAPIROunion,function nd relation re taught n elementaryourses across thediscipline,o thatthey onstitutehe common anguageof mathematics.23Also,as Maddy (p. 26) puts t, theforceof set-theoreticoundationss tobring surrogates or)all mathematical bjectsand (instantiationsf) allmathematicaltructuresntoone arena- the universe fsets which llowsthe relations nd interactions etweenthenito be clearlydisplayed ndinvestigated'.he toolsof ettheoryre welldesigned or his tudy.Advocates fcategoric oundationsmake differentutanalogousclaims,pointing o theubiquitous oleof function nd morphismnmathematics.Theyalso showhow functorsreusefulnstudyingelations etween iffer-ent structuresi.e.,categories). neo-logicist ight ry odevelop similarline, mphasizingheubiquityfabstractionnmathematics,ut, s far s Iknow, ucha thesishas notbeenputforward. or are similar laimsmadeconcerningmathematical enefitsframifiedypetheorywhatever hosemight e).There s roomfor ispute mongour actualandpossible oundationalists.Some category heoristsrguethat the staticnotionsfrom ettheory restifling,nd theiropponentsclaim that categorytheory'sfoundationalaccomplishmentsre overblown. he debatesare legitimate,utfrom heperspectivefmathematicaloundations,e need not nsisthat heyhave awinner. here is no reasontoexpect single, niquefoundation. gain eta thousand lowersry o bloom.Maybemore hanone does.I shallconsider, inally, addy's epistemic laim p. 26): 'ifyouwant toknow f given tatementsprovable rdisprovable, oumean ultimately),from heaxiomsofthetheoryf ets'.Here I shallnotbroachthe nalogousissues fortheotherproposedfoundations. n a ruthlesslyiteral eading,Maddy can be takento endorsetheabove goal ofprovidinghe ultimatejustificationormathematical ropositionsfrom II above). Ifone actuallymeanshat he ssueofprovabilityssettledytranslationnto ettheory,henpresumablyettheory oes givethe ultimate round r ustificationor heoriginaltatement.owever, hiswouldbe too uncharitable reading,inceMaddy explicitlyejects uch ultimate oundationalssues.Mathematicians,as such, re notusuallynterestednquestions f ultimate'ustificationorin epistemic edigree.When theywonderwhether givenstatements atheorem,r can be refuted,hey re concernednot so much withhow it sultimatelyrounded, ut withwhether he stated xioms of given heory)and thepremises nd conditions re sufficientorthetheorem. here aregoodmathematical easonsfor his ogicalconcern. tbears on thegeneral-ity ftheresults,or xample.

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FOUNDATIONS F MATHEMATICS 35A set-theoreticnterpretations, in effect, translation rom heoriginallanguage nto thatof settheory. his typicallyorces he mathematicianomakeall definitionsnd reasoning xplicit. o anyhidden ssumptionsndlemmata re brought o light nd acknowledged. or example,one oftheso-calledgapsinEuclid'sElementss theassumptionhat fa linegoesfromthe nteriorf a circle o theexterior,hen tmust ntersecthe circle.Onemight hink hatthere s no needto state his:how couldanyonedoubt t?However, nce theprimitivesre translated'nto he anguage f ettheory,we see the theorem oesnotfollow rom heoriginal xioms and premises.If theproof s to be rigorous,t shouldnotrely n geometricntuition,ndwe see the need forthe explicit xiom of continuityuppliedby later

geometers.I propose qualificationoMaddy's ifyouwantto know f given tate-ment sprovableor disprovable, oumean(ultimately),rom heaxiomsofthetheory fsets'.The qualifications surelyn thespirit f mathematicalfoundations,nd isperhapsobvious.Supposea mathematician,orkingnhisowntheory,singhis ownmethods nd language, ives n argumentora proposition,laimingt to be a proof.Others n the field hallenge his,claiming hat here re ormaybe fallaciesn thetext.A historicalxampleis theperiod fterWiles'originalnnouncement fhisproof fFermat's asttheorem,nd before hegapswereplugged nd consensus eached. Somemathematiciansuestioned heproof.If one takesMaddy's suggestionoo literally,ne would think hat thedispute hould be settled ytranslatingheargumentntothe anguageofsettheory,nd thencheckingo see ifthe results formallyalid. This is afullymechanical rocess,mountingono morethancheckinghesyntax fthe formulaee.g.,countingeft nd right arentheses), aking ure hat hecitedaxioms likethe instances fthereplacement cheme)have therightform,ndthat nstances f modusonensrecorrect. here s notmuchroomfordoubthere. ndeed,that s thepoint supposedly.fthedispute otthisfar,the disputants ould even hire clerksto typethe formulae nto anelectronicmedium,nd thenprogram computero check ts orrectness.This isreminiscentfLeibniz'sdreamof universalharacteristic:

Whatmust e achieved sinfact his: hat very aralogisme recognizeds an errorof alculation,ndthat very ophismhen xpressednthisnewkind fnotation.. becorrectedasily ythe awsofthis hilosophicalrammar....nce this sdone, henwhen controversyrises, isputation illno morebe neededbetween wophilo-sophershanbetween wocomputers.twill ufficehat, eninhand, hey itdownandsay o eachotherLet uscalculate'.2424Leibniz, UniversalScience: Characteristic IV, XV', MonadologyndOtherhilosophicalEssays1686), r.P. SchreckerIndianapolis: obbs-Merrill,965), p. 11-21, tp. 14.

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36 STEWART SHAPIROMaddy did not have anythingike this n mind. f there s a dispute boutthe correctnessf a proof n the realworldofprofessionalmathematics,translationnto settheorywouldnothelp. fthere s anycomplexityn theoriginal rgument and theremust e ifthere s actualcontroversyver tscorrectness then ranslatingt nto ettheorys not at all straightforward.There are likely o be disputesover that.Moreover,the resultinget-theoreticrgumentwillalmost ertainlyot be a derivation.tepsmustbefilledn. The author ftheproofwill laimthat hegapsare allobvious, ndsome willbe. But therewillbe a lot ofsteps, nd itmaynotbe obvious hatall ofthegapsareindeedobvious.Also thefinal esult,f nyone ctually otthatfar,wouldbe horrendouslyong.No one could followt. f theywentthe route ftheclerks-and-computerheck,wewouldworrybouttheposs-ibility f humanerroron the part of the clerks nd the possibilityf asoftwarerror r a hardwaremalfunction.In general,mathematicianslwaysprefer n intuitive roof,where thetrainedreader can see what is goingon, rather han a derivation hat sformally orrect but opaque. Human mathematicians re not usuallyconvinced yan argumentnless hey nderstandt. This isnotpossible orlongand tedious et-theoreticranslations.A formalderivationn the languageof ZFC (or any otherformalizedtheory) s itself mathematical bject. Logicianshave become good atstudyingheseobjectsand proving hings bout them. The completenesstheorem,he ncompletenessheorem nd thousands fotherresults omeimmediatelyo mind. n contrast,roof,roperlyo called,is a rationallyconvincing iscourse howing hat a given propositions true,or followsfromccepted xioms ndpremises. here sindeedgoodreason o dentifyZFC-derivations ithproofs, roperlyo called, n some senseof identify'.The thesis s that ll genuinemathematical roofs ave counterpartsn thelanguageof settheory.t is the same sortofthing s Church'sthesis, ndthe evidencefor he two theses s similar:manytest aseshavebeen estab-lished,no counter-examplesreknown, here redirect rguments,tc. Wedo havegoodreasontoholdthatone does not need togo beyond hebasicprinciples fsettheoryn orderto carry ut ordinarymathematicalrgu-ments, t least nprinciple. o a largeextent,he branches fmathematicsare actuallyxpounded romhoseprinciples,tleastofficially.The questionhere s whatepistemic onclusions an be drawnfrom heidentificationfZFC-derivations ith roof, roperlyo-called.Asbefore,tis not feasible o adjudicatea real-world ase of a purported roofwitha translationnto thelanguageof settheory. imilarly,t is rareto showthat given ext onstitutescomputation yshowinghat tcorrespondsoa Turing-machineomputationr a recursive erivation.fthere eallys a? The Editors of ThePhilosophical uarterly,00oo4



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FOUNDATIONS OF MATHEMATICS 37dispute ver whether hetextdescribes computation,here s likely obea dispute oncerning hetherhe translation'nto heTuring or recursive)formalizations faithful.

The real value of Church'sthesis s its use in showingwhatcannotecomputed.Everyonenow knows hat t is futile o searchfor n algorithmto computethe haltingproblemor to decide first-orderalidity.Why?Because the sets re notrecursive,nd we havegoodreasonto believethatall computablefunctions re recursive. imilarly,he value of the identi-ficationf ZFC-derivationwithproof s in showingwhatcannotbe proved.This is one area whereMaddy's claimgains purchase.We knowthattheparallelpostulates formallyndependentf the other xiomsofgeometryand weknow hat hecontinuum ypothesissindependentftheaxiomsofZFC. So it would be irrational or a mathematician o keep lookingforproofsor refutations)fthesepropositionsnthebasisofthese xioms.25

IV. SUMMARY AND CONCLUSIONTo sum up, I return o the originalquestions.What is a foundation fmathematics?What is a foundation or? have pointedto three sensesof foundation':metaphysical,pistemic,ndmathematical, ith ubcases feach. First,a metaphysical oundationreveals the underlying atureofmathematicalbjects.Mathematics s such does notrequire his.All thatmatters s structure.t is a disputedphilosophical uestionwhether t isinterestingnd importantor philosopherfmathematicso enquire ntometaphysical atters,nyway. econdly,n epistemicoundationeveals hetrueproofs r ustificationsormathematicalropositions,r provides newayinwhichmathematical ropositionsan become known. leave it foranotherdayto delvefurtherntothequestionswhethermathematicalro-positions aveultimateustifications,nd howmuch ight ne can shedon thesubjectby ooking or hem.Thirdly, mathematicaloundations a theoryintowhich llmathematicalheories,efinitionsndproofsanbe translated,at east nprinciple. do notknowwhethermathematics eedsfoundationnthis ense.Perhapsnot.Butthefact s thatmathematics asone,andperhapsmorethanone. I think hesubjects better ff or hat, ut these re mattersonwhich venmathematiciansiffer.Ohio tate niversityUniversityf tAndrews

25For a similar oint, eeJ. Burgess,Proofs boutProofs: Defenseof ClassicalLogic', inM. Detlefsened.),Proof ogicndFormalizationLondon:Routledge, 992),pp. 8-23-? The EditorsfThe hilosophicaluarterly,200oo4

Foundations of Math_Metaphysics, Epistemology, Structure

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