PURE REALISM: PLATONISM AS A SERIOUS CONTEMPORARY ALTERNATIVE. Keywords: pure realism, Platonism, metaphysics, natures

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1 PURE REALISM: PLATONISM AS A SERIOUS CONTEMPORARY ALTERNATIVE Samuel C. Wheeler III Department of Philosophy University of Connecticut U-54 Room 103 Manchester Hall 344 Mansfield Road Storrs, CT USA Abstract. When it becomes a subject for a deep and systematical analysis, Platonism proves to be a serious rival for contemporary realisms in providing a modern metaphysics. Not only that pure Platonic realism has the advantage of elegance and clarity, but it also show at least three less abstract advantages: first, the Parmenidean principle is adhered to rather than being accepted just most of the time. Second, pure realism avoids the category of metaphysical necessity and third, the ontology becomes more simple, consisting only of natures Keywords: pure realism, Platonism, metaphysics, natures Contemporary realisms postulate natures relatively sparingly. Many apparently general terms, that is, terms that are formally predicates, are not said to be natures at all. Contemporary realists are thus non-realists 1 about such general terms as identity, non-identity, being a particular, being an entity, and being one thing. On the other hand, a pure or Platonic realism, following Parmenides, posits a nature for every general term needed in an account of the world. According to Platonic realism, natures are the fundamental entities from which any other entities, including particulars, are derived. There is one central difficulty in a pure realism, but that central difficulty is mirrored precisely in impure contemporary natures realisms such as Armstrong s 2 and Olson s. 3 Some features of the Platonic theory I ascribe to Plato fit well with contemporary metaphysical views. For example, the basic physical objects turn out to be entities almost Many of the ideas in this paper developed while auditing David Armstrong s seminar in the Spring of I would like to thank him for his tolerance of a non-believer. I would also like to thank Don Baxter for his puzzled responses to earlier confusing drafts. 1 This non-realism probably should not be construed as nominalism, since Armstrong does not claim that particulars are all particulars in virtue of being called particulars. Rather, it seems to me, Armstrong takes a Davidsonian line about is a particular. Is a particular is true of an entity if and only if it is a particular. 2 Armstrong, David, A World of States of Affairs, Cambridge UP, See also his Universals and Scientific Realism (two volumes) Cambridge UP, 1978, and A Combinatorial Theory of Possibility, Cambridge UP, Kenneth Russell Olson, An Essay on Facts, Center for the Study of Language and Information, Stanfrod, It is worth noting that, on a pure Platonic theory, there is an entity for every Olsonian fact and every state of affairs of Armstrong s. These entities could be regarded as the truth-makers for the corresponding sentence. 91

2 exactly like the temporal parts we find in Armstrong and Lewis. Plato, as a Heraclitean about the physical world, regards any enduring physical object as a temporal sequence of instantaneous entities. On several points, the contemporary Platonism theory differs from Plato s. The centrality of mind that we find in Plato doesn t correspond to anything plausible as a contemporary view. Also, the idea that there will turn out to be natures corresponding to persons, organisms, and other medium-sized objects is compatible with Platonism, but not implied by the general theory. 4 The problems with such alleged natures that lead Armstrong to characterize them as second class properties are problems for Platonism as well. Finally, the imperfection of the physical in relation to the Forms, which Plato emphasizes in the early dialogues, 5 does not obviously conform to contemporary physical theory. 6 I call this theory Platonism because I think, but will not argue, that Plato holds its central theses. On occasion I will refer to passages in dialogues that can indicate that important components of the view at hand were Plato s. My project is to find one theory that will accommodate these Platonic requirements and also be a viable competitor with contemporary realisms. Whether the theory is Plato s in detail is another matter. I argue that the Platonic theory is as good as Armstrong s at providing a modern metaphysics. 7 I also claim for it several advantages: that it draws a natural line between the logical and the metaphysical, that it therefore can treat all non-formal modal truths 8 as due to natures and that it is ontologically more economical. I hold no variety of realism, and would 4 Plato at least has a way to say what it is for some complexes of micro-particles to constitute single individuals, as we will see below. 5 The later dialogues seem to emphasize continuity between Forms and instances, rather than conceiving of the difference as a sharp difference in realms. Roughly, the most general forms are differentiated into more and more particular forms, and finally into instances. The relation between the Platonism developed here and the view sketched in the Philebus is obscure. In the Philebus, Plato seems to be presenting an account in the way Aristotle describes it in Metaphysics A6, as the successive application of number to the Great-and-Small, yielding first the Forms, and then the sensible instances. I think that, if Forms are formulae, and the Great-and-Small is differentiation, the theories may come out close to the same. The main difference would be that spatio-temporality would be a species of differentiation. The material in the Philebus and Aristotle s reports is too sparse, however, for me to have any theory of precisely how one account fits or fails to fit the other. 6 The imperfection of the physical world in relation to the world of Forms is compatible with Platonism, but not required by theories that are Platonist by the account given here. Several considerations are relevant: 1) The Platonic versions of micro-particles need not be imperfect relative to their natures, although one could assign a corrupting influence to the nature spatio-temporality. However, it appears empirically that micro-particles are completely uniform, so such corruption is not a likely hypothesis. 2) Medium-sized object natures may or may not be part of a Platonism. If there is a Form of Man, the relationship between that nature and individual men could be treated in a number of ways. Individual men could be natures that are parts of all men, so that everyone would have a nature that was a differentiation of the nature of Man. Individual men would be unifications of the right sequences of temporal parts of sequences of collections. For Plato, as we will see, unification is not automatic, so that individual men (as people with personal histories) would be distinguished from competing complexes by really being one thing. 3) Since all there is to the world, ultimately, is natures, and since individuals-at-instants (temporal parts of individuals) will be just very specific natures, there will be no difference in kind between the Forms, construed as general natures, and such individuals. 7 This is very high praise for Platonism indeed, as anyone who has studied Armstrong s work will know. 8 The treatment of mathematical truth will be dealt with in only the sketchiest way, following Armstrong. It is quite clear that mathematical entities are natures, but the theory of their composition leaves too many choices to pick one. 92

3 argue that Davidson has shown how to avoid realism. 9 However, I am interested in the question: Given that one is inclined to be a realist, why not be a pure, Platonic realist? The advantages of elegance and clarity seem to me to be clearly on the side of pure realism. I will begin by sketching the outlines of the theory, citing lightly the Platonic texts that inspire various points. I will then discuss various advantages of the theory and show how the theory addresses a number of apparent problems. On occasion, I will compare the theory with one contemporary alternative metaphysics, that of David Armstrong s A World of States of Affairs. I choose Armstrong s theory as a foil for Platonism because Armstrong s theory seems to me the most carefully developed and viable of the nature theories currently being entertained. The central argument for the adequacy of the theory discusses a central passage in the most thorough and careful account of facts I know of, Kenneth Olsen s Essay on Facts, which is also a most thorough discussion of Bradley s paradox, the central issue with realisms that posit natures. I) Outlines of the Theory A) Parmenides Principle The central feature of the theory is the principle of Parmenides, as Plato understands it: This is the principle that every aspect of what is the case is a being. I understand the one over many argument for the existence of natures 10 to be a special application of this general consideration. The thought is that there is something about two frogs, for instance, which is the same. That feature which is the same must be an entity or nothing. To deny that things in common or things ascribed are entities by appeal to ways, for instance, appeals to something partway between things and nothing. Parmenides point could be expressed by saying that there is nothing between being and non-being. Either commonalities are entities or they are nothing at all. The apparent options, given that frogs are the same, are either to postulate two irreducible notions of sameness and identity, the numerical and the qualitative, or to take the prima facie more elegant tack of reducing qualitative to numerical identity. The basic nominalist 11 tactic postulates two irreducible notions of sameness. Two frogs are both frogs, and are the same in that respect, but there is no entity that is their frogness. What they share, if such is demanded, is that they are both frogs. To reduce qualitative to numerical sameness is to postulate some variety of universal, and to analyze predication as a relation of particular to universal. 12 As has been widely observed, this apparently leaves a residue, since cases of instantiation, although they are the 9 I would deny, therefore, that there is any real need for a metaphysics for contemporary science. Consider explanations of what it takes for Fred is a frog to be true. One the one hand, there is the trivial remark that Fred is a frog is true just in case Fred is a frog. The right-hand side specifies conditions in the world that suffice for that truth. On the other hand, there is the indefinitely complicated scientific account, which would go into what sorts of things amphibians are, what kind of planet and environment make it possible for such amphibians to live, and so forth. I would deny that there is a third account, intermediate between the trivial and the scientific, which provides the metaphysical underpinnings for Fred being a frog or for the scientific correlate. 10 I do not call Platonic natures universals because that term s correlate, particulars corresponds to something very different in Plato. 11 The Davidsonian alternative I would endorse is not a nominalism in anything like the etymological sense. For Davidson, what two frogs share is not the name frog but that they are both frogs. 12 The most frugal entities to be referents for general terms are sets. They are also the least explanatory. Lewis theory, for instance, finds referents for general terms, but gives an account of causal laws that is essentially Humean. Plato and Armstrong want explanations of physical necessities that are more robustly explanatory. 93

4 same in being cases of instantiation, seem not to themselves be instances of universals. Instantiation, as we will see below, requires a relation with some special talents. In a modern theory of universals, several kinds of general terms are excluded from the argument that general terms name universals. First, the categorical general terms, such as is an entity, is a particular, is one thing and is a universal, are excluded because of the apparently paradoxical results of having universals for such terms. Second, certain general relational terms are regarded as logical, such as is a part of, is identical to, and is distinct from and are treated as part of a logical apparatus the necessity of whose principles is too transparent to need explanation by natures. One reason for the exclusion of some of the ubiquitous general terms from the domain of natures is that the particulars they would attach to seem already to have such alleged natures. Is a particular, if a universal, would attach to a particular. But the particular it would attach to must already be a particular. Thus no explanation of how it happens that this particular shares being a particular with another seems to be offered by positing the nature is a particular. Similarly, is one and exists have difficulty finding subjects to attach to, since every single existing subject would already have them. A further prima facie difficulty with exists and is one is that they would have to apply to themselves. B) Natures On the Platonic theory I propose, the fundamental entities are natures, the Platonic Forms. In fact, all entities are natures, perhaps very detailed ones, and nothing else. Since, as I will argue, spatio-temporal locations are natures, natures with spatio-temporal locations as components are so detailed that they can have only one instance. Particular physical individuals are such natures. Natures are entities that can be parts of distinct other entities. When they are parts of other entities, they are wholly present in those other entities. A spatio-temporal individual, that is, has natures themselves as parts, and as parts of those parts. The nature Sameness, explicated below, is the relation in virtue of which entities are wholly present in distinct entities. 13 A Platonic theory has no categorial features, but rather postulates one kind of beings, natures. The mega -natures Being, Unity, Sameness, Difference, and Spatio-temporality 14 are natures among the others. 15 Natures are thus not abstract, but rather are the only real components of what exists and is the case. Natures are typically unities made up of other natures, which are themselves made up of further natures. The parts of a nature are its proximal constituents. Predication is this part-whole relation arising out of Sameness and Difference, as we will see below. 16 To begin to see how this works, let me first discuss the mega -natures: 1) Unity: A) Unity and the unified: the fundamental difficulty illustrated and accepted. Unity is that in virtue of which every single entity is single. Unity is a part of anything that is one. In 13 Entities are distinct in virtue of difference, itself a nature, as we will see below. 14 I hypothesize that spatio-temporality is named motion and rest in the Sophist. 15 The Sophist, st is my source for the mega -natures. 16 As we will see, predication is a part-whole relation that differs from Lewis and Armstrong s mereology. The first part of Peter Simons Parts (Oxford UP, 1987) has an excellent discussion of the various choices available to the formalizer of the various part-whole relations. Parts of the Parmenides also seem to express Plato s view that there are many distinct senses of part, so that the One is divided in one sense (same as distinct entities) but not in another (one and the same entity in many entities). 94

5 the case of Unity Itself, Unity is an improper part. In the case of composite entities, unity is a part of the composite. Here we have a case of the fundamental difficulty with the theory. As Aristotle points out, in Metaphysics Z 17, if the thing that makes an entity one thing is a component of the complex, then some binder is needed to weld that component to the others. If that binder is an entity, a regress results since the binder itself would need to be bound. The alternative, that unity binds itself to the others, strikes Aristotle as paradoxical. How could Unity be a component in the collection and also the explanation of why the collection is a unity? The combination of a part-whole account of predication and Unity as that in virtue of which an entity is one gives the paradoxical result that a component of a composite is that in virtue of which the whole composite is a single thing. That is, unity s being a component of a compound makes the compound, including unity itself, be a single thing. The binding of elements in a composite entity is effected by one of the very entities to be bound together. For purposes of this paper, 17 my response to Aristotle s criticism is that it is a variant of the difficulty Armstrong and Olson face in the case of the unification effected by a state of affairs or fact coming to exist. For Armstrong and Olson, the difficulty takes the following form: The universal and the particulars that make up a state of affairs exist whether or not the state of affairs obtains. If an instantiation relation is invoked to describe the difference, that relation, if a normal relation, is itself an entity that exists whether or not the state of affairs obtains. Thus, an explanatory regress would emerge: We can only get Instantiation to obtain between universal F and particular a by positing another relation to bind the three. That further binding relation, call it Obtaining, would itself require binding to the previous three entities. This regress is genuine, if the following two principles obtain: I: Any relation must be instantiated in order to bring about a state of affairs. II: Instantiation is a relation. Somehow, the binding into a single entity that instantiation effects must be identified with the product of that binding, the fact-ness or state of affairs-ness of the resulting entity. Olson, in An Essay on Facts, page 61, accepts this awkward consequence by saying that the fact itself is the binding relation that unifies the particulars and the universal into a fact. What is really at stake here is the ontological status of facts. Are they analyzable without residue into their components, or must they be acknowledged as irreducible entities in their own right? What distinguishes a fact from a mere assortment of separate items is the connection existing among its constituents. Indeed nothing seems to stand in the way of our identifying the fact itself with this connection. But we must not think of the latter as an abstract relation like the eating that unites not only this wolf and that lamb but any pair consisting of eater and eaten. The word relation is ambiguous as between this and the more concrete sense of a being related. It is in the latter sense that the judgment the wolf is eating the lamb may be said to affirm a relation between the wolf and the lamb; it does not affirm the abstract relation of eating, whatever that might entail. In contrast, connection can only mean a being connected. There is not the same temptation to think of it as an additional thing that does the connecting. The connection is not a constituent of the fact; it is the fact itself. For Olson, then, the unification of a universal and a particular in a fact is not an instance of a universal, even though all facts share this unification. But since there being a unification is the difference between particular a and universal U co-existing and there being a 17 I take Aristotle s remark to be a compelling reason to abandon realism and to abandon the idea that sentences refer. This essay is a defense of Platonism given that one is determined to be a realist, not a defense of Platonism. Aristotle s objection is really a different form of David Lewis objection that Armstrong, among others, needs a non-mereological kind of composition. See his review of A World of States of Affairs in Times Literary Supplement, February 13,

6 fact that a is U, Olson has to say that that difference is not something. So the difference between a) Joe being a frog and b) Joe and frogness both existing comes down to the existence of a fact in case a) and the lack of existence of that fact in case b). The fact itself is the addition of being which both makes a fact exist and constitutes the resultant object. The above Olson passage is crucial, both for Olson and Armstrong and for the argument that Platonic realism is no more paradoxical than contemporary realisms. 18 In effect the solution Armstrong and Olson endorse makes instantiation a counter-example to principle I, which their account would then treat as true of all relations except for the instantiation relation. Olson and Armstrong would resist the idea that the connection is a relation, because making it a relation seems to undermine their idea that facts or states of affairs are ontologically basic. The difficulty in this passage is that it is the same connection that binds Fred and green into a fact as binds Joe and red. Everything about the situation makes the connection indeed an abstract relation as well as a concrete relation. In short, there is no significant difference between denying that instantiation is a relation and asserting that it is a relation that can be a constituent of the compound it unifies. There is little difference in oddity in the following two choices: 1) Instantiation is a relation such that, when instantiated, it is itself a component of the state of affairs it brings about. 2) Instantiation is not a relation at all, even though it constitutes an addition to being. That addition of being is the fact itself, and is the same kind of addition of being wherever it occurs. The fact itself, the result, is the addition of being that converts a pair of entities into a distinct third entity. Instantiation, from a Platonic perspective, is unification. A green frog is a unity of Unity, Green, Froghood, and a region. The whole discussion of truth-makers, whether termed facts or states of affairs, would in Platonic terms be a discussion of unifications. Generally speaking, to every state of affairs there corresponds a unification, and to every unification there corresponds a state of affairs. Plato, since his ontology has nothing but beings, is constrained to treat any connection, relation or non-relation, as an entity. Thus it seems that Plato s requirement that Unity unify the entities of which it is a component is forced upon him. A mark of this unification is that parts (in the predication sense) of the entity apply to the whole entity. Green Itself, if a part of a composite, makes the whole composite green, just as unity makes the whole composite one. The oddity that the unification of natures in an individual includes Unity Itself as both unifier and component is precisely parallel to the fact being both the addition to being that distinguishes facts from aggregates of universal and particular. It is no more an oddity than the ontological status of facts as self-makers in Olson and Armstrong. B) Predication and Unity. Unity has some further special characteristics that show the nature of Platonic predication. Every entity has unity as a component. But consider a composite nature, such as being a Man. Each of the components will have unity as a part of it, given that the component natures are single entities. The nature Unity is present to each of the parts, and to each of the parts of the parts David Armstrong, in A World of States of Affairs, page 118, having cited Olsen, goes on to say that the non-mereological binding which makes facts out of universals and particulars is a non-relation. It is difficult to see how this differs from saying that the instantiation relation is a very special relation with a special property of being itself a component of the unity it brings about. 19 We need not think of property-instances but need only have the concept of a nature as what can be, as a whole, part of many distinct entities. (See the discussion of Sameness below.) Instances of unity in the sense of the Tallness in Phaedo are just Tallness Itself. Instances as diverse from natures are individuals such as Phaedo. Another Platonic notion of part (see below) makes instances arts of natures. 96

7 The picture of a composite entity, then, is at least a picture of unity being a part of the object at each level of parts. Implicit in this picture is the idea that the presence of unity in a part (the unity of a part) is not sufficient to unify the whole entity. On a modern mereological version of the part-whole relation, there would be no difference between an aggregate with a unified part and a whole unified entity. That is, since the modern mereological part-whole relation is transitive, if Unity were a part of a part, it would be a part of any whole of which the part was a part. If participation is a part-whole relation, the part-whole relation in question cannot be the normal modern mereology. Rather, the part-whole relation that is predication must be akin to the relation of sets to their members. That is, just as a set can have sets as members but need not have members of those sets as members, so an entity made of parts can have composite parts without having parts of those parts as parts. For some parts of parts of the composite, for instance the Unity part and the Being part, those parts will also be parts of the composite. The natures that an entity has are its proximal parts. A diagram would make this clearer: Suppose that entity E is composed of natures C and D. If we represent unification by parentheses, then E = (C, D). Then if C and D are both composite natures themselves, i.e. if C=(A,F) and D=(B,G), then A and B are not parts of E. The participation relation, the sameness of nature and the entity that has the nature, is this kind of part-whole relation. A spatio-temporal individual, then, is a unity of natures and a spatio-temporal region. 20 Unlike set-theory, of course, the existence-condition for a composite will be that is has Being and Unity. Among the other modern mereological truths that are not true of the Platonic part-whole relation that is predication 21 are that not every two entities have a sum. The treatment of spatially and temporally located objects must wait until we treat the entity Difference Itself and discuss chora, below. For the moment, we can say that such entities will have spatio-temporal regions as proximal components. The spatio-temporal regions are a domain of entities with the structure of the number-volume. 22 Because every region has a sub-volume of the number volume as proximal part, regions are natures that have as part of their natures all of their relations to other regions. That is, just as every number necessarily bears all its relations to other numbers, where necessarily means in virtue of its nature, so regions do as well. Note that an entity of which a region is a proximal part need not have relations to other regions or their occupying entities as its parts. So while John s location bears necessary relations to other locations, John does not, even though John s location is a proximal part of John. 20 The account of individuals will be somewhat analogous to a trope-theory, but with the special features that unity and existence, etc., are tropes. The important difference is that the Platonic account has only numerical identity, so that, really, individuals are coalitions of natures. Another key difference is that spatio-temporality itself is a nature whose sub-natures are individual regions. 21 Something close to the modern mereological part-whole relation will obtain for spatio-temporal entities, but the holding of this relation will be a special circumstance due to the mathematical structure of space and time. That is, regions, by their nature, are such that parts of parts of them are parts of them. The existence axiom for modern mereology, however, is not Platonic. Plato is not committed to the view that given any entities a and b there is an entity c that is their sum. As we will see, there is no shortage of entities on the Platonic view, but not all mereological sums exist. For an entity to exist, it must be one and a being, so the natures being and existence must be proximal parts of it. 22 Plato generates the numbers from the differentiation of Unity and Being in the Parmenides 143b- 144a10. Platonists can generate numbers from natures as they choose, appealing to further primitive numerical natures, if need be. 97

8 Not all imaginable compound natures exist as single natures, even though we can think of such compounds. 23 Plato recognizes that we can work out the consequences of mathematical hypotheses that are not real. Mathesis, the second division of the divided line, calculates the consequences of hypotheses that may not be true. Such hypotheses, which would be legitimate branches of mathematics, might be such possible domains as the nonnatural numbers. 24 Being distinguishes the composite natures that exist from the composites that are merely possible. 25 The merely possible configurations are proper objects of a kind of knowledge, which can be very detailed, even though the components lack being and unity. The selectivity of Unity, the fact that, generally speaking, there does not exist a composite nature just because the components exist, is crucial for the understanding of natural laws. Snow and cold are a single nature, but there is no nature snow and hot. Likewise, there may be an event-nature, ((A+B+t1)(C+t2)), but no other event-nature with (A+B+t1) as a component. In such a situation, A and B could not occur together without C occurring. Natural laws thus reflect the selectivity of Unity and Being, as we will see below. 2) Being The nature Being is coextensive with the nature Unity. The same feature of making a being out of components of which it is a member obtains for Beings as for Unity. The difficulty in treating being as a nature, and in supposing that entities in general exist in virtue of having being as a part, is that the other parts of a being would seem to be left out of existence. A Platonist needs to think of the nature Being as that part of everything in virtue of which things are real. If Being is a part of everything, then nothing lacks it, and every attempt to get to, for instance Unity Itself apart from being ends up with nothing. For a Platonist, however, this just means that, while unity and being are different natures, they are inseparable natures. 26 That Being Itself is a nature is a consequence of the Parmenidean principle, so, although Being Itself might be dispensed with, it is prima facie a nature. It might be wondered what makes Being Itself distinct from Unity Itself, since they seem to be co-extensive, and since each has the other as component. The answer is that Difference makes them distinct. That is, entities are distinct in virtue of difference, not by having some entity pertain to one that does not pertain to the other. 3) Difference Difference is the nature in virtue of which distinct things are different, the nature corresponding to non-identity. Difference is a component of other relations that entail distinctness of relata. Several issues arise in understanding how non-identity can be a nature, while the theory retains an ontology of beings only. 23 Platonists could differ on whether such natures are possible. Physical laws are argued below to be compound natures. Are they contingent? There is a reason that unity doesn t join primeness and four, and one could hold that the compounds there are have a similar necessity. Plato seems to think that the Good and the Beautiful provide such constraints. 24 Non-natural numbers, for instance, even though we have very detailed information about their properties, have no reality and so are objects of mathesis but not nous. To see how much we can know of alleged entities that may not exist, see John Kemeny s paper, Undecidable Problems of Elementary Number Theory, Math. Annalen, Bd. 135, 1958, pp My thanks to Scott Lehmann for bringing this article to my attention. 25 One Platonic theory of merely possible mathematical entities and merely possible natures would be a version of Armstrong s combinatorial theory. The components of merely possible natures are real, but there is no nature with those components. That is, Being and Unity are lacking to the aggregate, as it were, of those natures. Exactly the same account would apply to merely possible frogs. 26 Plato discusses existence as a nature in the Sophist 248ff., as well as in the Parmenides. 98

9 What sort of entity corresponds to the fact that A is different from B? The natural idea is to postulate an entity whose components are A, Difference, Unity (because it is one thing), Being, and B. For this entity to exist, Difference must be distinct from each of the other components. Thus, there is, for instance, the entity that is A being distinct from Difference. For this difference to exist, Difference must differentiate itself from A. Thus Difference is a component of differences, just as unity and being are components of unities and beings. What makes A different from Difference, that is, what makes that composite entity a difference, is one of its components. The composite nature Different from A can be spoken of as a part of Difference, following Plato s Sophist 27, but this sense of part is compatible with Difference being a single nature wherever it is, as we discuss below. 28 Difference differentiates itself from other entities because Difference is that in virtue of which all distinct entities are different. It is also the key to Plato s way out of the regress cited by Aristotle in Metaphysics B. 29 Thus every fact of difference is a composite entity of which Difference Itself is a part. Difference is a component of every non-reflexive n-ary relation among entities. The parts of the different referred to in the Sophist will include natures (is 5 feet longer than, is longer than, are separated by 5 feet) as well as the different from Fred. Difference has parts in the sense that there are entities (all particular relations of difference and all instances of difference) that contain difference as a part but are not identical with difference. In this sense, Being, Unity, and sameness all have infinities of parts distinct from them, even though they are one thing. Plato plays with these paradoxes generated by the part-whole ambiguities in his system in the hypotheses of the Parmenides. The plurality of instances of a nature with many instances can be termed parts of that nature. The nature is both the same as and different from its parts. Which brings us to Sameness. 4) Sameness The nature Sameness is the nature common to both numerical identity and qualitative identity. The fundamental idea is having a nature is common. Since natures are numerically identical whatever they are natures of, this means that different things are also the same. There are several cases to list: 1) Two frogs are the same, since what they are is froghood, but the are also different frogs. 2) Froghood itself and a given frog are the same, since what they both are is froghood, but they are different entities. 3) The froghood in a given frog is the same as froghood itself and in no way different. 4) Froghood itself is the same as froghood itself and in no way different. 5) Fred the frog is the same as Fred the frog and in no way different. Sameness, like love, is a relation that sometimes holds between an entity and itself and sometimes holds between distinct entities. Since natures are the fundamental entities, having a common nature and not having any diverse natures is being strictly identical. 27 Sophist 258c 28 Briefly, for the moment, since a given natures is a component of distinct entities, there is a sense in which the nature is divided. But that is compatible with it being numerically the same entity. The discussion of this point I take to be a topic of the Parmenides. 29 The regress (B 1001a30ff) is an argument that if being or unity were natures, then there would be no other entities. Suppose that being were a nature. Then another nature, say being a frog, would have to have being in order to be at all. But consider what is left when the being is abstracted from being a frog. That remnant, if real, is also a being. So, since nothing is a non-being, being is the only feature. Difference, though, as a nature that distinguishes itself, allows things to be distinct from being while yet being beings. 99

10 Sameness is not quite strict identity, but strict identity can be defined in terms of it. If A and B have a common part and no different parts, then A=B. There is strict identity between the froghood in Fred and Froghood itself, but, while Fred is the same as froghood, Fred is not strictly identical with froghood, since Fred is different from froghood, even though he is a frog. Platonism takes natures to be the only entities. So, given that qualitative identity is numerical identity of natures, the qualitative identity sense of is is as basic as numerical identity. If Fred and Joe are both frogs, Fred and Joe can be said to be the same, in the qualitative sameness sense. There is a nature they both are. The same relation can then obtain between the Form Itself and the instances. They both have a nature in common. The numerically identical things throughout are the nature and the instances, that is, the unities of that nature with other natures. Fred is the same as himself and Froghood is the same as itself. 30 Plato s account assimilates the is of predication to the is of identity. The fundamental fact about entities having natures is that the nature is what they are. If natures are fundamental, then the whole nature is present in any instance. If an entity has a nature, the whole entity has that nature. That nature is what the entity is. 31 So a green frog is green, not a part of green. 32 Only thus can Fred be wholly a frog, yet also wholly green. If natures are wholly present in instances, then what the instance is, is identical with the nature. The whole idea that a frog has the entire nature froghood pervading the frog requires that the frog be, in the relevant sense, identical with the nature. The nature is all there. But, since the instance is an instance of several natures, it must be identical with them all. But this means that identical is not strictly identical. Sameness is this non-strict identity, which allows an entity to be the same as distinct things (Sophist). To summarize: If A is the same as B, then all of A is B. If Joe is a frog, then the whole of Joe is a frog. The whole nature is present in all of Joe. Joe is the same as froghood. However, A may also be the same as C, even though C is different from B. Joe is also green, and greenness is distinct from froghood. So Joe is both the same as and different from froghood. Sameness is thus the basic nature in predication, and so is the nature in virtue of which predication-type parts and wholes are parts and wholes. If the unity with which a nature is the same is spatio-temporal, as we will see below, sameness is the participation of physical entities in the Forms. Sameness with difference is the predication part-whole relation. Since Joe is a frog, is green, is hoppy, and a variety of other non-identical things, Joe can be said to have those natures as parts. Self-predication, then, comes out automatically, given that predication is sameness of subject and predicate and improper parts are parts. The feature of the predication part-whole relation that it is set-to-member-like rather than mereological reflects the fact that 30 Why think that the whole nature is present in an instance, according to Plato? In Parmenides 131a-c the young Socrates proposes that natures are like the day, which is present as a whole to everyone. Parmenides replies that this is just like saying that a sail covers everyone, so that different parts of the sail cover different individuals. This patently non-equivalent replacement indicates to me that Plato is saying that the young Socrates hasn t thought the various senses of part, whole and sharing through very thoroughly, and that the first analogy is right. 31 Just as Aristotle regards the essence of an object as being what the whole object is, so Plato regards the nature of an object as what the whole object is. And just as an Aristotelian entity is identical with its essence, so an instance of a Platonic nature is identical with that nature. However, for Plato, an entity can be identical with several distinct natures. 32 Plato allows that there is a sense of part in which the various things that are green are parts of Greenness Itself. See the Parmenides 131 a-c. 100

11 predication is sameness. If A is the same as B, all of B is A. But if a mereological part of B is the same as C, not all of B need be the same as C. So, Fred s heart is pulsating, but Fred is not. Fred s heart is not the same as Fred s skin, even though both of them are through and through parts of Fred. 33 Is it just confusing to call this a part-whole relation? Plato argues that there are many different sense of part-whole (see below) and the distinct natures with which an entity is the same is one of them. It is a part-whole relation because the instance is identical with many distinct natures, which can be spoken of as parts. The things it is the same as can be spoken of as parts of it, because, although they are not identical with everything it is identical with, they are identical with it. So, in the special case where A is the same as B but not also different from B, sameness is strict identity. Every entity is strictly identical with itself. Sameness without difference is transitive, reflexive, and symmetric. 34 For composite entities, that is, while they are the same and different from their components, the unification of them all is the same as itself and not different from itself. So sameness without difference is identity. 35 5) Composite and Primitive Natures Sameness as identity raises another quite ancient difficulty with natures, which we are now in a position to state and resolve. Consider a mega-nature, Unity Itself. Unity Itself has Being as a component, but it is something other than Being. If it is a composite of Unity and Being, then how can it be an improper part of itself? It would seem to be both a proper and an improper part of itself. That is, if we try to diagram the nature Unity, we seem to get U + B, since Unity exists, and so has Being as a component. This gives the impression that there is a residue, what is left after we subtract the Being component from the whole, which residue is the Unity component. But, as the Parmenides points out, such a residue would have to be, and to be one itself. As the Parmenides illustrates, such a process would make Unity Itself infinitely complex. Furthermore, the infinity would not be productive, since in fact the process never arrives at a residue that is Unity apart from Being. The problem is a general one for any nature that is not entirely composed of other natures, that is, every primitive nature. Since being and unity are components of every entity and so every nature, it seems that the factor that makes the nature the nature it is must be what is left after being and unity are subtracted. This difficulty rests on the same misconception about natures as the Third Man objection. Such a residue itself must be an entity with Being 33 Aristotle s requirement that no being contains another as actual part reflects the combination of the realization that a predicate applies to the whole of something and his denial that unity is a feature. For Aristotle, what makes something one is having an essential feature. That feature is what makes the entire being exist. An actual part within an actual being would have its being constituted by two essences. 34 Since any feature A had that B lacked would be a difference between A and B, sameness without difference between A and B also entails that any feature of A is a feature of B. 35 There is a multiplication of complex entities entailed by treating sameness as an entity. Since Armstrong regards identity as a non-entity, he needs no states of affairs consisting of a thing s identity with itself. Its self-identity follows from itself. Platonists must have a complex consisting of unity, sameness, and the entity in question for each fact of self-identity. Given the rich population of relational entities already required, though, such entities do not increase the absolute numbers of entities infinity added to infinity is infinity. What is important is that the entities are no different from differences, so not a new kind of complex. 101

12 and Unity, and must therefore have a residue itself, it would seem. 36 Thus the famous regress ensues. The dilemma of the Third Man reflects a mistake about the nature of natures. Natures are that in virtue of which things that have them have that nature. That in virtue of which Unity is Unity is Unity; that in virtue of which it is a being is being. In general, for any primitive nature, that in virtue of which it is what it is is itself. Every primitive nature is what it is in virtue of itself. There is no component that adds the special something that makes it the nature it is. If there were, it would have to be a nature itself. So, there is no entity that is the result of subtracting Being Itself from the One Itself. Any primitive nature explains itself, and is itself in virtue of itself. This means that the primitive natures are improper parts of themselves, even though they have components that are distinct from them, and even though in a diagram explaining features, the nature appears to appear twice. Since a nature is what it is in virtue of itself, Unity has Unity as an improper part, not as a proper component. So, the infinite complexity that the first hypothesis of the Parmenides generates from Unity has exactly one unity. The entities that have unity are distinct, but the unity in them is not. Anything is one in virtue of Unity Itself and Unity Itself, whatever its structure, is one in virtue of itself. What is true of Unity is true of any primitive nature. If being Orange were a primitive nature, it would have being and unity, but there would be no residue that is the result of subtracting unity and being from Orange Itself. The whole nature makes Orange Itself Orange Itself. The important point is that primitive natures are not differentiated by their components. What differentiates distinct primitive natures is Difference Itself. Otherwise, as Aristotle pointed out, one could prove that Being was the only nature and that Unity was the only nature. 37 Each nature is itself in virtue of itself. 6) Spatio-Temporality One could suppose that the generation of entities that takes place in the Parmenides is a sketch of how Plato wants to generate the whole universe from Being, Unity, Difference, and Sameness. The Parmenides seems to claim that some features of spatio-temporal existence are implied by the consequences of there being one thing. It might seem that Plato thought that the generation of numbers 38 was sufficient for the generation of space and time. The problem with such a project is that, as Plato surely realized, many different entities could have the 36 The first hypotheses of the Parmenides can thus be seen as addressing the difficulty in the opening section. 37 Aristotle Metaphysics (B 1001a30ff) cited and discussed above. 38 The generation of numbers requires some parenthetical discussion. Plato shows that, given a Unity that has Being, and is Different from Being, and so forth, the natures of the natural numbers are all already present. He seems to regard it as undecided what the internal constitution of a composite number is. A Platonist, if anyone, is in a position to generate numbers of all sorts in the familiar ways. One would like a Platonic reply to the analog of Benacerraf s What Numbers Could Not Be, (Benacerraf, Paul, "What Numbers Could Not Be," in Readings in the Philosophy of Mathematics, edited by Benacerraf and Putnam, Cambridge UP, 1983) but since Armstrong and others modern property-theorists have essentially waved their hand at a derivation of mathematical necessities, we can follow suit. What is clear is that, for Plato, mathematical necessities are a species of natural necessity. He hopes, of course, that they are the underlying necessities. One can be a Platonist whatever one thinks about deriving physics from mathematics. 102

13 properties of the real number volume without being space and time. 39 The central dialogue on spatio-temporal existence is the Timaeus. The Timaeus supposes a craftsman confronting the chora or receptacle, an entity that is chaotic and unstructured, and structuring it by imposing Forms. The puzzle for me has always been what to make of chora. Chora, since it has being and is different from the Forms, already participates in Forms. The correct interpretation of chora is just like the correct interpretation of other primitive natures: Chora is spatio-temporality itself. The likely story, which cannot be literally correct, imagines it a nature apart from Unity, Being, and Difference that is structured by number. If we set aside the residue story, it is still true that, while chora must be, must have difference, and must be structured by the numbers, it is a nature in itself. But it is a primitive nature whose nature is due to the whole of itself, with its necessary components. Spatio-temporality is spatio-temporality in virtue of itself. There is more to two objects being five meters apart than just fiveness and difference, and that more that makes being five meters apart spatial separation rather than some other five-unit separation is spatiotemporality. Briefly chora is spatio-temporality. It is that nature which, structured by geometry and succession, is space and time rather than some other domain with the structure of the numbervolume. Spatio-temporality is not reducible to Difference, Unity and Being, however unfolded they are according to the Parmenides. Rather, there is something about spatio-temporality that makes it space and time. Given this nature, what is it for that this nature to be structured by the numbers? Corresponding to points and regions in the geometrical volume there are points and regions in the spatial volume. Likewise, corresponding to the points in the number line, there are temporal points. Spatial and temporal points bear relations that are spatial and temporal correlates of the numbers that are parts of them. Alternatively, and more realistically, the number-volume can be treated as a four-dimensional number volume, consisting of spatiotemporal points. Spatio-temporal points themselves are entities constituted by chora, unity, and a four-dimensional number-nature. That is, they are unifications of number-nature complexes and spatio-temporality. The population of the space-time volume is thus points and regions, related in the familiar ways. Spatio-temporal relations are relations among space-time points and regions, primarily. It is entirely compatible with pure realism that there be other basic natures, or that chora in fact turn out to be a general category for a number of distinct natures that constitute spatiotemporality. In addition, a pure realism has no difficulty with there being natures that determine fundamental properties of matter and energy. Just as Plato did not think that he was in a position to say how the world actually operates in many areas, so Platonism need not suppose either that there are or are not a limited number of further primitive natures. 7) Spatio-Temporal particulars Given space-time points and regions, the constitution of spatio-temporal particulars is straightforward. A minimal spatio-temporal particular is the unification of a space-time point and a number of other natures. A medium-sized object is the unification of a number of natures and regions. The regions themselves are similar unifications of sub-regions, down to minimal regions, possibly points or atomic minima. At each level, if there is an entity, there is 39 In the Sophist, the further natures Motion and Rest are available to be intrinsically indeterminate natures that the numbers make determinate. Motion and Rest could be read as spatio-temporality, which would then be conceived as the nature that, when combined with the geometry generated from the One, Being and Difference, yields spatio-temporal locations. Motion and rest thus might be another term for chora. 103