Benacerraf, P., Mathematical Truth

7/27/2019 Benacerraf, P., Mathematical Truth

http://slidepdf.com/reader/full/benacerraf-p-mathematical-truth 1/20

Journal of Philosophy Inc.

Mathematical TruthAuthor(s): Paul BenacerrafSource: The Journal of Philosophy, Vol. 70, No. 19, Seventieth Annual Meeting of theAmerican Philosophical Association Eastern Division (Nov. 8, 1973), pp. 661-679Published by: Journal of Philosophy, Inc.Stable URL: http://www.jstor.org/stable/2025075 .

Accessed: 26/01/2014 22:31

Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at .http://www.jstor.org/page/info/about/policies/terms.jsp

.JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of

content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms

of scholarship. For more information about JSTOR, please contact [email protected].

.

Journal of Philosophy, Inc. is collaborating with JSTOR to digitize, preserve and extend access to The Journal

of Philosophy.

http://www.jstor.org

This content downloaded from 132.248.9.8 on Sun, 26 Jan 2014 22:31:11 PMAll use subject to JSTOR Terms and Conditions

http://www.jstor.org/action/showPublisher?publisherCode=jphil

http://www.jstor.org/stable/2025075?origin=JSTOR-pdf

http://www.jstor.org/page/info/about/policies/terms.jsp





http://www.jstor.org/stable/2025075?origin=JSTOR-pdf

http://www.jstor.org/action/showPublisher?publisherCode=jphil



THE JOURNALOF PHILOSOPHYVOLUME LXX, NO. I9, NOVEMBER 8, I973

MATHEMATICAL TRUTH *

A THOUGH this symposiums entitled MathematicalTruth, I will also discuss issues which are somewhatbroader but whichnevertheless ave the notionof mathe-

maticaltruth t theircore,whichthemselves epend on how truthin mathematics s properly xplained.The most mportant f theseis mathematicalknowledge. t is mycontention hattwoquite dis-tinctkinds of concernshave separatelymotivatedaccountsof thenature of mathematical truth: (1) the concern for having a

homogeneous emanticaltheoryn whichsemanticsforthe propo-sitions of mathematicsparallel the semanticsfor the rest of thelanguage,' and (2) the concernthat the account of mathematicaltruthmeshwith a reasonableepistemology.t will be mygeneralthesis that almost all accounts of the concept of mathematicaltruth an be identifiedwithserving ne or anotherofthesemastersat theexpense of the other.Since I believe further hatboth con-cerns mustbe met by any adequate account, findmyself eeply

* To be presented at a symposiumon Mathematical Truth, sponsored ointlyby the American Philosophical Association,Eastern Division, and the Associa-tionforSymbolicLogic, December 27, 1973.

Commentatorswill be Oswaldo Chateaubriand and Saul Kripke; their com-ments renotavailable at thistime.Various segments f an early 1967) versionofthis paper have been read at Berkeley,Harvard, Chicago Circle, JohnsHopkins,New York University, rinceton, nd Yale. I am grateful or hehelp I received ontheseoccasions,as well as formany commentsfrommy colleagues at Princeton,both students and faculty. am particularly ndebted to Dick Grandy,HartryField, Adam Morton, and Mark Steiner.That these have not resulted in moresignificantmprovements s due entirelyto my own stubbornness.The presentversion is an

attempt to summarize the essentials of the longer paper whilemaking minor improvements long the way. The original versionwas writtenduring 1967/68 with the generous support of the John Simon GuggenheimFoundation and Princeton University. his is gratefully cknowledged.

1 I am indulging here in the fictionthat we have semanticsfor the rest oflanguage, or, more precisely, hat the proponentsof the views that take theirimpetus fromthis concernoften think of themselves s having such semantics,at least for philosophicallyimportantsegmentsof the language.

66i








662 THE JOURNAL OF PHILOSOPHY

dissatisfiedwith any package of semanticsand epistemology hatpurports to account for truth and knowledgeboth within and

outside of mathematics. or, as I will suggest, ccounts of truththat treatmathematical and nonmathematicaldiscoursein rele-vantly imilarwaysdo so at thecost ofleaving t unintelligible owwe can have any mathematicalknowledgewhatsoever;whereasthosewhich attributeto mathematicalpropositionsthe kinds oftruthconditionswe can clearly know to obtain, do so at theexpense of failing to connect these conditionswith any analysisof thesentenceswhichshowshow the assignedconditions re con-ditionsof theirtruth.What thismeansmustultimately e spelled

out in somedetail if I am to make out mycase, and I cannothopeto do thatwithin his imitedcontext.But I will try omakeit suffi-ciently learto permityou to udge whether r not there s likely obe anythingn theclaim.

I take it to be obvious that any philosophically atisfactoryc-count of truth, eference,meaning, and knowledgemustembracethemall and mustbe adequate forall the propositions o whichtheseconcepts pply.2An accountofknowledge hat eemstowork

for certain empirical propositions about medium-sizedphysicalobjectsbutwhichfails to accountformoretheoretical nowledge sunsatisfactory-not nly because it is incomplete,but because itmay be incorrect s well, evenas an accountofthethingst seems ocoverquite adequately. To thinkotherwisewould be, amongotherthings,to ignore the interdependence f our knowledge in dif-ferent reas. And similarly oraccountsof truth nd reference.theory f truthfor the language we speak, argue in, theorize n,mathematize n, etc., should by the same token

provide similartruthconditions for similar sentences. The truthconditionsas-signed to twosentences ontaining uantifiershouldreflectn rele-vantly imilarways the contributionmade by the quantifiers. nydeparture from a theorythus homogeneouswould have to bestronglymotivated o be worthconsidering. uch a departure, or

2 I shall in facthave nothingto sayabout meaning in thispaper. I believe thatthe concept is in much deserveddisrepute, but I don't dismiss it for all that.Recent work, most notably by Kripke, suggests that what passed for a long

time for meaning-namely the Fregean sense -has less to do with truththan Frege or his immediate followersthought it had. Reference is what ispresumablymost closely connected with truth,and it is for this reason thatI will limit my attentionto reference. f it is granted that change of referencecan take place without a corresponding hange in meaning, and that truth isa matter of reference, hen talk of meaning is largelybeside the point of thecluster of problems that concern us in this paper. These comments are notmeant as arguments,but only as explanation.








MATHEMATICAL TRUTH 663

example,mightmanifesttself n a theory hatgave an accountofthe contribution f quantifiersn mathematical easoningdifferent

fromthat in normal everyday easoningabout pencils,elephants,and vice-presidents.avid Hilbert urged such an account in OntheInfinite, which s discussedbriefly elow. Later on, I will trytosay more about whatconditions would expect a satisfactoryen-eral theory ftruth orour languagetomeet, s well as moreabouthow suchan account s to meshwithwhat take to be a reasonableaccountof knowledge.Sufficet to say here that, althoughit willoften be convenient o presentmydiscussion n termsof theoriesof

mathematical ruth,we should

alwaysbear in mind

that whatisreally at issue is our over-allphilosophical view. I will argue that,as an over-all iew, t is unsatisfactory-noto muchbecausewe lacka seemingly atisfactoryccountof mathematical ruthor becausewe lack a seemingly atisfactoryccount of mathematicalknowl-edge-as because we lack any account thatsatisfactorilyringsthetwo together. hope thatit is possibleultimately o producesuchan account; I hope further hat thispaper will help to bringoneabout by bringing nto sharperfocus some of the obstaclesthat

stand n itsway.I. TWO KINDS OF ACCOUNT

Considerthefollowing wosentences:(1) There are at least three arge citiesolderthanNew York.(2) There are at least threeperfect umbersgreater han 17.

Do theyhave the same logicogrammatical orm?More specifically,are theyboth of theform

(3) There are at leastthreeFG's thatbear R to a.

where There are at leastthree' s a numericalquantifier liminablein the usual way in favorof existential uantifiers, ariables,andidentity;F' and 'G' are to be replacedbyone-placepredicates,R'by a two-place redicate, nd 'a' by thename of an elementof theuniverseof discourseof the quantifiers?What are the truth ondi-tions of (1) and (2)? Are theyrelevantlyparallel? Let us ignoreboth thevaguenessof 'large' and 'older than' and thepeculiaritiesofattributive-adjectiveonstructionsn Englishwhich makea largecitynotsomething argeand a citybut more althoughnot exactly)like somethingarge for a city.With thosecomplications et aside,it seemsclear that (3) accurately eflectshe formof (1) and thusthat 1) will be true f and only fthe thingnamed bytheexpressionreplacing a' ('New York') bears therelationdesignatedby the ex-

3Translated and reprinted in Paul Benacerraf and Hilary Putnam, eds.,Philosophy of Mathematics (Englewood Cliffs,N.J.: Prentice-Hall, 1964).








664 THEJOURNAL FPHILOSOPHY

pression eplacingR' ('0 is olderthan0') toat leastthree lements(of the domain of discourseof the quantifiers)which satisfy he

predicatesreplacing 'F' and 'G' ('large' and 'city',respectively).This, I gather, s whata suitable truth efinition ould tellus. AndI think t's right.Thus, if 1) is true, t is because certain ities tandin a certainrelationto each other, tc.

But whatof (2)? May we use (3) in thesameway as a matrix nspellingout the conditionsof its truth?That sounds like a sillyquestiontowhichtheobviousanswer s Of course. Yet thehistoryofthesubject thephilosophy fmathematics) as seenmany otheranswers.Some (including one of my past and present selves ),

reluctant o face the consequencesof combiningwhat I shall dubsuch a standard semanticalaccount with a platonistic view ofthenatureofnumbers, ave shiedaway from upposing hatnumer-als are names and thus,by implication, hat 2) is of theform 3).David Hilbert (op. cit.) chose a different ut equally divergentapproach,in his case in an attemptto arrive at a satisfactoryc-countof the use of thenotionof infinityn mathematics. n oneconstrual,Hilbert can be seen as segregating class of statements

and methods,those of intuitive mathematics, s those whichneeded no furtherustification. et us suppose that theseare allfinitely erifiable n some sense that is not precisely pecified.

Statementsf arithmetic hatdo not sharethisproperty-typically,certainstatements ontainingquantifiers-are seen by Hilbert asinstrumental evices forgoing from real or finitely erifiablestatementso real statements,muchas an instrumentalistegardstheories n naturalscienceas a way ofgoing fromobservation en-tencesto observation entences. hese mathematically

theoreticalstatements ilbertcalled ideal elements, ikening their ntroduc-tion to the introduction fpoints at infinity n projectivegeom-etry:theyare introducedas a convenience to make simplerandmore elegant the theoryof the thingsyou really care about. Iftheir ntroduction oes not lead to contradiction nd if theyhavetheseotheruses, then it is justified:hence the searchfor a con-sistency roofforthefull system f first-orderrithmetic.

If this is a reasonable, f sketchy,ccount of Hilbert's view, it

indicatesthathe did not regardall quantifiedstatements eman-ticallyon a par withone another.A semantics orarithmetic s heviewed it would be veryhard to give. But hard or not, it wouldcertainlynot treatthe quantifier n (2) in the same way as the

4 See my What Numbers Could Not Be, Philosophical Review, LXXIV, 1(January1965): 47-73.











of thesevarious approachesto thetruth f suchsentences s (2). Atthis point I wish only to introduce hedistinction etween, n the

one hand, thoseviewswhich attribute heobvious syntax and theobvious semantics) o mathematical tatements, nd, on the other,thosewhich, gnoring he apparent yntax nd semantics,ttempt ostate truth onditions or to specifynd accountfortheexisting is-tribution f truthvalues) on the basis of what are evidentlynon-semantic yntacticonsiderations. ltimately will arguethateachkind of account has its merits nd defects: ach addresses tself oan important omponent f a coherent ver-allphilosophicaccountoftruth nd knowledge.

But what are thesecomponents, nd how do theyrelate to oneanother?

II. TWO CONDITIONS

A. The first omponentof such an over-all view is more directlyconcernedwiththe conceptof truth.For presentpurposeswe canstate t as therequirement hattherebe an over-all heory f truthin termsof which it can be certified hat the account of mathe-matical truth s indeed an accountofmathematical ruth.The ac-

count should imply truth onditionsformathematical ropositionsthat are evidently onditions of theirtruth and not simply, ay,of theirtheoremhoodn some formal ystem). his is not to denythat being a theorem f some system an be a truth ondition fora given propositionor class of propositions. t is ratherto requirethat any theory hat proffersheoremhood s a conditionof truthalso explain the connectionbetweentruth nd theoremhood.

Anotherwayofputtingthisfirst equirements to demand that

any theory f mathematical ruthbe in conformity itha generaltheoryof truth-a theory of truth theories, f you like-whichcertifieshat the property f sentences hat the accountcalls truthis indeed truth.This, it seems to me, can be done only on thebasisofsome generaltheory or t leastthe anguage as a whole I assumethat we skirt paradoxes in some suitable fashion). Perhaps theapplicabilityof thisrequirement o thepresentcase amounts onlyto a plea thatthe semantical pparatusof mathematics e seen aspart and parcel of that of the natural anguage in which t is done,and thus thatwhatever emantical ccount we are inclined to giveofnames or,more generally, fsingular erms, redicates, nd quan-tifiersn the mother ongue ncludethosepartsof the mother onguewhichwe classifysmathematese.

I suggest hat, f we are to meet thisrequirement, e shouldn'tbesatisfiedwith an account that fails to treat 1) and (2) in parallel













referential anguages. I assumethatthe truth onditionsforthelanguage (e.g., English) to which mathematese ppears to belong

are to be elaborated much along the lines thatTarski articulated.So, to some extent,the question posed in the previoussection-How are truthconditionsfor (2) to be explained?-may be inter-preted as asking whether the sublanguage of English in whichmathematicss done is to receive thesame sortof analysis s I amassuming s appropriateformuchof therestofEnglish. f so, thenthequalms I shall sketch n the nextsectionconcerning ow to fitmathematical knowledge into an over-all epistemologyclearlyapply-though they an perhapsbe laid to restby a suitablemodi-fication f theory.f, on the otherhand,mathemateses not to beanalyzed along referentialines, then we are clearly n need notonly of an accountof truth i.e., a semantics)for this new kind oflanguage,but also of a new theoryof truththeoriesthat relatestruthforreferentialquantificational) anguagesto truthforthesenew (newly nalyzed) anguages.Givensuch an account,thetaskofaccountingformathematicalknowledgewould stillremain;but itwould presumablybe an easier task, since the new semantical

pictureof mathematesewould in mostcases have been promptedby epistemologicalconsiderations.However, I do not give thisalternativeserious consideration n this paper because I don'tthinkthat anyonehas ever actually chosen it. For to choose it isexplicitlyto considerand reject the standard interpretationfmathematical anguage, despite its superficial nd initial plausi-bility, nd thento providean alternative emantics s a substitute.6The combinatorial theoristswhom I discuss or referto have

usuallywanted

to have theircake and eat it too: theyhave notrealized that the truthconditions that theiraccount supplies formathematical anguage have not been connectedto thereferentialsemanticswhichthey ssumeis also appropriateforthat anguage.Perhaps the closestcandidate for an exception is Hilbert in theview I sketchedbrieflyn theopeningpages of thispaper. But topursuethisfurther ere would takeus too farafield.Let us return,therefore,o our praise of the standardview.

One of its primary dvantages s that the truthdefinitions orindividualmathematical heories husconstruedwill have thesamerecursion clauses as those employedfor their ess loftyempiricalcousins.Or to put it anotherway,they an all be takenas partsof

s I sometimeshink his s oneofthethingshatHilaryPutnamwants odoin his stimulatingrticle Mathematics ithoutFoundations, hisJOURNAL,LXIV, 1 (Jan. 19, 1967): 5-22.









the same language for which we provide a single account for

quantifiersregardless of the subdiscipline under consideration.Mathematical and empiricaldisciplineswill not be distinguishedin point of logical grammar. I have already underscored theimportance f this advantage: it means that the logicogrammaticaltheorywe employ n less recondite nd more tractabledomainswillserveus well here. We can do withone, uniform,ccount and neednot invent another for mathematics.This should hold true onvirtually ny grammatical heory oupled withsemantics dequatetoaccountfortruth.My bias forwhat call a Tarskiantheorytems

simply from the fact that he has given us the only viable sys-tematicgeneral accountwe have of truth.So, one consequenceofthe economyattendingthe standardview is that logical relationsare subject to uniform reatment: heyare invariantwith subjectmatter. ndeed, theyhelp define the conceptof subjectmatter.The same rulesof inferencemaybe used and theiruse accountedforby the same theorywhich providesus with our ordinary c-count ofinference, husavoidinga double standard. fwe rejectthe

standardview,mathematical nferencewill need a new and specialaccount.As it is, standarduses of quantifiernferences re justifiedby some sort of soundness proof. The formalization f theoriesin first-orderogic requiresfor its justification he assurance pro-vided by the Completenesstheorem)that all the logical conse-quences of the postulateswill be forthcomings theorems.Thestandardaccount delivers these guarantees.The obvious answersseem to work. To reject the standard view is to discard theseanswers.Newoneswouldhaveto be found.

So much for the obvious virtuesof this account. What are itsfaults?

As I suggestedbove, the principaldefect fthe standard ccountis that it appears to violate the requirement hat our account ofmathematical ruthbe susceptibleto integrationnto our over-allaccount of knowledge.Quite obviously, o make out a persuasivecase to this effectt would be necessary o sketchthe epistemologyI take to be at least roughlycorrect nd on the basis of which

mathematical ruths, tandardlyonstrued, o notseem to constituteknowledge.This would require a lengthy etour through he gen-eral problemsof epistemology. will leave that to another timeand contentmyselfhere with presenting brief summary f thesalient features f that viewwhichbear mostimmediately n ourproblem.













tween p and justifying belief in p on those grounds cannotbe made. But for that knowledgewhich is properlyregardedas

some form f ustified rue belief,thenthelink mustbe made. (Ofcourse not all knowledgeneed be justified rue belieffor thepointto be a sound one.)

It will come as no surprise that this has been a preamble topointing out that combining this view of knowledgewith thestandard view of mathematical ruthmakes t difficulto see how

mathematicalknowledge s possible. If, forexample,numbersarethekindsof entitiestheyare normally aken to be, then the con-

nection between the truth onditions orthe statements fnumbertheory nd any relevanteventsconnectedwith the people who aresupposed to have mathematicalknowledgecannot be made out.9It will be impossible o account forhowanyoneknows ny properlynumber-theoreticalropositions.This second condition on an ac-count of mathematical ruthwill not be satisfied, ecause we haveno account of how we know that the truthconditionsfor mathe-matical propositions obtain. One obvious answer-that some ofthese propositions re true if and only if they are derivable from

certain xioms via certainrules-will not help here. For, tobe sure,we can ascertain that those conditionsobtain. But in such a case,what we lack is the link between truth nd proof,when truth sdirectly efined n thestandardway. n short, lthough tmaybe atruth ondition of certainnumber-theoreticropositions hattheybe derivable from certainaxioms accordingto certainrules, thatthis s a truth onditionmustalso followfrom heaccountof truthifthe conditionreferredo is tohelp connect ruth nd knowledge,

if it is by theirproofsthatwe know mathematical ruths.Of course,givensomeset-theoreticalccount of arithmetic,oththe syntax and the semanticsof arithmetic an be set out so assuperficially o meet the conditionswe have laid down. But theregress hat this nvites s transparent, orthe same questionsmustthenbe asked about theset theoryn terms f whichtheanswers recouched.

V. TWO EXAMPLES

There are manyaccountsofmathematical ruth nd mathematicalknowledge.The theses have been defending re intended o applyto them all. Rather than try o be comprehensive,owever, willdevote these ast few pages to the examinationof two representa-

9For an expressionof healthyskepticism oncerningthis and related points,see Mark Steiner, Platonism and the Causal Theory of Knowledge, thisJOURNAL,

LXX, 3 (Feb. 8, 1973): 57-66.









tive cases: one standard view and one combinatorial view.First thestandardaccount,as expressedbyone of its mostexplicit

and lucid proponents, urtGodel.Godel is thoroughlywarethaton a realist i.e., standard) ccount

of mathematical truth our explanation of how we know thebasic postulatesmust be suitablyconnectedwith how we interpretthe referential pparatus of the theory. hus, in discussing ow wecan resolvethe continuumproblem,once it has been shown to beundecidable by the accepted axioms, he paints the followingpicture:

. . . theobjects f transfiniteettheory.. clearlyo notbelong o thephysical orld ndeven heirndirectonnection ith hysicalxperi-ence s very oose ..

But, despite heir emotenessrom ense xperience, e do haveaperceptionlso oftheobjects f et heory,s isseenfrom hefact hatthe xioms orce hemselvespon us as being rue. don't eewhyweshouldhave ess confidencen thiskindofperception,.e., nmathe-matical ntuition, han in sense perception, hich nducesus tobuild up physical heories nd to expectthat future enseper-

ceptions ill greewith hemnd,moreover,o believe hat questionnot decidable ow has meaning nd maybe decided n the future.10

I find hispictureboth encouraging nd troubling.What troublesme is that without n accountofhow the axioms force hemselvesupon us as being true, the analogy with sense perceptionandphysical science is withoutmuch content.For what is missing spreciselywhat my second principle demands: an account of thelink between our cognitivefaculties and the objects known. In

physical sciencewe have at least a starton such an account,andit is causal. We accept as knowledge only those beliefs which wecan appropriately elate to our cognitive faculties. Quite appro-priately,our conceptionof knowledge goes hand in hand withour conception fourselves s knowers. o be sure,there s a super-ficial nalogy. For, as Godel points out, we verify xioms bydeducing consequences fromthem concerning reas in whichweseemto have moredirect perception clearer ntuitions).But weare nevertold how we knoweven these,clearer,propositions. orexample,the verifiable onsequences f axioms of higher nfinityare (otherwise ndecidable) number-theoreticalropositionswhichthemselves re verifiable ycomputationup to any given nteger.But the story, o be helpfulanywhere,must tell us how we know

1O What Is Cantor's Continuum Problem? revisedversionin Benacerraf andPutnam,op. cit., p. 271.









statements f computational arithmetic-if theymean what thestandard ccountwould have themmean.And thatwe arenot told.

So the analogy s at best superficial.So muchfor the troubling spects.More importantperhapsand

what I find encouraging is the evident basic agreementwhichmotivatesGodel's attempt to draw a parallel between mathe-matics and empirical cience.He sees, think, hatsomethingmustbe said to bridgethechasm,createdbyhis realistic nd platonisticinterpretation f mathematicalpropositions,between the entitiesthat form the subject matter of mathematicsand the human

knower. nsead of tinkeringwith the logical formofmathematicalpropositions r with thenatureof theobjectsknown,he postulatesa specialfaculty hroughwhichwe interact withtheseobjects.Weseem to agree on the analysisof the fundamentalproblem,butclearlydisagreeabout the epistemological ssue-about what ave-nues are open to us throughwhichwe maycome to knowthings.

If our accountofempiricalknowledge s acceptable, tmustbe inpartbecause it tries to make the connection vident n the case ofour theoreticalknowledge,where it is not prima facie clear how

thecausal account s to be filled n. Thus, whenwe come tomathe-matics, he absenceof a coherent ccountofhow our mathematicalintuition s connectedwith the truth fmathematical ropositionsrenders heover-all ccountunsatisfactory.

To introducea speculativehistoricalnote, with some founda-tion n thetexts,tmightnotbe unreasonableto supposethatPlatohad recourse o theconceptofanamnesis t leastin partto explainhow,giventhenatureof theforms s he depictedthem, ne could

everhaveknowledge fthem.11The combinatorial view of mathematical ruthhas epistemo-logical roots. t starts romthepropositionthat,whatevermay bethe objects of mathematics, ur knowledge is obtained fromproofs.Proofs are or can be (forsome,mustbe) writtendown orspoken; mathematicians an surveythemand come to agree thatthey re proofs. t is largely hrough heseproofs hatmathematicalknowledge is obtained and transmitted. n short,this aspect ofmathematicalknowledge-its (essentially inguistic)means of pro-duction and transmission ives their mpetusto the class of viewsthat I call combinatorial.

Noticing the role of proofs n the productionof knowledge, t

1L The soul, then, as being immortal,and having been born again manytimes, and having seen all things that exist,whether in this world or in theworldbelow,has knowledgeof themall (Plato, Meno, 81).









seeksthegrounds of truth n theproofsthemselves. ombinatorialviews receive additional impetus from the realization that the

platonist casts a shroud of mystery ver how knowledgecan beobtainedat all. Add that realizationto thebelief thatmathematicsis a child of our own begetting mathematicaldiscovery,n theseviews, s seldomdiscovery bout an independentreality), nd it isnot surprising hatone looks foractsof conceptionto account forthe birth.Many accounts of mathematicaltruthfall under thisrubric.Perhapsalmostall. I have mentioned everal n passing, ndI discussed Hilbert's view in On the Infinite verybriefly. he

finalexample I wishto consider s thatof conventionalistccounts-the cluster f views thatthe truths f logic and mathematicsretrue or can be made true) n virtue of explicit conventionswheretheconventionsn question are usually thepostulates f thetheory.Once more,I will probablydo themall an injusticeby lumpingtogether numberof viewswhich theirproponentswould mostcertainlyiketokeepapart.

Quine, in his classic paper on this subject,12 as dealt clearly,

convincingly,nd decisivelywiththe view that the truths f logicare to be accounted foras the productsof convention-farbetterthan I could hope to do here.He pointedout that, ince we mustaccount for infinitelymany truths, the characterization f theeligible sentences s truthsmust be wholesale ratherthan retail.But wholesale characterizationan proceed only via general prin-ciples-and, ifwe are supposed not to understand ny logic at all,we cannot extractthe individual instancesfromthe generalprin-ciples:wewould need logic for uch a task.

Persuasive s thismaybe, I wishto add anotherargument-notbecause I thinkthis dead horse needs further logging, ut bothbecauseQuine's arguments limitedto thecase oflogic and becausetheprincipalpoints wish to bring out do not emerge ufficientlyfrom t. Indeed,Quine grants heconventionalist ertainprinciplesI should like to deny him. In restinghis case againstconvention-alismon the needfor wholesalecharacterization f nfinitely anytruths, uine concedes thatwerethereonlyfinitelymanytruths o

be reckonedwith, the conventionalistmight have a chance tomakeouthiscase.He says:

If truth ssignmentsouldbe madeone by one,rather han aninfiniteumberta time, he bovedifficultyoulddisappear; ruthsof ogic .. would imply e assertedeverallyyfiat, nd theproblem

12 W.V. Quine, TruthbyConvention, eprintedn BenacerrafndPutnam,Op. cit.









of inferringhemfrommoregeneral onventions ouldnot arise.(p.344).

Thus, if some way could be found to make sentences f logicweartheir truth values upon theirsleeves,the objections to the con-ventionalist ccount of truthwould disappear-for we would havedetermined ruthvalues for all the sentences,which s all thatonecould ask.

I wonder,however,what such a sprinkling f the word 'true'would accomplish.Surely t cannot sufficen orderto determineconcept of truth o assign values to each and every entence f the

language [suppose now that the language is set theory,n somefirst-orderormalization] let those with an even number of horse-shoes be true ).

What would make such an assignment f thepredicate true' thedetermination fthe conceptoftruth? imply he use of thatmono-syllable? Tarski has suggestedthat satisfaction f ConventionTis a necessary nd sufficientondition on a definition f truthfora particular anguage.13A mere (recursive)distribution f truthvalues can be parlayed nto a truth heory hat satisfiesonventionT. We can rest withthatprovided we are prepared to beg what Ithink s the main question and ignorethe concept of translationthat occurs in its (ConventionT's) formulation.What would bemissing, ard as it is to state, s the theoretical pparatusemployedby Tarski in providing truth definitions, .e., the analysis oftruth n termsof the referential onceptsof naming,predica-tion, satisfaction, nd quantification.A definitionthat does notproceedbythecustomaryecursion lauses forthe customary ram-

matical formsmaynot be adequate, even f tsatisfies onventionT.The explanation mustproceed throughreference nd satisfactionand, furthermore, ust be supplementedwithan account of refer-ence itself. ut thedefense fthis ast claim is too involved matterto take up here.'4

13 Alfred Tarski, The Concept of Truth for Formalized Languages, re-printed in Tarski, Logic, Semantics, and Metamathematics New York: Oxford,1956).ConventionT is statedon pp. 187/8 as follows:

CONVENTION T. A formally orrectdefinitionof the symbol Tr', formulated

in the metalanguage, will be called an adequate definitionof truth if ithas thefollowing onsequences:

(a) all sentenceswhich are obtained from the expression xeTr if andonly if p' by substituting orthe symbol x' a structural-descriptiveame ofany sentence of the language in question and for the symbol p' the ex-pressionwhich forms he translation f this sentence nto themetalanguage;

(,B)the sentence foranyx, ifxeTr thenxeS' (in otherwords Tr cS').14For an excellent presentationof a similarview, see HartryField, Tarski's

Theory of Truth, this JOURNAL, LXIX, 13 (July13, 1972): 347-375.










ARISTOTLE'S CONCEPTION OF MATTER 679

cases in which it provides for t fromthosein which it does not?Consistencyannot be the answer.To urgeit as suchis to miscon-

strue the significance f the fact that inconsistencys proof thattruthhas not been attained.The deeperreason once more is thatpostulational stipulationmakes no connectionbetween the propo-sitions and their subject matter-stipulation does not providefortruth.At best, t limits he class oftruth efinitionsinterpretations)consistentwith the stipulations. ut that s not enough.

To clarify he point, consider Russell's oft-cited ictum: Themethod of 'postulating'what we want has many advantages; theyare the same as the advantagesof theft ver honest toil. 6 On theview I am advancing, hat's false. For with theft t least you comeaway with the loot, whereas implicit definition,conventionalpostulation, nd their ousinsare incapable of bringing ruth. heyare not only morallybut practicallydeficient s well.

PAUL BENACERRAF

PrincetonUniversity

MATTER *

SCULPTORS aresometimesaidtowork nthis rthatmaterial-marble or wood or terracotta.The GreeksculptorMyron,accordingto ancienttestimony, orkedalmostexclusivelyn

bronze.But it could be misleading to put it thisway.The fact sthatMyron's statueswere made of bronze: his famousDiscoboluswas a bronze tatue.But it is unlikely hatMyrondid much actualworkon,orwith, hebronzeof whichtheDiscobolus wasmade. In-deed, it is unlikelythat thatbronzeeven existed at the time thatMyron was doing his main work on the statue. The Discobolusitselfhas long since ceased to exist; no doubt some barbarian in-vader had it melted down and used its bronze to make a shield.But we know a good deal about it, owing to the descriptions fLucian and Pliny,and to Roman copies of it, in marble, everalofwhichhave survived.Thus we know that it was hollow and thatit was cast by the so-called lost-wax process.This means that

Myronwould have begunbymodeling, omewhat oughly, figurein clay. He would thenhave coveredthisclaymodel with a thin

16 Bertrand Russell, Introduction to Mathematical Philosophy (London:Allen & Unwin, 1919), p. 71.

* To be presented n an APA symposiumon Aristotle'sConception of Matter,December 27, 1973. Commentatorswill be John M. Cooper and Russell M.Dancy; see thisJOURNAL, this issue,696-698 and 698-699, respectively.

Date post:	13-Apr-2018
Category:	Documents
Upload:	mortdei-jarlson
View:	236 times
Download:	0 times

Benacerraf, P., Mathematical Truth

Documents