On the Nature of Mathematical Truth CG Hempel

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On the Nature of Mathematical TruthAuthor(s): C. G. HempelSource: The American Mathematical Monthly, Vol. 52, No. 10 (Dec., 1945), pp. 543-556Published by: Mathematical Association of America

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ON THE NATURE OF MATHEMATICAL TRUTH

C. G. HEMPEL, QueensCollege

1. The problem. It is a basic principle f scientific nquirythat no proposi-tionand no theory s to be accepted without dequate grounds. n empirical ci-ence,which ncludes boththenaturaland thesocial sciences, thegroundsfor he

acceptance ofa theory onsist n theagreement f predictions ased on the the-ory with empirical evidence obtained eitherby experimentor by systematicobservation.But what are the groundswhich sanctionthe acceptance of mathe-matics?That is the question I proposeto discuss in the presentpaper. For rea-sons which willbecome clear subsequently, shall use the term"mathematics"here torefer o arithmetic, lgebra,and analysis-to the exclusion, n particular,ofgeometry 1].

2. Are the propositions f mathematicsself-evident ruths?One of the sev-eral answerswhichhave been given to our problem asserts that the truths of

mathematics, n contradistinction o the hypothesesof empirical science, re-quireneitherfactual evidencenorany other ustification ecause theyare "self-evident." This view, however, which ultimately relegates decisions as tomathematicaltruthto a feeling fself-evidence, ncountersvarious difficulties.First ofall, many mathematicaltheorems re so hard to establishthat even tothe specialist in the particularfield theyappear as anythingbut self-evident.Secondly, it is well knownthat some of the most interesting esults of mathe-matics-especially in such fields s abstractset theory nd topology-run coun-ter to deeply ingrained ntuitions and the customarykind of feelingof self-

evidence. Thirdly,the existenceof mathematicalconjecturessuch as those ofGoldbach and of Fermat,whichare quite elementary n contentand yet un-decided up to this day, certainlyshows that not all mathematicaltruthscanbe self-evident.And finally, ven if self-evidencewere attributedonly to thebasic postulatesofmathematics, romwhichall othermathematicalpropositionscan be deduced, it would be pertinentto remarkthat judgmentsas to whatmaybe considered s self-evidentre subjective; they may varyfromperson topersonand certainly annot constitute n adequate basis fordecisionsas to theobjective validityof mathematicalpropositions.

3. Is mathematics he mostgeneral empirical cience? According o anotherview,advocated especially by JohnStuartMill,mathematics s itself n empiri-cal sciencewhichdiffers romthe other branchessuch as astronomy,physics,chemistry, tc.,mainly n two respects:its subject matter s moregeneralthanthat of any other fieldof scientific esearch,and its propositionshave beentested and confirmed o a greaterextent than those of even the most firmlyestablishedsectionsofastronomy rphysics. ndeed, accordingto thisview,thedegree to whichthe laws ofmathematicshave been borne out by the past ex-periencesof mankind s so overwhelminghat-unjustifiably--we have come to

think of mathematical theoremsas qualitatively different rom the well-con-firmed ypothesesor theoriesofotherbranchesof science: we considerthemas

543







544 ON THE NATURE OF MATHEMATICAL TRUTH [December,

certain,while other theories re thoughtof as at best 'very probable" or veryhighly onfirmed.

But this view,too, is open to serious objections.From a hypothesiswhich sempiricalin character-such as, forexample,Newton's law ofgravitation-it

is possible to derive predictions o the effect hat under certainspecified ondi-tionscertainspecified bservable phenomenawill occur. The actual occurrenceof these phenomenaconstitutes onfirming vidence, their non-occurrence is-confirmingvidencefor hehypothesis. t follows nparticularthat an empiricalhypothesis s theoreticallydisconfirmable;.e., it is possible to indicate whatkind of evidence, if actually encountered,would disconfirm he hypothesis. nthe lightof this remark, onsidernow a simple "hypothesis"from rithmetic:3+ 2 =5. If this s actually an empirical generalizationofpast experiences, henit must be possible to state what kind of evidence would oblige us to concede

the hypothesiswas not generallytrue after all. If any disconfirmingvidencefor hegivenproposition an be thought f,the followingllustrationmightwellbe typicalof it: We place some microbeson a slide, puttingdown first hree ofthem and then another two. Afterwardswe count all the microbes to testwhether n this instance3 and 2 actually added up to 5. Suppose now that wecounted 6 microbesaltogether.Would we considerthis as an empiricaldiscon-firmation fthe given proposition, r at least as a proofthat it does not applyto microbes?Clearly not; rather,we would assume we had made a mistakeincountingor that one of themicrobeshad split in two between the first nd the

secondcount.But under no circumstances ould thephenomenon ust describedinvalidatethe arithmetical ropositionnquestion;for he atter sserts nothingwhatever bout the behavior ofmicrobes; tmerely tatesthatanysetconsistingof3+2 objects may also be said to consistof 5 objects. And this is so becausethesymbols"3+2" and "5" denote thesame number:theyare synonymousbyvirtueof thefactthatthesymbols"2," "3," "5," and a+" aredefined or tacitlyunderstood) n such a way thattheabove identityholdsas a consequenceof themeaningattached to the conceptsinvolved in it.

4. The analyticcharacterofmathematicalpropositions.The statementthat

3+2 = 5, then, s true for imilarreasons as, say, the assertionthat no sexage-narian is 45 years of age. Both are true simply by virtue of definitions r ofsimilarstipulationswhich determinethe meaningof the key termsinvolved.Statementsofthis kindshare certain mportant haracteristics: heirvalidationnaturallyrequiresno empirical vidence; theycan be shown tobe truebya mereanalysis of themeaningattached to the termswhichoccur in them.In the lan-guage of ogic,sentencesof thiskind are called analyticor truea priori,which sto indicatethat theirtruth s logically ndependentof,or logicallypriorto, anyexperiential vidence [2]. And while the statementsofempiricalscience,which

are synthetic nd can be validated only a posteriori, re constantly ubject torevision n the lightof new evidence,the truthofan analytic statementcan beestablisheddefinitely,nce and for ll. However, this characteristic theoretical







1945] ON THE NATURE OF MATHEMATICAL TRUTH 545

certainty"ofanalyticpropositionshas tobe paid for t a highprice:Ananalyticstatementconveysno factual information. ur statement bout sexagenarians,forexample, assertsnothingthat could possiblyconflictwithany factual evi-dence: it has no factual implications,no empiricalcontent; and it is precisely

for hisreason thatthe statement an be validatedwithoutrecourse o empiricalevidence.

Let us illustrate this view of the nature of mathematical propositionsbyreference o another,frequently ited, example of a mathematical-or ratherlogical-truth, namelythe proposition hatwhenever = b and b= c thena = c.On what grounds can thisso-called "transitivity fidentity" be asserted? Is itof an empiricalnature and henceat least theoretically isconfirmable y empiri-cal evidence?Suppose, for xample,that a, b, c, are certain hades ofgreen, ndthat as far as we can see, a=b and b=c, but clearly a0#c.This phenomenon

actuallyoccursunder certainconditions; do we consider t as disconfirmingvi-dence for the propositionunder consideration?Undoubtedly not; we wouldargue that ifa 0 c, it is impossiblethat a = band also b= c; betweenthetermsofat least one oftheselatterpairs,theremustobtain a difference,houghperhapsonlya subliminalone. And we woulddismiss thepossibility fempiricaldiscon-firmation,nd indeedtheidea that an empiricaltest shouldbe relevanthere,onthegroundsthat identity s a transitive elationbyvirtueof tsdefinition rbyvirtueofthe basic postulatesgoverning t [3]. Hence, the principle n questionis truea priori.

5. Mathematics as an axiomatized deductive system. I have argued so farthat thevalidityofmathematicsrestsneither n itsalleged self-evidential har-acternoron any empiricalbasis, but derivesfrom hestipulationswhichdeter-mine the meaningof the mathematicalconcepts,and that the propositionsofmathematics are therefore ssentially "true by definition."This latter state-ment,however, s obviouslyoversimplifiednd needs restatement nd a morecareful ustification.

For therigorousdevelopmentof a mathematical theoryproceeds notsimplyfrom set of definitions ut ratherfrom set of non-definitionalropositions

which re notprovedwithin hetheory; hese are thepostulates or axiomsof thetheory[4]. They are formulatedn termsofcertainbasic or primitive onceptsforwhichno definitionsre providedwithin hetheory. t is sometimes ssertedthat the postulatesthemselvesrepresent implicitdefinitions" fthe primitiveterms. Such a characterizationof the postulates,however, s misleading.Forwhilethepostulatesdo limit, n a specific ense, themeaningsthatcan possiblybe ascribedtotheprimitives, nyself-consistentostulatesystem dmits,never-theless,manydifferentnterpretations ftheprimitive erms thiswill soon beillustrated),whereasa set ofdefinitionsnthe strict ense oftheworddetermines

themeaningsofthedefiniendan a unique fashion.Once theprimitive erms nd thepostulateshave been laid down, theentiretheory s completelydetermined; t is derivablefrom ts postulationalbasis in








thefollowing ense: Every termof the theory s definable n termsof the primi-tives,and everyproposition f the theory s logicallydeduciblefrom hepostu-lates. To be entirelyprecise, t is necessary lso to specify heprinciples f ogicwhich re to be used in theproofof thepropositions,.e. in theirdeductionfrom

the postulates. These principlescan be stated quite explicitly.They fall intotwo groups: Primitive entences,or postulates,of logic (such as: If p and q isthe case, then p is the case), and rulesofdeductionor inference including,forexample, the familiarmodus ponens rule and the rules of substitutionwhichmake it possibleto infer, rom generalproposition, ny one of ts substitutioninstances). A more detailed discussion of the structureand content of logicwould,however, ead too farafield n the contextof this article.

6. Peano's axiom systemas a basis formathematics.Let us now considerapostulate systemfromwhichthe entire arithmetic f the natural numberscan

be derived.This systemwas devised by the Italian mathematician nd logicianG. Peano (1858-1932). The primitives f thissystemare the terms "0," 'num-ber," and "successor." While, of course,no definition f these terms s givenwithinthe theory, he symbol "0" is intendedto designatethe number0 in itsusual meaning,whilethe term "number" s meantto refer o the naturalnum-bers 0, 1, 2, 3 - ** exclusively.By the successorof a naturalnumbern, whichwill sometimes briefly e called n', is meant the natural number mmediatelyfollowing in the naturalorder.Peano's systemcontainsthefollowing postu-lates:

P1. 0 is a numberP2. The successorofany number s a numberP3. No twonumbershave the same successorP4. 0 is not thesuccessorofany numberPS. If P is a property uch that (a) 0 has the property , and (b) whenever

numbernhas theproperty , thenthesuccessorofn also has thepropertyP, theneverynumberhas the propertyP.

The last postulate embodies the principleof mathematical inductionand

illustrates n a veryobviousmannerthe enforcementf a mathematical"truth"by stipulation.The construction felementary rithmeticon this basis beginswiththe definition f the various natural numbers.1 is defined s the succes-sorof0, or brieflys 0'; 2 as 1', 3 as 2', and so on. By virtueof P2, this processcan be continued ndefinitely; ecause ofP3 (in combinationwithP5), it neverleads back to one of thenumberspreviouslydefined, nd in view ofP4, it doesnot lead back to 0 either.

As thenextstep,we can set up a definition fadditionwhichexpresses n apreciseformthe idea that the addition of any natural number to some given

number may be consideredas a

repeatedaddition of

1;the latteroperationis

readily xpressiblebymeansofthesuccessorrelation.This definitionfadditionrunsas follows:








Dl. (a) n+O=n; (b) n+k'=(n+k)'.

The two stipulationsof this recursivedefinitionompletelydetermine he sumof any two integers. Consider, forexample, the sum 3+2. Accordingto thedefinitions f thenumbers2 and 1,we have 3+2=3+1'=3+(0')'; by DI (b),3+ (0') = (3+0')' = ((3+0)')'; butbyD1 (a), and by the definitions f the num-bers4 and 5, ((3 +0)')' = (3')' =4'=5. This proof lso rendersmoreexplicit ndprecise thecommentsmade earlier n thispaper on the truthofthe propositionthat3+ 2=5: Withinthe Peano systemofarithmetic,tstruthflowsnotmerelyfrom he definition f the concepts involved,but also from he postulates thatgovernthese various concepts. (In our specific xample,the postulates P1 andP2 are presupposedtoguaranteethat 1, 2, 3,4, 5 are numbers nPeano's system;thegeneralproof hat Dl determines he sum ofanytwo numbers lso makes useof P5.) If we call the postulates and definitions f an axiomatized theorythe"stipulations"concerning heconceptsofthattheory, henwemay say now thatthe propositions f the arithmetic f the naturalnumbers re trueby virtueofthe stipulationswhich have been laid down initiallyforthe arithmetical on-cepts. (Note, incidentally, hat our proofof the formula"3+2 = 5" repeatedlymade use of thetransitivity f dentity; he latter s accepted here as one of therules of ogicwhichmay be used in theproofofany arithmetical heorem; t is,therefore,ncludedamong Peano's postulatesno more thanany otherprincipleof ogic.)

Now, themultiplication f natural numbersmaybe definedbymeans of the

following ecursivedefinition, hichexpresses n a rigorous orm he idea that aproductnkof two ntegersmay be considered s the sum of k terms ach ofwhichequals n.

D2. (a) n 0=0; (b) n*k'=n*k+n.

It now is possibleto provethe familiargeneral aws governing ddition andmultiplication,such as the commutative,associative, and distributive laws(n+k=k+n, n*k=k n; n+(k+l)=(n+k)+l, n.(k.l)=(n-k).l; n.(k+l)= (n. k) + (n l)).-In terms of addition and multiplication, he inverse opera-

tionsofsubtraction nd divisioncan thenbe defined.But it turnsout that these"cannot always be performed"; .e., in contradistinction o the sum and theproduct,the difference nd the quotient are not definedforevery couple ofnumbers;forexample, r- 10 and 7 10 are undefined.This situation suggestsan enlargement f the numbersystem by the introduction f negative and ofrationalnumbers.

It is sometimes held that in order to effect his enlargement,we have to"assume"yr else to "postulate" theexistenceofthedesiredadditional kindsofnumberswithproperties hat make themfitto fill he gaps ofsubtraction nddivision.This method ofsimplypostulatingwhat we want has its advantages;but,as Bertrand Russell [5 puts it, theyare the same as theadvantages of theftover honesttoil; and it is a remarkablefact that thenegativeas well as the ra-








tional numberscan be obtained fromPeano's primitivesby the honest toil ofconstructing xplicitdefinitions orthem,withoutthe introduction f any newpostulatesor assumptionswhatsoever.Every positiveand negativeinteger-incontradistinction o a natural numberwhichhas no sign-is definable s a cer-

tain set of orderedcouples ofnaturalnumbers;thus,theinteger+2 is definableas the set of all orderedcouples (m,n) ofnaturalnumberswherem=n+2; theinteger -2 is the set of all orderedcouples (m, n) of natural numberswithn ==m+2.-Similarly, rationalnumbers re defined s classes of orderedcouplesof ntegers.-The various arithmetical perationscan then be definedwithrefer-ence to thesenewtypesof numbers, nd the validityof all the arithmetical awsgoverning hese operations an be provedbyvirtueofnothingmorethanPeano'spostulatesand the definitions f the various arithmetical oncepts involved.

The much broadersystemthusobtained is still ncomplete n thesense that

not every numberin it has a square root, and more generally,not every alge-braic equation whose coefficientsre all numbersof the systemhas a solutionin the system.This suggestsfurther xpansionsof the numbersystemby theintroduction freal and finally f complexnumbers.Again, this enormousex-tensioncan be effected y mere definition,without the introduction fa singlenewpostulate [6]. On thebasis thusobtained,the variousarithmetical nd alge-braicoperationscan be defined or. he numbers f the newsystem, he conceptsoffunction, f imit, f derivativeand integralcan be introduced, nd the famil-iar theoremspertaining o these conceptscan be proved,so thatfinally hehuge

systemof mathematicsas here delimitedrestson the narrow basis of Peano'ssystem: Every concept of mathematicscan be definedby means of Peano'sthreeprimitives, nd everypropositionof mathematicscan be deduced fromthe fivepostulatesenrichedby the definitions fthe non-primitiveerms [6a].These deductions can be carriedout, in most cases, by means ofnothingmorethan the principlesofformal ogic; the proofofsome theorems oncerning ealnumbers,however, equiresone assumptionwhich s notusuallyincluded amongthe latter.This is the so,-called xiom ofchoice. It asserts thatgivena class ofmutuallyexclusive classes,noneofwhich s empty, hereexistsat least one classwhich has exactly one elementin commonwith each of the given classes. Byvirtueof thisprinciple nd the rulesofformal ogic,the contentofall ofmathe-matics can thusbe derivedfromPeano's modestsystem-a remarkable chieve-ment n systematizinghe contentof mathematics nd clarifyinghefoundationsof its validity.

7. Interpretations f Peano's primitives.As a consequenceof thisresult,thewhole systemofmathematicsmightbe said to be truebyvirtue ofmere defini-tions (namely,ofthe non-primitivemathematicalterms)providedthat thefivePeano postulatesare true.However, strictly peaking,we cannot, at this unc-

ture,refer o the Peano postulatesas propositionswhich are eithertrue orfalse,for heycontainthreeprimitive ermswhichhave not been assignedany specificmeaning.All we can assertso far s thatany specific nterpretationf theprimi-








tives whichsatisfies hefivepostulates-i.e., turnsthem nto truestatements-will also satisfy ll the theoremsdeduced from hem.But forPeano's system,there are several-indeed, infinitelymany-interpretationswhich will do this.For example,let us understandby 0 the originofa half-line, y the successor of

a point on that half-line he point 1 cm. behind it, countingfromthe origin,and by a number ny point which s eitherthe originor can be reached fromtbya finiteuccessionof steps each of which eads from ne pointto its successor.It can thenreadilybe seen that all the Peano postulatesas well as the ensuingtheorems urn nto truepropositions, lthoughthe interpretation iven to theprimitives s certainlynot the customaryone, which was mentioned earlier.More generally, t can be shownthat every progression f elements of any kindprovidesa true nterpretation,r a "model,"ofthe Peano system.This exampleillustratesour earlierobservation that a postulate system cannot be regarded

as a set of"implicitdefinitions" ortheprimitive erms:The Peano system per-mits of many differentntetpretations,whereas in everyday as well as in sci-entific anguage,we attach one specificmeaningto the concepts of arithmetic.Thus, e.g., in scientific nd in everydaydiscourse,the concept 2 is understoodin such a way that from he statement "Mr. Brown as well as Mr. Cope, but noone else is in theoffice,nd Mr. Brown s not the same personas Mr. Cope," theconclusion "Exactly two personsare in the office"maybe validly inferred. utthe stipulations aid down in Peano's systemfor the natural numbers, nd forthe number2 in particular,do not enable us to draw this conclusion; theydo

not "implicitlydetermine" the customarymeaningof the concept 2 or of theotherarithmeticalconcepts. And the mathematiciancannot acquiesce at thisdeficiency y arguingthat he is not concerned withthe customary meaningofthemathematicalconcepts; for nproving, ay, thatevery positivereal numberhas exactlytwo real square roots,he is himself singtheconcept2 in itscustom-arymeaning, nd hisverytheorem annot be provedunlesswe presupposemoreabout the number2 than is stipulated in the Peano system.

If thereforemathematics s to be a correct heoryofthe mathematical con-cepts in theirintendedmeaning,it is not sufficientor ts validation to haveshownthat the entiresystem s derivablefrom he Peano postulatesplus suit-able definitions; ather,we have to inquirefurther hether hePeano postulatesare actually true whenthe primitives re understood n theircustomarymean-ing.This question,ofcourse,can be answeredonlyafter hecustomarymeaningof the terms 0," "natural number,"and "successor" has been clearlydefined.To this task we now turn.

8. Definitionof the customarymeaning of the concepts of arithmetic npurely ogical terms. At first lush, itmight eema hopelessundertaking o tryto define hesebasic arithmetical onceptswithoutpresupposing thertermsof

arithmetic,which would involve us in a circular procedure. However, quiterigorousdefinitions f thedesiredkind can indeedbe formulated,nd it can beshownthat for heconceptsso defined, ll Peano postulatesturn nto true state-










niceties to be stated here, s a careful xpressionof a simple idea which s illus-trated by the following xample: Consider the number 5, i.e., the class of allquintuplets.Let us select an arbitrary ne of thesequintupletsand add to it anobject which s not yetone of its members.5', the successor of5, may then be

defined s the number pplyingto the set thus obtained (which,of course, s asextuplet).Finally, t ispossibleto formulate definitionf the customarymean-ing of the concept of natural number; this definition,which again cannot begiven here,expresses, n a rigorousform, he idea that the class of the naturalnumbers onsistsof the number0, its successor,the successor ofthat successor,and so on.

If the definitions ere characterized re carefullywritten ut-this is one ofthecases where the techniquesofsymbolic,or mathematical, ogicprove indis-pensable-it is seen that thedefiniens feveryone of them contains exclusively

termsfrom he fieldofpure logic. In fact, t is possible to state the customaryinterpretationf Peano's primitives, nd thus also the meaningofevery conceptdefinablebymeans of them-and that includesevery conceptof mathematics-in termsofthefollowing expressions, n addition to variables such as "x" and"C": not,and, if-then; for every bjectx it is the case that . . ; there s someobjectx such that ... ; x is an element f class C; the class of all thingsx suchthat ... And it iseven possibleto reduce the number f ogical conceptsneededto a mere four:The first hreeof the concepts ust mentioned re all definablein termsof "neither-nor," and the fifths definableby means ofthe fourth nd

"neither-nor."Thus, all the conceptsof mathematicsprove definable n termsoffourconceptsofpure logic. (The definition fone of the morecomplexcon-cepts of mathematics n termsof the fourprimitives ust mentionedmay wellfillhundreds or even thousands of pages; but clearlythis affectsn no way thetheoretical mportanceof the result ust obtained; it does, however,show thegreat convenience nd indeedpractical ndispensability ormathematicsof hav-inga large systemofhighly omplex defined onceptsavailable.)

9. The truth of Peano's postulates in theircustomary nterpretation. hedefinitions haracterized n the preceding ection may be said to renderprecise

and explicitthe customarymeaningofthe conceptsof arithmetic.Moreover-and this is crucial forthe question of the validity ofmathematics-it can beshownthatthe Peano postulatesall turn nto true propositions fthe primitivesare construed n accordance withthe definitionsust considered.

Thus, P1 (O is a number) s true because theclass of all numbers-i.e., natu-ral numbers-was defined s consisting f0 and all its successors.The truthofP2 (The successor of any number s a number)follows rom he same definition.This is truealso ofP5, the principleof mathematical nduction.To prove this,however,we would have to resort to the precise definition f "integer" rather

thanthe loose descriptiongivenof thatdefinition bove. P4 (Ois not thesucces-sor ofany number) is seen to be true as follows:By virtue of thedefinition f"successor,' a number which is a successor ofsome numbercan apply only to








classes which contain at least one element; but the number0, by definition,applies to a class if and only ifthat class is empty.-While the truth of PI, P2,P4, P5 can be inferred rom he above definitions imply by means of the prin-ciples of logic, theproofof P3 (No two numbershave the same successor) pre-

sentsa certain difficulty. s was mentioned n the preceding ection, the defini-tionofthe successor of a numbern is based on theprocess ofadding,to a classofn elements,one element not yet containedin that class. Now ifthere shouldexist only a finitenumber of things altogether then this process could not becontinuedindefinitely,nd P3, which (in conjunction withP1 and P2) impliesthat the integersform n infinite et, would be false. Russell's way ofmeetingthis difficulty9] was to introducea special "axiom of infinity,"which stipu-lates, in effect, he existence of infinitelymany objects and thus makes P3demonstrable.The axiom of infinityan be formulated n purely ogical terms

and may therefore e considered as a postulate of logic; however, t certainlydoes notbelongto thegenerallyrecognizedprinciples f ogic; and it thus intro-duces a foreign lement into the otherwiseunexceptionablederivationof thePeano postulates frompure logic. Recently,however, t has been shown [to]that a suitablesystemof ogical principles an be set up which s even less com-prehensive han the rules oflogic which are commonlyused [It], and inwhichthe existence of infinitelymany objects can be proved withoutthe need for aspecial axiom.

10. Mathematics as a branchof ogic.As was pointed outearlier, ll thethe-oremsofarithmetic, lgebra,and analysiscan be deduced from hePeano postu-lates and the definitions f thosemathematical termswhichare not primitivesin Peano's system.This deductionrequiresonly the principlesof logic plus, incertaincases, the axiom ofchoice. By combining hisresultwithwhat has justbeen said about the Peano system,the following onclusionis obtained,whichis also known as thethesisof ogicism oncerninghenatureofmathematics:

Mathematics is a branchof ogic. It can be derived fromogic in the follow-ingsense:

a. All theconceptsofmathematics, .e. ofarithmetic, lgebra,and analysis,can be defined n termsof fourconcepts of pure logic.

b. All the theoremsofmathematicscan be deduced from hose definitionsby means of theprinciples f ogic (including he axiomofchoice).

In this sense it can be said that the propositionsof the system of mathe-matics as heredelimited re truebyvirtueofthe definitions fthe mathematicalconcepts nvolved,or thattheymake explicitcertaincharacteristicswith whichwe have endowedourmathematicalconceptsbydefinition. he propositions fmathematicshave, therefore,he same unquestionablecertaintywhich s typicalof such propositions s "All bachelors are unmarried,"but theyalso share the

complete ack ofempiricalcontentwhich s associated with that certainty:Thepropositionsofmathematics are devoid of all factual content; they conveynoinformationwhatever on any empirical subject matter.








11. On theapplicabilityf mathematicso empirical ubjectmatter. hisresult seems to be irreconcilablewith the fact that afterall mathematicshasprovedto be eminently pplicable to empirical ubject matter, nd that indeedthe greater part of present-dayscientificknowledge has been reached only

throughcontinual reliance on and application of the propositionsof mathe-matics.-Let us tryto clarifythis apparent paradox by reference o some ex-amples.

Suppose thatwe are examining certainamountofsomegas,whosevolumev,at a certainfixed emperature,s foundto be 9 cubic feetwhenthepressurepis 4 atmospheres.And let us assume further hat thevolume of the gas forthesame temperatureand p =6 at., is predictedby means of Boyle's law. Usingelementary rithmeticwe reason thus: For correspondingvalues of v and p,vp= c,and v= 9 when p = 4; hencec = 36: Therefore,when p = 6, thenv= 6. Sup-

pose that thisprediction s borneout by subsequenttest. Does that show thatthearithmetic sed has a predictivepowerof ts own,that itspropositionshavefactualimplications?Certainlynot. All the predictivepowerheredeployed,allthe empirical contentexhibitedstems fromthe initial data and fromBoyle'slaw,which assertsthat vp= c for nytwo corresponding alues ofv and p, hencealso forv= 9, p = 4, and forp = 6 and thecorresponding alue ofv [12 . The func-tionof themathematicshereapplied is notpredictive t all; rather, t is analyticor explicative: it renders explicit certain assumptions or assertionswhich areincluded n thecontentof the premisesof theargument in our case, thesecon-

sist of Boyle's law plus the additional data); mathematical reasoningreveals

that those premisescontain-hidden in them,as it were,-an assertionaboutthe case as yetunobserved.In acceptingourpremises-so arithmetic eveals-we have-knowingly or unknowingly-already accepted the implicationthatthep-value in question is 6. Mathematical as well as logical reasoning s a con-ceptual technique of makingexplicitwhat is implicitly ontained in a set ofpremises.The conclusionsto whichthis technique leads assert nothingthat istheoreticallyew in the sense of not being contained in the content of thepremises.But theresultsobtainedmaywellbe psychologicallyew:we maynothave been aware,beforeusingthetechniquesof ogicand mathematics,whatwe

committedourselvesto in acceptinga certainset of assumptionsor assertions.A similar analysis is possible in all othercases of applied mathematics, n-

cludingthoseinvolving, ay, the calculus. Consider,for xample,thehypothesisthata certainobject,moving na specified lectricfield,will undergo constantaccelerationof5 feet/sec2. or the purposeoftestingthishypothesis,we mightderivefromt,bymeansof twosuccessive ntegrations,heprediction hatiftheobject is at rest at thebeginning fthe motion,thenthedistancecoveredby itat any timet is 4t2feet.This conclusionmay clearlybe psychologicallynewtoa personnot acquainted with the subject, but it is not theoreticallynew; the

contentoftheconclusion salreadycontained nthat ofthehypothesis bout theconstantacceleration.And indeed,hereas wellas inthecase ofthecompressionofa gas, a failureof the prediction o come truewould be considered s indica-








tive of the factual ncorrectnessf at least one ofthe premises nvolved f.ex., ofBoyle's law in its application to the particulargas), but never as a sign that thelogical and mathematicalprinciples nvolvedmightbe unsound.

Thus, in the establishment f empiricalknowledge,mathematics as well as

logic) has, so to speak, thefunction f a theoretical uice extractor: the tech-niques ofmathematical and logical theory an produce no more uice of factualinformationhan is contained nthe assumptionsto which theyare applied; butthey may produce a great deal more uice of thiskind than mighthave been an-ticipated upon a first ntuitive nspectionofthose assumptionswhich form heraw material forthe extractor.

At thispoint, t may be well toconsiderbriefly he status of thosemathemat-ical disciplineswhich are not outgrowths farithmetic nd thus of logic; theseinclude in particular topology,geometry, nd the various branches of abstract

algebra, such as the theory ofgroups, lattices, fields,etc. Each of these disci-plines can be developed as a purelydeductive system on the basis of a suitableset ofpostulates. If P be the conjunction of the postulates fora given theory,then the proof fa propositionT of that theory onsists ndeducingT from bymeans of theprinciples f formal ogic. What is established by theproof s there-forenot thetruth f T, but rather he fact thatT is trueprovidedthatthepostu-lates are. But since both P and T contain certainprimitive ermsof the theory,towhichno specificmeaning s assigned, t is notstrictly ossible to speak of thetruth f eitherP or T; it is thereforemore dequate to state thepoint as follows:

Ifa propositionT is logicallydeduced from , then every specificnterpretationof the primitiveswhich turns all the postulates ofP into truestatements,willalso render T a truestatement.-Up to this point, the analysis is exactlyanalo-gous to that ofarithmetic s based on Peano's set of postulates. In the case ofarithmetic, owever, t provedpossible to go a step further, amelytodefine hecustomarymeaningsof theprimitivesn termsofpurely ogical conceptsand toshow that the postulates-and therefore lso the theorems-of arithmetic reunconditionally rue by virtueof these definitions.An analogous procedure snot applicable to those disciplineswhich are not outgrowths farithmetic:Theprimitives f thevariousbranches of abstractalgebra have no specific custom-ary meaning";and ifgeometryn its customary nterpretations thoughtof as atheory of the structure of physical space, then its primitiveshave to be con-struedas referringo certaintypesofphysicalentities, nd the question ofthetruthofa geometrical heory n this nterpretationurns nto an empiricalprob-lem [13]. For the purpose of applying any one ofthese non-arithmetical isci-plines to some specificfieldofmathematicsor empirical science, it is thereforenecessaryfirst o assign to the primitives ome specificmeaningand then toascertain whether nthis nterpretationhepostulatesturn ntotruestatements.If this s thecase, thenwe can be sure that all thetheorems re truestatements

too, because theyare logicallyderived from he postulatesand thus simplyex-plicatethe contentofthe atter nthegiven nterpretation.-In their pplicationto empirical ubject matter, herefore,hese mathematicaltheoriesno less than








those whichgrowout ofarithmetic nd ultimatelyout of pure logic,have thefunctionof an analytic tool, whichbringsto lightthe implicationsof a givenset ofassumptionsbut adds nothing o theircontent.

But while mathematics nno case contributes nything o thecontentof our

knowledgeof empiricalmatters, t is entirely ndispensableas an instrument orthe validation and even for the linguisticexpressionof such knowledge: Themajorityof the morefar-reaching heories n empirical science-including thosewhich end themselvesmosteminently o prediction r topractical application-are stated withthehelpof mathematical oncepts;the formulation fthesetheo-riesmakesuse, in particular,of the number ystem, nd of functionalrelation-ships amongdifferent etricalvariables. Furthermore,he scientific est ofthesetheories, he establishmentof predictionsby means of them,and finally heirpractical application, all require the deduction, from he general theory, f cer-

tain specific consequences; and such deduction would be entirely mpossiblewithout the techniques of mathematics which reveal what the given generaltheory mplicitly sserts about a certain special case.

Thus, the analysis outlined on these pages exhibits the system of mathe-matics as a vast and ingenious conceptual structurewithoutempirical contentand yet an indispensableand powerful heoretical nstrument or the scientificunderstandingnd masteryoftheworld ofour experience.

References

1. A discussion f the tatus fgeometrys given n my rticle, eometrynd Empirical ci-ence,AmericanMathematicalMonthly,ol. 52,pp. 7-17, 1945.2. The objectionssometimesaised hatwithout ertain ypes f xperience,uch s encoun-

tering everal bjectsofthesame kind, he ntegers nd the arithmeticalperationswith hemwouldnever ave been nvented,nd that hereforehepropositionsf rithmetico have an em-pirical asis.Thistype f rgument,owever,nvolves confusionf he ogical ndthepsychologi-icalmeaningftheterm basis.' It may verywellbe the case thatcertain xperiencesccasionpsychologicallyheformationf rithmeticaldeas and in this enseform n empirical basis"forthem; ut thispoint s entirelyrrelevantor he ogical uestionss to thegroundsnwhich hepropositionsf arithmetic ay be accepted s true.The pointmade above is that no empirical"basis"orevidencewhateversneeded o establish hetruth fthepropositionsf rithmetic.

3. A precise ccount fthedefinitionnd theessential haracteristicsfthe dentity elationmaybe foundnA. Tarski, ntroductiono Logic,New York,1941,Ch. III.4. For a lucid nd concise ccount fthe xiomaticmethod, ee A. Tarski, .c.,Ch. VI.5. Bertrand ussell,ntroductiono Mathematical hilosophy, ewYork nd London, 919,

p. 71.6. For a more etailed ccount f the onstructionf thenumber ystem nPeano'sbasis, f.

Bertrand ussell, .c., sp. Chs. and VII.-A rigorousnd concise resentationfthat onstruc-tion, eginning,owever, ith he et of ll integers ather han hatofthenatural umbers, aybe found nG. Birkhoffnd S. MacLane,A Survey f Modern lgebra, ewYork1941,Chs. , II,III, V.-For a general urvey f the constructionf the number ystem, f. lso J. W. Young,Lectures n theFundamental oncepts f Algebra nd Geometry, ew York, 1911, sp. ectures

X, XI,XII.6a. As a result fvery eep-reachingnvestigationsarried ut by K. Godel t s known hatarithmetic,nd a fortiori athematics,s an incompleteheoryn the followingense:While llthosepropositions hich elong o the classical systems f arithmetic,lgebra, nd analysis an







556 ON THE NATURE OF MATHEMATICAL TRUTH

indeedbederived,n the ense haracterizedbove,from he Peano postulates,here xistnever-theless therpropositions hich an be expressedn purely rithmeticalerms,nd which retrue, utwhich annot e derived romhePeanosystem. ndmore enerally:or any postulatesystem farithmeticor ofmathematicsor hat matter)whichs notself-contradictory,hereexistpropositionshichre true, ndwhich an be stated n purelyrithmeticalerms, utwhichcannotbe derived rom hat postulate ystem. n otherwords,t is impossible o constructpostulate ystemwhichs not self-contradictory,nd which ontains mong ts consequenceslltrue ropositionshich an be formulatedithin he anguage f rithmetic.

This factdoesnot,however, ffecthe result utlined bove,namely, hat t is possible odeduce, romhe Peanopostulatesnd the dditional efinitionsfnon-primitiveerms,ll thosepropositionshich onstitutehe lassical heoryf rithmetic,lgebra,nd analysis; nd it is tothesepropositionshat referboveandsubsequentlys thepropositionsfmathematics.

7. For a more etailed iscussion,f.Russell, .c., Chs. II, III, IV. A completeechnical e-velopmentf the idea can be found n the great tandardwork n mathematicalogic,A. N.WhiteheadndB. Russell, rincipiaMathematica, ambridge,ngland, 910-1913.-For veryprecise ecent evelopmentfthe theory, ee W. V. 0. Quine,Mathematical ogic, NewYork1940.-A specific iscussionfthe Peano system nd its interpretationsrom he viewpointfsemanticss includedn R. Carnap,Foundations fLogicand Mathematics,nternationaln-cyclopediaf Unifiedcience, ol. I, no.3, Chicago, 939;especiallyections 4, 17,18.

8. The assertion hat hedefinitionsiven bovestatethe"customary" eaningfthe rith-metical ermsnvolveds to be understoodnthe ogical,not the psychologicalense f theterm"meaning."t would bviouslyeabsurd o claim hat he bove definitionsxpresswhat very-bodyhas nmind"when alkingbout numbersndthevarious perationshat an beperformedwith hem.What sachieved ythose efinitionss rather "logical econstruction"fthe onceptsof rithmeticn the ense hat fthedefinitionsreaccepted, hen hose tatementsnsciencendeverydayiscourse hich nvolve rithmeticalermsan be interpretedoherentlynd systemati-cally nsuch mannerhat hey recapableofobjective alidation. he statementbout thetwopersonsn theofficerovides very lementaryllustrationfwhat s meant ere.

9. Cf.Bertrand ussell, .c.,p. 24 and Ch.XIII.10. This result asbeenobtained y W. V. 0. Quine;cf.hisMathematicalogic,NewYork,

1940.11. The principlesf ogicdeveloped n Quine'sworknd nsimilarmodernystemsfformal

logic mbody ertain estrictionss comparedwith hose ogicalruleswhich adbeenrather en-erallyccepted s sounduntil boutthe urn f he20th entury. t thattime, hediscoveryf hefamous aradoxes f ogic, speciallyfRussell's aradox cf.Russell, .c.,Ch.XIII) revealed hefact hat the ogicalprinciplesmplicitncustomarymathematicaleasoningnvolved ontradic-tions ndthereforeadtobecurtailedn one mannerranother.

12. Note that wemay say "hence"by virtue ftheruleofsubstitution,hich s oneofthe

rules f ogical nference.13. For a more etailed iscussionfthispoint, f. he rticlementionedn reference.

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On the Nature of Mathematical Truth CG Hempel