Per Martin Löf, Informal Notes on Foundations

8/13/2019 Per Martin Lf, Informal Notes on Foundations

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t / . About the s imples t mathematical l n ~ u g e fo r which a non-t r i v i n l theory of meaning can be formulated i s the l n g u ~ e inwhich the only objec ts and func t ions dea l t with are the na tu ra lnumbers and the pr imi t ive recurs ive func t ions . Tr i v i a l as t h i slanguage may seem t i s neve r the le s s r i c h e n o u ~ h t0 al low us toframe answers to such ques t ions as

What i s a na tu ra l number?

What i s zero?

What i s the successor of a na tu ra l nuillber?

Moreover the answers tha t we sha l l give to these quest ions w i l lserve as paradigms fo r our answers to the more general qnest ions

What i s a type?

What i s a mathematical objec t of a given type?

which we sha l l pose l a t e r fo r the theory of types .Charac te r i s t i c of many inves t iga t ions o f the concept of

na tu ra l number i s tha t they at tempt to def ine na tu ra l number interms of o ther , more pr imi t ive , concep ts . Thus Frege def ined anumber as an extens ion of a proper ty , Russe l l as a c las s of s imi -l a r c lasses , and von Neumann as a s e t in the cumulat ive hie ra rchy.According to the l a t t e r , now orthodox d e f i n i t i o n , the na tu ra lnumbers 0 , 1 2 are defined to be the s e t s ,{ , { . { } \ ,However these def in i t ions only re


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What i s a c la s s?

What i s a s e t in the cumulat ive hiera rchy?

For Dedekind, on the othe r hand, the na tu ra l numbers were theelements of a simply i n f i n i t e system ( t ha t i s , ~ s y s t e m s a t i s f y -ing the Peano axioms) whose pa r t i c u l a r na t ure we have di sre ?;ardedby our f acu l ty of ab s t r ac t i o n . And he bel i eved he was ab le toprove the ex i s tence o f i n f i n i t e systems and t he re fo re a l so o fs imply i n f i n i t e ones. Of y e t ano ther kind i s Witt ?;enstein 's answeri n the Trac ta tus : A number i s the exponent of an opera t ion . Itcon ta ins in embryo the ana lys i s t ha t we sha l l give , not only fo rna tu ra l numbers, but fo r mathematical objec t s of any type.

The m ~ h e m t i c l symbols tha t we s h a l l use, wil l be div idedi n to funct ion cons tan t s

o s , + , . , . ,and numerical Yariables

x, y 'I The func t ion cons tan t s a re fu r t h e r divided i n to p r imi t ive andl.j def ined ones . Only 0 and s a r e ~ All the r e s t a re de-l f ined . An express ion (meaningless , in ?;eneral) i s simply a s t r i n g

o f symbols , l i k e

0 , XY+X, X +s(O),To t a l k about t he l n ~ u g e as we s h a l l do when d e s c r i b i n ~ i t ssyntax and semant ics , we a l so need s y n t ac t i c a l v a r i ab l e s . These


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are

f g

which stand for function constants ,

v w which stand for numerical variables , and

a , b ,

which stand for expressions. When we want to indicate that anexpression may contain the var iables v1, . . , vk and no others,we shal l denote i t by a(v1 , . ,vk) . This allows us to denote t heresul t of subst i tut ing the expressions a 1 , , ak for v1 , . , vkin ~ . . . ,vk) by a(a 1 , . . , ak) . Linguis t i c ex press i ons , considered in the or d i nary w y, are not mathematical objects . Only

*. in metamath ematics are th ey t reated a s such. But, of course, when

III i

ta lking about expressions as metamathematical objects, we usel inguis t ic expressions in the ordinary sense. They can never be

ltl(dispensed with. Likewise, the subst i tut ion operat ion, which takesa 1 , , ak and a(v 1 , ,vk) into a(a 1 , . . . , ak) i s not a funct ionin the mathematical sense. Only when the expressions are t reatedas metamathematical objects, does subst i tu t ion become a function,which i s defined by recursion on the expression as metamathemat-ica l object a(v1 , ,vk).

The rules of the language of primitive recursive funct ionsproduce statements of the f ive forms


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f (x 1 ,x , . ,x) = y y i s the value of the r ecur s ive ly definedfunc t ion f fo r the subordinate arguments x 1 . . x, and the.{pr inc ipa l argument x)a = x (a denotes x , o r x i s the value of a)

x i s a na tu ra l number

a i s a numerica l funct ion

a = bora b ( the func t ions a and b a r e de f in i t i ona l lyequal , o r have the same value)

The phrases fo l lowing f ( x i . . ,xk ,x) = y , a= X and a= binparantheses only show how to read these s ta tements in n ~ l i sThey must in no way be regarded as explanat ions of the meaningsof these s ta tements . As such, they would have to be unders tooda l ready, and it would be absurd of us to assume t h i s , because t h et a sk t ha t we have se t ourse lves i s prec i se ly to exp la in the mean-ings of these s t a tements , it being t o t a l l y i r r e l e v a n t whetherthey a re expressed in mathemat ica l nota t ion o r i n Engl ish . How-ever , s i nce we sha l l formulane our theory of meaning in n ~ l i sit wi l l be convenient to ha ve the opt ion o f express ing the s t a t e -ments whose meanings we a r e t o exp la in in Engl i sh as wel l .

When the express ion occur r ing in the l e f t hand member of adenotat ion s ta tement a (v i , . ,vk) = x con ta ins var i ab les , it wi l lalways be derived from c e r t a i n s s i ~ n m e n t s vi xi , . . vk = xko f values to the var i ab les . e sha l l ind ica te t h i s by a f igureof the form


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and say that a(v1 . . . ,vk) X for Vi= x1 . . k = Xk or,a l te rna t ive ly that xis the value {denotation) of a{v1 . . ,vk)for the arguments x1 . . . xk.

Rules of computation. The recursion scheme s t ipu la tes thatgiven numerical functions a(v 1 ,vk) and b(v1 ,vk,v,w), wemay define a (k+1)-place funct ion f by the computation rules

a(v1 . . . ,vk) xf(x 1 . . ,xk,o) x

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V = w yf{x 1 . . ,xk,x) y b(v1 . . ,vk,v,w) = z

f (x1 . . xk s( x )) zThe symbol f i s to be uniquely associa ted with the expressionsa(v1 ,vk) and b(v1 . ,vk,v,w), disregarding the choice of thevariables that v1 vk, : a n d w stand for {natura l ly , since achange of those variables does not af fec t the way in which f i scomputed). As already indicated, in a computat ion statement

x1 xk are cal led the subordinate arguments, x the principalargument, andy the value of the funct ion f for these arguments.

Rules . of denotat ion. The computation ru les re fe r back tostatements of the form a(v1 . ,vk) = x, and hence they do not


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allow us to determine the value of a recursively defined funct ionunt i l the denotation rules have been given. They read

V X assignment)

0 = 0a = x

sraT = s x)- = x

In the l a s t rule , f is a {k+i)-place function defined by recursion.Fi r s t and second Peano axioms. When formulated as rules,

they read0 i s a natural number

x i s a natural numbers{x) is a. natural numberRules of function formation. These are the rules p r o d u i n ~

statements saying of an expression that i t i s a numerieal func-t ion, namely

v is a numerical function0 i s a numerical funct ion

a is a numerical functions a) is a numerical functiona 1 . ak, a are numerical functionsf a 1 ,ak,a) i s a numerical funct ion


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IIcre , of course , v i s a numerica l var iab le , and f i s a k +1 ) -p l acefunct ion def ined by recurs ion .

Rules of d e f i n i t i o n a l equa l i ty . Using the nota t ion a= br a the r than s b, they read

a 1 , . . , ak a re numer ical func t ionsf a 1 , , ak ,o ) = a a 1 , , ak )

a 1 , , ak, a a re numerica l func t ions

a l = c l ak = ckb a 1 , ,ak = b c 1 , ,ck)a i s a numerica l func t ion a= b

b = ab = c- -a = c

In t h e f i r s t two.of t h ese ru l e s , f i s a func t ion def ined by r e -c u r s ion from ~ h nume r i c a l fu n c t ions a v 1 , ,vk a n db v1 , ,vk ,v ,w ) , and, in the t h i rd ru l e , b v 1 , . . ,vk) i s assu medto be a numerica l func t ion .

With t h i s , the f o r mula t ion of the syn tax o f the l n ~ u ~ e o fpr imi t ive r e c u r s ~ v e func t ions i s complete . Note t h a t each o f i t sru les r e f e r s s o l e l y to the syn tac t i ca l forms of the s ta t ementsoccurr ing as premises and conclus ion. This i s what makes it i n toa formal language. It remains f o r us to exp la in the meani ngs o ft h e s ta tements t h a t can be der ived by means of the formal ru l e sor , wha t amounts to the same, how t h ey a re unders tood) because

unders tanding a language, even a formal one, i s not mere ly to; .unders tand i t s ru les as r u l e s of symbol manipu la t ion . Del iev in ;


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t h i s s the mistake of f o r ma l ism .To unders tand a l n ~ u ~ e i s to unders tand i t s ru l e s , and

to understand a ru le i s to unders tand the conclus ion under theassumption t h a t the premises have been unders tood.

The meaning of a computat ion s ta tement f x 1 , ,xk ,x) = yis the way in which it i s der ived . Simi la r ly , the m e n i n ~ of adenotat ton s ta tement a v 1 , ,vk} = X i s the way in which it i s-derived from the assignments v i = xi , ... vk = xk. Thus, tounder s t a nd a computat ion o r deno t a-tion s ta tement , we must know

nothing beyond how it i s der ived . This means t ha t , fo r the com -puta t ion and deno ta t ion s ta tement s , the fo rma l i s t i n t e rp r e t a t i o nis correc t For example, we unders tand t h a t 0 + s{s O)) = s s O))from the de r iva t ion

- = 0 z = 00 0 0 srzy s O) - s O)= z =

0 s O) = s O) TZT = s s O))0 s s O)) = s s O))

Since to unders tand a ru le i s to unders tand the conclus ion underthe assumption t ha t the premises have been unders tood, t l1ere i snothing t o unde rs t and about the ru les of computat ion anrl denota-t ion . I f we know how to der ive the p remises , we know of coursehow to derive the conclus ion, namely, by apply ing the ru le inques t ion . This i s the g roun d f o r ca l l ing them mer e s t i p u l t i o n

The meaning of a s ta tement o f the form


i s the way in which a na tu ra l number i s -detcrminect as the va lue


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of a recurs ive ly def ined func t ion fo r x as pr inc ipa l and na tu ra lnumbers as subordinate arguments. Thus to undersLand tha t x i sa na tu ra l number we mu s t k now ho w to determine a na tu r a l number ys uch tha t f (x 1 , . . ,xk ,x) = y under the assumption t h a t f i s a(k+1)-p lace funct ion def ined by recurs ion and x 1 , . , xk arena tu ra l numbers. What determines the value of a r ecur s ive ly de-f ined funct ion fo r given arguments i s the scheme of recurs ion .and hence we cannot unders tand a pa r t i c u l a r s ta tement o f the formt ha t we are consider ing without looking back on t h i s scheme. Asi s imp l i c i t in what we have j u s t sa id , a na tu ra l number i s anexpress io n fo r wh i c h we have understood the s ta tement abo:ve . Or .as we may say , a na tu ra l number i s an express ion which we haveunderstood ( in te rp re ted ) as a n a t u r a l number. T hi s i s our answerto the quest ion

What i s a na tu ra l number?posed in the beginning. I t s r e l a t i o n to W it tg en s t e in s answer:A na tu ra l number i s the exponent of an opera t ion , becomes c l e a ri we reformulate it thus: A na t ura l number i s the pr i n c i r a la r gument of func t ions def i ned by recurs ion . Indeed the i t e ra t i o n

f schemei

f (y ) = z

i s nothing but a spec ia l form of the recurs ion scheme the expo-nent n being the pr inc ipa l and x the subordinate argument of thebinary func t ion fn (x ) . So our fo rmula t ion i s obta ined from Wit t -g e n s t e i n s by consider ing the recurs ion scheme in i t s general

..J Rc : : . ciJ t v ~ t . u . . ~ t.l. 4 < > . \ . . ~ < - t vv _ v a . . l . . ~ s - 4 - c : > o J lt t


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have understood i t . And, to understand i t w merely have tounderstand that the expressions a v 1 . . . ,vk) and b{v1 . ,vk,v,w),which in a purely formal way are associated with the symbol fare numerical funct ions.

Given these explanations, it is clear that what allows us tounderstand determines the meaning of) the f i r s t Peano axiom

0 is a natural numberi s the f i r s t clause

a{v 1 vk) xf x 1 . . . ,xk,o) x /

of t h ~ recursion scheme. Indeed, that f i s a k+1)-place functiondefined by r e c u r s ~ o n means that the expressions a v1 . . ,vk) andb v 1 . . ,vk,v,w) in terms of which i t i s defined are numericalfunct ions. In ~ a r t i c u l a r so i s a v 1 . . ,vk). Hence, givennatura l numbers x 1 . xk, w know how to determine a natura lnumber x such that a{v1 . . ,vk) = x for ~ = x 1 . vk = xk.The f i r s t clause of the recursion scheme s t ipula tes that th is

number x i s to be the value of f for the subordinate argumentsx 1 xk and the principal argument 0. And, of course, therei s no other rule which allows us to derive a statement of theform f x 1 . ,xk,O) = x. This i s the meaning of the f i r s t Peanoaxiom, that i s the way in which a natura l number is determinedas the value of a recursively defined funct ion for the principalargument 0 and natural numbers as subordinate arguments. I t i s


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also our answer to the ques t ion

What i s zero?

because, as s t a t ed in the opening sen tence of Frege ' s Grundlagenthe quest ion what the numbe r z e ro i s , come s to the same as th equest ion what the symbol 0 mean s

Our explanat ion of the meaning of the symbol 0 i s typ ica lof how substance can be given to Wit tgens te in ' s s l o ~ n tha tmean i ng i s use, or , more e xp l i c i t l y , t ha t the m e n i n ~ of an ex-press ion i s determined by the ru les t ha t govern i t s use in the14nguage of which t forms a par t . I t would be absurd to i n t e r -pre t t ha t s logan as saying t ha t meaning i s confer red au tomat ic -a l ly upon t h e express ions o f a language by i t s ru l e s . There wotil dthen be no need fo r a theory of meaning. I f we l ay down any oldse t of ru les , l i k e those of se t theory with the u n re s t r i c t e d( incons i s ten t ) comprehension axiom o r t hose of the type- f reeca lculus of lambda-conversion, the express ions der ivable by meansof t hose ru les al low in genera l no othe r than the fo rmal i s ti n te rp re ta t ion , according to which the meaning of an express ion

~ the way in which t i s der ived. The d i f f i c u l t y in expla ining

Lhe meaning of an express ion of a language i s to discern what areru les of the language t ha t determine i t s meaning. Fo r ins tance ,in the language of pr imi t ive recurs ive func t ions , the symbol 0occurs e xp l i c i t l y , not only in the f i r s t c lause o f the recurs ionscheme, but a l so twice ) in t h e deno ta t ion ru le = 0 , in t h ef i r s t Peano axiom, in one of the ru les of func t ion format ion,and in one . of the ru les of d e f i n i t i o n a l equa l i ty . I t i s by nomeans automatic tha t t i s the f i r s t c lause of the r ecur s io n

t


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scheme and none of the other rules tha t determines the meaning ofthe symbol as a natura l number.

I f somebody suggests tha t the funct ion of the spark plugs inan in ternal-combustion engine is to be connected by cables viathe dis t r ibu tor to the bat tery w must explain to him how theengine works, thereby convincing him that the funct ion of thespark plugs i s to igni te the mixture of petrol and a i r which issucked into the cylinders from the carbure t tor . I t i s not that i tis wrong to say that the spark plugs are connected by cables viathe dis t r ibu tor to the bat te ry: they cer ta in ly are . But that i snot what w should primari ly pay our a t t en t ion to in order tounderstand the funct ion of the spark plugs in the running of theengine. Similar ly , in a r i thmet ic i f somebody suggests that theI meaning of the symbol i s determined by the f i r s t Peano axiom,

I he is wrong, not because i s not a natura l number: i t ce r t a i n ly1 i s bu t because t h e f i rs t Peano axiom is not the rule from which

the meaning of the symbol i s l ea rn t . Instead, w must di rec tftI his a t t en t ion to the f i r s t clau se of th e r ecurs ion scheme, becausetha t is the rule which t e l l s h im how a na tu ra l number i s dete r -mined as the va l ue of a re cu rs iv ely defined funct ion when i sinser ted into i t s pr inc ipa l argument place together with natura l

f

flf

numbers as subordinate arguments.Recall that to understand a ru le w must understand the

conclusion under the assumption tha t the premises have been

It

understood. Hence, to understand the second Peano axiom

x i s a natural numbers x) i s a natura l number


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we must know how to eva lua te a r ecur s ive ly def ined func t ion fo rthe pr inc ipa l argument s x ) , given t ha t we know how to eva lua teit fo r the pr inc ipa l argument x . So l e t f be a k+1)-place func-t ion def ined by recurs ion and x 1 xk na tu ra l numbers. By

a ~ s u m p t i o n we know how to determine a na tu ra l number y such t ha tf x 1 ,xk,x) = y . Since f i s a {k+1)-place func t ion defined byrecurs ion , the second of the express ions assoc ia ted with the symbol f , ca l l it b{v 1 ,vk ,v ,w) , i s a numerical func t ion . Hence,x 1 . . xk, a n d y being na tu ra l numbers, we know how to de te r -mine a na tu ra l number z such t ha t b v 1 ,vk ,v ,w) = z fo rv 1 = x 1 vk = xk, v = x a n d ~ = y . The second c lause of therecurs ion scheme

V = X \ = yf{x 1 . . ,xk ,x) = y b v 1 ,vk ,v ,w) = z

f x 1 . . ,xk ,s{x)) = zs t i pu l a t e s t ha t t h i s number z is to be the value of f fo r thesubordinate arguments x 1 . xk and the pr inc ipa l argument s x ) .This i s the meaning of the second Peano axiom, or , what amountsto the same, the meaning of the symbol s . Our explana t ion of t h emeaning of s{x) in terms of the meaning of x , i s a t the same t imeour answer to the ques t ion

What i s the successor of a na tu ra l number?

because, to ask f o r t h ~ meaning of the express ion s{x) , i s thesame as to ask what s x) i s .

I t i s of utmost importance to rea l ize tha t the f i r s t two


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Peano axioms do not s t ipu la te what a natural number Thatwould be to say that a natural number is an expression for whichwe have derived ins tead of understood) the statement

x i s a natural numberThis is to confuse natura l numb ers wi th nume r a l s which i s theformalis t mis i nterpretat ion of the concept of natural num ber.What the f i r s t two Peano axioms show, is the form of a naturalnumber. But an expression i s a natura l number, not in vir tue ofi t s form, but in vir tue of i t s funct ion. And th i s function is toserve as the pr inc ipa l argument of functions defined by recursion.

Having explained the meanings of the f i r s t two Peano axioms,we now turn to the rules of funct ion formation. Remember tha t .to understand a statement of the form

a v1 . . ,vk} i s a numerical funct ionwe must know given natura l numbers x 1 xk, how to determinea natura l number X SUCh that a v 1 . . ,vk) =X for v1 = x1 .vk = xk. In par t icu la r we understand the statement

v i s a numerical functionfor a numerical variable v, by looking back on the denotat ionrule which says that

v x;

provided x is assigned as value to the variable v. I t is th iss t ipulat ion which allows us to in te rpre t the symbol as a numer-ical funct ion. The statement

l


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IIfftlI

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0 i s a numerica l funct ion

i s understood from the s t i p u l a t i o n

0 = 0and the f i r s t Peano axiom. Thus we cannot i n t e rp re t the symbol 0as a numerical funct ion u n t i l a f t e r we have in te rp re ted t as ana tu ra l number. The ru le

a i s a numerical func t ions a ) i s a numer ical func t ion

i s understood from the s t i p u l a t i o n

srar = s x )and the second Peano axiom. Suppose namely t ha t a i s a numerica lfunc t ion , t h a t i s , t h a t we know how to determine a na tu ra l num-be r X such t ha t a = X. Then s x ) i s a na tu ra l number by thesecond Peano axiom, and the denotat ion ru le above s t i p u l a t e s t h a tstaT s x ) . This i s the meaning of the above r u l e of func t ionformat ion, t ha t i s , the way in which a na tu ra l number namely,s x ) ) i s determined as the va lue of s a ) under the assumptiont h a t a i s a numerica l func t ion . Not e tha t we can und e r s tand t h isr ule only a f t e r we have und e r s tood the second Peano axiom. Thef ina l ru le of func t ion format ion i s

a 1 . ak , a are numerica l funct ionsf , ak , a ) i s a numerica l funct ion


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where f is a k+1) - place function defined by recursion. To under stand i t , w must know how to evaluate f (a 1 , . . ,ak ,a) under theassumption that w know how to evaluate a 1 , . . , ak and a , tha ti s , that w know how to determine natural n u m ~ e r s xi , . . , xk

-and x such that i 1 = x i , . , ak = xk and a = x. But th is i sprecisely what the denotat ion rule

a = x

t e l l s us. Indeed, since f i s a k+1)-place function defined byrecursion and the subordinate arguments x 1 , , xk are naturalnumbers, w know that a natura l n u ~ b e r i s determined as i t s valuewhenever a natural number i s inserted into i t s pr inc ipa l r ~ u r n e n tplace. In par t icu la r , th i s i s so for x, that i s , w know how todetermine a natural number y such that f(x 1 , . . ,xk,x) = y. Thedenotation rule s t ipu la tes that th i s number y is to be the valuedenotat ion) of f(a 1 , . . ,ak ,a) . This is how the meaning of theexpression f (a 1 , . . . ,ak,a) as a numerical funct ion is determinedin terms of the meanings of the expressions a 1 , . , ak and a.

ItI

Composition of funct ions. I f a 1 , . . , ak and b v1 , . . . ,vk)are numerical funct ions, then so is b a 1 , . ,ak}, andb{a

1, . . ,ak} = y provided a1 = xi , ... ak = xk and

b v1 , . ,vk) = y for v i = x i ... vk = xk.This cannot be proved, t has to be understood. And what

has to be understood i s the way in which b a 1 , . ,ak) i s evaluated, and_ that i t s value i s the same as i s obtained by f i r s t

e v l u t i n ~ a 1 , . , ak and _then evaluat ing b v 1 . . ,vk) for thesevalues as arguments. So suppose ~ a 1 , . , ak are numerical: 1:


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funct ions, that i s that w know how natural numbers x 1 , . , xkare determined as thei r values

Suppose fur ther that b v 1 , .-. . ,vk) i s a numerical funct ion. Then,x 1 , . , xk being natura l numbers, w }{now how to determine anatural number y as the value of b v 1 , . . . ,vk) for these arguments

Replacing the var iables v1 , . , vk in th is evaluation by thee x p r e s s i o ~ s a 1 , . . , ak and attaching to i t the evaluat ions ofthese

w see that the natural numb.er y is determined as the value ofb a1 , ,ak). This is how w understand that b a 1 , . . . ,ak) is anumerical f u n ~ t i o n

The l a s t part of the statement about composition of func-t ions s ta tes that the value of a composite funct ion is determinedby the values of i t s components. Reading Bedeutung for value,w recognize th i s as one of the t h s ~ that was set for th byFrege in t iber Sinn und Bedeu t ung. Remember that i t i s not some-thing which can be pr ove d , but mus t be understood. Hence the word


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t hes i s r a t he r than theorem.The meaning of a de f i n i t i ona l equa l i ty , tha t i s , a s ta tement

o f the form

i s the way in which a na tu ra l number i s determined as the commonva lue of a (v 1 . . ,vk and b(v 1 ... ,vk when na tu ra l numbers areass igned as va lues to the var iab les v 1 . vk. Thus, to under-stand a sta tement of t h i s form, we must know, fo r a rb i t r a r i l ygiven na tu ra l numbers x 1 . xk how to determine a na tu ra lnumber x such t ha t

fo r v 1 = x 1 . . vk = xk. Al te rna t ive ly , we might have sa id t ha ttwo express ions are de f in i t i ona l ly equal i f they a re both n u ~ ri c a l funct ions , and, moreover, they take the same value fo ra rb i t r a r i l y given na tu ra l numbers as arguments.

Consider now the f i r s t ru le of def in i t iona l equa l i ty

a 1 . ak are numerica l funct ionsf ( a 1 ... ,ak ,o) = a{a i , ... ,ak

To unders tand it we must understand the conc lus ion under theassumption t ha t the premises have been understood. So supposet ha t we know how to determine na tu ra l numbers x 1 ... xk sucht ha t a 1 = x 1 . . ak = xk. By composi t ion of func t ions ,a (a 1 . . , ak i s a numerica l func t ion , and a (a 1 ... ,ak = xwhere xis the na tu ra l number such tha t a (v i , . . ,vk = x fo rv i = x i , . . vk xk. On the o ther hand, the f i r s t c lause of


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the recurs ion scheme and the denotat ion ru les y ie ld

I

0 = 0 a{v1 v,) =xK

This i s how we determine the na tu ra l number x such tha t

The second ru le of d e f i n i t i o n a l equa l i tya 1 ak, a are numerica l func t ions

20

i s unders tood in a s imi la r way. Assume t ha t we know how to d e t e r -mine na tura l numbers x 1 . Xk and X such t ha t a1 = x 1ak = xk and a = x . Then, s ince f i s assumed to be a k+1)-p lacefunct ion def ined by recurs ion , we can determine, f i r s t , a na tu ra lnumber y such t ha t f x 1 . ,xk ,x) = y and, second, a na tu ra lnumber z such t ha t b v 1 . ,vk ,v ,w) = z fo r = x 1 vk = xk,v = and w= y . By the deno ta t ion ru le

a = x

and composit ion of func t ions ,

n the othe r hand, t h i s number z i s determined as the value o f


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vi = xi vk = xk V =X. . . . . . . .a = x f xi , . . ,xk,x) = y b v1-----:-:--. vk v w = z

= x1 ak = xk sr r = s x) f X Xk, S X) ) = ZfT a 1 ;-:-- , ak, s a) } = z

Fig. 1

w = y. .... .


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f a 1 , . , ak , s a ) ) by the second c lause of the recurs ion schemeand the dP.notation ru les , as shown in F ig . 1 . This i s hO\V weunders tand the second ru le of d e f i n i t i o n a l equa l i ty . The t h i rdr u l e

a1 = c1 ak = ckb a 1 , ,ak) = b c 1 , . , ck)

i s the pr inc ip le t ha t d e f i n i t i o n a l equa l i ty i s preserved undersubs t i tu t ion . Assume t h a t i t s premises have been unders tood,

22

tha t i s , tha t we know how to determine na tu ra l numbers x 1 , . . , xksuch t ha t

Since , by assumpt ion, b v 1 , ,vk) i s a numerical func t ion , weknow how to de termine a na tu ra l number y such tha t b v 1 , . . ,vk) = yfo r v 1 = x 1 , , vk = xk. y composi t ion of func t ions ,

b a 1 , , ak) = y b c 1 , c k)which prec i se ly what \Ve had to see . Of the remaining ru les ofd e f i n i t i o n a l equa l i ty .

a i s a numerica l func t ion a = b a b b = cb a

the r e f l e x iv i ty and symmetry a re unders tood immedia te ly , and thet r a n s i t i v i t y as fo l lows. Suppose t ha t we know how to de terminena tu ra l numbers x and y such t ha t a x b and b y = Then,s ince both x and y a re determined as the value o f b, they mustbe the same express ions . Therefo re the same na tu ra l number x i s


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23

determined as the value of both a and c , which i s p r ec i s e ly whatwe must know in order to unders tand tha t a = c This f in i shes ourexplanat ions of the meanings of the ru les of the l n ~ u g e ofpr imi t ive recurs ive func t ions .

Statements of the f ive forms tha t we have been cons ider ingcannot be proved, only unders tood. There i s no ques t ion of t he i rbeing t rue o r fa l se , l i ke ord inary mathemat ica l o r metamathe-mat ica l ) propos i t i ons . They can only be meaningful o r meaning l ess .For example the sta tement

0 i s a na tu ra l number

i s not rue , but meaningful , and the sta temen t

i s a na tu r a l number

i s not f a l s ~ but m e a n i n ~ l e s s because t he re i s no way of eva lu -a t ing a r ecur s ive iy defined func t ion fo r the pr inc ipa l a rgument .Or as we may say, we cannot i n t e rp r e t the symbol as a na tu ra lnumber. I t i s m e n i n ~ l e s s as a na tu ra l number .

When we ha ve unders tood a language, t ha t i s , when we haveunders tood i t s ru les , we know . tha t the s t a tements which can b ejormal ly de r i v ed by means of those ru les are meaningful .

For an ord inary mathemat ica l propos i t i on , t he re remains thequest ion , even a f t e r it has been unders tood, t ha t i s , a f t e r ithas been unders tood t ha t it i s a propos i t i on , whether it i s t rueo r f a l s e . And t h i s ques t ion c ~ n only be se t t l ed by proving o rdisproving t ha t i s , proving the negat ion of) it However, t ha ta l i n ~ u i s t i c express ion i s a proof o f a c e r t a i n mathemat ica lpropos i t ion i s not s o m e t h i r i ~ t ha t we can again s e t about to


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24

prove: it has to be unders tood . When we reach the words Q.E.D. a tthe end of a proof of a theorem, we are supposed to have understood t h a t it i s a proof of the theorem in ques t ion .

The d i s t i nc t ion between s ta tements t h a t can be proved andthose tha t can only be unders tood, i s , e s s e n t i a l l y , Wit tgen-s t e i n s d i s t i n c t i o n in the Trac ta tus between what can be sa idand what can only be shown. It i s a l so c lose ly re l a t ed to hi sd i s t i n c t i o n between proper t i e s and r e l a t i o n s proper ( ex te rna lprope r t i e s and re l a t ions ) and formal ( i n t e rn a l ) prope r t i e s andr e l a t i o n s , and, somewhat l a t e r , between concepts proper andformal concepts . proper ty proper i s a propos i t i ona l funct ionwhich ass igns to an objec t the propos i t i on tha t the objec t hasthe proper ty in ques t ion . Th i s i s an ordinary mathematical pro-pos i t ion , which we may t ry to prove o r disprove. Bein1; an evennumber, a prime number, the sum of two prime numbers, and so on,are a l l proper t i e s proper (o f na tu ra l numbers . That s o m t h i n ~f a l l s under a formal concep t , on the othe r hand, cannot be proved:it has to be unders tood . Or, in W it tgens te in s words, it cannotbe sa id : it shows i t s e l f . formal concept cannot be repre-sentedby a propos i t i ona l func t ion . In the l n ~ u g e of pr imi t ive recur -s ive func t ions , the concepts of na tu ra l number and numerica lfunc t ion a re both formal concepts . And the r e l a t i o n of d e f i n i -t iona l equa l i ty i s a formal r e l a t i o n .

In the Trac ta tus , Wit tgens te in apparen t ly thought of aformal ( i n t e rn a l ) proper ty as a property of o b jec t s . This expla ins why he sa id t h a t a proper ty i s i n t e rn a l i f it i s unth ink-ab l e tha t i t s objec t should not possess it I f the s ta tement


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5

0 i s a na tu ra l number

i s const rued as saying of the objec t 0 t ha t t i s a n a tu r a lnumber, then we may indeed say t h a t t i s unthinkable tha t 0 werenot a na tu ra l number, because we cannot perce ive the objec t{unders tand the symbol 0 withou t perceiving unders tanding) tas a na tu ra l number. However, a formal proper ty i s not a proper tyof ob jec t s . Only proper t i e s proper are proJJert ies of ob jec t s .A formal proper ty i s a proper ty of express ions . Likewise , whatf a l l s under a formal concept i s an express ion , not an objPc t .The f i r s t Peano axiom says o f the symbol 0 t ha t t i s a n a tu r a lnumber, t h a t i s , t ha t t works as the pr inc ipa l argument of func-t ions defined by recurs ion .

Ins tead of saying t ha t we unders tand s ta tements of the forms


and

a v 1 ,vk i s a numerica l func t ion

we have sometimes sa id t ha t we unders tand i n t e rp r e t ) the expres-s ion x as a na tu ra l number and the express ion a v 1 . ,vk} as anumerica l func t ion . There i s no ques t ion of our u n d e r s t n d i n ~separa te ly the express ion x and the concept of na tu ra l numberand then proving t ha t x i s a na tu ra l number: when we unders tandthe express ion x , we unders tand t as a na tu ra l number. So we maysay t ha t , in the s ta tement t ha t x i s a na tu ra l number, the pred i -ca te i s contained in the subj .ec t provided, l i k e Wit tgens t e in ,we t ake the sub jec t to be the objec t and not the express ion x) .


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6

This is precisely the character is t ic of a statement which i sanalyt ic in Kant s terminology. Similarly, we do not understandthe expression a(v1 . ,vk) separately from the concept ofnumerical funct ion and then prove that i t fa l l s under th is concept: we understand i t as a numerical funct ion. So the statementthat a(v1 vk) i s a numerical function is also analyt ic . Onthe other hand, a l l mathematical proposi t ions in the ordinarysense, that i s proposi t ions which we prove, are according toKant, synthetic (and a pr ior i since the i r t ru th does not dependon experience}. The dis t inc t ion between statements that can beproved and th os e t hat can only be understood, may be regarded asa fomulat ion of the dis t inc t ion between synthetic and analy t i cjudgements which i s not l imited to statements of subject -predicateform. Ano.ther such formulation (used by Quine for example) i sthat a statement is analyt ic i f i t i s t rue by vir tue of i t smeaning. But that formulation i s l ess fortunate, because ananalyt ic judgement i s a statement which has been understood and for which there is no question of being t rue or fa lse .

In terms o knowledge, the dis t inc t ion s between knowledgeof t ru th and knowledge of m ~ n i n What we lmm when we haveproved a statement i s i t s t ru th and what we know when we haveunderstood a statement i s i t s meaning. That a person knows thet ru th of .a proposi t ion manifests i t s e l f in his ab i l i ty to provei t . In the terminology used by Dummett in his Bris to l paper,knowledge of t ru th i s verbalizable knowledge. Knowledge of themeaning of a l inguis t ic expression, on the other hand, can onlymanifest i t s e l f in the ab i l i t y to use the expression correct ly .


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7

Thus there i s no one l i ngu i s t i c a c t , l i ke t ha t of proving atheorem, which conc lus ive ly shows t ha t we know the m e a n i n ~ of al i ngu i s t i c express ion . In pa r t i c u l a r , t i s not in our a b i l i t y toexpla in i t s meaning t ha t our unders tanding of a l i n ~ u i s t i c ex-press ion mani fes t s i t s e l f . I f t hat we r e the cas e a lmost n orna heruatician could be sa i d to und e r s tand the pr imi t ive not io nsof mathematics. In Dummett s words, knowledge of m e n i n ~ i simpl i c i t knowledge.

Our understanding of the f ree var iab le s ta tements

a tv 1 . . ,vk) i s a numerical func t ion

and

i s necessa r i ly uniform in the arguments, because our lmowledgehow to determine a na tu ra l number x such t ha t

r espec t ive ly

fo r v 1 = x 1 ~ vk = x k comes so le ly from the knowledge t ha tx 1 xk a re na tu ra l numbers. There i s no ques t ion of ~ r o v i ns t a t ements of t h i s kind by n d u c i o n t ha t i s , by separa t in gcases according to the form of one of the arguments . That wouldbe to t r ea t them as metamathematical propos i t ions proper . Forexample, we cannot understand the f r ee va r iab le s ta tement


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28

as opposed to v + 0 = i ) because we cannot see from the definingequations of the addit ion function and the denotation rules howto determine a natural number x such that 0 + v= x fori= x.n the other hand, we can prove by induction as metamathematical

t heorems t ha t 0 + x = x i s derivable, and hence t ha t 0 + v = x i sderivab le fro m i = x , for a l l numerals x. But tha t i s someth ingent i re ly differen t from understanding tha t 0 + v = x for v = x.

When the language of primitive recursive functions i s t rea tedm e t a ~ a t h e m a t i c a l l y the expressions become rnetamathematicalobjects of an induct ively defined type, ahd the five di fferen t

'forMs of statements that we have been considering are turned intoi nductively de f i ne d metamathematical pro posit ions proper , that i s ,proposi t ions which can be proved and combined by means of thelogical connectives and quant i f iers . In part icular ,

x i s a natural numberis turned into a property proper, and

into a k+1 ) -place re la t ion proper between expressions as meta-mathematical objects. Therefore we can form the metamathematicalproposi t ion

\ lx 1 ) . . . \Jxk) x1 is a natural numberxk i s a natural number 3 x) x i s a natural numbera v i .. , vk) = X for Vi = Xi, , k = Xk))


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in which the quant i f iers r n ~ e over expressions as metamathemati-cal objects. I t expresses that the open expression as metamathematical object a v1 ,vk) is convertible in the sense ofTai t , and must in no way be confused with the statement

a v1 . . ,vk) is a numerical funct ionThe former is a proper rnetamathematical proposi t ion which w mayt ry to prove, whereas the l a t t e r can only be understood. Now tfollows direct ly from our understanding of the langua ge of prim-i t i ve recursive funct ions and the fact that a natural number musthave the form of a numeral, tha t , i f the statements

x 1 i s a natura l number

xk i s a natural numberand

a v1 ,vk) i s a numerical functioncan be derived, then an expression x is determined for which

x i s a natural number

and

can be derived, in the l a t t e r case, from the assignments ~ = x1 vk = xk. The metamathematical counterpart of th is i s thetheorem, easi ly proved by T a i t s method of conver t ib i l i ty , tha t ,


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i

a v 1 . ,vk i s a numerica l funct ion

can be der ived , then a v 1 . . ,vk i s conver t ib le . Two th ings mustbe born in mind in connec t ion with t h i s theorem. F i r s t , in botht h e fo rmula t ion and the proof of the theorem, we use l i ngu i s t i cexpress io n s i n t h e ord inary sense . Second, the proof of t h et h eorem has to be unders tood. Otherwise , it wo u ld have n o cog n i -t i v e va lu e . And unders tanding the metalanguage in which the proofof conver t ib i l i t y i s ca r r i ed out , i s a t l ea s t as d i f f i c u l t asunder s tanding the objec t lan guage, in t h i s case , the l n ~ u g e ofp r i mi t i v e rec u r s ive func t ions , fo r wh i ch conve r t i b il i ty i s p roved.I t would be absurd to mainta i n, a t l e a s t in the case of the Ian -g uage of pr imi t ive recurs ive func t ions , t ha t we do unders tand th emetalan guage but not the objec t langua e , and hence t ha t t h ere i sa gen uine n eed fo r the proof o f conver t i b i l i t y . But ; of course ,i t may be i n t e r e s t i n g fo r o t h e r reasons .

Per Martin Löf, Informal Notes on Foundations

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