the plight of the platonist

The Plight of the PlatonistAuthor(s): Philip KitcherSource: Noûs, Vol. 12, No. 2 (May, 1978), pp. 119-136Published by: WileyStable URL: http://www.jstor.org/stable/2214688 .

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The Plight of the Platonist PHILIP KITCHER

UNIVERSrIY OF VERMONr

Arithmetical Platonism is the thesis that statements of arithmetic are true or false in virtue of the properties of objects, the natural numbers, which do not exist in space-time, and which therefore deserve to be called "abstract objects". Many mod- ern philosophers find the thesis obvious. For some arithmetical statements are true. In these statements the numerals seem to function as singular terms. A statement can only be true if the singular terms which it contains refer to objects. Hence the numerals refer to objects. However, the numerals cannot refer to material objects because there are probably not enough material objects and, in any case, the truth of arithmetical statements does not depend on the fate of any material object. So the referents of the numerals, the numbers, must be abstract objects.

Faced with this argument, the anti-Platonist seems to have three options: (a) to give up the view that there are true arithmetical statements; (b) to abandon the connection between truth and reference which is central to Tarski's work on truth; (c) to reject the Fregean claim that numerals are singular terms. Neither of the first two alternatives is attractive. Charles Chihara has considered (a) ([3] Chapter II), but, as he concedes, a proponent of (a) would find it hard to account for the usefulness of arithmetic. Pursuing (b) would require us to give up Tarski's theory of truth, and there are impressive reasons for favouring that theory (see [5]). So the anti- Platonist is left with (c), the proposal that the clear and simple account of arithmetical language which Frege gave in [7] is incorrect and that a different view of the logical form of arithmetical sentences is required.

I shall try to show that there is good reason to look more closely at these apparently uninviting options. For arithmetical Platonism appears to lead to a difficult predicament.

NOUS 12 (1978) 119 ID 1978 by Indiana University



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In [1], Paul Benacerraf mounts an interesting attack on Frege's thesis that numbers are objects. Unfortunately it is not altogether clear what Benacerraf is attacking; his argument may be construed as a criticism of Frege, which attempts to show that Frege's logicism is untenable, or it may be seen as a broader objection to arithmetical Platonism in general. I shall reformulate it in the latter way. We should begin by noting the inconsistency of the following four statements.

(1) The numerals are singular terms which refer to abstract objects.

(2) If the numerals refer to objects then there are particular objects to which they refer.

(3) There are no particular sets to which the numerals refer.

(4) There are no abstract objects distinct from sets.

The problem is to resolve the inconsistency of (1)-(4) by find- ing an acceptable way to deny (at least) one of them.

Not any way of denying one of (1)-(4) is acceptable, for there are constraints imposed by post-Fregean mathematical practice. We can add to any standard set theory (e.g. ZF set theory) principles of the forms

ro = a', (X)(y)(S(X, y) = -(X, Y))

where a, F((x, y)' are set theoretic expressions, and so generate sentences which look like the Peano postulates. Let us fix our set theory as ZF, and consider the theories which result from the addition of such principles to ZF, theories which I will call ersatz arithmetics. (As Benacerraf pointed out, there are many ersatz arithmetics). Associated with each ersatz arithmetic is a sequence of sets which, from the point of view of the arithmetic, are its numbers, and a set-theoretic relation which, again from the point of view of the ersatz arithmetic, is the successor relation.

The anti-Platonist would like to show that (1) is at fault, by arguing that the constraints of the last paragraph do not allow for the denial of any of the other theses. His first task is to overcome an influential view of W. V. Quine. Quine proposes



THE PLIGHT OF THE PLATONIST 121

that (2) is not the banal consequence of our normal view of reference which it appears to be, but is in fact false. So he claims that (1), (3) and (4) are all true: numbers are sets but there are no particular sets to which the numerals (absolutely) refer ([12]).

Quine cannot, however, rest with the bare denial of the thesis (2), for talk of truth requires some positive account of reference. What Quine proposes is that semantic theorists must use a relativized notion of reference. Instead of asking for the reference of a singular term simpliciter it is only legitimate to demand the reference of a singular term relative to a translation manual.

Hartry Field has argued cogently that this relativized notion of reference will not suit the purposes of semantic theory ([6]: 206-9). The heart of the problem is this: to explain his notion of relativized reference Quine must offer some equivalence such as

(5) For every singular term a, object x and manual M, a refers to x relative to M iff M maps a into "x".

As it stands, (5) embodies a use-mention muddle which can be resolved only if the notion of unrelativized reference is al- ready available to us. Since Quine denies the legitimacy of this latter notion his own attempt to provide a useful approach to reference collapses, and the most he can achieve is a concept of coreferentiality which is useless for defining truth.

I conclude that Quine's attempt to solve Benacerrafs problem by denying (2) fails. We shall consider Field's own modification of (2) below.

Perhaps we can evade the difficulty by giving up (4) and claiming that numbers are abstract objects distinct from sets. (This approach is defended by Mark Steiner in [14]; Steiner also seems to accept a Quinean approach to (2) ([14]: 72)). To do so is to refuse to allow any ersatz arithmetic as genuine arithmetic, that is, to suppose that when we inspect the ersatz arithmetics, reading'0' as referring to zero and 'Sxy' as referring to the successor relation, none of the ersatz arithmetics, so interpreted, is true. Now there are bad reasons for wanting to accept some reduction of arithmetic to set theory. We cannot advocate reduction on straightforward epistemological grounds-for the reduction does nothing to increase the cer-



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tainty of our arithmetical knowledge ([14]: 71-2). However, there are also good reasons to block the denial of (4).

Claiming that there are abstract objects distinct from sets is uneconomical. Yet Steiner has urged that the set theoretic reduction of arithmetic is false economy. To reduce arithmetic to set theory is ". . . [to] achieve ontological gain without epistemological gain, indeed at epistemological loss." ([14]: 75). Steiner's point seems to be that, since set theory is less certain than arithmetic, the reduction of arithmetic to set theory sacrifices certainty in the pursuit of less important ontological benefits. But this is to present a false picture of the epistemological situation. By identifying numbers as sets we do not compel ourselves to adopt the derivation of arithmetic from set theory as the proper and preferred route to arithmetical knowledge. Reduction does not deprive us of our previous evidence for arithmetic. If, as Steiner suggests, our knowledge of arithmetic is based on our grasp of the intended model of the first-order Peano postulates ([14]: 76) then we may continue to justify arithmetical claims by means of this grasp, and arithmetic will be no less certain than before. The only difference will be that we now claim that the elements of this model are sets. (Should there be some cataclysm in the background set theory it will have no impact on the ordinary truths of arithmetic. We shall continue to believe these ordinary truths, denying the identification of numbers with sets if such modifi- cations are needed). Arithmetic remains epistemologically unaffected by the reduction, and we achieve ontological gain without epistemological loss.

Steiner supposes that we have some evidence for the truth of the axioms of arithemtic prior to our derivation of them as theorems of set theory. My claim is that, whether or not we take this evidence to consist in a "grasp of the standard model," we do not lose it when we reduce arithmetic to set theory. Even if a proof is available to us, the proof may not be our primary justification for accepting the theorem proved, and there are many cases in which we happily accept a proof while continuing to base our beliefs on the evidence that was previously available. A complex quantum-theoretic derivation of a description of an elementary law about chemical reactions does not supplant the evidence drawn from simple experiments. Similarly, as the history of mathematics shows, mathematicians are often prepared to concede that a new, rigorous proof is inferior as justification than old, intuitive




evidence. Bolzano and Cauchy, for example, recognized explicitly that loose geometrical arguments provide much better evidence for theorems about continuous functions than the long arithmetical derivations they hoped to produce. (For a discussion of Bolzano's attitude, see [8]: 229-3 1).

Mathematical proofs can serve other functions besides that of increasing the certainty of our knowledge. In some cases, as in the case at hand, they can be part of a scheme of ontological reduction in which our justifications for accepting the theorems proved remain unaffected. Moreover, without making us any more certain that a theorem is true, a proof can show us why it is true: proofs may yield explanatory dividends. (See ?V of [8]). Reducing arithmetic to set theory has explanatory, as well as ontological, value. For, in the light of the reduction, our understanding is advanced through ex- hibition of the kinship between theorems of arithmetic and theorems in other developments of set theory (in particular, branches of abstract algebra).

Unfortunately, the situation is not ideal. Identifying some ersatz arithmetic as real arithmetic would achieve ontological gain but its explanatory benefits would be achieved at the cost of generating explanatory puzzles. (What was.so special about the ersatz arithmetic we chose?). We have no grounds for distinguishing among the ersatz arithmetics in this way. To deny (4) and claim that numbers are abstract objects distinct from sets is even-handed, but leaves us puzzled as to why we can find set-theoretic substitutes for arithmetic. Intuitively, we would prefer an account of arithmetic which related arithmetic to each ersatz arithmetic, rather than an account which selects one ersatz arithmetic arbitrarily or an account which denies all identities between ordinary arithmetic and the many set theoretic substitutes.

In [4], Dedekind presents a way of developing arithmetic which seems to satisfy this intuitive preference. Dedekind's work has not been discussed in the context of Benacerraf's problem, but two recent attempts to solve that problem- those of Nicholas White and Hartry Field-develop Dede- kind's basic idea. I shall now consider these attempts.

II

In [16] Nicholas White sketches an account of arithmetic which does not construe the numerals as singular terms. Our



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intuitive preference demands that we relate the numbers to each co-sequence of sets (i.e. to the sequence of numbers of each ersatz arithmetic). White pursues this idea by showing how to parse sentences of arithmetic as statements about co-sequences. On White's view, the set theorist's explicandum is to be 'x is an n inp' (where 'p' ranges over c-sequences) rather than Frege's 'x = n'. But it seems easy to analyse 'x is an n inp' as '(3yi).. .(3yn-i)(Y precedesx inp &... .&Yn-1 precedesx inp & yl + Y2 &* * *& yl I Yn-1 & Y2 Y3 & & Yn-2 t Yn-1 & (Z)(Z precedes x inp D z = y1 v... .V z = yn-1))'. Similarly, we can give an obvious explication of 'x is the successor of y in p', rewrite the Peano postulates as statements about all c-sequences and derive them as set theoretic theorems.

This proposal will work only if we have first developed in our set theory the theory of co-sequences. Unfortunately, however, the identification of co-sequences with sets will be as problematic as the identification of numbers with sets. To see this, let us consider White's proposal in more detail.

To focus attention on the trouble spot I will deviate slightly from the usual formalism of ZF by using 'R' as a variable ranging over relations. Our aim will be to define the notion of an co-sequence using the notions of set and relation. (This approach is similar to that taken by Dedekind in [4]). We begin with the explication of'R is a 1-1 correspondence fromx toy':

(6) Corr(R, x, y) (z)(zsx D (3!w)(way & zRw)) &(z)(zey D (3!w)(wax & wRz)).

We now introduce the notion of a quasi-number set. A set x is a quasi-number set with respect to a relation R just in case there is a unique element 1 (x, R) s x such thatR is a 1-1 correspondence fromx tox\{ 1(x,R)}. A setx is a number set with respect to R iff it is a quasi-number set with respect to R and is the smallest quasi-number set with respect toR containing l(x,R). (The number sets are the c-sequences over the objects of the theory). More formally:

(7) Q(x, R) (Ty) c s x & Corr(R, x, x \{y}))

(8) l(x,R)-=y-Q(x,R) &y = (7z)(z sx & Corr(R,x,x\{z}))

(9) N(x,R)=Q(x,R) & (z)((Q(z,R) & 1(x,R) cZ) s X C z).




It is now easy to introduce the idea of the successor ofy in the number set x with respect to relation R.

(10) S(y, x, R) = z-N(x, R) & z = (w)(w e x & wRy).

The Peano postulates can now easily be rewritten as statements about number sets. For example, "Every number has a successor" becomes:

(1 1) (x)(y)(R)(Nx, R) D ( e x D (3z)(S(y, x, R) = z))).

The approach just sketched develops further White's semantics for arithmetic, but it falls short of a reduction of arithmetic to set theory. To complete the reduction we need to eliminate the references to relations. That can be achieved of course by identifying relations as sets of ordered pairs and, in turn, identifying ordered pairs as sets. Now the problem of the ordered pairs is exactly parallel to Benacerraf s problem. Can we solve it by using a strategy similar to that just developed?

The trick we used above was to take as our explicandum 'x is ann inp'; analogously, we should now try to analyse 'x is a (y, z) in w' where the variable 'w' ranges over sets containing all explicata of ordered pairs under the usual developments. If we call such sets OP-sets, then the set of all Wiener-ordered- pairs is an OP-set as is the set of all Kuratowski-ordered-pairs, and so forth.

An elementary use of the axiom of replacement shows that it is wrong to talk of OP-sets. Instead, we must talk of OP-classes if we are to pursue White's strategy. (ZF must thus be augmented with a theory of classes; see, for example, [ 15]).

But there is a more fundamental problem. There are no OP-classes simpliciter; a class is only an OP-class relative to a correlation of its elements with pairs of sets. To see this, consider the class of Kuratowski-ordered-pairs, { { {x }, {x,y } }: x, y eV}. This is an OP-class with respect to the uniform correlation of (x,y) with {{x}, {x,y}}, but not with respect to the uniform correlation of (x, y) with { {x} }. Moreover, consider any two arbitrary sets a, b. Let f be the function of two arguments such that f(x,y) = {{x}, {x,y}} (x,y e V). We can now define a function g(x, y) thus:



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g(x,y) =f(x,y) (x,y e V,x $ a,x $ b)

g(a,y) =f(b,y) (yeV)

g(b,y) =f(a,y) (yeV)

The class of Kuratowski-ordered-pairs is an OP-class with respect to g as well as with respect to f. Since a, b are arbitrary, there is no saying absolutely which element of the class is a (y, z). Any member of the class can be a (y, z) if we choose our correlation function correctly.

The reduction of arithmetic to set theory given by (6)-(10) seems to avoid the arbitrariness of the usual approaches but it does so at the cost of quantifying over relations as well as over sets. To identify relations with sets of ordered pairs and ordered pairs with sets in one of the usual (arbitrary) ways would subvert White's programme because the extension of the predicate 'N(x, R)' would vary with the particular explication of 'ordered pair' chosen and because the problem of the ordered pairs is exactly parallel to the problem of the numbers. If there were only one explication of 'ordered pair' then we could identify relations with sets of ordered pairs and so achieve a unique specification of the extension of 'N(x, R)'. Iterating White's original strategy at the level of the ordered pairs we attempted to regard the appropriate explicandum as 'x is a (y, z) in w' rather than 'x = (y, z).' Feeding this back into the reconstruc- tion of arithmetic, the appropriate predicate to define would be 'N(x, y, w)' where 'x', 'y' range over sets and 'w' over OP- classes. (Here y is a relation, that is, a set of ordered-pairs as defined byw) . The failure of White's strategy at the level of the ordered pairs prevents us from completing this reconstruc- tion successfully.

Robert Cummins has suggested that the situation is not as black as I have painted it. He proposes that we begin with the class of Kuratowski-ordered-pairs, call its members proto-pairs, and use the proto-pairs to define a notion which will satisfy White's needs. A proto-function will be a class of proto-pairs satisfying the usual uniqueness condition that a class of pairs is a function only if it contains no members (x,y ), (x, z ) withy ? z. Then we can explicate 'x is a (y, z) in w with respect tof' as follows:

x is a (y, z) in w with respect to f just in case f is a proto-function which is one-one onto the class of proto-




pairs and which maps x onto the proto-pair of x, y (i.e. {{f}, {y, Z}}).

To the charge that the choice of proto-pairs is arbitrary Cummins responds that the proto-pairs are "simply machin- ery." Arbitrariness in choosing them is no more worrying than arbitrariness in one's choice of notation. However, I do not think that this reassurance is warranted. Consider now what happens when we try to do arithmetic in set theory. We begin with the notion of a relation. A set is a relation if it is a set of ordered pairs, . . . , that is, if it is a set of ordered pairs with respect to an OP-class, . .. , that is, if it is a set of ordered pairs in an OP-class with respect to a proto-function. Our initial task, then, is to define a predicate 'Rel(x, zv,f)'. Continuing to develop arithmetic in parallel fashion to (5)-(9) we would eventually define the predicate 'N(x,y, zv,f)'. But now we are no better off than when we started with the predicate 'N(x,R)'. The variable f' ranges over an arbitrarily chosen class of sets just as the variable 'R' does.

Standard reductions of arithmetic to set theory seek to define the natural numbers. Once we recognize that there are many set-theoretic explications of number it is natural to try to relate the natural numbers to all the set-theoretic explications. We cannot pick out sets of natural numbers simpliciter; rather, a set is a set of numbers with respect to a relation. So we try to define 'N(x, R)'; this can be done, but only at the cost of either talking about unreduced relations or identifying relations arbitrarily. Our next tack is to try to identify relations as sets of ordered pairs and relate the ordered pairs to all the various explications. But there are no OP-classes simpliciter; a class is an OP-class with respect to a proto-function. So we try to define 'N(x, y, z,vf)'; this can be done, but only at the cost of either talking about unreduced proto-functions or identifying proto-functions arbitrarily. And so it goes.

Application of White's strategy solves the particular problems with which we began but it fails to avoid parallel problems. However many times we use White's approach we shall always end up with a reduction of arithmetic to set theory which generates a problem parallel to that from which we started.

Hartry Field's attempt to develop a notion of partial reference ([6]) which can be used to solve Benacerrafs problem



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encounters similar troubles. In general, Field suggests that we can overcome difficulties with "defective singular terms" by allowing the reference relation between names and objects to be many-many. Thus a name may partially refer to a number of objects. Since some sets of terms can be seen as "working together," it is useful to generalize the relation of partial reference by taking it to be a relation between sequences of terms and sequences of objects. In particular, we can now propose that the sequence of numerals partially refers to each co-sequence.

As it stands, Field's approach gives us no way of constructing object-language derivations of arithmetic within set theory. However, even if we postpone that issue, there are complications as soon as we try to work out the details of the semantical theory. If we are constructing a theory of truth for arithmetic we will advance, in the metalanguage, some such clause as

(12) ('1', 'S', 'NN') partially refers to (a, /3, y) iff. y is an co-sequence, ,/ is the ordering relation of the sequence and a is the first member of the sequence.

Apparently, our semantical theory commits us to the existence of sequences. We can avoid this metalinguistic commitment by identifying ordered pairs (and, derivatively, ordered n-tuples) with sets in one of the usual arbitrary ways. This would seem to show that Field's programme of using the notion of partial reference to avoid such commitment cannot be carried through completely. The alternative is to try to apply the programme directly to the ordered pairs. However, since the ordered pair terms "work together" (in Field's phrase), we cannot try some specification of the form

(13) '(a, b)' partially refers to a iff....

Rather, we must seek a clause of the form

(14) '{z: (3x)(3y)(z = (x, y))}' partially refers to a iff....

We have seen that this type of specification cannot be given. Classes are not OP-classes simpliciter. So either functions (ordered pairs, relations) must be admitted as irreducible entities or they must be eliminated arbitrarily.




The problem is exposed even more clearly when we try to elaborate Field's approach so as to permit us to do arithmetic within set theory. The standard derivations of the Peano postulates will no longer be acceptable because we have con- cluded that some of their premisses (e.g. '1 = {0}', '(x)(Sx = x U {x})') are not true (on Field's theory they are neither true nor false). The difficulty arises because the standard derivations thrive on the alleged co-reference of set-theoretic terms and numerals, expressed in the object-language by means of the identity predicate.

We can remedy the situation by introducing terminology which mirrors the relation of partial co-reference in the way that identity mirrors co-reference. Let us enlarge the class of atomic wffs of ZF by taking all expressions of the form rt 1 -ri tj (where tl, t2 are terms) as atomic wffs. Add to the

semantics the rule that rt, t22 is true iff there is ad such that t1 partially refers to d and t2 (fully) refers to d. (What we really need, of course, is an addition to the definition of 'satisfaction' which yields this as a rule of truth. Producing such an addition is a complicated business. I assume, for the sake of argument, that it can be done).' We add an analogue of Leibniz' law:

(15) (x)(()(x -r-Ly D'I(y)) D 4)(x))

and derive arithmetic from the principle

(16) (x)()(z)(((1, { (u, v): v is the successor of u}, NN) (x, y, z)) D (z is an o-sequence & y = {(zv, t): t

immediately succeeds wv in z } & x is the first member of z)).

If we can analyze away the uses of the notions of sequence and ordered pair in (16) then, with the help of (15), we can obtain the usual Peano postulates.

However, Field's approach founders on the problem of the ordered pairs. What set is identified with (0, {0}) de- pends on the identifications of the other ordered pairs, and conversely. So we must specify the ordered pairs all at once. We could mimic Field's approach by correlating sequences of ordered pairs with sequences of sets-if we had the notion of sequence at our disposal.



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If we try to develop the theory of ordered pairs without assuming the notion of sequence, the closest we can get to something like (15) is

(I17) (x) (I(y ,z): y ,z sVI}x D :(Dx))

for some appropriately chosen '4:>(x)'. However, it is hard to see how (17) could enable us to recover the individual ordered pairs. Furthermore, as we saw above, OP-classes are only OP-classes relative to an appropriate correlation of the ordered pairs with their members. But from this it follows that there is no way to select '?I(x)' without invoking the notion of function or something equivalent.

I conclude that neither Field nor White provides a way of satisfying the intuitive preference uncovered in ?P. Benacer- raf s problem remains as a puzzle for the Platonist. I shall now try to diagnose the source of the trouble.

III

I have suggested that the Platonist faces recurrent problems of a similar type when he comes to give a theory of truth for arithmetic. A natural response is to point out that Platonism offers the only available theory of arithmetical truth, that I have not shown how to avoid Benacerraf's problem by aban- doning Platonism and that, in any case, the blemishes noted in the previous sections are insignificant in comparison with the mutilations which nominalists and constructivists inflict on mathematics. I shall try to give a brief counter to this response.

Consider first an idea which was quickly rejected, the idea that we might give up the claim that arithmetical statements are either true or false. The problem with this idea is that it affords us no account of the usefulness of arithmetic. But does Plantonism, with its claim that arithmetic reports the properties of abstract objects, do any better? Why should these abstract objects be so important to us? Why is it that by study- ing them we improve our ability to describe and explain the behaviour of more familiar objects? Why do premisses about abstract objects play such an important role in our reaonings about physical things? Such questions expose our underlying reasons for taking arithmetic to be true. Platonism appears to give an account of arithmetical truth-but does it fit with these reasons?




A remark of Russell's suggests an answer: arithmetic is concerned with "the more abstract and general features" of the world. Russell's suggestion is not precise, but perhaps the Platonist gives us a way to sharpen it. The study of abstract objects discloses the structural properties of ordinary things. (Thus, for example, the sequence of natural numbers serves as an abstract representation of our counting of ordinary things). Platonism, so interpreted, espouses the thesis that arithmetic is true in virtue of the structural features of the world, elaborating this vague thesis as the idea that arithmetic is true in virtue of the properties of abstract objects.

Problems begin to arise when the Platonist tries to produce the abstract objects which embody the structure in virute of which arithmetic is true. His theory gives him too many abstract instantiations of that structure, the von Neumann numbers, the Zermelo numbers, and so forth. Sensitive to the original idea that talk of abstract objects is a way of talking about structure, he tries to isolate what is common to these various sequences of abstract objects, arriving at a view which is like that of White or Field. However, since he regards functions as abstract entities he has no way of resisting demands to reduce his ontology by identifying functions with sets. Canons of economy and explanatory unification direct the Platonist to see his realm of Platonic entities as a universe of sets. But the notion of set is insufficient to yield a mathematical theory which can be interpreted as describing the abstract structure of the world. As we have seen, purely set theoretic developments of mathematics must arbitrarily assign a special status to particular instantiations of mathematical structures.

One way to react is to suppose that there are two fundamental types of mathematical entities-sets and functions, or, perhaps, sets and ordered pairs. Significantly, those mathematicians who insist most strongly that mathematics is concerned with structure, adopt this position (see Bourbaki [2]: 72-4; also Dedekind [4]). However, parallel considera- tions to those urged above against Steiner's theory of arithmetic seem to tell against it.

Here then is the plight of the Platonist. He must either flout principles of economy and explanatory unification or he must opt for some arbitrary set-theoretic development of mathematics, which vitiates any claim that talk of abstract objects is a method of talking about abstract structure.2



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I think that we can do better. We can try to articulate Russell's dark utterance by thinking of the structural properties of ordinary objects as determinants of the ways in which we can operate upon those objects. Arithmetic can be re- garded as a science of ideal operations. Its utility arises from the fact that operations we perform on ordinary objects approximate the characteristics of these ideal operations.

A crude paradigm may help. In teaching children the meaning of 'set' we get them to grasp the notion of a collection via an understanding of the notion of collecting. Engaged in collective activity, they learn what it is to "make three" and to add. Arithmetic should not be about such operations in the sense that the interpretation of '+' is "heaping up"-but I suggest that we can regard arithmetic as being about operations, and one way to construe those operations is to view them as the simple types of physical operations to which I have just alluded. The relation between arithmetic and these operations is rather like that between geometry and the contours of physical objects.

Can we rewrite sentences of arithmetic in a way which will avoid reference to abstract objects? Let us take as primitive the notions of a unitary collecting, the notion of one collecting being a successor of another and the notion of collectings being matchable. (These notions are easily interpretable in terms of our crude paradigm: the first corresponds to "making one", the second to "adding one", while two collectings are matchable if it is possible to follow their performance with an operation correlating the objects collected). We can now reformulate first- order arithmetic. For example, the second Peano postulate, asserting that every number has a successor, becomes the assertion that every collecting has a successor collecting. Using some intuitive principles about matchability of collectings, and transcribing the usual axioms, we obtain a theory which yields analogues of the usual theorems.

There is an apparently devastating objection: when the arithmetical operations are construed in terms of the crude paradigm suggested above, various important existence as- sumptions (e.g. our transcription of the second Peano postulate) are probably false. I claim that this point is unworrying. Arithmetic may or may not be true of the physical manipulations which we actually perform. However, there are possible worlds at which arithmetic is true of our physical collectings and we can legitimately regard our own world as an approxi-




mation to such ideal worlds.3 (We can even help our own world to approximate the ideal worlds more closely, by engag- ing in further collective activity!).

If this should seem bizarre, it may help to recall the motivation for regarding some arithmetical statements as true: we wanted to be able to explain the usefulness of arithmetic. My proposal provides an explanation. Arithmetic is useful because the manipulations we perform on physical objects approximately satisfy the constraints which the arithmetical axioms place on the arithmetical operations, just as Euclidean geometry is useful because the contours of physical objects approximately satisfy the constraints which the Euclidean axioms place on geometrical figures. (Of course, there may also be other, less crude, ways of construing the arithmetical operations).

Arithmetical operations can be characterized quite easily in terms of more general notions. Taking the ideas of collecting and correlating as primitive and also using the idea of an operation's acting on certain objects we can develop a version of set theory from which my version of arithmetic flows. Filling out the details is a long story, but one important feature needs to be made explicit. To generate an analogue of standard set theory, we shall have to move away from our original paradigm with its idea of simple operations on objects. I shall account for abstract mathematics by supposing that some mathematical operations apply to previously performed operations: collectings on collectings serve as substitutes for sets of sets, and so forth. The usual Zermelo hierarchy is thus replaced by a hierarchy of collectings, similar to the heirarchy of constructions described in the usual versions of constructivism.

This naturally raises the question of whether I can account for classical mathematics. I would claim that adopting the ontological view that mathematics is the science of ideal operations does not commit me to the epistemological theses of constructivism. From Kant on, constructivist philosophies of mathematics have supposed that we know a priori what constructions we can and cannot perform. I have argued elsewhere ([9], [10]) that these epistemological claims fail. With their demise, we can adopt a more pragmatic attitude to the question of which mathematical operations are possible, using the success of classical mathematics to argue for a hierarchy of collectings rich enough to accommodate it.



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Obviously much more needs to be said. To articulate my approach fully I would need to show how it can handle a number of important ontological and epistemological problems. My aim here has been to sketch an alternative to Platonism, an alternative which can avoid a specific difficulty for Platonism.

How does the approach solve Benacerraf s problem? The answer comes from shifting our view of the content of mathematics. So long as we view sets and functions as Platonic entities there is no way to resist the economic pressure to identify functions as sets. From our new perspective things are different. Operations of correlating are to be kept separate from operations of collecting because we want to construe those operations in physical terms and the physical operations of collecting and correlating are dJfferent. We want our theory to be applicable to these two distinct types of physical operations. Hence we must distinguish two basic kinds of ideal mathematical operations. By doing so we neither violate the canons of economy nor fail to achieve explanatory unification. Our experience presents us with two different types of physical operations whose difference is reflected in the mathematical theory which we apply to those operations. In claiming that that theory is (approximately) true of those operations we only acknowledge the existence of entities which we would have had to accept in any case.

My solution to Benacerraf's problem motivates the mathematicians' distinction between sets and functions by regarding mathematics in general as a system, like geometry, which we apply to explain our experience. To say that Eucli- dean geometry is (approximately) true is to avow the ability of Euclidean geometry to explain and describe, inter alia, the contours of physical objects. To say that set theory and arithmetic are (approximately) true is to vouch for the ability of set theory and arithmetic to explain and describe, inter alia, our manipulations of physical objects. If these latter theories are to serve that purpose they must differentiate correlative and collective operations. But, just as acceptance of the approximate truth of geometry entails no acceptance of the actual existence of ideal geometrical objects, so too, our acceptance of the approximate truth of set theory and arithmetic does not fill our ontology with Platonic entities.

My sketch of an alternative approach to arithmetical and set theoretical truth may help us to see that Benacerraf s




problem touches on a fundamental weakness of Platonism: the inability of the Platonist to advocate the view that talk of abstract objects is a way of talking about abstract structure, while simultaneously bowing to the legitimate demands of economy and explanatory unification. We can abandon Platonism, and, if we do, I think we can escape the plight of the Platonist.4

REFERENCES

[1] Paul Benacerraf, "What Numbers Could Not Be," Philosophical Review 74(1965): 47-73.

[2] N. Bourbaki, Elements of Mathematics: Theory of Sets (Reading, Mass., 1968). [3] Charles Chihara, Ontology and the Vicious-Circle Principle (Ithaca, 1973). [4] R. Dedekind, The Nature and Meaning of Numbers in W. Beman (ed. and trans.)

Dedekind's Essays on the Theory of Numbers (New York, 1901). [5] Hartry Field, "Tarski's Theory of Truth," Journal of Philosophy 69(1972): 347-

75. [6] Hartry Field, "Quine and the Correspondence Theory," Philosophical Review

83(1974): 200-28. [7] G. Frege, TheFoundations ofArithmetic translated byJ. A. Austin (Oxford, 1951). [8] Philip Kitcher, "Bolzano's Ideal of Algebraic Analysis," Studies in the History and

Philosophy of Science 6(1975): 229-69. [9] , "Kant and the Foundations of Mathematics," Philosophical Review

84(1975): 23-50. [10] , "Hilbert's Epistemology'" Philosophy of Science 43(1976): 99-115. [11] Charles Parsons, "Ontology and Mathematics," Philosophical Review 80(1971):

151-176. [12] W. V. Quine, "Ontological Relativity," in Ontological Relativity and Other Essays

(New York, 1969). [13] Michael Resnik, "Mathematical Knowledge and Pattern Cognition," Canadian

Journal of Philosophy 5(1975): 25-39. [14] Mark Steiner, Mathematical Knowledge (Ithaca, 1975). [15] G. Takeuti and W. Zaring, Introduction to Axiomatic Set Theory (New York, 1971) [16] Nicholas White, "What Numbers Are" Synthese 27(1974): 111-24.

NOTES

'The relation which serves as our substitute for idenity will either collapse into the idenity relation (if all denoting terms denote fully) or will be asymmetric (if there are some partially denoting terms). We could try to mirror the phenomena of partial co-reference in the object language in a different way, by introducing wffs of form rt, .n t2',and by taking rt,I- t2' to be truejust in case there is some d such that both t,

and t2 partially denote d. The substitute for identity produced by this approach will be symmetric, but will be intransitive if there are partially denoting terms.

2In an interesting recent paper ([ 13]), Michael Resnik develops a position, which he calls "structuralism", whose main thesis is the idea that mathematical statements are true or false in virtue of the properties of structures. As Resnik points out, structuralism is a variant of Platonism, (structures are abstract objects) but the struc- turalist is apparently able to evade Benacerraf's problem, since he takes mathematical objects to be "nodes or positions in certain patterns" ([13]: 35). I have two worries about Resnik's approach. Firstly, as he himself admits, it is not clear how to specify identity conditions for the primitive notion of patterns. Secondly, and more crucially



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for the issue at hand, it is not clear how Resnik proposes to give a detailed semantics for arithmetic, and, in consequence, it is not obvious that he will avoid the basic Platonist mistake of talking about structure by producing one (abstract) instantiation of structure. My own account, in the remainder of this paper, attempts a non- Platonistic way of talking about structure, which seems to me to be consistent with Resnik's basic goals and to receive support from many of his arguments.

3In ?III of [11], Charles Parsons considers theories of arithmetical truth which explicitly introduce modal notions into the logical form of arithmetical statements. My proposal is slightly different. The sentences of arithmetic we usually accept are approximately true of the physical operations we perform, that is, there is a close possible world at which those sentences are true of our physical manipulations. The difference lies in the place at which the modal element is introduced: on the theory Parsons considers, it occurs in the logical form of the sentences; on my proposal, modality is relevant to the analysis of the notion of approximate truth.

4This paper grew out of a paper, "What Ordered Pairs Are Not," which was presented to the Pacific Division of the American Philosophical Association in the Spring of 1976. 1 am grateful to my commentator, Robert Cummins, who made some helpful suggestions. I also owe a long-standing debt to Paul Benacerraf, who first stimulated my interest in the problem. The valuable comments of an anonymous referee for Noiis have enabled me to improve the final version.

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