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Chapter 16 Nominalism
CHAPTER 16
NOMINALISM 146.
INTRODUCTION- THE PROBLEM OF UNIVERSALS
Before giving a survey of the discussion between nominalism and platonism in our special domain, it will be helpful to point out briefly in which sense the traditional terms "platonism" and "nominalism" should be taken in this context. Both directions are concerned with the problem of finding out what object or objects are denoted by a universal term such as "good" or "red" or "horse" or "natural number". (1) Neglecting such extremistic versions of platonism as the view ascribed to William of Champeaux, which seems rather to be related to the monism of the Megarics or of Spinoza, we can say that, for a platonist, each universal term, in much the same manner as a singular term, denotes a certain substance. The platonist will not deny that a universal term can also be used collectively to denote a multitude of concrete objects, but he claims that this collective usage is derivative and presupposes a certain substantial unity which, so to speak, is hidden in and behind that multitude. (2) The nominalist, on the other hand, does not wish to be committed to gratuitously presupposing the existence of such a substantial unity; he feels that the collective usage of universal terms for multitudes of concrete objects corresponds to their original sense and needs not be justified by the existence of a substantial unity or abstract entity.
In the past, discussions on this subject had a purely academic character, as the difference of opinion hardly implied any divergence on the technical level. In the theory of the syllogism, platonists and nominalists accepted the same modes, though their way of justifying them might slightly differ; and platonists and nominalists would hardly disagree as to the validity or non-validity of a given mathematical proof. The debate was restricted to the domain of the speculative foundations of logic and had no repercussions in the infinitely larger field of its applications. 464
THE COMPREHENSION AXIOM
465
The present debate, however, is concerned both with the pure theory of logic and with its applications, for nowadays there is no clear-cut separation between the two domains. Before we consider the situation in more detail, a terminological remark must be made. In the following discussion, the terms "class" and "set" will sometimes be used in the technical sense they were given in Chapter 14, and sometimes in a rather loose manner; whenever it. is necessary to stress non-technical usage, the term "multitude" will be used. 147.
THE COMPREHENSION AXIOM
The crucial point in the original constructions of Frege and Cantor was the application of a principle which has been called the Axiom of Comprehension and which we now state as follows (ct. Section 112): (i) The mathematical entities which share a certain property constitute a set of which they are the elements and which is uniquely determined by the characteristic property; (ij) Each set is a mathematical entity and hence may appear in its turn as an element of a set; (iij)
Two sets which contain the same elements are identical.
If the characteristic property is expressed by a formula "U(x)", then the corresponding set is denoted as ExU(x); in "ExU(x)", the variable "x" is bound, hence the same set can be written as "E1IU(y)" , provided "y" does not occur in "U (x)" . The platonistic inspiration underlying the Comprehension Axiom is quite obvious. Sets appear at first as multitudes of mathematical entities sharing a certain property. But nevertheless a set is also considered as a unity which is capable in its turn of appearing as a member of another multitude. This transformation of a multitude into a unity is called compression (H. Hermes). However, it is exactly this compression which has brought about the paradoxes of logic and set theory (the semantic paradoxes belong to a separate category; ct. Chapter 17) and therefore these paradoxes have made numerous logicians and mathematicians turn away from platonism; at any rate, they show the need for a renewed critical examination of platonistic conceptions.
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148.
NOMINALISM EXTENSION AND INTENSION
It may, on the other hand, be asked whether the Comprehension Axiom really reflects an authentic platonism. For it explicitly excludes the distinction between intension ("sense", "Sinn") and extension ("denotation", "Bedeutung"). When we say that a set is determined by a certain characteristic property of its elements, it is clearly this property which constitutes the intension, the substantial unity, in accordance with platonistic views; this unity is supposed to be given in advance and hence cannot result from "compression". On the other hand, our declaration that two sets having the same elements are identical clearly makes the set depend upon its extension, upon the multitude of its elements, and this rather looks like nominalism. It must be granted that identity of extension not always implies identity of intension. The set of all morning stars and the set of all evening stars have the same extension, but their intension is different. The same can be said of the sets:
However, though the distinction between extension and intension certainly constitutes one of the main tenets of historic platonism, it does not follow that it ought to playa role in our present discussion. In fact, it would be very difficult to apply the distinction in the domain of pure logic and mathematics. On the level of everyday discourse the distinction is to a certain degree indispensable, as certain qualities may happen to inhere "by accident" in one and the same individual object. For instance, only by accident the same physical object which sometimes appears as morning star also appears as evening star. The similarity of the case of the sets E,,(x = 2 + 2) and E,,(x = 2.2) is only apparent: the fact that 2 + 2 is identical with 2·2 cannot by any means be considered as accidental. It will be clear that in pure logic and mathematics no reasonable criterion for the identity of intensions is available, except the identity of the corresponding extensions, and that is exactly the criterion which is incorporated into the Comprehension Axiom. R. Carnap (1947) and A. Church (1951) have attempted to construct formal systems in which the distinction between extension and intension is explicitly taken into account. These attempts have met with
467
REVISION OF THE COMPREHENSION AXIOM
nearly insurmountable complications which - in the present author's opinion - are not counter-balanced by considerable advantages; but this issue is not yet closed. Nevertheless, in the present discussion the distinction between extension and intension will not be systematically considered. We shall see that, in spite of this simplification, no final decision in favour of either platonism or nominalism can be taken. 149. REVISION OF THE COMPREHENSION AXIOM In Chapters 13 and 14, we have discussed the reVISIOns of the Comprehension Axiom which have been proposed in connection with the various attempts at eliminating the paradoxes of logic and set theory. This subject will be taken up again in Chapter 17. In the context of our present discussion the situation can be summed up as follows. We make a distinction between arbitrary multitudes, or classes, and multitudes which can be compressed, or sets. Then the following statement of a revised Axiom of Comprehension is possible: (i) The mathematical entities which share a certain property constitute a class of which they are the elements and which is uniquely determined by the characteristic property; (ij) Classes satisfying such and such conditions can be compressed and are called sets; each set is a mathematical entity and hence may appear as an element of a class; (iij) Two classes which contain the same elements are identical. Each of the various systems of set theory or logic which have been constructed by Zermelo, Russell, Fraenkel, von Neumann, Quine, and others (and several of which are described in other parts of this book) is characterised by a specific choice of the conditions for compression under (ij), This choice is influenced in each case by two opposite considerations: on the one hand, compression is to be admitted whenever it is required in view of retaining the constructions of Cantor and Frege; on the other hand, the paradoxes of logic and set theory ought to be avoided. In addition to the (revised) Axiom of Comprehension, the above-mentioned systems (which we shall refer to as RZ systems) presupposes some version of elementary logic (or many-sorted quantification theory; Section 79), of the axiom of choice, and of the axiom of infinity; sometimes, still other axioms are required.
ct.
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NOMINALISM
It should be noted that the indeterminacy which Godel (ct. Section 130) pointed out in the naive usage of the word "set" also affects the naive usage of such terms as "property" and "multitude". Again we may construe such terms to refer, either to such properties and multitudes as can be defined by means of suitable expressions, or to arbitrary properties and multitudes irrespectively of, it and how they can be defined. If we adopt the above revision of the Comprehension Axiom as final, then the indeterminacy is transferred to the terms "class" and "set". The first interpretation will be preferable from a nominalistic point of view, but the second one is more in keeping with platonism. This indeterminacy can be (and has been) eliminated, to a certain extent, by another revision of the Comprehension Axiom. This axiom is then made to express conditions, not only for the compression of a given class, but also for the existence of classes. From a nominalistic point of view, it will be preferable to restrict the axioms of class existence as much as possible. But on the other hand, if we are to retain the constructions of Frege and Cantor, we have to adopt a rather liberal policy in this respect and so it seems that we are committed to a platonistio attitude.
150.
IMPREDICATIVE DEl!'lNITIONS
Hence, even though the revision of the Comprehension Axiom tends to attenuate the platonistic element in set theory and logic, its influence is far from being eliminated; this influence appears, for instance, from the manner in which the various revised versions of the Comprehension Axiom still enable us to introduce impredicative definitions. (I) Let us first consider the multitude R=E.,(x EX) which in the original systems of Cantor and Frege gave rise to the Russell paradox. In these systems, no distinction between sets and other multitudes is made; so the definition of R is clearly impredicative, as R appears among the values of the variable "z". Mter the above revision, however, the situation is completely different. The variable "x" is clearly one which ranges over sets, as only for sets a supposition "x E ... " makes sense. Now the definition of R will be impredicative, if R is still a value of "x", that is, if R is a set. But if R is a set, then we clearly obtain the Russell paradox.
IMPREDICATIVE DEFINITIONS
469
Hence, provided the revision of the Comprehension Axiom has been carried out in an efficient manner, R cannot be a set and so does not appear among the values of "x"; hence the definition of R is no longer impredicative. (2) As all paradoxes of logic and set theory derive from impredicative definitions, one might consider the possibility of eliminating all impredicative definitions by simply adopting the following condition for the compression of a class: a class X can be compressed, if it can be defined without bound variables ranging over sets. However, this drastic solution of the difficulty cannot be accepted, as its introduction would make it impossible to establish abstract set theory as developed in Chapter 14. For instance, already the compression of S(x) or: E.[(Ew)(z E W & W EX)], would be impossible (ct. Section 170). (3) If one (tentatively) adopts a genetic or constructivistic point of view, then the following consideration presents itself. A set is defined in an impredicative manner, if it results from the compression of a class the definition of which contains a bound variable which ranges over sets. Now this definition presents a circulus vitiosus. From an extensional point of view, the definiendum appears at first as a multitude. This multitude will be determinate, if for any set we know whether or not it belongs to the multitude. However, by compressing the multitude, we obtain a "new" set, which the definition could not possibly take into account. Hence, the multitude to be compressed not being determinate, the definition is clearly circular. From an intensional point of view, the definiendum at first appears as a unity which only afterwards produces a certain multitude of elements; these elements are those sets which satisfy a certain condition which is part of the definition. Again, the definition fails to determine this multitude; but this can only happen, if already the underlying unity was left indeterminate. To sum up: a definition which contains a bound variable has a meaning which depends on the range of that variable. On the .other hand, this range is meant to include the definiendum and thus depends upon the meaning of the definition. Hence, impredicative definitions are circular and so cannot be expected always to fulfil their purpose.
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NOMINALISM
If, for the moment, we stick to our genetic or constructivistic point of view, the above discussion suggests an alternative interpretation of definitions containing bound variables, as follows. We start from a certain initial stock of sets; this class, to be called M o' may consist, for instance, of all finite ordinals: I?J,
1 = {I?J}, 2={1?J, I}, 3={1?J, 1, 2}, ....
Now the variables which appear in definitions are, once for all, interpreted as ranging over M o' and hence each definition determines a certain subclass X of M o which, if certain conditions are satisfied, may be compressed into a set. Such a set may turn out to appear already in M o ; otherwise, it is not added to the range of the set variables. It will be clear that in this manner the above objections are met; on the other hand, this interpretation does not enable us to define certain notions which we do not wish to give up (cj. Section 170).
(4) From the point of view of a stolid platonist, the above objections are devoid of any foundation. A definition is not meant to introduce anything "new", but only to single out a certain element in a given domain of previously existing elements. The class of all sets is a domain of this kind; at the same time, it provides the range of all set variables; therefore, no indeterminacy can result from the fact that a definition contains a bound set variable. I think that a thoroughgoing platonist would ascribe the paradoxes to the fact that sets are defined in a roundabout manner. For him, a multitude presupposes a substantial unity, the straightforward manner to define a multitude would, therefore, be first to point out that substantial unity or set which then would not fail to produce the corresponding multitude of elements. But, instead, we first define this multitude and then try to compress this multitude into a unity and thus to obtain a set. For the platonist, it remains to be seen whether, by describing a certain characteristic property, we always obtain a well-defined multitude; and the so-called "conditions jor compression" are, from his point of view, rather to be construed as existence conditions for multitudes. On the other hand, the possibility of compressing any multitude into a set is for the platonist a matter of course. In fact, there is nothing in the various RZ systems which would contradict this interpretation.
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OONTEMPORARY NOMINALISM
151.
CONTEMPORARY NOMINALISM-THE
RICHARD
PARADOX AND
GODEL'S THEOREM ON RELATIVE CONSISTENOY
Most contemporary logicians and mathematicians are inclined to accept the various RZ systems, at least for the time being, as a sound basis for pure mathematics. Some of them are also willing to accept their platonistic background, while many are not at all interested in the ontological aspects of the problem. But there is also a certain feeling of uneasiness about the dependence of pure logic and mathematics upon platonistic ontology, and this explains the development of contemporary nominalism. This tendency can be traced back to Russell's first attempts at eliminating the paradoxes (c/. Section 170) and to Poincare's wellknown maxims, as stated in his Dernieres pensees: 1°. Ne jamais envisager que des objets suscept.ibles d'etre definis en un nombre fini de mots; 2°. Ne jamais perdre de vue que toute proposition sur l'infini doit etre la traduction, I'enonce abrege de propositions sur Ie fini; 3°.
Eviter les classifications et les definitions non predicat.ives.
The nominalistic tendency was strongly defended by L. Chwistek and S. Lesniewski and is now represented by Quine, Tarski, Henkin, Nelson Goodman, R. M. Martin, J. H. Woodger, and others. Whereas modern platonism consciously ties on to historical platonistic ontology, contemporary nominalism has hardly any connections with traditional nominalism. It rather has arisen, so to speak, as a spontaneous reaction to the platonistic elements in the systems of Frege and Cantor and as a result of the discovery of the paradoxes. We can distinguish three elements in contemporary nominalism, namely: (i) Criticism of the RZ systems and their platonistic background; (ij) Attempts at constructing a system for the foundations of pure logic and mathematics in agreement with nominalistic views; (iij) Nominalistic re-interpretation of the RZ systems. ad (i) Nominalists agree, of course, with the constructivistic (or conceptualistic) objections which we discussed in Section 150, and they will not at all be convinced by the platonist's reply. Quine has emphasised that, for a nominalist, the objectionable point in the RZ systems is not so much the compression of certain multitudes into a unity (for this unity might remain purely nominal) but rather the admission of this unity as a value of certain bound variables;
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NOMINALISM
for this seems to entail the attribution to the compressed multitude of a degree of substantiality to which a nominalist does not wish to commit himself. ad (ij) Nominalistic systems for the foundations of mathematics have been constructed by Leon Chwistek and by S. Lesniewski. In this connection, we should also mention the system of Quine's New Foundations and P. Lorenzen's constructive analysis. All these systems differ rather strongly from the more current RZ systems and hence it is not possible to discuss them here (ct. Sections 175 and 176).
ad (iij) The platonistic features to which the nominalists object do not properly inhere in the RZ systems themselves, they rather appear in the current interpretations of these systems. Though, of course, the RZ systems have been constructed in view of a platonistic interpretation, it does not follow that no other interpretation is possible. Accordingly, much thought has been given recently to the problem of giving an interpretation of the RZ systems which can be accepted from a nominalistic point of view. Strangely enough, the most profound ideas in this direction have been developed by Kurt Godel, a logician whose sympathies are rather with platonism. In the following discussion I take, in addition, advantage of work by Henkin, Quine, and Tarski. (I) A nominalistic interpretation of an RZ system requires that all entities to which such a system refers (in other words; all entities which appear in a model of the system) can be considered as concrete objects. Now this condition will be satisfied (in first approximation) if we interpret such a system as referring to such entities as can be defined in it. For we can, in that case, identify each entity with one of its defining expressions, and these expressions can certainly be considered as concrete objects (the distinction between use and mention will be observed if we adopt the device described in Section 87). However, it remains to be seen whether the entities definable in an RZ system provide a model for that system. (II) As each natural number can be defined in all RZ systems, the elements of the set N of all natural numbers are "faithfully" represented in our tentative nominalistic model. N itself will also be represented by some defining expression, and so will be ~(N). (III) We now turn to the elements of ~(N). In our tentative model, we only have the definable subsets of N, of which there are
CONTEMPORARY NOMINALISM
473
only denumerably many. However, on account of the theorem of Cantor (ct. Section 114), $(N) must contain more elements than N itself, and so it seems that our tentative model fails to fulfil Cantor's theorem; if that is indeed the case then our tentative model is not acceptable. However, we must beware of premature conclusions. (a) By the completeness theorem of Lowenheim-Skolem-GodelHenkin, every consistent RZ system must have a denumerable model. In this model, both Nand $(N) are represented by denumerable sets. Nevertheless, the theorem of Cantor is fulfilled, for the relation F which establishes a one-to-one correspondence between the elements of N and those of $(N) cannot be defined in the RZ systems (this follows from an analysis of the Richard paradox). (b) It does not follow, however, that our tentative model fulfils Cantor's theorem. This will be the case, if and only if the solution of the problem, whether or not the RZ systems contain a definition of a relation which establishes a one-to-one correspondence between the elements of N and the definable subsets of N is negative. This problem is equivalent to the unsolved problem of Tarski, as stated in Section 207. (c) It can be objected that both our tentative model and the Lowenheim-Skolem-GOdel-Henkin model are non-standard models. However, for most RZ systems the existence of a standard model is at any rate dubious; for the system of Quine's New Foundations the non-existence of standard models was actually proved by Rosser and Wang (1950). (IV) Real trouble arises if we pass on to the set $($(N)). For by a result of Tarski's (ct. Section 207), the class ilRD of all definable subfamilies of $(N) cannot be definable in an RZ system. As $($(N)) is, of course, definable in any RZ system, it cannot be identified with ilRD • So at this point our tentative nominalistic model fails to serve its purpose. (V) It has been pointed out by Godel that, on account of his theorem on relative consistency (ct. Section 129), another model can be found which serves the purpose of nominalism, though, presumably, it would be more to the taste of a conceptualist. We have to replace, in the above model construction, definable by constructible entities. We then obtain a model for the RZ system under consideration in which only definable entities appear; thus in a sense, it can be considered as consisting of concrete objects.
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NOMINALISM
Example 1. Give an analysis of the Richard paradox (ct. Section 162) as suggested under (III), (a). - Hint: suppose the relation R to be defined by an expression Ut», Y) of the RZ system under consideration; consider the set: Ex[x eN & (Y){U(x, Y) .....x e Y}].
Example 2. Show that the problem under (III), (b), is equivalent to Tarski's problem.
152.
THE AXIOM OF INFINITY
In the above method of interpretation, there remains one difficulty to be considered. All RZ systems contain some Axiom of Infinity. Therefore, our interpretation seems to depend upon our willingness to suppose the existence of infinitely many concrete objects. Can such a supposition be accepted from a nominalistic point of view l With respect to this question of conscience, two attitudes are possible. (i) We can accept the existence of infinitely many concrete objects as a cosmological hypothesis which has a certain likelihood and which to a sufficient extent agrees with our present knowledge in the field of natural science. However, this attitude would make pure logic and mathematics dependent on the sciences of nature; in view of some future development in this domain, our cosmological hypothesis might very well become rather improbable. One might even suspect this way of justifying the Infinity Axiom to be circular, as natural science in its turn depends upon pure mathematics; but this last objection is not valid, as we shall see under (ij). (ij) We drop, from all RZ systems, the Infinity Axiom A. Then, instead of any theorem X of the original system, we shall (by the deduction theorem) still be able to prove a theorem A ~ X. And, if X is of such a kind as to play actually a role in practical computation, then we shall even have a theorem: An~X,
where An expresses the existence of at least n concrete objects and n is some (possibly astronomically large) natural number. As natural science depends only upon mathematical theorems of this special kind and as, whenever such a theorem happens to be applied, the corresponding hypothesis An will be plausible from the context, it will be clear that natural science does not presuppose the Infinity Axiom; thus this axiom may, without circularity, receive support from natural science.
CONCLUDING REMARKS
475
Example. Give a precise statement of the metamathematical theorem used under (ij). Try to prove it with respect to some RZ system.
153.
CONCLUDING REMARKS
It will be clear that the above arguments do not settle the dispute between nominalism and platonism. The platonist will observe that the nominalistic re-interpretation of RZ systems including an Infinity Axiom depends upon the consistency of these systems, and that any attempt at proving their consistency by elementary methods must fail on account of Godel's incompleteness theorem. Now the platonist is ready to take this consistency for granted in view of the agreement between the RZ systems and his intuitive conceptions. But for a nominalist this is no reason to be convinced of the consistency of the RZ systems. And even if we do not raise this point, the nominalistic re-interpretation is of an extremely artificial nature. This observation does not apply to RZ systems without Infinity Axioms, but in this case there is no point in the nominalistic objections and hence there is no need for a re-interpretation. Moreover, the nominalist's willingness to drop the Infinity Axiom witnesses a misappreciation of the mathematical value of the intuitive conceptions which are at the bottom of the constructions of Cantor and Frege. If the Infinity Axiom is dropped, then only a rather trivial fragment is left, for instance, of Cantor's theory of ordinals; and, if the Infinity Axiom were false, how could the conception of transfinite ordinals have arisen? The nominalist's reply can, perhaps be stated briefly as follows. The platonist's last remark is no more conclusive than the ontological argument; moreover, if the nominalist does not wish to accept the Infinity Axiom as a basic principle, this is because he is convinced neither of its truth nor of its consistency with the more elementary axioms underlying the RZ systems, it is not because he believes it to be false. Therefore, dropping the Infinity Axiom means a loss in logical strength but not a loss in mathematical value, for the hypothetical theorems A -+ X are not meant as trivial implications. If the nominalistic re-interpretation of the RZ systems is more or less artificial and if their consistency cannot, from a nominalistic point of view, be considered as a matter of course, this is to be explained by reference to the historical fact that the construction of these systems
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NOMINALISM
was strongly influenced by platonistic notions. It is the more significant that, in spite of this influence, a nominalistic re-interpretation is possible. And it follows that one may accept the main results of logic and set theory and try to develop these fields without being committed to platonistic beliefs. It seems to me that the dispute between platonism and nominalism is highly important from the point of view of general philosophy. It is a pity that the subject is not more accessible. Although it would seem that there is a certain affinity between nominalism and intuitionism, Brouwer and Heyting, in dealing with universals, adopt an attitude which rather recalls platonism. Freudenthal, on the other hand, seems to represent a nominalistic tendency. It is interesting to observe that, on the basis of Brouwer's Fundamental Theorem ic]. Section 140), it is possible to give a nominalistic interpretation of intuitionistic mathematics. Heyting, in his defense of the notion of a choice sequence, observes that only those questions concerning choice sequences make sense which refer to all possible extensions of a given initial segment. The meaning of the phrase:
All choice sequences x in a spread M have the property A, is explained as follows:
we know in advance that, whenever an element x 0/ M is generated by a sequence 0/ tree choices, it must always turn out after finitely many choices have been made that the element x has the property A. Now suppose that all choice sequences z in a finitary spread M have a certain property A. Then we can associate with each element x of M a certain natural number n"" namely, the number of choices which has to be made before x turns out to have the property A. It is easy to see that every finitary spread M can be represented as a binary tree T; the elements x of M will appear as the branches in the tree T. If, after n", choices, the element x turns out to have the property A, then with the corresponding point on the branch in T we connect the formula A(n",). Then, by the Fundamental Theorem, T can be decomposed into finitely many subtrees TCv) such that with each vertex p a certain formula A(n) is connected. In this case, by the definition (3), sub (vij), we have agreed to say that the formula (Ex)A(x) is valid on the tree T. Thus, the above phrase concerning
BIBLIOGRAPIDCAL NOTES
477
all choice sequences in a given finitary spread M can be restated as a phrase which refers to the corresponding finitary tree T as a whole; from a nominalistic point of view, this seems to be much more satisfactory. The above ideas are of some importance in connection with the completeness proof for intuitionistic elementary logic in Section 145; but it is not possible now to return to this matter. BIBLIOGRAPHICAL NOTES My understanding of the subject is based to a considerable extent on a disoussion with Quine and Tarski, at a meeting in Amersfoort on August 31st and September Iat, 1953. Nominalistic views are defended by QUINE [4], HENKIN [4], MARTnl [1], MARTnl-WOODGER [1], and WOODGER [1]. Platonism is represented by GODEL [4], [5] and by CHURCH [5]. CARNAP [3], [4] differs both from CHURCH and from QUINE. On medieval nominalism, one may consult E. A. MOODY, Truth and Consequence in Mediaeval Logic (Studies in Logic), Amsterdam 1953. On platonism and nominalism in connection with intuitionism, see HEYTING [3], [4], and FREUDENTHAL [1]; ct. E. W. BETH [10], where full references are given. On the Infinity Axiom as a cosmological hypothesis, ct. KUSTAANHEIMO [1].
NOMINALISM 146.
INTRODUCTION- THE PROBLEM OF UNIVERSALS
Before giving a survey of the discussion between nominalism and platonism in our special domain, it will be helpful to point out briefly in which sense the traditional terms "platonism" and "nominalism" should be taken in this context. Both directions are concerned with the problem of finding out what object or objects are denoted by a universal term such as "good" or "red" or "horse" or "natural number". (1) Neglecting such extremistic versions of platonism as the view ascribed to William of Champeaux, which seems rather to be related to the monism of the Megarics or of Spinoza, we can say that, for a platonist, each universal term, in much the same manner as a singular term, denotes a certain substance. The platonist will not deny that a universal term can also be used collectively to denote a multitude of concrete objects, but he claims that this collective usage is derivative and presupposes a certain substantial unity which, so to speak, is hidden in and behind that multitude. (2) The nominalist, on the other hand, does not wish to be committed to gratuitously presupposing the existence of such a substantial unity; he feels that the collective usage of universal terms for multitudes of concrete objects corresponds to their original sense and needs not be justified by the existence of a substantial unity or abstract entity.
In the past, discussions on this subject had a purely academic character, as the difference of opinion hardly implied any divergence on the technical level. In the theory of the syllogism, platonists and nominalists accepted the same modes, though their way of justifying them might slightly differ; and platonists and nominalists would hardly disagree as to the validity or non-validity of a given mathematical proof. The debate was restricted to the domain of the speculative foundations of logic and had no repercussions in the infinitely larger field of its applications. 464
THE COMPREHENSION AXIOM
465
The present debate, however, is concerned both with the pure theory of logic and with its applications, for nowadays there is no clear-cut separation between the two domains. Before we consider the situation in more detail, a terminological remark must be made. In the following discussion, the terms "class" and "set" will sometimes be used in the technical sense they were given in Chapter 14, and sometimes in a rather loose manner; whenever it. is necessary to stress non-technical usage, the term "multitude" will be used. 147.
THE COMPREHENSION AXIOM
The crucial point in the original constructions of Frege and Cantor was the application of a principle which has been called the Axiom of Comprehension and which we now state as follows (ct. Section 112): (i) The mathematical entities which share a certain property constitute a set of which they are the elements and which is uniquely determined by the characteristic property; (ij) Each set is a mathematical entity and hence may appear in its turn as an element of a set; (iij)
Two sets which contain the same elements are identical.
If the characteristic property is expressed by a formula "U(x)", then the corresponding set is denoted as ExU(x); in "ExU(x)", the variable "x" is bound, hence the same set can be written as "E1IU(y)" , provided "y" does not occur in "U (x)" . The platonistic inspiration underlying the Comprehension Axiom is quite obvious. Sets appear at first as multitudes of mathematical entities sharing a certain property. But nevertheless a set is also considered as a unity which is capable in its turn of appearing as a member of another multitude. This transformation of a multitude into a unity is called compression (H. Hermes). However, it is exactly this compression which has brought about the paradoxes of logic and set theory (the semantic paradoxes belong to a separate category; ct. Chapter 17) and therefore these paradoxes have made numerous logicians and mathematicians turn away from platonism; at any rate, they show the need for a renewed critical examination of platonistic conceptions.
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148.
NOMINALISM EXTENSION AND INTENSION
It may, on the other hand, be asked whether the Comprehension Axiom really reflects an authentic platonism. For it explicitly excludes the distinction between intension ("sense", "Sinn") and extension ("denotation", "Bedeutung"). When we say that a set is determined by a certain characteristic property of its elements, it is clearly this property which constitutes the intension, the substantial unity, in accordance with platonistic views; this unity is supposed to be given in advance and hence cannot result from "compression". On the other hand, our declaration that two sets having the same elements are identical clearly makes the set depend upon its extension, upon the multitude of its elements, and this rather looks like nominalism. It must be granted that identity of extension not always implies identity of intension. The set of all morning stars and the set of all evening stars have the same extension, but their intension is different. The same can be said of the sets:
However, though the distinction between extension and intension certainly constitutes one of the main tenets of historic platonism, it does not follow that it ought to playa role in our present discussion. In fact, it would be very difficult to apply the distinction in the domain of pure logic and mathematics. On the level of everyday discourse the distinction is to a certain degree indispensable, as certain qualities may happen to inhere "by accident" in one and the same individual object. For instance, only by accident the same physical object which sometimes appears as morning star also appears as evening star. The similarity of the case of the sets E,,(x = 2 + 2) and E,,(x = 2.2) is only apparent: the fact that 2 + 2 is identical with 2·2 cannot by any means be considered as accidental. It will be clear that in pure logic and mathematics no reasonable criterion for the identity of intensions is available, except the identity of the corresponding extensions, and that is exactly the criterion which is incorporated into the Comprehension Axiom. R. Carnap (1947) and A. Church (1951) have attempted to construct formal systems in which the distinction between extension and intension is explicitly taken into account. These attempts have met with
467
REVISION OF THE COMPREHENSION AXIOM
nearly insurmountable complications which - in the present author's opinion - are not counter-balanced by considerable advantages; but this issue is not yet closed. Nevertheless, in the present discussion the distinction between extension and intension will not be systematically considered. We shall see that, in spite of this simplification, no final decision in favour of either platonism or nominalism can be taken. 149. REVISION OF THE COMPREHENSION AXIOM In Chapters 13 and 14, we have discussed the reVISIOns of the Comprehension Axiom which have been proposed in connection with the various attempts at eliminating the paradoxes of logic and set theory. This subject will be taken up again in Chapter 17. In the context of our present discussion the situation can be summed up as follows. We make a distinction between arbitrary multitudes, or classes, and multitudes which can be compressed, or sets. Then the following statement of a revised Axiom of Comprehension is possible: (i) The mathematical entities which share a certain property constitute a class of which they are the elements and which is uniquely determined by the characteristic property; (ij) Classes satisfying such and such conditions can be compressed and are called sets; each set is a mathematical entity and hence may appear as an element of a class; (iij) Two classes which contain the same elements are identical. Each of the various systems of set theory or logic which have been constructed by Zermelo, Russell, Fraenkel, von Neumann, Quine, and others (and several of which are described in other parts of this book) is characterised by a specific choice of the conditions for compression under (ij), This choice is influenced in each case by two opposite considerations: on the one hand, compression is to be admitted whenever it is required in view of retaining the constructions of Cantor and Frege; on the other hand, the paradoxes of logic and set theory ought to be avoided. In addition to the (revised) Axiom of Comprehension, the above-mentioned systems (which we shall refer to as RZ systems) presupposes some version of elementary logic (or many-sorted quantification theory; Section 79), of the axiom of choice, and of the axiom of infinity; sometimes, still other axioms are required.
ct.
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NOMINALISM
It should be noted that the indeterminacy which Godel (ct. Section 130) pointed out in the naive usage of the word "set" also affects the naive usage of such terms as "property" and "multitude". Again we may construe such terms to refer, either to such properties and multitudes as can be defined by means of suitable expressions, or to arbitrary properties and multitudes irrespectively of, it and how they can be defined. If we adopt the above revision of the Comprehension Axiom as final, then the indeterminacy is transferred to the terms "class" and "set". The first interpretation will be preferable from a nominalistic point of view, but the second one is more in keeping with platonism. This indeterminacy can be (and has been) eliminated, to a certain extent, by another revision of the Comprehension Axiom. This axiom is then made to express conditions, not only for the compression of a given class, but also for the existence of classes. From a nominalistic point of view, it will be preferable to restrict the axioms of class existence as much as possible. But on the other hand, if we are to retain the constructions of Frege and Cantor, we have to adopt a rather liberal policy in this respect and so it seems that we are committed to a platonistio attitude.
150.
IMPREDICATIVE DEl!'lNITIONS
Hence, even though the revision of the Comprehension Axiom tends to attenuate the platonistic element in set theory and logic, its influence is far from being eliminated; this influence appears, for instance, from the manner in which the various revised versions of the Comprehension Axiom still enable us to introduce impredicative definitions. (I) Let us first consider the multitude R=E.,(x EX) which in the original systems of Cantor and Frege gave rise to the Russell paradox. In these systems, no distinction between sets and other multitudes is made; so the definition of R is clearly impredicative, as R appears among the values of the variable "z". Mter the above revision, however, the situation is completely different. The variable "x" is clearly one which ranges over sets, as only for sets a supposition "x E ... " makes sense. Now the definition of R will be impredicative, if R is still a value of "x", that is, if R is a set. But if R is a set, then we clearly obtain the Russell paradox.
IMPREDICATIVE DEFINITIONS
469
Hence, provided the revision of the Comprehension Axiom has been carried out in an efficient manner, R cannot be a set and so does not appear among the values of "x"; hence the definition of R is no longer impredicative. (2) As all paradoxes of logic and set theory derive from impredicative definitions, one might consider the possibility of eliminating all impredicative definitions by simply adopting the following condition for the compression of a class: a class X can be compressed, if it can be defined without bound variables ranging over sets. However, this drastic solution of the difficulty cannot be accepted, as its introduction would make it impossible to establish abstract set theory as developed in Chapter 14. For instance, already the compression of S(x) or: E.[(Ew)(z E W & W EX)], would be impossible (ct. Section 170). (3) If one (tentatively) adopts a genetic or constructivistic point of view, then the following consideration presents itself. A set is defined in an impredicative manner, if it results from the compression of a class the definition of which contains a bound variable which ranges over sets. Now this definition presents a circulus vitiosus. From an extensional point of view, the definiendum appears at first as a multitude. This multitude will be determinate, if for any set we know whether or not it belongs to the multitude. However, by compressing the multitude, we obtain a "new" set, which the definition could not possibly take into account. Hence, the multitude to be compressed not being determinate, the definition is clearly circular. From an intensional point of view, the definiendum at first appears as a unity which only afterwards produces a certain multitude of elements; these elements are those sets which satisfy a certain condition which is part of the definition. Again, the definition fails to determine this multitude; but this can only happen, if already the underlying unity was left indeterminate. To sum up: a definition which contains a bound variable has a meaning which depends on the range of that variable. On the .other hand, this range is meant to include the definiendum and thus depends upon the meaning of the definition. Hence, impredicative definitions are circular and so cannot be expected always to fulfil their purpose.
470
NOMINALISM
If, for the moment, we stick to our genetic or constructivistic point of view, the above discussion suggests an alternative interpretation of definitions containing bound variables, as follows. We start from a certain initial stock of sets; this class, to be called M o' may consist, for instance, of all finite ordinals: I?J,
1 = {I?J}, 2={1?J, I}, 3={1?J, 1, 2}, ....
Now the variables which appear in definitions are, once for all, interpreted as ranging over M o' and hence each definition determines a certain subclass X of M o which, if certain conditions are satisfied, may be compressed into a set. Such a set may turn out to appear already in M o ; otherwise, it is not added to the range of the set variables. It will be clear that in this manner the above objections are met; on the other hand, this interpretation does not enable us to define certain notions which we do not wish to give up (cj. Section 170).
(4) From the point of view of a stolid platonist, the above objections are devoid of any foundation. A definition is not meant to introduce anything "new", but only to single out a certain element in a given domain of previously existing elements. The class of all sets is a domain of this kind; at the same time, it provides the range of all set variables; therefore, no indeterminacy can result from the fact that a definition contains a bound set variable. I think that a thoroughgoing platonist would ascribe the paradoxes to the fact that sets are defined in a roundabout manner. For him, a multitude presupposes a substantial unity, the straightforward manner to define a multitude would, therefore, be first to point out that substantial unity or set which then would not fail to produce the corresponding multitude of elements. But, instead, we first define this multitude and then try to compress this multitude into a unity and thus to obtain a set. For the platonist, it remains to be seen whether, by describing a certain characteristic property, we always obtain a well-defined multitude; and the so-called "conditions jor compression" are, from his point of view, rather to be construed as existence conditions for multitudes. On the other hand, the possibility of compressing any multitude into a set is for the platonist a matter of course. In fact, there is nothing in the various RZ systems which would contradict this interpretation.
471
OONTEMPORARY NOMINALISM
151.
CONTEMPORARY NOMINALISM-THE
RICHARD
PARADOX AND
GODEL'S THEOREM ON RELATIVE CONSISTENOY
Most contemporary logicians and mathematicians are inclined to accept the various RZ systems, at least for the time being, as a sound basis for pure mathematics. Some of them are also willing to accept their platonistic background, while many are not at all interested in the ontological aspects of the problem. But there is also a certain feeling of uneasiness about the dependence of pure logic and mathematics upon platonistic ontology, and this explains the development of contemporary nominalism. This tendency can be traced back to Russell's first attempts at eliminating the paradoxes (c/. Section 170) and to Poincare's wellknown maxims, as stated in his Dernieres pensees: 1°. Ne jamais envisager que des objets suscept.ibles d'etre definis en un nombre fini de mots; 2°. Ne jamais perdre de vue que toute proposition sur l'infini doit etre la traduction, I'enonce abrege de propositions sur Ie fini; 3°.
Eviter les classifications et les definitions non predicat.ives.
The nominalistic tendency was strongly defended by L. Chwistek and S. Lesniewski and is now represented by Quine, Tarski, Henkin, Nelson Goodman, R. M. Martin, J. H. Woodger, and others. Whereas modern platonism consciously ties on to historical platonistic ontology, contemporary nominalism has hardly any connections with traditional nominalism. It rather has arisen, so to speak, as a spontaneous reaction to the platonistic elements in the systems of Frege and Cantor and as a result of the discovery of the paradoxes. We can distinguish three elements in contemporary nominalism, namely: (i) Criticism of the RZ systems and their platonistic background; (ij) Attempts at constructing a system for the foundations of pure logic and mathematics in agreement with nominalistic views; (iij) Nominalistic re-interpretation of the RZ systems. ad (i) Nominalists agree, of course, with the constructivistic (or conceptualistic) objections which we discussed in Section 150, and they will not at all be convinced by the platonist's reply. Quine has emphasised that, for a nominalist, the objectionable point in the RZ systems is not so much the compression of certain multitudes into a unity (for this unity might remain purely nominal) but rather the admission of this unity as a value of certain bound variables;
472
NOMINALISM
for this seems to entail the attribution to the compressed multitude of a degree of substantiality to which a nominalist does not wish to commit himself. ad (ij) Nominalistic systems for the foundations of mathematics have been constructed by Leon Chwistek and by S. Lesniewski. In this connection, we should also mention the system of Quine's New Foundations and P. Lorenzen's constructive analysis. All these systems differ rather strongly from the more current RZ systems and hence it is not possible to discuss them here (ct. Sections 175 and 176).
ad (iij) The platonistic features to which the nominalists object do not properly inhere in the RZ systems themselves, they rather appear in the current interpretations of these systems. Though, of course, the RZ systems have been constructed in view of a platonistic interpretation, it does not follow that no other interpretation is possible. Accordingly, much thought has been given recently to the problem of giving an interpretation of the RZ systems which can be accepted from a nominalistic point of view. Strangely enough, the most profound ideas in this direction have been developed by Kurt Godel, a logician whose sympathies are rather with platonism. In the following discussion I take, in addition, advantage of work by Henkin, Quine, and Tarski. (I) A nominalistic interpretation of an RZ system requires that all entities to which such a system refers (in other words; all entities which appear in a model of the system) can be considered as concrete objects. Now this condition will be satisfied (in first approximation) if we interpret such a system as referring to such entities as can be defined in it. For we can, in that case, identify each entity with one of its defining expressions, and these expressions can certainly be considered as concrete objects (the distinction between use and mention will be observed if we adopt the device described in Section 87). However, it remains to be seen whether the entities definable in an RZ system provide a model for that system. (II) As each natural number can be defined in all RZ systems, the elements of the set N of all natural numbers are "faithfully" represented in our tentative nominalistic model. N itself will also be represented by some defining expression, and so will be ~(N). (III) We now turn to the elements of ~(N). In our tentative model, we only have the definable subsets of N, of which there are
CONTEMPORARY NOMINALISM
473
only denumerably many. However, on account of the theorem of Cantor (ct. Section 114), $(N) must contain more elements than N itself, and so it seems that our tentative model fails to fulfil Cantor's theorem; if that is indeed the case then our tentative model is not acceptable. However, we must beware of premature conclusions. (a) By the completeness theorem of Lowenheim-Skolem-GodelHenkin, every consistent RZ system must have a denumerable model. In this model, both Nand $(N) are represented by denumerable sets. Nevertheless, the theorem of Cantor is fulfilled, for the relation F which establishes a one-to-one correspondence between the elements of N and those of $(N) cannot be defined in the RZ systems (this follows from an analysis of the Richard paradox). (b) It does not follow, however, that our tentative model fulfils Cantor's theorem. This will be the case, if and only if the solution of the problem, whether or not the RZ systems contain a definition of a relation which establishes a one-to-one correspondence between the elements of N and the definable subsets of N is negative. This problem is equivalent to the unsolved problem of Tarski, as stated in Section 207. (c) It can be objected that both our tentative model and the Lowenheim-Skolem-GOdel-Henkin model are non-standard models. However, for most RZ systems the existence of a standard model is at any rate dubious; for the system of Quine's New Foundations the non-existence of standard models was actually proved by Rosser and Wang (1950). (IV) Real trouble arises if we pass on to the set $($(N)). For by a result of Tarski's (ct. Section 207), the class ilRD of all definable subfamilies of $(N) cannot be definable in an RZ system. As $($(N)) is, of course, definable in any RZ system, it cannot be identified with ilRD • So at this point our tentative nominalistic model fails to serve its purpose. (V) It has been pointed out by Godel that, on account of his theorem on relative consistency (ct. Section 129), another model can be found which serves the purpose of nominalism, though, presumably, it would be more to the taste of a conceptualist. We have to replace, in the above model construction, definable by constructible entities. We then obtain a model for the RZ system under consideration in which only definable entities appear; thus in a sense, it can be considered as consisting of concrete objects.
474
NOMINALISM
Example 1. Give an analysis of the Richard paradox (ct. Section 162) as suggested under (III), (a). - Hint: suppose the relation R to be defined by an expression Ut», Y) of the RZ system under consideration; consider the set: Ex[x eN & (Y){U(x, Y) .....x e Y}].
Example 2. Show that the problem under (III), (b), is equivalent to Tarski's problem.
152.
THE AXIOM OF INFINITY
In the above method of interpretation, there remains one difficulty to be considered. All RZ systems contain some Axiom of Infinity. Therefore, our interpretation seems to depend upon our willingness to suppose the existence of infinitely many concrete objects. Can such a supposition be accepted from a nominalistic point of view l With respect to this question of conscience, two attitudes are possible. (i) We can accept the existence of infinitely many concrete objects as a cosmological hypothesis which has a certain likelihood and which to a sufficient extent agrees with our present knowledge in the field of natural science. However, this attitude would make pure logic and mathematics dependent on the sciences of nature; in view of some future development in this domain, our cosmological hypothesis might very well become rather improbable. One might even suspect this way of justifying the Infinity Axiom to be circular, as natural science in its turn depends upon pure mathematics; but this last objection is not valid, as we shall see under (ij). (ij) We drop, from all RZ systems, the Infinity Axiom A. Then, instead of any theorem X of the original system, we shall (by the deduction theorem) still be able to prove a theorem A ~ X. And, if X is of such a kind as to play actually a role in practical computation, then we shall even have a theorem: An~X,
where An expresses the existence of at least n concrete objects and n is some (possibly astronomically large) natural number. As natural science depends only upon mathematical theorems of this special kind and as, whenever such a theorem happens to be applied, the corresponding hypothesis An will be plausible from the context, it will be clear that natural science does not presuppose the Infinity Axiom; thus this axiom may, without circularity, receive support from natural science.
CONCLUDING REMARKS
475
Example. Give a precise statement of the metamathematical theorem used under (ij). Try to prove it with respect to some RZ system.
153.
CONCLUDING REMARKS
It will be clear that the above arguments do not settle the dispute between nominalism and platonism. The platonist will observe that the nominalistic re-interpretation of RZ systems including an Infinity Axiom depends upon the consistency of these systems, and that any attempt at proving their consistency by elementary methods must fail on account of Godel's incompleteness theorem. Now the platonist is ready to take this consistency for granted in view of the agreement between the RZ systems and his intuitive conceptions. But for a nominalist this is no reason to be convinced of the consistency of the RZ systems. And even if we do not raise this point, the nominalistic re-interpretation is of an extremely artificial nature. This observation does not apply to RZ systems without Infinity Axioms, but in this case there is no point in the nominalistic objections and hence there is no need for a re-interpretation. Moreover, the nominalist's willingness to drop the Infinity Axiom witnesses a misappreciation of the mathematical value of the intuitive conceptions which are at the bottom of the constructions of Cantor and Frege. If the Infinity Axiom is dropped, then only a rather trivial fragment is left, for instance, of Cantor's theory of ordinals; and, if the Infinity Axiom were false, how could the conception of transfinite ordinals have arisen? The nominalist's reply can, perhaps be stated briefly as follows. The platonist's last remark is no more conclusive than the ontological argument; moreover, if the nominalist does not wish to accept the Infinity Axiom as a basic principle, this is because he is convinced neither of its truth nor of its consistency with the more elementary axioms underlying the RZ systems, it is not because he believes it to be false. Therefore, dropping the Infinity Axiom means a loss in logical strength but not a loss in mathematical value, for the hypothetical theorems A -+ X are not meant as trivial implications. If the nominalistic re-interpretation of the RZ systems is more or less artificial and if their consistency cannot, from a nominalistic point of view, be considered as a matter of course, this is to be explained by reference to the historical fact that the construction of these systems
476
NOMINALISM
was strongly influenced by platonistic notions. It is the more significant that, in spite of this influence, a nominalistic re-interpretation is possible. And it follows that one may accept the main results of logic and set theory and try to develop these fields without being committed to platonistic beliefs. It seems to me that the dispute between platonism and nominalism is highly important from the point of view of general philosophy. It is a pity that the subject is not more accessible. Although it would seem that there is a certain affinity between nominalism and intuitionism, Brouwer and Heyting, in dealing with universals, adopt an attitude which rather recalls platonism. Freudenthal, on the other hand, seems to represent a nominalistic tendency. It is interesting to observe that, on the basis of Brouwer's Fundamental Theorem ic]. Section 140), it is possible to give a nominalistic interpretation of intuitionistic mathematics. Heyting, in his defense of the notion of a choice sequence, observes that only those questions concerning choice sequences make sense which refer to all possible extensions of a given initial segment. The meaning of the phrase:
All choice sequences x in a spread M have the property A, is explained as follows:
we know in advance that, whenever an element x 0/ M is generated by a sequence 0/ tree choices, it must always turn out after finitely many choices have been made that the element x has the property A. Now suppose that all choice sequences z in a finitary spread M have a certain property A. Then we can associate with each element x of M a certain natural number n"" namely, the number of choices which has to be made before x turns out to have the property A. It is easy to see that every finitary spread M can be represented as a binary tree T; the elements x of M will appear as the branches in the tree T. If, after n", choices, the element x turns out to have the property A, then with the corresponding point on the branch in T we connect the formula A(n",). Then, by the Fundamental Theorem, T can be decomposed into finitely many subtrees TCv) such that with each vertex p a certain formula A(n) is connected. In this case, by the definition (3), sub (vij), we have agreed to say that the formula (Ex)A(x) is valid on the tree T. Thus, the above phrase concerning
BIBLIOGRAPIDCAL NOTES
477
all choice sequences in a given finitary spread M can be restated as a phrase which refers to the corresponding finitary tree T as a whole; from a nominalistic point of view, this seems to be much more satisfactory. The above ideas are of some importance in connection with the completeness proof for intuitionistic elementary logic in Section 145; but it is not possible now to return to this matter. BIBLIOGRAPHICAL NOTES My understanding of the subject is based to a considerable extent on a disoussion with Quine and Tarski, at a meeting in Amersfoort on August 31st and September Iat, 1953. Nominalistic views are defended by QUINE [4], HENKIN [4], MARTnl [1], MARTnl-WOODGER [1], and WOODGER [1]. Platonism is represented by GODEL [4], [5] and by CHURCH [5]. CARNAP [3], [4] differs both from CHURCH and from QUINE. On medieval nominalism, one may consult E. A. MOODY, Truth and Consequence in Mediaeval Logic (Studies in Logic), Amsterdam 1953. On platonism and nominalism in connection with intuitionism, see HEYTING [3], [4], and FREUDENTHAL [1]; ct. E. W. BETH [10], where full references are given. On the Infinity Axiom as a cosmological hypothesis, ct. KUSTAANHEIMO [1].