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THE NATURE OF NUMBERPeter Forrest a & D. M. Armstrong ba University of New England,b University of Sydney,

Available online: 20 Jan 2010

To cite this article: Peter Forrest & D. M. Armstrong (1987): THE NATURE OFNUMBER, Philosophical Papers, 16:3, 165-186

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Philosophical Papers Vol. XVI (1987), No. 3

THE NATURE OF NUMBER’

Peter Forrest University of New England

D.M. Armstrong University of Sydney

In the first section we consider a view that makes numbers properties of properties. (The view is less orthodox than this description may suggest.) After various modifications, the theory is rejected. In the second section the view is transformed into something that seems more satisfactory: the theory that numbers are relations between properties. (Here we build on work by Glen Kessler.) In the third section we show that, given this view, natural, rational and real numbers are all numbers in exactly the same sense. The fourth section considers whether we should split the difference between the theories of the first two sections and take numbers to be relational properties of properties. A fifth and final section asks whether our account of number casts light on the nature of classes.

I

Properties of Properties? Suppose that being an electron is a universal. If it is, it is a sort which ‘divides its extension’. (The phrase echoes Quine’s semantic notion of an expression which ‘divides its reference’.) Its instances may be said to be an electron, and are such that the electrons do not overlap. We call such universals ‘strongly particularizing’ universals. Now consider aggregates (not classes) of electrons. There are one-electron aggregates, two- electron aggregates, three-electron aggregates, ... N-electron aggregates. Taken as types, not tokens, these yield a series of structural universals. It is not a necessary truth that N is restricted to finite numbers. There is, or may be, the type

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166 FORREST AND ARMSTRONG

aleph-null-electron aggregate. There are other similar series of universals. Perhaps a one-

proton aggregate, a two-proton aggregate ... is such a series. We are now in a position to make a first suggestion as to

what a number is. Let us take 19 as an example. Consider the class of universals which can all be described as 19-P aggregate structures, where ‘P’ ranges over the universals of the sort described. It would appear that each such universal is a member of this class in virtue of a certain property (a second- order universal). Perhaps this common property is the number 1 9?

A general objection to a definition of 19 along these lines may be that it involves universals. Many philosophers in the empiricist tradition are suspicious of them. But we suggest that our (final) account of number will itself be some reason to believe in universals. One of us (Armstrong) would make the same point about universals in connection with laws of nature. We would both make it in connection with causality. Universals, indeed, are ontological maids of all work. They get things done. They enable good explanations to be given.

In any case, many philosophers who do not believe in universals recognize the need to provide a substitute for them. (For instance, equivalence-classes of exactly resembling property-instances or ‘tropes’.) Perhaps these substitutes can be plugged into our account. Objection 1. The number of electrons, protons, etc. may be limited. The universe may be quite small. But the natural numbers are not limited, and beyond them are the transfinite numbers. Reply. One of us (Forrest) is prepared to admit uninstantiated universals into his ontology. With this price paid, aggregate- universals can be provided for any number, however large. However Armstrong does not wish to admit such universals. He therefore must modify the theory. He would replace the uninstantiated universals with (merely) possible universals. (And would hope to avoid an ontology ofpossibiliu by means of a combinatorial theory of possibility.)Z So instead of a number being a property common to certain classes of universals, it will be a property common to certain classes of possible universals.

Objections 2. The number 0 is a number. But being no

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T H E NATURE OF NUMBER 167

electrons, being no protons, etc. are not aggregating universals. Indeed, if negative universals are denied, as we think that they should be, they are not even universals at all.

Replv. (Pro tern.) 0 is a very special number. We might well need a special account to deal with it. An account will be given in Section 11.

Objection 3. We can number things that do not fall under the same universal. Let ‘is an A’ abbreviate ‘is either an electron or a proton’. A certain aggregate may be an aggregate of just 19 As, because it is made up of 10 electrons and 9 protons. Yet we, at any rate, would be very reluctant to admit being an A as a universal. In general, we wish to deny that thereare disjunctive universals. There is no genuine one (only a mere Cambridge one) that runs through the individual As.

Reply. Nevertheless, the type, aggregate of 10 electrons and 9 protons, is a perfectly good structural universal (if being an electron and being a proton are universals). It could be identically instantiated in many instances. And it would seem to share a property with aggregate-universals such as being a 19-electron aggregate, being a 19-proton aggregate, and being an 1 1-electron and an 8-proton aggregate.

What must be admitted, however, is that in the case of the aggregate of 10 electrons and 9 protons the unit property: either an electron or aproton, is not a universal. And since the account to be advanced in Section I1 makes essential use of unit-properties, this is an issue that we require to address.

Contrary to the position taken by one of us in the past (Armstrong), we wish to distinguish between universals and properties. All monadic universals are properties, but not all properties are universals. In particular, we wish to admit disjunctive properties while still denying that they are universals.

We believe that we can do this without ontological cost because it seems that disjunctive properties are supervenient upon universals. Suppose that being an electron and being a proton are universals. Any world which contains these universals will automatically, superveniently, contain the property of being an electron orproton, a property which will automatically be a property of each electron and proton. But what is thus supervenient, it is plausible to say, is not an ontological addition.3 We may therefore speak of disjunctive

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168 FORREST AND ARMSTRONG

properties in good conscience. If genuine universals are hard to come by,4 and identifiable

only at the end of arduous, normally at least a posteriori, investigation, as we believe to be the case, then most of the unit-properties involved in number and counting will be of this relaxed, disjunctive, sort. (Forrest has suggested that we call them ‘multiversals’.)

Objection 4. Suppose that the king has just nineteen horses. Being a 19-horse aggregate is a universal, or, more probably, a very complex disjunction of universals. (Being a horse is unlikely to be a universal.) 19 may be thought to be a property of these universals. The trouble, however, lies with the unit- property of being a horse of the king’s. Since the king is a particular, this is not even a disjunction of universals. It is a ‘property’ in a sense not yet provided for.

Reply. We can nevertheless admit such properties freely, because they, too, are supervenient entities. However, what they are supervenient upon is not universals, but what we call states of affairs (others speak of ‘facts’). States of affairs are a matter of a particular’s having a property, or two or more particulars standing in a relation. If the king owns a horse, then there is a state of affairs of the king and a certain horse standing in the relationship of owner to owned. Supervenient upon this state of affairs is the (multi-instantiable but not universal) property of being a horse of the king’s.

Objection 5. Anything at all can be numbered. But for a token aggregate to instantiate a property, there had better be an aggregate! But do all the things that can be numbered form an aggregate?

One recent author who raises such doubts is Peter Simons. Criticizing Kessler5 he writes:

Some things, one wants to say, are just too disparate to constitute a single individual; not just because they are scattered, because many scattered things are individuals,but because they come from different categories. How could a thing, a property, an event, a number, a color, a university, a syllogism and an angel fuse into one?6

Reply. Simons admits that his intuition here is not

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THE NATURE OF NUMBER 169

conclusive. We think that wherever a class can be formed, as it can in this case, there an aggregate of these things also exists. The matter strikes us as a reasonable case for ‘spoils to the victor’. If what on other ground seems to be our best theory of number should require such aggregates, then so beit. It is to be noted in any case that Simons is guilty of rhetorical exaggeration when he speaks of the parts of the putative aggregate as ‘fusing into one’.

Objection 6. An objection of Simons’ in the same paper that seems more important is this. Consider the following figure d taken as a type:

d

We have been working with properties that ‘divide their extension’. Given this, how many (marked off) squares does d contain? One apparently has to say just one, on the ground that when any one complete square has been cut out, there is not enough left for another square. But could it not also be thought of as a two-square aggregate (a&b) and a three-square aggregate (a&b&c)?

Reply. This points to the need to modify our account. What we must do is to drop the demand that the constituent properties which are found in the structural property instantiated by the aggregate, and which provide the basis for the unit-properties, are in every case properties which divide their extension. The properties must be particularizing, but they may be weakly particularizing, where the instances falling under the properties may be partially identical, as in Simons’ diagram. In that diagram, we can say, one, two or three squares are contained depending on what specifications are taken to constitute a square. If no overlap is allowed, then there is only one square, although three candidates. If overlap is allowed, but not a part-whole relation between squares, then there are two squares. If overlap and part-whole relations are acceptable, there are three squares.

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170 FORREST A N D ARMSTRONG

So the diagram, taken as a type, is a one-square, two-square and three-square aggregate, though not in exactly the same sense of ‘a square’. If number is a property of properties, this property will have the property of being a one or being a two and being a three.

Lewis has argued with us that to move to weakly particularizing properties is to move to any property at all, at any rate if disjunctive properties (disjunctions of universals) are admitted. For instance, if determinate extensions are accepted as universals, then being extended is automatically a property because it is a disjunction of all determinate extensions. Furthermore, it particularizes (weakly). A diagram-type, like that considered by Simons, may truly be said to be three overlapping squares. But equally a determinate extension, such as being two inches in length is made up of huge multitudes of extensions, many of which overlap.

We accept Lewis’s point. What it seems to show is that, for the case of weakly particularizing properties, aggregate- properties involving such properties are only of interest where there are not too many numbers associated with the aggregate- property. The situation with regard to Simons’ diagram is tolerable. But where being extendedis taken as a disjunction of all determinate extensions it will carve up a two inches in length aggregate in too many ways to be interesting.

Nevertheless, even in the latter case there will be numbers associated with aggregate properties. After all, there could be a calculation (partly dependent upon the actual nature of the extension) to determine the maximum number of extended things, overlap permitted, which aggregate to a two-inch extension. This number, and all lesser numbers, could then be taken to be properties of being two inches in length.

Objection 7. But now for the body-blow to our property of properties analysis. The structural property which is being N Fs will, in general, also be being A4 Gs, where M f N, and F + G. Consider the structure-type e:

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THE NATURE OF NUMBER 171

Working the whole time with strongly particularizing properties, so that source of ambiguity is ruled out, even so e is both a structure of 4 squares and 2 two-by-one oblongs. So in terms of our theory it is (at least) both a four and a two. Other structural properties will exhibit far worse multiplicity.

This point (originally made by Forrest) will remind us of Frege’s argument that numbers cannot be properties of objects, because any object will have innumerable numbers depending upon how it is ‘sliced’. By moving from objects to properties, we move to something ‘thinner’, which is not quite so easy to slice. But now it appears that the properties will nevertheless have an embarrassingly large number of numbers.

Reply. This objection is not absolutely fatal to the suggestion that numbers are properties of our structural properties. Provided that the class of all and only the 4- properties is not the same as the class of all and only the 2- properties, it will perhaps not matter that they have common members, as the property eabove is a member of both the class of the 4s and the class of the 2s. For it can still be argued that the properties which determine the two overlapping classes of structural properties are distinct properties, and that they are the numbers 4 and 2 respectively.

Objection to reply. The situation is becoming very artificial though. The huge majority of the structural properties involved are numerically ambiguous in the way that Forrest has pointed out. It is required that associated with each number there is its own set of structural properties, a different set for each number, even though the sets exhibit much overlap. We find it hard to see how one might meet this demand except by finding, for each number, a special structural property which has that number and no others. But then we cannot think of plausible candidates for such properties.

All in all, it seems best to turn to a new theory.

I1

The Natural Numbers as Relations The initial suggestion was that the number N was aproperty which all N-P aggregate-structures have in common. But,

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largely because of Objection 7, we have come to consider that an adequate account of the number N must exhibit the way being an N- P aggregate is relative to the particularizing property being P. Now Kessler7 has provided just such an account. On it, the number N is identified with a relation, which we shall call K N , between the aggregate of N Ps and the unit property of being a P. So on Kessler’s account a number is a relation between a particular, namely some aggregate of N Ps, and a property, namely being a P.

We endorse his account, judging it correct as far as it goes. But we have an objection to treating it as the most basic one. Consider an aggregate of 19 electrons. On Kessler’s account it is related by K,, to the property being an electron. We ask: why is it so related ? If his account were treated as the ultimate or basic account of the number 19, there would be no answer. But intuitively we can answer the question. The aggregate is related by K,, to the property of being an electron because of its structure. Its structure is that of something which is the sum of distinct* parts each of which is an electron. In order to avail ourselves of this intuitively plausible explanation we need to consider the structure of the object related by K,,, or in general KN, to the property of being a P. As realists about structural properties we take these structures to be properties characterised recursively thus:

(i) being the sum of one (distinct) part which is a P just is being a P; (ii) being the sum of N + I (distinct)parts which are Ps is being the sum of two distinct parts one of which has the property of being a P , the other of which has the property of being the sum of N (distinct) parts which are Ps.

In order to avail ourselves of an intuitively acceptable explanation of why the aggregate stands to the property in the relation KN we have introduced the relevant structures. But now we notice - notice rather than hypothesise -an internal relation I N between the structure of the aggregate and the particularising property. By an internal relation we mean one which is supervenient on those properties of the relata which are themselves universals.9 For example being bigger than is an

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T H E NATURE OF NUMBER 173

internal relation. That one thing is bigger than another is necessitated by their sizes, which are naturally taken to be genuine universals. By constrast, being the cause of is more naturally taken as an external relation. This is especially plausible if the causation is itself of a probabilistic kind.

Because internal relations are supervenient on the properties of the relata it is often possible to notice the internal relation simply by considering the relata. For instance once you know the sizes of two objects, no further experience is required to determine whether one is bigger than the other. Likewise we claim to notice the internal relation 1 N rather than hypothesise it. At least, we notice it for small integers. We extend the result to large integers by straightforward induction.

We now have an ontological account of the integers, greater than one. (We shall come to zero and one soon.) For each N and for each particularising property being a P, there are three items of interest, the first is the Kessler relation K N , the second is the recursively characterised structural property of being the aggregate of NPs, the third is the internal relation I N between that structural property and being a P. The connection between the three is that K N holds between an object x and a particularising property being a P just in case x has a property itself related to being a P by I N . We could not identify the structural property with the number N, because it depends on the property of being a P. But we may identify N with either K N or I N . Because I N is the more basic relation we shall stipulate that N is to be identified with I N .

We have described I N and K N as relations. Are they universals? We say I N is a universal because we take it that being an aggregate of; say, 19 efectrons stands to being an electron in the same way as being the aggregate of 19 apples stands to being an apple. This consideration has the peculiar consequence that a genuine universal can relate properties which are not themselves universals. This peculiarity we allow only for internal relations, where we are comforted by the way the relation is supervenient on the relata. However, if this peculiarity is taken as a reductio of the claim that I N is a universal, this will not damage the rest of our account. Similar considerations apply to K N.

We shall now review two of the previous objections and

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consider three new ones. Objection 2 concerns the number zero. What we are to say of the relations KO and Io? When we introduced K N and IN we described some of their relata. No doubt what we said suggested these were the only relata. We no explicitly remove this suggestion. We propose that the structural property of being the aggregate of Ndisjoint Ps and, in addition, a furtherpart containing no Pis, like the structural property of being just the aggregate of Ndisjoint Ps, related by I N to being a P. Likewise we propose that an object which is an aggregate of N Ps and a part which contains no P is related by K N to being P. This generalised account now applies quite straightforwardly to N = 0. KO holds between an object which contains no P and the property being a P. Likewise 10 holds between the structural property of being an object containing no P and the property of being a P. (If there are no Ps at all then see the reply to Objection 1.)

It is to be noted that our account requires the introduction of negative properties. These involve more difficult problems than disjunctive properties. However, we do reject negative universals, and we hold that a thing’s negative properties are supervenient upon the universals which it instantiates together with, perhaps the ‘totality’ state of affairs that these are the totality of the universals that the thing instantiates.

Objection 6, due to Simons, requires us to extend the relations K N and IN to cover the case where the unit property being P is a weakly particularising property. In that case we can distinguish between various different structural properties which may stand to being P i n the relations IN for different N. Returning to the Simons example of the squares, the diagram has the property of being the sum of one square and a disjoint part containing no square. It also has the property of being the sum of two squares neither of which is apart of the other. Yet again it has the property of being the sum of three non- identical squares. These properties are related by I,, I,, and I, to being a square. This nicely reflects the intuitive ambiguity in the questions ‘How many squares are there in the diagram? The answer might be ‘One’, because once we have taken away a complete square there is not enough left over to make another. Or it could be ‘Two’ because we can form two squares neither of which is a part of the other. Or, perhaps most naturally, the

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T H E NATURE OF NUMBER 175

answer could be ‘Three’. As a consequence we can say that the one diagram stands in all three relations K,, K, and K, to being a square. It should be noted that if we took Kessler’s account as fundamental, it would not be possible to explain the initially surprising result that three numbers relate the one object to the one property.

We now turn to some new objections. Objection 8. On our account I , should be a relation between

being a P and being a P, or between being a Pplus a further part containing no P and being a P. This relation would seem to be either identity or partial identity. Armstrong has previously argued against identity being a universal.10

Reply. Our reply is a ‘strategic withdrawal’. One of Armstrong’s arguments was that identity is knowable a priori and so suspect as a universal. The other was that it did not contribute any casual powers. Both these arguemnts would apply to all internal relations. We now consider that internal relations can be knowable a priori and that they need not contribute casual powers. The justification for this amendment is that internal relations are supervenient on their relata. We are prepared,then, to treat I , as identity or partial identity, and take it to be an internal relation.

Objection 9. We have not proposed numbers as hypothetical entities, like electrons. We should, therefore, give due attention to the way numbers are experienced. Suppose one sees four parrots on the tree. Of what is one aware? Of the parrots, of their properties, and that there are four of them. The objector denies that one is immediately aware of a relation between the aggregate of the parrots and the property being a parrot.

Reply. The phenomenology of numbers is tricky. But we shal agree, at least for the sake of argument. Our knowledge of the relation I,, is not immediate. But it is based on an immediate awareness. There are four parrots on the tree because the property of being the aggregate of four parrots is instantiated by something on the tree. Further, to be aware of a property or relation is to be aware of it as $instantiated. (Even in a hallucination if one is aware of a property or relation which is not instantiated where one thinks it is, at least one is aware of it as instantiated - it seems to be intantiated.) Hence to be aware

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176 FORREST AND ARMSTRONG

of the structural property of being the aggregate of four parrots just is to be aware that there are four parrots. One is immediately aware, then, of the structural properties. The internal relation I, is only noticed by reflection, by thinking about the property one is aware of.

Objecion ZO. There is a well-known recursive account of the natural numbers.1’ It goes like this:

(i) ‘There is at least one F‘ is analysed as ‘There is an ‘F‘, (ii) ‘There are at least (n + I ) Fs’ is analysed as ‘There is an x which is an F, and there are at least n Fs not identical to x’ and (iii) ‘There are n Fs’ is analysed as ‘There are at least n Fs but not at least (n+l) Fs’.

In view of this it might be objected that our account is uneconomic. Why introduce the relations IN when we already have a satisfactory account?

Reply. In the first place, our account will generqlise to negative numbers, rational numbers, and real numbers, but the recursive analysis given above does not. But in any case, we reject the charge of lack of economy. We did not posit or hypothesise various relations, we discovered or noticed them. Eighteenth century Europeans might as well have charged Captain Cook which lack of economy in positing or hypothesising New Zealand. In detail, our reply is that the structural properties such as being the aggregate of N disjoint Ps are already part of Armstrong’s theory of universals and are not required solely for present purposes. Moreover, as we indicated in the reply to Objection 9, for small integers N we are aware of those properties when we are aware of the number of Ps. And the relation IN, being internal, is noticed once one reflects on their relata. What would be uneconomical would be a property or an external relation posited solely for the purpose of accounting for the integers.

I11

The Univocity of Numbers Thus far we have attempted to give a more fundamental

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THE NATURE OF NUMBER 177

account than Kessler’s without disagreeing with him. However in a footnote’2 he extends his account to negative integers and to rationals by taking, say, the rational number m/n to be a relation which holds between natural numbers. On this account, rn/ n holds between x and yjust in case nx = my. Here we disagree. For the number 19, say, is not just a natural number, it is also a rational number. So it would have to be construed not merely as the relation K,, between an aggregate and a property, but also as a relation between such relations. Thus the rational 19 would relate the integer 2 (i.e. K,) to the integer 38 (i.e. K38). Yet again, 19 is a real number, and Kessler suggests reals are suitable aggregates of rationals. (19 would be the aggregate of all the rationals m/ n where m < 19n). That the number 19 should have a quite different ontological analysis as a natural number, a rational and a real is not acceptable to us.13 Consider the sentence, ‘There are 19 cherries in the bowl and each weighs 19 grams.’ The numeral ‘19’ is, we claim, univocal in its two occurences. If it were not, the calculation that the total weight of the cherries in the bowl is 361 grams would not be an application of the multiplication tables for the natural numbers. Accordingly we extend our account of natural numbers to include rationals and reals.

Consider again an example described in the reply to Objection 6, that of a 2 kilogram mass of lead. We say that it stands in the relation K, to the property being a I kilogram mass, and that being a 2 kilogram mass stands in the relation I, to being a I kilogram mass. The same account of the number 2 applies to the 2 in being 2 kilograms as in being the aggregate of 2 electrons. Quite generally we submit that IN can hold between various magnitude properties being P and being Q. If I N does hold between being P and being Q then K N holds between an instance of P on the one hand, and Q on the other. But this account now extends to the case where N is not a natural number. Thus being of mass 5 kilograms stands to being of mass 2 kilograms in the internal relations 12.5 And being of length 4acm stands to being of length 2cm in the internal relation I,, Even negative and complex numbers can be included. For not all magnitudes are extensive, and intensive magnitudes admit of negative and complex degrees. Consider electric charge. Being of charge 8 coulombs stands in the

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relation I, to being of charge 2 coulombs. Likewise being of charge 8 coulombs stands in an internal relation I - I to being of charge -8 coulombs. In this way all numbers, not just natural ones, are exhibited as internal relations between properties. 14

The chief objection to this account would seem to be a continuation of Objection 10. The objector denies the need to introduce the various relations. As before we reply by claiming in many cases to notice internal relations rather than hypothesise them. And we claim further that they add nothing to the ontological cost of a theory because of their supervenience. But, although it is not strictly necessary, there is an additioinal reply to the objection. It is that the ‘nominalist’ who sees no need for the relations we posit has a much harder task in accounting for magnitudes than the natural numbers. The best attempt we know of this is Field‘sls. As a specimen of his strategy consider an object with a mass twice of that object b. He handles the putative relation of having-mass-t wice-of by considering a four-term congruence relation. If u, v, w and x stand in this relation then we would say that the ratio of u’s mass to v’s is the same as w’s to x’s. So we can replace a’s having mass twice that of b’s by a, b, w and x standing in this relation for various w and x. Field’s strategy is open to an objection like one of those against Resemblance Nominalism. For example, the perturbation of Uranus’s orbit by Neptune is explained by the fact that the mass of Uranus and Neptune stand in a certain ratio. Intuitively, small masses which stand in the same ratio, or even large masses in that ratio which are far away from the Solar System, are quite irrelevant to the explanation. Yet on Field’s account it is Uranus, Neptune, w and x standing in a certain congruence relation which provides the explanation of perturbation, where w and x are objects standing in the same mass ratio as Uranus does to Neptune. This incorporation of the irrevelant into explanations is a serious defect in Field’s strategy. Accordingly we reject it.

IV

Numbers as Relational Properties. It may seem a bit strange to say that numbers are relations,

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THE NATURE OF NUMBER 179

whether relations that hold between a unit-property and an aggregate (the Kessler relations), or internal relations that hold between a unit-property and structural properties of aggregates (the I relations). Is it not more natural to think of numbers as properties of something?’6

There are those who are unsympathetic to such intuitions. They produce definitions of this sort: the number 4 is the class of all classes similar to a given class (some given quadruple). What matter if this is quite unintuitive? It will, it is claimed, do the mathematicaljob, and any other jobs involving the number 4.

We, however, are more tender-minded. We would like to satisfy the property-intuition if we can. And, indeed, we can do something. We can at lease substitute a relational property for a relation.

Suppose that a man fathers a child. Fathering is a relation. But the man himself is a father, and that is a property which, has, not a relation, albeit a relational property and supervenient upon the relational state of affairs of a man fathering a child. Let us now take as example a case involvinga an internal relation. Oxford blue is darker than Cambridge blue. That is an internal relation which holds between those two shades. But Oxford blue has the relational property of being darker than Cambridge blue. Again, if some surface is Oxford blue in colour, then it will have the relational property of being darker in colour than Cambridge blue.

There is an internal relation which holds between any structural property of the 19-P aggregate sort and being a P. But any such structural property has a relational property: bearing the 119 relation to being a P. Let us call this relational property “19’. Again, if some aggregate has a structural property of the 19-P sort, then this object will have the relational property of bearing the KI9 relation to being a P. Call it ‘MI<. Both Nl9 and MI9 have quite strong claims to be the entity that ‘19’ really names.

We readily concede, of course, that such a structural property, and such an aggregate, will have many other relational properties involving N!, and MI9 respectively, not to speak of relational properties involving other Ns and Ms. But often the structural property and the unit-property will be

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salient in the situation, thus fixing the relational property. If there are just four parrots in the tree, then it may well be that the flock is perceived as a four-parrot aggregate, and very likely that the unit-property focussed on is that of being a parrot. The relational property of bearing I4 to being a parrot will then be a salient property of the salient property being a four-parrot aggregate. Equally, the relational property of bearing K4 to being a parrot will be a salient relational property of the flock (in virture of its 4-parrot structural property).

V

What are Classes? Finally, something about classes. One view concerning the natural numbers is that they are properties of classes. We take this to be a more plausible view than that they are classes of classes, but we have proposed an account of numbers that makes no direct appeal to classes at all. At the same time, however, it is clear that there is some close connection between classes and numbers. So it may be wondered what light our account of numbers sheds on the notion of class. On the Kessler account a number is a relation between an aggregate and a unit property. If numbers are also to be associated with classes, then a class should itself be constructed in some fashion from an aggregate and a unit property.

What, then, is the relation between a class and the corresponding aggregate, that is, the mereological sum of its members? (We maintain, of course, against Peter Simons in particular, that such a sum always exists.) We have been told on high authority that the identity-conditions for classes are ‘crystal clear’, and we accept that classes are identical if and only if their members are identical. But compared to aggregates, classes are somewhat insubstantial and mys- terious. They are ‘abstract’, as the unhelpful jargon supplied by the same high authority goes.

There are different ways of dissecting an aggregate: different ways that it is divisible into parts. Every such dissection of an aggregate into parts (parts which may or may not be wholly disjoint but whose mereological sum is always the aggregate) stands in very close relation to a certain class. In symbolic

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T H E NATURE OF NUMBER 181

terms, instead of having the aggregate [a 4- b 4- c ...I we have the class {a , b, c, ... 1 . What would turn this aggregate into this class?

It is here, we suggest, that the Kessler relations, in particular, cast helpful light on the situation. These relations hold between an aggregate and some unit property P, strongly or weakly particularizing. Now where such relations hold we can also think of P as operating upon the aggregate to produce one or more classes, but where each class is such that the mereological sum of the members of that class is the aggregate. P thus carves up, as it were, or acts as a cookie-cutter on, the aggregate. One such ‘product’ of this ‘process’ may be { a, b, c

This in turn suggests that we should in the first instance identify that class with a conjunction of states of affairs: Pa & Pb & Pc. The class is more complex than the aggregate [a + b + c....] because in the class the parts of the aggregate are embedded in states of affairs. Notice that these states of affairs, like classes, are particulars rather than universals.

This identification can be in the first instance only. For there will be other particularizing properties besides P which operate upon the same aggregate to give the very same class. So, to consider only the class-member a, we need to involve in the analysis not only the state of affairs Pa, but also Qa ... and all the other particularizing properties which carve out just that class out of just that aggregate.

Should we then take the conjuction of Pa, Qa ... and similarly for b, c ... ? That conjunction exists, of course. But it seems to us better to capture the ‘Idea’ of a class to disjoin the particularizing properties. We will have a disjunction of the states of affairs Pa, Qa ... conjoined with the disjunction of Pb, Qb ... and so on, through all theclass-members. Alternatively, one could form the disunctive property (P V Q ...), predicate it of each member of the class and then conjoin the result.

One advantage in considering the disjunction rather than the conjuction is that it enables us to characterise class membership without undue complexity. First we say that a state of affairs is about a particular a if it is of the form Pa. For example, Rab is about both a and b. Then a particular is a member of some class, here identified with a state of affairs,

...) .

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just in case either that state of affairs or some part (i.e. conjunct) of it is about the particular in question.

Our construction may be compared to Russell’s so-called ‘no-class’ theory.” The most obvious difference is that Russell is giving an account of classes in terms of propositional functions. We reject the nominalism implicit in this. So we shall reconstruct Russell, replacing propositional functions by properties. Hence for each class there will be a family of co- extensive properties any one of which might have been used as a replacement for the class. Then Russell’s claim is that in asserting that a class has a certain property one is asserting that some member of the family of co-extensive properties has that property. 18

The major difference between our account and the reconstruction of Russell is that we insist classes are states of affairs rather than properties. l9 We have several reasons for this. First, classes are intuitively particulars rather than properties. Otherwise they would not be so easy to confuse with aggregates. Second, Kessler’s account of numbers, which we have built upon, involves the relation between an aggregate and a property. If numbers are also to be associated with classes, we require both the aggregate and the property to be involved in the class. Third, a given class could not have had different members, but a given property could have20 and this will inevitably complicate any attempt to replace classes by properties.

The theory of classes we have sketched would fit in quite well with an account of classes recently proposed by Lewis.21 He suggests that classes may be treated as mereological sums of unit-sets. His problem is to get from the members of the unit- sets to the unit-sets themselves. Clearly, that is not a mereological operation. On our view the unit-set will be a state of affairs: say, the state of affairs of the member of the set having that disjunctive property formed by disjoining those properties which pick out the member as one P, one Q, etc. Moving from the particulars and properties in a state of affairs to states of affairs is not a mereological operation.22 But if a set has a plurality of members, then it is a conjuction of states of affairs involving the individual members. The conjoining of states of affairs is mereological. The parallel with Lewis is quite close.

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T H E NATURE OF N U M B E R 183

Even so, there would remain a discrepancy between our original account and this melding of Lewis’s view with our own. Given a class with a plurality of members, we demanded that each member instantiate each of the particularizing properties involved. This demand will not necessarily be satisfied if plural classes are constructed from unit-classes in the Lewis manner. Our feeling, however, is that this is a relatively minor difference. We prefer our own view, but do not know of any argument which seems to us to settle the matter in our favour.23

Three matters now require to be discussed: higher-order classes, ordered n-tuples and the null-class.

Higher-order classes. Let us begin by considering a first- order class, but, to make it difficult for ourselves, a very heterogeneous one. Peter Simons’ example, considered in Section I , Objection 5 , will do nicely. Suppose that (contra Simons) one can take a certain thing, a certain property, a certain event, a certain number, a certain colour, a certain university, a certain syllogism and a certain (existent) angel as constituting an aggregate. There is at least one strongly particularizing property which carves out the corresponding class from the aggregate (and has K, to the aggregate). The property is: either a thing or aproperty or an event or a number or a colour or a university or a syllogism or an angel. These disjuncts themselves are unlikely to be universals for many at least of the 8. As a result, there will be still more disjunctivity concealed in the disjuncts. But once disjunctive properties have been admitted, as we have admitted them, there are, we hope, no further problems of principle.

This and perhaps other properties operate upon the aggregate to yield a first-order class, a class which will be a certain complex state of affairs. Each member will have the same disjunctive property - the disjunction of all the unit- properties for the class. These states of affairs are conjoined in a conjuctive state of affairs. This is the class.

So far nothing new. This conjunction is a token: a certain particular state of affairs. But it is a token of a certain type. Hence we can abstract from the particular particulars involved, substitute unbound variables, and reach the notion of that type of state of affairs.

Now suppose that we have certain other first-order classes.

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They too will be conjunctive states of affairs, each of them a state of affairs of a certain type. We take the Simons class and these other classes and form a class of these classes. What is a unit-property which will operate upon the corresponding aggregate to produce that particular class of classes? The answer is: a disjunctive property whose disjuncts are the state- of-affairs types associated with each of the first-order classes. To this must be added any property co-extensive with this disjunctive property. Given these properties, form the higher- order class in the same way that the first-order classes were each formed, using a disjunction of the unit-properties.

Ordered n-tuples. We would like to exorcize the ghosts of Kuratowski and Wiener. We do not think of ordered classes as mere unordered classes of classes. Rather, as Russell pointed out long ago, the notion of order is the notion of certain sorts of relation. The n-tuple will be a conjunction of states of affairs involving each member of the n-tuple. But in addition we require relations to order the members.

Here is a way of thinking of the situation. We have an n- tuple <a,, ... an>. As usual, take the corresponding aggregate. For an unordered class the aggregate is cut up by a single unit- property: a single cookie-cutter. For the n-tuple the aggregate is cut up by a different unit-property for each member: a cookie-cutter that changes its shape.

These unit-properties will be relational properties of the members. The relations, R, which help to constitute these properties must be of the following nature. Rs hold between the integers and the members of the n - t ~ p l e . ~ ~ They can be internal relations. Then for each integer m, m will have a particular R to a certain member of the n-tuple, which is thus constituted the rn th member of the n-tuple. If the integer K is not identical with m, then m will fail to have that particular R to the mth member.

Writing RPm for the relational property that picks out the mth member via a relation to the integer rn, the ordered n-tuple becomes the conjunctive state of affairs RP,a, & RP,a, ... RPmam ... RPnan. To get the requisite extensionality of the n-tuple, each RPm must be expanded into a disjunction of all the RPs that do the job of picking out the mth member.

The null-class. Finally, what of the null-class? If we follow

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out the analogy of unit-properties as cookie-cutters, then for the case of this special class we will be interested in properties which fail to carve any units out of any aggregate. (For Forrest these properties will be uninstantiated properties. For Armstrong they will be merely possible properties, and so strictly not properties.)

Now consider the disjunction of all those particularizing properties which fail to pick out an object from any aggregate whatsoever. That all these disjuncts so fail is a state of affairs. Admittedly, it is a very special state of affairs, involving as it does negation. We hold that it is supervenient upon the conjunction of all positive states of affairs plus, perhaps, the ‘totality’ state of affairs that these are all the positive states of affairs. But it is a state of affairs. Given our general approach, can it not be identified with that very special class: the null- class?

We note that if our view of classes is along the right lines, then any nominalistic attempt to give an account of properties in terms of classes must be rejected. We make bold to think that this is a difficulty for that species of Nominalism rather than for ourselves.

NOTES

I . This paper was presented to the A.A.P. Conference at Monash University in August 1986. We have benefited from contributions to the discussion there by John Bigelow, John Fox, Adrian Heathcote and David Lewis. Wearealso greatly indebted to subsequent correspondence with Lewis. In addition Forrest would like to acknowledge the influence on his thought of remarks by Chris Mortensen and Charles Pigden. 2. Although admitting uninstantiated universals, Forrest is sympathetic to the view that all such universals may be ‘combinatorially derived‘ from instantiated universals. 3. One of us (Armstrong) holds that what is supervenient is never an ontological addition to what it supervenes on. Forrest does not share this view. But he considers none of the superveniences mentioned in this paper to be an ontological addition. 4. Or, at any rate, non-supervenient universals are hard to come by. We shall argue shortly that certain internal relations are universals. They are also easy to come by. But they are supervenient upon their terms. 5. G. Kessler ‘Frege, Mill and the Foundations of Arithmetic’, The Journal of Philosophy. 77,2 (Feb 1980), pp.65-79. 6. P. Simons ‘Against the Aggregate Theory of Number’, TheJournalo/Philosophy, 79,3 (March 1982), p.164. 7. G . Kessler ‘Frege, Mill and the Foundations of Arithmetic’.

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8. Because the property of being an electron is strongly particularizing we need not be careful here about whether we mean ‘non-identicar or ‘disjoint’ by ‘distinct’. It will matter when we consider weakly particularizing properties. 9. D.M. Armstrong, A Theory of Universals: Universals and Scientific Realism, Volume 11, Cambridge University Press. Cambridge, 1978, pp.84-6. 10. D.M. Armstrong A Theory of Universals, p. I I . 1 1 , See, for example. H. Field, Science Without Numbers, Princeton University Press, Princeton, 1980, pp.21-2. 12. G. Kessler, ‘Frege, Mill and the Foundations of Arithmetic’ n.18, p.78. 13. We would like to thank Charles Pigden for alerting us to the peculiarity of different ontological analyses for 19 as an integer. 19 as a rational and 19 as a real. 14. Or relations. There can be polyadic magnitudes. such as being a certain distance apart. 15. H. Field, Science Wirhour Numbers, Princeton University Press, Princeton, 1980. 16. On Bealer’s neo-Fregean account [G. Bealer, Qualit.v and Concept, Clarendon Press, Oxford, 1982 Ch.61 a number is property of a property. As an attempt a t a fundamental account this is open to the objection that it does not extend to the rationals and reals. 17. B. Russell, Inrroducrion to Mathematical Philosoph.v, George Allen and Unwin, London 1919, Ch. XVII. 18. B. Russell, Introduction to Mathematical Philosophy pp. 187-9. 19. Bealer, like Russell, replaces classes by properties. See G. Bealer, Quality and Concept. pp. I 11-9. 20. D.M. Armstrong, Nominalism and Realism, Universals and Scientific Realism. Volume 1. Cambridge University Press, Cambridge, 1978. p.38. 21, D. Lewis. ‘Against structural Universals’, Australasion Journal of Philosophy, 64,l (March 1986). p.37. 22. D.M. Armstrong ‘In Defence of Structural Universals’, Australasian Journd of Philosophy 64,l (March 1986), pp.56-86, and P. Forrest ‘Neither Magic Nor Mereology: A Reply to Lewis’, Australasian Journal of Philosophy 64.1 (March

23. For suitably homogeneous classes there will be no difference between the two accounts. 24. We see no need, at least in the finite case, to posit special ordinal numbers. We leave the complexities of the infinite ordinals to the mathematicians.

1986). pp.89-91.

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