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The Razor Argument of Metaphysics A.9

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  • 1 Instituto de Investigaciones Filosóficas, Universidad Nacional Autónoma de México, Circuito Mario de la Cueva s/n, 04510, Ciudad de México, Mexico
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Abstract

I discuss Aristotle’s opening argument against Platonic Forms in Metaphysics A.9, ‘the Razor’, which criticizes the introduction of Forms on the basis of an analogy with a hypothetical case of counting things. I argue for a new interpretation of this argument, and show that it involves two interesting objections against the introduction of Forms as formal causes: one concerns the completeness and the other the adequacy of such an explanatory project.

Abstract

I discuss Aristotle’s opening argument against Platonic Forms in Metaphysics A.9, ‘the Razor’, which criticizes the introduction of Forms on the basis of an analogy with a hypothetical case of counting things. I argue for a new interpretation of this argument, and show that it involves two interesting objections against the introduction of Forms as formal causes: one concerns the completeness and the other the adequacy of such an explanatory project.

1 Introduction

Aristotle’s examination of Plato’s Forms or Ideas in Metaphysics A.9 opens up with a general criticism. Aristotle criticizes the introduction of Forms on the basis of an analogy with a hypothetical case of counting things (the ‘Countability analogy’). According to Aristotle, those who posit the Forms in seeking to explain ‘these things here’ posited ‘others equal in number to these, as if someone who wanted to count things thought that he would not be able because they were too few, so having added more things he would count’ (990a34-b4). The suggested conclusion is that, just as it seems unreasonable to introduce more things to count the original things, so it is similarly unreasonable to introduce Forms to explain ‘these things here’. This implicit recommendation of ontological parsimony in formal explanation is the reason why the argument has been labelled ‘Aristotle’s Razor’ (Frede 2012). The argument is puzzling. One puzzle relates to the identification of the entities Aristotle calls ‘these things here’. The most natural interpretation is that ‘these things here’ are sensible individual objects, what I call the ‘Individuals interpretation’ (Robin 1908; Ross 1924; Cherniss 1944; Frede 2012). The main problem with this is that it is implausible that the number of Forms is approximately equal to the number of individuals. By contrast, a different kind of interpretation, what I call the ‘Types interpretation’, holds that ‘these things here’ are types (sets or classes) of individuals (Alexander of Aphrodisias, In Metaph. A; Bonitz 1848; Apostle 1966; Annas 1976). This interpretation can, apparently, handle the equality claim, for it seems plausible that the number of Forms is approximately equal to the number of types (sets or classes) of sensible things that share a qualification. But, as the defenders of the Individuals interpretation argue, Aristotle cannot by ‘these things here’ be referring to types (sets or classes), for these are supposed to be abstract things and, hence, cannot be the object of demonstrative reference.

In this paper, I offer a new interpretation of the Razor, the ‘Pluralities interpretation’, according to which Aristotle means, by ‘these things here’, the diverse pluralities of sensible things that share a qualification: the many Fs, the many Gs, and so on. I defend this, first (Section 3 below), drawing on Plato’s own way of introducing Forms—in particular, in Republic 10 (596a5-b2), where he introduces Forms in relation to ‘the many things’ (τὰ πολλά), which, I argue, should be understood as the plurality of pluralities of sensible things that share a qualification. I argue then (Section 4 below) that the Pluralities interpretation provides a more satisfactory interpretation of the Razor. First, I argue that pluralities can be called ‘these things here’, for, as I show in Section 3, pluralities are not, for Plato, abstract objects (sets or classes) that exist over and above their members: the plurality of F things just is (or, better, are) the F things. Secondly, I show that this interpretation is the only available way of making sense of the equality claim, and that, actually, it receives direct support from Aristotle’s own explanation of this thesis (990b6-8). Furthermore, there is a less well discussed puzzle, namely, in what sense, if any, the counting case and the case of introducing Forms could be similar. I show in Section 5 below that the two cases are appropriately similar and that, on this basis, the Razor suggests two interesting objections against the introduction of Forms as formal causes: one concerns the completeness and the other the adequacy of such an explanatory project.

2 The Razor: The Text and the Debate

Aristotle begins his critique of Plato’s Forms in Metaphysics A.9 with the following argument (990a33-b8):1

[I] Περὶ µὲν οὖν τῶν Πυθαγορείων ἀφείσθω τὰ νῦν (ἱκανὸν γὰρ αὐτῶν ἅψασθαι τοσοῦτον)· οἱ δὲ τὰς ἰδέας τιθέµενοι πρῶτον µὲν ζητοῦντες τωνδὶ τῶν ὄντων λαβεῖν τὰς αἰτίας ἕτερα τούτοις ἴσα τὸν ἀριθµὸν ἐκόµισαν, [II] ὥσπερ εἴ τις ἀριθµῆσαι βουλόµενος ἐλαττόνων µὲν ὄντων οἴοιτο µὴ δυνήσεσθαι, πλείω δὲ ποιήσας ἀριθµοίη· [III] σχεδὸν γὰρ ἴσα (ἢ οὐκ ἐλάττω) τὰ εἴδη ἐστὶ τούτοις, περὶ ὧν ζητοῦντες τὰς αἰτίας ἐκ τούτων ἐπ’ ἐκεῖνα προῆλθον. καθ’ ἕκαστον γὰρ ὁµώνυµόν τι ἐστί· καὶ παρὰ τὰς οὐσίας τῶν τε ἄλλων ἐστὶν ἐπὶ πολλῶν ἕν, καὶ ἐπὶ τοῖσδε καὶ ἐπὶ τοῖς ἀϊδίοις.

[I] Then, let us leave now the Pythagoreans, for it is enough to have engaged with them this much. But, as for those who posit the Ideas, first, seeking to grasp the causes of these things here they introduced others equal in number to these, [II] as if someone who wanted to count things thought that he would not be able because they were too few, so having added more things he would count. [III] For the Forms are approximately equal to (or not less than) these things, from which, seeking their causes, they advanced to the Forms. For, in each case, there is something homonymous; also besides the substances and in the case of the other things there is a one over many, both over these things and over the eternal things.

The text divides neatly into three parts. Part I (990a34-990b2) introduces what I call Explanation and Equality:

1. Explanation: The Platonists introduce Forms as causes of these things here.2 [Assumption]

2. Equality: The Forms are equal in number to these things here. [Assumption]

Part II (990b2-4) introduces the Countability analogy:

3. Countability: The introduction of Forms is analogous to the case where someone who wants to count some things thinks that she is unable to do this because they are too few and so introduces more things in order to be able now to count. [Assumption]

The conclusion that is suggested by this analogy is as follows:

4. Conclusion: The introduction of Forms is an unreasonable multiplication of entities similar to the multiplication of things in order to count. [From 1-3]

Part III (990b4-8) comes back to both Explanation and Equality, adding a qualification to the latter: Forms are ‘approximately’ (σχεδόν) equal ‘or not less’ than these things, and introducing, as we will see later, an explanation of Equality. Parts II and III are reproduced almost identically in Μ.4 (1078b34-1079a4). The only important changes in Μ.4 are a revision of Equality and a more precise identification of ‘these things here’. We will return to them in a moment.

Explanation presents the Forms as causes or explanatory entities. But the Razor is very cryptic concerning the explananda: they are simply called ‘these things here’, something which has led to controversy. What kind of explanation was Plato pursuing, according to Aristotle, when he introduced Forms? In Metaphysics A.7 (988a34-b5), Aristotle describes the Platonists’ introduction of Forms as part of an explanatory formal project. He says that they introduced Forms as causes but, unlike most of their predecessors, they did not take them to be causes of change or material causes, but formal causes, causes of ‘the essence (τὸ τί ἦν εἶναι) of each of the other things’ (988b4-5).3 That is, what the Platonists wanted to explain is not why x changes, or what x is made of, but what makes x F, what makes x be qualified in that way. And their answer is that in order to respond to this question we need to postulate a Form. It is the Form of F-ness that makes any F thing F. For example, the Form of Beauty makes any beautiful thing beautiful. Each beautiful thing is beautiful in virtue of being related in some suitable way to the Form of Beauty, by ‘partaking’ of such Form.

Countability articulates the analogy that is at the basis of the argument. According to this, the introduction of Forms involves some sort of objectionable duplication or multiplication of entities. The analogy is puzzling, but it has not been, to my knowledge, sufficiently discussed. Frede even considered it a mere joke, for ‘the idea that counting gets easier when numbers are larger’ seems to her ‘ludicrous’ (2012, 352). So, it seems, Aristotle cannot be seriously comparing the introduction of Forms to that case of counting. However, as I will argue in Section 5 below, the two cases are sufficiently similar to make the analogy, and the Razor, appropriate and worthy of consideration.

Equality is the premise that has been most discussed. At first sight, it may seem like a gratuitous addition to the argument. However, we will see that this is not the case. Explanation, Countability and Equality are bound up with one another in the following way. In fulfilling his explanatory project, Plato introduced in each case a Form in correlation with the things he wanted to explain, ‘these things here’; so he ended up introducing Forms (approximately) equal in number to ‘these things here’; and this is why there is some sort of duplication or multiplication (which is objectionable, according to Countability): for each of ‘these things here’ Plato introduced one Form.

There are two main kinds of interpretation of the Razor: the Individuals interpretation and the Types interpretation. Both are mainly geared towards dealing with the puzzle of what are ‘these things here’ and with Explanation and Equality. According to the Individuals interpretation, ‘these things here’ are individual objects, or substances, like Socrates or the moon.4 Thus, according to this interpretation, the Razor would be saying that the Platonists advanced to the Forms in order to give a formal explanation of individual objects, that Forms are approximately equal in number to them and that their introduction involves some sort of objectionable multiplication of such entities. However, the Individuals interpretation does not deal satisfactorily with Explanation and Equality. In a sense, Plato does not want to explain individual objects as such, for he does not postulate a Form corresponding to each individual object. In fact, in his lost treatise Peri ideōn, Aristotle seems to have acknowledged this, since he directed an objection against a certain Platonic argument for the existence of Forms, the Object of Thought Argument, to the effect that it would have the unwanted consequence of introducing Forms of individuals like Socrates or Plato (Alexander, In Metaph. 82.1-6 Hayduck with Fine 1993, 126-8). This also makes the accommodation of Equality difficult, for it is hard to see how Forms could be approximately equal in number to individuals, if there are no Forms of individuals. As Frede 2012, 352 acknowledges, it seems more plausible to say that there are many more individuals than Forms:

While the number of individuals is indefinitely large, the type of unities (ἓν ἐπὶ πολλῶν) they participate in must be finite, even if there are Forms of all of their properties, so that one individual partakes of many Forms. Because the number of individuals is unlimited, not even a rough numerical equality between Forms and their participants would result.

Supporters of the Individuals interpretation are forced to mitigate the fact that Equality turns out to be false according to them. Frede dismisses Equality as a deliberate exaggeration (2012, 352). But, why would Aristotle deliberately exaggerate in this way? This can only make his critique less compelling. Even if the Razor were, at bottom, a joke, as Frede thinks, doing such a thing would make the joke seem less fair and, perhaps, less effective. Ross, for his part, agrees that Equality is not strictly true, but seems to think that it is at least close to the truth. According to Ross, ‘ἴσα is not to be taken very strictly. One Idea was common to many particulars; but, on the other hand, one particular shared in many Ideas, so that, speaking very roughly, Aristotle says that numbers are equal’ (1924, 191). However, Ross’s idea is flawed. The fact that you can establish a one-many correspondence in both directions between two groups of things tells you nothing about their relative cardinality, simply because you can do such a thing concerning groups that differ very widely in cardinality, such as persons and movies: one movie is seen by many persons and one person sees many movies. Again, Cherniss claims that ‘neither the ἴσα of A nor the πλείω of M can be taken quite strictly, since there could be no way of calculating the relative number of ideas and individual sensibles’ (1944, 200 n. 118). But, if Cherniss were right that we could not calculate the relative number of Forms and individuals, then the most reasonable thing to do would be to abstain from making any claim about it, rather than to say that they are equal or that there are more Forms. Thus, the Individuals interpretation does not seem to be able to deal adequately with Equality.

The Types interpretation purports to avoid the difficulties that the assumption that ‘these things here’ are individuals generates. Thus, Alexander (In Metaph. 77.3-6 Hayduck) says:

<σχεδὸν> δὲ εἶπεν <οὐκ ἐλάττω> εἶναι τὰ εἴδη <τούτοις,> οὐ τοῖς καθ’ ἕκαστα καὶ ἀτόµοις λέγων, ἀλλὰ τοῖς ἐπ’ αὐτοῖς εἴδεσιν. οὐδὲ γὰρ Σωκράτους καὶ Πλάτωνος ἐζήτουν τὰς αἰτίας, ἀλλ’ ἀνθρώπου καὶ ἵππου. τούτοις γὰρ σχεδὸν ἰσάριθµα τὰ εἴδη κατ’ αὐτούς.

In saying that the Forms are approximately, not less, than these things, he is not speaking about particulars and individuals, but about kinds over these. For, they did not seek the causes of Socrates or Plato, but of man and horse. For, according to them, the Forms are approximately equal in number to these.

Different authors have called the entities to which the Types interpretation appeals different things. Ross 1924, 191 and Cherniss 1944, 199 n. 118 call them ‘classes’ when they describe Alexander’s position; and that is how Dooley translates εἴδη (‘kinds’) in the passage just quoted (1989, 112). Frede 2012, 352 and Annas 1976, 155 call them ‘types’, and Robin 1908, 121 n. 150 calls them ‘universals’. I have preferred the term ‘types’ in labelling the interpretation, but nothing hangs on that. They may be taken to be universal types, classes or sets of things. What is crucial is that all authors agree that they are something abstract, non-sensible, over and above the sensible particulars. Accordingly, what the Razor would be saying is that the Platonists appealed to Forms in order to give a formal explanation of types (classes or sets) of sensible things, that the Forms are approximately equal in number to them and that their introduction involves some sort of objectionable multiplication of such entities. This interpretation seems prima facie better able to explain Explanation and Equality, since it seems reasonable to say that the Forms are introduced to explain types (classes or sets) of things and that, hence, they are approximately equal in number to them.

However, this interpretation has been dismissed on textual grounds. Ross, commenting on 990b2, says: ‘That τούτοις means individual things, not, as Alexander and Bonitz suppose, classes of things, is shown by τωνδὶ τῶν ὄντων’ (1924, 191). Likewise, Frede agrees that this ‘is indeed the natural reading of the text, because of the demonstrative article at the beginning’ (2012, 352). So, the idea is that it cannot be types (sets or classes) that are referred to demonstratively by the phrase ‘these things here’ (τωνδὶ τῶν ὄντων) or by ‘these’ (τούτοις), because types (sets or classes) are abstract entities and one cannot refer demonstratively to abstract entities.5 In addition, there is an even more troubling textual problem for the Types interpretation in the version of M.4. There, besides modifying Equality, Aristotle identifies the things in question as particular sensible things: ‘for the Forms are, so to say, more than the particular sensibles (τῶν καθ᾽ ἕκαστον αἰσθητῶν)’ (1078b36-1079a1).6 These textual difficulties are, I think, decisive.

Thus, neither the Individuals interpretation nor the Types interpretation seems satisfactory. Each appears to get something right and something wrong. This should lead us, I think, to look for a different interpretation.

3 Pluralities

When Aristotle calls the things in search of whose causes Plato advanced to the Forms ‘these things here’, he does not refer by this term to all the sensible individuals there are or to types (sets, classes) of sensible things. So, what are ‘these things here’? In order to answer this question, it may be helpful to look at Plato’s own way of introducing Forms in the middle dialogues. We will see that Plato does not introduce Forms in relation to types (sets, classes) or individuals, but in relation to pluralities of sensible things that share a qualification. Furthermore, we will consider in detail how Plato understands pluralities so as to be able to see in Section 4 below that an interpretation that takes ‘these things here’ to be pluralities can offer a more satisfactory account of the Razor.

Forms are generally introduced in the middle dialogues in relation to collections of sensible things, the many F things, the many G things and so on. Thus, when in Republic 5 (475e-476d, 479ab) Socrates introduces the Forms for the first time in the dialogue, he distinguishes, most conspicuously, the Form of Beauty, which is one, from ‘the many beautiful things’ (τὰ πολλὰ καλά) and, in general, he distinguishes pluralities qualified by opposite predicates, like ‘just’ / ‘unjust’, ‘good’ / ‘bad’, ‘big’ / ‘small’, from the corresponding Forms. Likewise, in the Phaedo (74a-c), Socrates introduces the Forms by distinguishing the Form of Equality from the equal things. And there are many other similar passages.7 But the most important one for our purposes is in Republic 10. There Socrates again introduces the Forms, asking Glaucon (596a5-b2):

Bούλει οὖν ἐνθένδε ἀρξώµεθα ἐπισκοποῦντες, ἐκ τῆς εἰωθυίας µεθόδου; εἶδος γάρ πού τι ἓν ἕκαστον εἰώθαµεν τίθεσθαι περὶ ἕκαστα τὰ πολλά, οἷς ταὐτὸν ὄνοµα ἐπιφέροµεν … Θῶµεν δὴ καὶ νῦν ὅτι βούλει τῶν πολλῶν. οἷον, εἰ ’θέλεις, πολλαί πού εἰσι κλῖναι καὶ τράπεζαι … Ἀλλὰ ἰδέαι γέ που περὶ ταῦτα τὰ σκεύη δύο, µία µὲν κλίνης, µία δὲ τραπέζης.

Do you want us then to begin inquiring at this point by our usual procedure? For we are used to positing, I think, each one Form in relation to each of the many things to which we give the same name … Now let’s take any of the many things you want. For example, if you wish, there are many beds and tables … But regarding these utensils there are, I think, two Ideas, one of bed and one of table.

Here, Plato’s way of introducing Forms in relation to ‘the many things’ is called the ‘usual procedure (µεθόδου)’ when they begin inquiring or investigating (ἐπισκοποῦντες). Such ‘usual procedure’ is what is called Plato’s ‘One Over Many’ argument or principle. There is a controversy about the translation and interpretation of this passage and, consequently, about how such argument is to be understood. However, the main points I want to make here do not depend on the resolution of that controversy, so, I will leave it aside for the moment.

It should be uncontroversial that in this passage Plato establishes a global correlation between Forms and the things with respect to which he posits Forms, designating the latter with a single abstract term: ‘the many things’ (τὰ πολλά). And given that this procedure is characterized as the usual procedure for beginning an inquiry, I think it is safe to say that here Plato establishes a global correlation between the two elements of his project of formal explanation: the explanans, the Forms, and the explananda, ‘the many things’. But, what exactly are ‘the many things’? I take it that ‘the many things’ does not designate a collection or plurality of sensible individuals, e.g. all the sensible individual objects there are, but a certain collection or plurality of pluralities of sensible things. That is, ‘the many things’ designates a higher-level plurality, a plurality that has pluralities as members. And Plato makes clear that he is not concerned with all the pluralities of sensible things, but just with those pluralities ‘to which we give the same name’. How to understand the term ‘the same name’ is part of the controversy concerning this text, but at the very least these pluralities will be those that appear in this and the other passages just mentioned. Thus, if we could give a list of the things that the term ‘the many things’ picks out, the list would not go: Socrates, the Parthenon, Mount Olympus,…; rather, it would go: the many beautiful things, the many just things, the many good things, and so on. The following confirms this interpretation. First, when Plato establishes a correlation between Forms and the many things he says: ‘we use to posit, I think, each one Form in relation to each of the many things (περὶ ἕκαστα τὰ πολλά)’ (596a6-7). The translation may seem to admit of a reading where ἕκαστα distributes over individual objects, so that an individual object may be one of the many things. But such a reading is unavailable: τὰ πολλά is not in the genitive, it is in the accusative, just as ἕκαστα, and it is likewise governed by περί. Hence, it distributes over ‘manys’, pluralities. So, a more literal translation would be: ‘in relation to each many’. That is, the members of the plurality designated by ‘the many things’ are manys, pluralities of sensible things to which we give the same name.

Secondly, when Socrates spells out what he means by the phrase ‘take any of the many things you want’ (596a8) (here τὰ πολλά is indeed in the genitive), he offers as examples not individuals (the Parthenon or Mount Olympus), but two different pluralities of things to which we give the same name: ‘there are many beds and tables.’ That is, among all the many things, Socrates chooses to take as examples the many beds and the many tables.8

Scholars have usually assumed that, when Plato talks about ‘the many things’ in Republic 10, he is talking about pluralities, groups or collections, plural things of some sort.9 However, the main points on which I will focus here have not been sufficiently considered: (a) what exactly a plurality could be for Plato; and (b) how any such global correlation between Forms and ‘the many things’ could be established given Plato’s view of pluralities. I will devote most of the remainder of this section to (a) and the next to (b), where I discuss how the Razor could be explained assuming that a global correlation between Forms and ‘the many things’ is involved in it. But before that it is necessary to say something about what some scholars have deemed controversial concerning this passage and the correlation established therein.

The controversy revolves around the following lines: εἶδος γάρ πού τι ἓν ἕκαστον εἰώθαµεν τίθεσθαι περὶ ἕκαστα τὰ πολλά, οἷς ταὐτὸν ὄνοµα ἐπιφέροµεν (596a6-7). The translation I have given tried to be as literal and neutral as possible: ‘For we are used to positing, I think, each one Form in relation to each of the many things to which we give the same name.’ But there are two aspects of it that are deemed controversial: (a) how to understand the expression ‘the same name’ and (b) how to understand the direction of the correlation between ‘each Form’ and ‘each of the many things’. On the traditional interpretation of this phrase, ‘the same name’ is understood as ‘a name common to all members of a plurality’ and the direction of the correlation is understood as going from each plurality to one Form. So, this interpretation takes the phrase as enunciating a principle to the effect that we should posit one Form for each plurality of things to which we apply a common name or predicate.10 Such a One Over Many argument has sometimes been considered too profligate, for some think it is not reasonable to posit Forms, conceived as universals, whenever we apply a predicate to a plurality of things (think of disjunctive and negative predicates, for example). For universals should be posited only to account for the real features or common nature of things. That is why Fine 1993, 112 has proposed a restricted version of this traditional interpretation, according to which the use of ‘name’ in the argument is technical: it does not mean any name, but just those names that denote real properties, ‘property-names’.11 However, an alternative construal of the passage was suggested by Smith 1917. He questioned the view that the passage contains the principle that one Form is posited corresponding to each plurality to which a common name is applied, and suggested that the term ‘the same name’ might be understood as ‘the same name as the Form’. According to him, the passage contains only a statement of the uniqueness of Forms. Smith’s specific reasons in support of his interpretation seem to be unsound, but Sedley 2013 has recently defended this interpretation with new and very subtle arguments.12 Sedley’s main argument is that translators and commentators have overlooked the significance of the double ‘each’ of our passage. On the basis of some examples of (allegedly) similar phrases with double ‘each’ in Plato, Sedley argues that the linguistic force of this device is such that ‘the function of the first “each” is to generalize, but that of the second “each” merely to correlate’ (126).13 So, he concludes that the same happens in our Republic 10 passage; only ἕκαστον generalizes, while ἕκαστα merely correlates. Accordingly, Sedley proposes the following translation: ‘It is, I take it, our custom to posit each Form as one single item in relation to a corresponding plurality’ (128). Thus he claims that we should infer from this that ‘Socrates is not asserting the principle of one Form per plurality, just the more modest principle of one plurality per Form’ (128). Sedley goes on to defend Smith’s other insight regarding the expression ‘the same name’—but we need not enter into the details of this argument (fortunately, we do not have to take sides here with any of the available interpretations).

My main concern here is with the application of the correlation between Forms and pluralities of sensible things to Aristotle’s Razor argument. Under any of the three interpretations on offer, the essential point about the correlation that I want to use to interpret the Razor holds. All three interpretations take the passage as establishing a correlation between Forms and the pluralities of sensible things that Plato wants to explain (the target pluralities), and so assume that there are, as far as this passage is concerned, as many Forms as there are target pluralities. This is the main point I will use in Section 4. The interpretations disagree on the scope of these target pluralities and on what this passage tells us about why Plato targets, specifically, these pluralities. The traditional interpretation holds that the target pluralities are all the pluralities of sensible things that have the same name, so that there are as many Forms as these. The restricted traditional interpretation holds that the target pluralities are all the pluralities of sensible things that have the same property-name so that there are as many Forms as these. And both claim that the disputed passage contains an argument for the existence of Forms: it is because these pluralities have this feature (that they have the same name or that they have the same property-name) that Plato introduces a corresponding Form. The alternative Smith / Sedley interpretation holds that the target pluralities are all the pluralities of sensible things that have the same name as the Forms, so that there are as many Forms as these pluralities. But it claims that the passage does not tell us which pluralities these are or why they are targeted: the answers to these questions should be sought in other passages that introduce Forms. On the other hand, Aristotle clearly thinks that Plato has a One Over Many argument for the existence of Forms: he explicitly says so in Metaphysics A.9, just a few lines after the Razor (990b8-9, 13-14). And it is very likely that he has in mind the most conspicuous passage of the middle dialogues where Plato appears to introduce such argument. Thus it is very plausible that Aristotle assumes either the traditional interpretation or the restricted traditional interpretation of the disputed passage.14 We should remember this when we examine the Pluralities interpretation of the Razor, which holds that Aristotle has in mind there the correlation mentioned in Rep. 10. (We need not assume that Plato himself intended the disputed passage in either of these ways.) So I am going to use from now on the term ‘the pluralities of things that share a qualification’ (or the higher-level version of it) to designate Plato’s τὰ πολλά. You can cash out this according to any of the three interpretations of τὰ πολλά discussed: the overall result, I take it, should not be affected.

Now let us turn to the question of what exactly a plurality could be for Plato, so that we are able to see later how the Pluralities interpretation of the Razor actually works.

There are, in the history of philosophy, two opposed conceptions of pluralities. As Russell pointed out, we can take a plurality as one or as many.15 To take a plurality as one is to commit ontologically to the existence of pluralities as single entities. That is, it is to accept the truth of the following claim of singular existential quantification:

(∃x), x is a plurality.

According to this view, plural terms like ‘the bachelors’ refer to a single entity, for the plurality of bachelors is one single entity besides the many bachelors. There are two main options for those who hold this view. One is to identify pluralities with abstract objects such as sets or classes. So, for example, the plurality of bachelors would be, in this alternative, the set or class of bachelors, a single abstract object different from its members, the individual bachelors.16 Another option is to identify pluralities with wholes or mereological sums. According to this, the plurality of bachelors would be a whole, a single individual non-abstract object composed of the individual bachelors, its parts.17

By contrast, to take a plurality as many is to eschew ontological commitment to pluralities as single entities. The plurality of bachelors, according to this view, just is (or, better, are) all the individual bachelors. That is, for this view, plural terms like ‘the bachelors’ do not refer, singularly, to an entity, but only, plurally, to all the many bachelors at once. For, there is not one thing that is the plurality of bachelors. Thus, the use of the singular term ‘plurality’, according to this view, should be taken as a convenient way of referring plurally to many things. That is, the singular quantification over a distinctive kind of entity, plurality, that the use of ‘plurality’ may apparently imply should be taken only as a syntactic abbreviation of a distinctive kind of quantification, plural quantification, over certain things, the things that are the ‘members’ of the plurality: (∃xx)Fxx. In the case of the example, this would be an instance of plural quantification over the individual bachelors.18

How does Plato understand pluralities? This issue does not seem to receive much explicit discussion among Plato scholars, who have focused mostly on the question of how he understands Forms and, to a lesser extent, on the question of how he conceives of particular sensible things. Yet, there seems to be a tendency to attribute to Plato the view that a plurality is a one, at least in the context of the discussion of the One Over Many principle. For some scholars tend to take such a principle as a correlation between a Form of F-ness and the set (or class) of F things, even if they translate πολλά as ‘plurality’ or as ‘many’. For example, Fine 1993 uses ‘set’ much more often than ‘plurality’ to account for the different versions of the Third Man argument that appear in Plato and in Alexander. And, sometimes, she shifts from one word to the other in the same paragraph, as if these designated the same thing (362-3 n. 40, my emphasis):

But he [Plato] might none the less believe that a plurality of things, all of which are F in the same way, are F in virtue of something that is not a member of the plurality. For if they were all F in virtue of one of the members of the set, say in virtue of F1, then they would not all be F in just the same way, since F1, but nothing else in the plurality, would be F in virtue of itself. Further, what grounds would there be for singling out just one member of a plurality … as the one thing in virtue of which they are all F? Alternatively, if each member of the set is F in virtue of another member of the set.

Perhaps you think that this is just a loose use of ‘set’—what Black 1971, 616 calls ‘the lowbrow’ conception of sets, which takes sets as collections (that is, as many)—so that Fine is not taking sets as abstract single entities over and above their members (‘the highbrow’ conception of sets, according to Black). But it seems to me that she is adopting the highbrow conception and, hence, the plurality-as-one view. For example, she says (351 n. 51):

Whereas P-TMA asks what many F things have in common, the Resemblance Regress begins by considering a single F thing. However, of course, it is a single F thing rather than the set of F things that is like a Form of F … or participates in it, so perhaps Plato assumes that a is a member of some suitable set.

It is only if you adopt the highbrow conception of sets that it makes sense to deny that, for Plato, the set of F things is like the Form of F, since the set of F things, according to this conception, is not, in general, itself F. For, when you take the lowbrow conception of sets, and take them as collections according to the plurality-as-many view, it is reasonable to say that the set (or plurality) of F things is like the Form of F and participates in it, for the set (or plurality) of F things just is, in this view, all the F things. Thus, I think Fine is implicitly assuming the plurality-as-one view, taking pluralities as sets according to the highbrow conception of sets.

And she is not alone in this.19 Here is another example. Discussing Plato’s Third Man Argument, F. Lewis says that the ‘conventional assumption that our initial plurality has at least two sensibles as members begins with Plato’. However, he adds (1991, 17 n. 9) that there seems—

… no reason why there should not be a one-member plurality from which we can generate a regress of forms along the lines of the TMA. Alex-PL itself, then, does not rule out the possibility that there exists one-member pluralities.

It is clear that Lewis is assuming here the plurality-as-one view that takes them to be sets or classes (according to the highbrow conception). For, it is only if you assume this view that it makes sense to talk about a one-member plurality. For a plurality is, in this view, a further abstract thing over and above its members, which can have many things, one thing, or even none, as members. One-member pluralities, according to this view, just are singleton sets or classes. By contrast, if you take the plurality-as-many view, it simply does not make any sense to say that there can be a plurality of one thing. A plurality, according to this view, is not one thing different from its members: the plurality of F things just are the F things. Thus, for this view, it is necessary that there are at least two F things for it to be true that there exists the plurality of F things. A one-member plurality would not, for this view, be a plurality at all: it would be just one F thing.20

However, I think it is a mistake to attribute the plurality-as-one view to Plato. To begin with, a plurality-as-many view seems to be the most natural interpretation of the Republic passage we have considered, for there Plato uses plural terms to designate pluralities—‘many’, ‘the many things’—never suggesting that, for example, the many beautiful things may be or compose in some relevant sense, one single individual thing.21 Plato does have a singular word to designate a plurality: πλῆθος. He uses the word most often non-philosophically to designate simply a multitude or great number of things: men, for example. But in the Parmenides, Plato uses it several times to designate plurality as opposed to the One. Yet, as we saw F. Lewis 1991 notes, Plato assumes that there can be a plurality only if there are at least two things. For example, he says: ‘being a plurality, they would partake of a number greater than the one’ (Πλῆθος δὲ ὂν ἀριθµοῦ πλείονος ἂν µετέχοι ἢ τοῦ ἑνός) (Parmenides 153a4-5).22 This assumption goes directly against the view that Plato takes pluralities to be sets or classes—for there are one-member sets or classes—though not necessarily against the view that pluralities are, at bottom, wholes. For Plato could assume that wholes need to have at least two parts. But there are other reasons against supposing that Plato held either version of the plurality as one view: either that pluralities are sets / classes, or that they are wholes.

With regard to the former, it is very doubtful that Plato countenanced sets / classes, conceived according to the highbrow conception, as singular abstract entities different from their members. Although the exact origins of the modern highbrow notion of set / class are difficult to pin down, it is generally agreed that it emerged only in nineteenth-century Europe, especially in Germany.23 Kanamori, for example, says that the empty set, the singleton and the ordered pair ‘are the simplest building blocks in the abstract, generative conception of sets’ (2003, 273). That is, these notions are the simplest necessary components of the modern highbrow conception of sets / classes as single abstract entities. And he argues that the three notions first appeared in the nineteenth century. He cites, for instance, Boole as someone who was already conscious of a shift from a traditional view of collections as many to a modern conception of collections as abstract classes, when he said: ‘By a class is usually meant a collection of individuals … but in this work the meaning of the term will be extended so as to include the case in which but a single individual exists … as well as the cases denoted by the term “nothing”’ (1854, 28).24 So, it is more plausible to assume that Plato, and other ancient writers, held the more traditional view of collections or pluralities-as-many, against which modern authors were working. Furthermore, to suppose that Plato assumed that besides the many sensible F things there is one abstract thing such as the set or class of the F things would go against the theoretical role that Plato intends Forms to perform. For, in that case, there would be already a non-sensible one over the many sensible F things, the set of F things—which may cast doubts on the need for a second non-sensible One over the many, the Form of F. Therefore, I think that it is very doubtful that sets or classes, as envisaged by the highbrow conception, are within the inventory of Plato’s ontology. Thus, Fine, Lewis and other scholars who interpret Plato’s pluralities of sensible F things as sets, or classes, seem to be anachronistically attributing a modern concept to an ancient author. But, there is no need to introduce such modern notion even for exegetical purposes. All the illumination that introducing set theory to the discussion of Plato can bring can equally be had talking about pluralities (as many), that is, using plural quantifiers to talk about sensible things (and Forms).25

But Plato certainly countenanced wholes, so it may perhaps look more plausible to suppose that Plato held the plurality-as-one view, which takes a plurality to be one individual, a whole composed of many things, its parts. Supposing so would have the consequence of committing Plato to a certain response to what Van Inwagen calls ‘the Special Composition Question’: ‘When is it true that ∃y the xs compose y? … Less formally, in what circumstances do things add up to or compose something? When does unity arise out of plurality?’ (1990, 30-1). Under the present supposition, Plato would respond: ‘Always.’ For whenever there were many things there would be a one, a whole, which they compose. That is, Plato would be committed to the thesis of Unrestricted Composition. This thesis seems prima facie like a substantive ontological commitment.26 There are two main ways of adopting this version of the plurality-as-one view and dealing with such commitment. One is to deny, like D. Lewis 1986 and 1991, that there is any further ontological commitment when, having accepted the existence of many things, we accept the existence of one thing, the whole they compose. For, according to Lewis, composition, the part-whole relation, or, in general, the many-one relation, is just like identity: the whole is its many parts and the many parts are the whole (1991, 81-2). This thesis, known as Composition as Identity, is controversial, although it has several contemporary defenders.27 It is motivated by intuitions derived from cases like the one introduced by Baxter 1998, 579: if a farmer has a field that consists of six plots it seems that the field, as one thing, is nothing but the six plots. The farmer cannot sell the six plots and keep the field.28 Several other philosophers are not persuaded, for they balk at the idea that one thing, the whole, is identical to many things.29 They want to preserve the distinction between one and many, so that the relation of identity can only be one-one or many-many. Thus, the other alternative for those who share this conviction and want to hold this version of the plurality-as-one view is to admit that composition is not ontologically innocent, that the whole is really a different entity over and above its parts. A problem with this view is that the resultant notion of whole, of an individual composed of parts, would perhaps be too weak, since there would be too many wholes, too many individuals. For example, this view would have difficulties in handling the case of the field that consists of six plots. Given that there are no restrictions for the existence of a whole beyond the existence of many things, it becomes hard to say what the farm could be over and above the six plots.

So this version of the plurality-as-one view involves accepting either of two controversial claims about composition. It would be nice if we did not have to saddle Plato with either of these. And in fact the evidence seems to go in the other direction. There is a relatively recent study of Plato’s views on wholes and parts that claims that Plato rejected both Composition as Identity and Unrestricted Composition. Harte 2002 forcefully argues that, in the Theaetetus and Parmenides,30 Plato examines the thesis of Composition as Identity in order to highlight the problems to which it leads and flag it as an understanding of composition that ought to be rejected. According to Harte (2002, 50), these negative considerations form the backdrop against which he stages his own account of composition in Parmenides, Sophist, Philebus and Timaeus.31 In this account, as reconstructed by Harte, composition is restricted: ‘it is ontologically committed or creative; and it centrally involves the existence of certain structural relations between the parts of a whole’ (268). The upshot is that wholes turn out to be ‘contentful structures’ in which structural relations are essential both to the identity of the whole and the identity of its parts: ‘the identity of the parts is determined only in the context of the whole they compose’ (268-9). It is beyond the scope of this work to defend Harte’s interpretation, but that it is a cogent and available option should give us pause in attributing too lightly to Plato a conception of plurality that blocks it from the outset. Thus, I think it is preferable, for this reason, to assume that Plato did not accept the version of the plurality-as-one view that takes pluralities to be wholes.32

But you might think that, since the evidence on which Harte’s interpretation is based comes mostly from late dialogues, Plato might still have held this version of the plurality-as-one view in the middle dialogues. However, there is evidence that Plato rejected both versions of the plurality-as-one view, and so took pluralities as many, in a dialogue usually considered earlier than the Phaedo or Republic: Hippias Major. In 300b-303d, a passage to which Scaltsas 2016 draws attention, Socrates argues against Hippias’ view that, in Scaltsas’ words, ‘the many are f if and only if each of the many is f’ (2016, 5). Socrates argues that there are cases like that, but that there are others where a predicate is predicated of many things but not of each of the many and, likewise, cases where a predicate is predicated of each of the many things but not of the many. Socrates defends this concerning predicates like ‘fine’ and ‘one’, but it is the latter which is of interest in this context. Socrates says to Hippias (301d6-e3):

ὥστε δόξαν εἴχοµεν περὶ ἐµοῦ τε καὶ σοῦ ὡς ἑκάτερος ἡµῶν εἷς ἐστι, τοῦτο δὲ ὃ ἑκάτερος ἡµῶν εἴη οὐκ ἄρα εἶµεν ἀµφότεροι—οὐ γὰρ εἷς ἐσµεν, ἀλλὰ δύο—οὕτως εὐηθικῶς εἴχοµεν· νῦν δὲ παρὰ σοῦ ἤδη ἀνεδιδάχθηµεν ὅτι εἰ µὲν δύο ἀµφότεροί ἐσµεν, δύο καὶ ἑκάτερον ἡµῶν ἀνάγκη εἶναι, εἰ δὲ εἷς ἑκάτερος, ἕνα καὶ ἀµφοτέρους ἀνάγκη.

And so we had an opinion about me and you that each of us is one, but that this which each of us would be we would not both be, for we are not one, but two—this we foolishly thought. But now, we have been taught by you that, if we both are two, each of us must be also two, and if each is one, we both must be also one.

And Socrates concludes (302b1-3):

Οὐκ ἄρα πᾶσα ἀνάγκη, ὡς νυνδὴ ἔλεγες, ἃ ἂν ἀµφότεροι καὶ ἑκάτερον, καὶ ἃ ἂν ἑκάτερος καὶ ἀµφοτέρους εἶναι.

Then it is not entirely necessary … that whatever we both are each also is, and that whatever each is we both also are.

What is interesting here is that Plato’s treatment of the predicate ‘one’ is suggestive of an objection against views that take a plurality as one. According to this objection, if we really took a plurality as one, it should then be possible to count a given plurality as one, but it is not possible. As Moltmann 2016, 107-8 argues:

Why can’t number-related predicates count the entire plurality as ‘one.’ That is, why is (32a) impossible, as opposed to (32b):

(32) a. John counted the ten children—he counted one.

b. John counted Mary—he counted one.

The problem is a serious one for both approaches of Reference to a Plurality, which both treat pluralities as single entities. The solution, in my view, can only be to conceive of pluralities not as single entities, but as ‘multitudes’, that is, as pluralities ‘as many’.

I think that, similarly, we should take Socrates’s insistence that we cannot say that the relevant many individuals, Socrates and Hippias, are one as evidence that Plato is embracing here the plurality-as-many view. Thus, I will assume from now on that Plato held this view and consequently that he thought that all the many things that share a qualification do not compose one thing.33

4 The Pluralities Interpretation

In the previous section we saw that Plato posits Forms in relation to ‘the many things’ (τὰ πολλά), and that ‘the many things’ are not all the individuals there are, but the plurality of pluralities of sensible things that share a qualification. That is, a list of ‘the many things’ would not go: Alcibiades, the Moon, Bucephalus … Rather, it would go: the many good things, the many big things, the many beautiful things, and so on. We also saw that pluralities are not single entities, such as sets, classes or wholes. The plurality of F things just is (or better are) the F things. I suggest that we should hold that the same global correlation between Forms and ‘the many things’ is at play in the Razor argument. That is, that ‘the many things’, the many pluralities of sensible things that share a qualification, are the things that Aristotle calls ‘these things here’ in the Razor, and in relation to which, he says, Plato posited Forms. The first reason for holding this Pluralities interpretation is ‘external’: it is that Aristotle knew Plato well, so he knew that he posited Forms in relation to ‘the many things’. Thus, it is likely that when he describes Plato’s view in the Razor he has in mind this global correlation between Forms and ‘the many things’. The second reason is, crucially, ‘internal’: there is in the Razor textual evidence that Aristotle holds the Pluralities interpretation, as can be gathered from his own explanation of Equality. And the third reason is that the Pluralities interpretation gives, as we will see, a solution to the problems that affect both the Types and the Individuals interpretations and gives, in general, a better interpretation of the Razor.

As we saw, the Types interpretation holds that ‘these things here’ are types, sets or classes, of sensible things. This hypothesis apparently allows their defenders to make sense of Explanation (the fact that ‘these things here’ are what Plato wants to explain), and Equality (the fact that they are roughly equal in number to the Forms). However, this view has decisive textual disadvantages. Types (sets or classes) are abstract objects that cannot be referred to demonstratively as ‘these things here’, and neither can they be called ‘sensible particular things’, as in the Μ.4 version of the Razor. However, ‘the many things’, the pluralities of sensible things that share a qualification are not sets, types or classes, so they are not abstract things. Hence, they are apt to be referred to with demonstrative terms such as ‘these things here’. In fact, Plato does just this. For example, in the Phaedo, Socrates says: ‘then, these (ταῦτα) equal things and the Equal itself are not the same’ (74c4). And they can, of course, be called ‘sensible particulars’, as in Μ.4. For, as we saw, the plurality of sensible particular F things just is, for Plato, the sensible particular F things; it is nothing over and above the F things.34 The Individuals interpretation, that ‘these things here’ are individuals (Socrates, Bucephalus, the Moon etc.), did not have the textual disadvantages of the Types interpretation, but it faced the difficulties of not being able to account for Equality and Explanation. The Pluralities interpretation does not face these difficulties. Concerning the latter, as Alexander says, Plato did not want to explain Socrates or Bucephalus; he did not want to explain just single things. But, as we have seen, Plato introduces Forms to explain pluralities: the many just things, the many beautiful things, the many equal things, and so on. This is what Socrates in Republic 10 calls the ‘usual method’ when inquiring about things. And there are passages like Phaedo 100b-101e where Socrates clearly characterizes the relation between a Form of F-ness and the many F things as explanatory: ‘it is safe for me or for anyone else to answer that it is through the Beautiful that beautiful things are beautiful’ (100e1-3).

We will see in a moment how the Pluralities interpretation explains Equality. Before we do this, it may be useful to deal with a very natural objection. You might think that if the plurality of sensible F things is nothing over and above the many sensible F things, and if these are just sensible individuals, then the Pluralities Interpretation does not seem to give a distinctive ontological solution to the Razor. If so, it collapses into the Individuals interpretation and does not offer an alternative to it. The following example may help to see the force of the objection. Suppose there were just seven sensible individuals: a, b, c, d, e, f and g. The Individuals interpretation would hold that ‘these things here’ are: a, b, c, d, e, f, g. The Pluralities Interpretation, on the contrary, would deny this and say that ‘these things here’ are pluralities, say, the plurality of a, b and c, the plurality of c, d and e, the plurality of a, g and b, the plurality of b, f and g—and so on. But then the worry is: if pluralities are nothing over and above individuals, then if the elements out of which the various latter pluralities are constructed, that is, the ur-elements of the plurality of these pluralities, are listed without repetition, do we not end up with a list of all the individuals there are (a, b, c, d, e, f, g)? If so, there would be no ontological difference between these interpretations and, a fortiori, no difference at all.

However, even if we assumed that Plato’s sensible F things are individuals, there would be a crucial difference between the two interpretations. There is, in general, an expressive and explanatory difference between theories which appeal to pluralities—that is, which use distinctive plural quantifiers over certain objects—and theories which appeal just to these objects—that is, which use singular quantifiers over them, even if the objects they quantify over are the same. For, such theories differ structurally in the way they organize the same ontological base. Take the case of the famous sentence: ‘There are some critics who admire only one another.’ It is a well-known fact that this sentence cannot be formalized as talking about individuals in singular terms, that is, by using singular quantifiers over individuals.35 However, it can be captured using plural talk about individuals. Informally, what the sentence says is that there is a plurality of things, critics, such that no member of this plurality admires herself or anyone outside of this plurality. This can be formalized using plural quantifiers over individuals:

xx {Cxx & ∀yz [(y is one of the xx & Ayz) → (yz & z is one of the xx)]}.

So, likewise, even if it were true that there is, at bottom, no ontological difference between the Individuals interpretation and the Pluralities interpretation, there still would be a difference in the expressive capacities of each interpretation, and hence, in their respective explanatory powers. For the Pluralities interpretation can deal more adequately with Explanation, as we have seen, and with Equality, as we will see in a moment.36

Yet, it is far from uncontroversial that the ur-elements of the plurality of pluralities of sensible things that share a qualification, τὰ πολλά, amount for Plato to the plurality of all particular individuals. Gosling 1960 influentially argued that the contrast between the Form of Beauty and the many beautiful things (τὰ πολλὰ καλά) that Socrates introduces in Republic 5 (475e-476d, 479ab), and similar contrasts in other passages, was not a contrast between the Form of Beauty as a universal and particular sensible beautiful individuals. According to Gosling, τὰ πολλὰ καλά are rather sensible low-level universal properties, such as a certain colour or a certain shape. For Socrates distinguishes such things from the non-sensible Form of Beauty in that the former are beautiful and not beautiful, in the sense that in some cases they seem to explain why a particular individual is beautiful, while in other cases they seem to explain why some other particular individual is ugly. Thus, such properties are inadequate candidates for defining what it is to be beautiful. These things are particular only in the sense that they are particular types, particular ways of being beautiful; but they are universal because they are repeatable: they have instances, some of which are beautiful, while some others are not. Gosling may be right about some passages, but there are others where some of the many sensible F things should be construed as particular individuals, as when Socrates talks about τὰ πολλὰ καλά in the Phaedo (78d10-e4), for he gives as examples of these things ‘men, horses, clothes’.37 Or, even more clearly, in Rep. 10 some of the pluralities that are members of τὰ πολλά are pluralities of sensible individuals, such as the many beds and the many tables. Thus it is perhaps more plausible to hold that the ur-elements of τὰ πολλά include both sensible universal properties and particular sensible individuals.38 However, this is just one possible version of the Pluralities interpretation. A second version holds that the ur-elements of τὰ πολλά are only sensible universal properties.39 A third version holds that they are only sensible individuals.40 Since the explanation of Equality, and of the Razor in general, requires only a generic application of the Pluralities interpretation, I will continue to proceed in this fashion—although everything I will say can be accommodated, I hope, under any of these three possible versions.41

As we have seen, Aristotle claims that the Forms are ‘approximately’ (σχεδόν) equal in number to ‘these things here’, the things Plato wanted to explain by the introduction of Forms. This is the thesis I call Equality. The Individuals interpretation and the Types interpretation fail to explain Equality because both attempt to provide an ontological solution to the Razor, the puzzle of what ‘these things here’ are. By this I mean that both attempt to locate a certain domain of entities and identify them as ‘these things here’. Consequently, Equality turns out to be a matter of there being (roughly) a one-to-one correspondence between things in that domain and things in the domain of Forms. But such a procedure is doomed to failure, since it is not possible to establish a one-to-one correspondence between Forms and things that are within the inventory of Plato’s ontology. The Individuals interpretation characterizes the relevant domain as all the individuals there are, which makes Equality indefensible, for there is not even a rough one-to-one correspondence between Forms and individuals. By contrast, the Types interpretation identifies the relevant domain as certain abstract entities, types, sets or classes. However, it is very doubtful that types, sets or classes, conceived as single abstract entities over and above the many sensible things, are within the inventory of Plato’s ontology. So, Equality cannot be explained either way.

But the Pluralities interpretation does not attempt to offer an ontological solution to the Razor, since it claims that ‘these things here’ are pluralities, and the plurality of F things is nothing over and above the Fs. However, for this reason, and even before considering how this interpretation purports to explain Equality, you may think that there is an insurmountable difficulty here. Equality establishes a numerical comparison, so it seems to imply that we can count ‘these things here’. But counting implies ontological commitment: existential quantification over the things you count. Since the Pluralities interpretation takes ‘these things here’ as pluralities, but pluralities are not single entities, how can we count pluralities and say that there are roughly as many Forms as pluralities of sensible things that share a qualification?

The answer is that, although we cannot count pluralities as single entities (ontologically), we can count pluralities in some sense, plurally, because we can count sensible things plurally; and counting sensible things plurally only involves plural existential quantification over these things. Saying that there are n pluralities, n many things, is just an abbreviated way of putting a claim of plural existential quantification. You can say, for example, ‘I am thinking about three things, the records, the books and the magazines I lost in the fire’. This seems to involve only plural existential quantification over individuals:

  • There are the xx such that the xx are records and got lost in the fire.

  • There are the yy such that the yy are books and got lost in the fire.

  • There are the zz such that the zz are magazines and got lost in the fire.

  • And I am thinking about the xx, the yy and the zz.

Thus, all that is involved in this apparent singular quantification, that there are three pluralities I am thinking about, is these various instances of plural quantification over individuals.42 But, if we can count pluralities in this attenuated sense we can also make comparative numerical claims about them.

So the fact that the Pluralities interpretation does not give an ontological solution to the Razor is no objection at all. On the contrary it is an advantage. By avoiding an ontological solution, the Pluralities interpretation avoids taking Equality as a matter of establishing a one-to-one correspondence between the Forms and a certain domain of objects. Instead, it takes Plato’s One Over Many principle literally as involving a one-many correspondence, between one Form and many sensible things. However, this one-many correspondence validates Equality because the correspondence is in each case between a Form and a plurality of sensible things that share a qualification and so are homonymous to the Form—namely the plurality that each Form is posited to explain. That is, it is a correspondence between the Form of Justice and the many just things, the Form of Beauty and the many beautiful things, the Form of Bed and the many beds, and so on. Therefore, the claim that Forms are approximately equal in number to ‘these things here’ amounts to the claim that there is in each case a Form corresponding to a plurality of sensible things that share a qualification, and this is just an abbreviated way of saying the longer plural quantified claim:

  • There are the xx such that the xx are just and to the xx there corresponds the Form of Justice.

  • There are the yy such that the yy are large and to the yy there corresponds the Form of Largeness. (And so on.)

And such a procedure makes sense of Equality because there would be one Form corresponding to a many in all the relevant cases.43

So far, my case for the Pluralities interpretation has been based on external evidence, Plato’s dialogues, and the fact that it is able to deal with Explanation and Equality. Now we will see that this interpretation receives direct internal confirmation from a part of the Razor that, although often discussed, has not been properly identified as the place where Aristotle actually justifies Equality (990b4-8):44

Σχεδὸν γὰρ ἴσα (ἢ οὐκ ἐλάττω) τὰ εἴδη ἐστὶ τούτοις, περὶ ὧν ζητοῦντες τὰς αἰτίας ἐκ τούτων ἐπ’ ἐκεῖνα προῆλθον. καθ’ ἕκαστον γὰρ ὁµώνυµόν τι ἐστί· καὶ παρὰ τὰς οὐσίας τῶν τε ἄλλων ἐστὶν ἐπὶ πολλῶν ἕν, καὶ ἐπὶ τοῖσδε καὶ ἐπὶ τοῖς ἀϊδίοις.

For the Forms are approximately equal to (or not less than) these things, from which, seeking their causes, they advanced to the Forms. For, in each case, there is something homonymous;45 also, besides the substances and in the case of the other things there is a one over many, both over these things and over the eternal things.46

As we saw, this can be considered a third part to the Razor, following Part I, where Aristotle states Explanation and Equality (990a34-990b2), and Part II, where he formulates Countability (990b2-4). This third part can also be divided in three sections. The first one (III.1) is: ‘for the Forms are approximately equal to (or not less than) these things, from which, seeking their causes, they advanced to the Forms’ (990b4-6). Here Aristotle repeats Explanation and Equality, adding a certain qualification to the latter (to which we will return later). Evidently, the repetition is needed to fulfil some purpose. Given the γάρ, ‘for’, I think that the purpose of this section is explanatory. Having just stated the main objection, Countability, according to which the introduction of Forms involves some sort of unreasonable multiplication of entities, Aristotle, as I suggested in Section 2 above, justifies this claim by showing how it derives from Plato’s own explanatory project. In Aristotle’s conception of this project, Explanation and Equality are inextricably joined; that is why, when they appear in the argument, twice, they do so together. So according to III.1, in pursuing his explanatory project Plato introduced in each case a Form in correlation to the things he wanted to explain, ‘these things’; so he ended up introducing Forms approximately equal in number to ‘these things’—and this is why there is some sort of duplication (which is objectionable according to Countability): for each of ‘these things’, Plato introduced one Form.

In the remainder of Part III, Aristotle seems to delve deeper into this issue, also telling us why Equality holds.

Just after III.1, Aristotle introduces a further explanation. III.2: ‘for, in each case, there is something homonymous’ (990b6-7). Again, the γάρ suggests that this is an explanation, this time of what precedes it, Equality. What is ‘something homonymous’? It is a Form: there is a homonymous Form in each case. Joining III.1 and III.2 together, what Aristotle is saying is that the Forms are approximately equal in number to ‘these things here’ because there is a homonymous Form in each case. But what does ‘in each case’—or alternatively, ‘with respect to each thing’ (καθ’ ἕκαστον)—mean here, and to what is the Form homonymous? A third section, III.3, provides an answer to these questions: ‘also, besides the substances and in the case of the other things there is a one over many’ (990b7-8). That this is a third section, separate from III.2, is something for which Primavesi 2012a argues in detail. Following Alexander’s interpretation (In Metaph. 77.20-2 Hayduck) that the phrase ‘the other things’ (τῶν τε ἄλλων) means the non-substantial categorical entities, Primavesi argues that the καί in 990b7 is ‘responsive (“also”)’ not copulative, as Ross reads it (2012a, 422). His first reason for this is that ‘παρὰ τὰς οὐσίας must be separated from ὁµώνυµόν τι ἐστί and go with τῶν τε ἄλλων, since οὐσίας does not refer to first ousiai … but to substantial things as opposed to non-substantial ones; thus παρὰ τὰς οὐσίας prepares, and makes intelligible, τῶν τε ἄλλων’ (loc. cit.). The second reason is that ‘ἐστὶν ἐπὶ πολλῶν ἕν holds for substantial things as well as for non-substantial ones’ (ibid.), so that this phrase can only be said of substantial things if the καί is not copulative bur responsive. I agree fully with these reasons. If this is correct, then III.2, ‘in each case there is something homonymous’, is further clarified by saying that both in the case of substances and in the case of non-substances there is a one-over-many. This cannot but mean that there is a one, a Form of F-ness, corresponding to many F things, for each substantial qualification and for each non-substantial categorical qualification (quality, quantity etc.).47 So joining the three sections (III.1-3) together, Aristotle is saying explicitly that Equality holds because in each case there is one homonymous Form corresponding to many things that share a qualification, both in the case of substantial and of non-substantial categorical qualifications.

Thus Aristotle’s own explanation of Equality confirms the Pluralities interpretation. For Aristotle makes clear here that the basis of his claim of numerical comparison between Forms and ‘these things here’ is that in all the relevant cases there is a One corresponding to many individuals. That is, Aristotle appeals to Plato’s ‘usual procedure’, the One Over Many argument. It is because of this One-Many global correspondence that Equality is true. Hence, it seems reasonable to conclude that, when Aristotle is comparing Forms and ‘these things here’ numerically, he is numerically comparing ‘Ones’, Forms, and ‘Manys’, pluralities. So Equality is the claim that there are roughly as many Forms as pluralities of sensible things that share a (substantial or non-substantial categorical) qualification. Therefore, the thesis that ‘these things here’ are Plato’s ‘many things’ (τὰ πολλά) of Republic 10, that is, pluralities, is based also on ‘internal’ evidence.48

In Section 5 below, we will address the other main puzzle that the Razor presents: Why is the Countability analogy an objection to Plato? But before we do that, there are two main things left to explain about Equality. The first is: why does Aristotle qualify it saying that ‘the Forms are approximately equal to (or not less than) these things’ (990b4-5)? Aristotle seems unsure about what to say. He seems confident that there are at least as many Forms as pluralities of sensible things that share a qualification, but hints that the former may be even more. In any case, he does not appear to think that the consideration he has in mind here affects much Equality, hence the qualification ‘approximately’ or ‘almost’ (σχεδόν). The second puzzle concerns the relation between the two versions of the Razor in A.9 and M.4: why does Aristotle reject Equality in M.4 and says directly that there are, so to say, more Forms than particular sensibles?

Commentators have focused more on the second puzzle, for the differences between A.9 and M.4 are well known. In several places in A.9 Aristotle uses the first person plural pronoun, ‘we’, when describing Platonic theses, while in M.4 he changes to the third person plural, ‘they’. Jaeger 1912 famously explains these differences by the hypothesis that A is an early work, where Aristotle still considers himself a Platonist, while M is a late work, where he has already abandoned Platonism. Thus, for example, Ross, following Jaeger, says concerning the two versions of the Razor: ‘In M Aristotle no longer speaks of himself as a Platonist, and permits himself at one point (1078b36) … to exaggerate an objection which was stated more moderately in A’ (1924, 190). As Annas puts it, our second puzzle is usually explained in this way ‘as an indication of different degrees of politeness towards the Academy’ (1976, 155). However, this interpretation needs to explain this change in degrees of politeness. If Aristotle is being more polite in A.9 it is because he is not pointing there to a problematic feature of Plato’s doctrine of Forms, which he now suggests in M.4. Annas 1976 mentions an explanation suggested to her by G. E. L. Owen that may serve this purpose. According to this, Aristotle says in M.4 that there are more Forms than pluralities of things that share a qualification49 because Aristotle is thinking of the Third Man Argument there, where for each plurality of sensible F things successive infinite Forms of F-ness have to be introduced. If so, then the second puzzle may be solved thus: in A.9 Aristotle is more polite to Plato, and so does not take into account there this problematic feature, and that is why he affirms Equality between Forms and pluralities of sensibles that share a qualification. But in M.4 he is less polite and so he points to that problematic feature and affirms that Forms are even more than pluralities of sensible things.

Owen’s hypothesis may help to solve our second puzzle. But, as such, it does nothing to meet the first puzzle. For it does not tell us why Aristotle says ‘approximately equal to (or not less than)’, suggesting that there may, perhaps, be more Forms. It seems implausible that he is also thinking about the Forms generated by the Third Man, for consideration of those Forms would severely affect Equality: he would not be able to affirm even an approximate claim of equality. However, I think we can have a unified account of both puzzles. First, we need not assume the controversial claim that the general difference between A.9 and M.4 reflects different stages in the intellectual development of Aristotle. We can rely on a more moderate explanation, as that of Primavesi 2012a. Primavesi explains the difference in the use of personal pronouns between A.9 and M.4 as simply reflecting different perspectives concerning Plato’s views. According to Primavesi, Aristotle in A offers an introductory, historical presentation of his first philosophy. That is why he provisionally adopts, in A.9, an ‘internal’ perspective concerning Plato’s views, and hence expresses himself as a Platonist. But once he has presented in full his ideas about first philosophy, he can adopt, in M.4, an external perspective and dissociate himself from Platonism. As Primavesi puts it, Aristotle ‘could associate himself formally and provisionally with the Academy … until he would have established an alternative ontology’ (2012a, 413).

Building on the ideas of Owen and Primavesi we arrive at the following unified explanation of our two puzzles. Aristotle is in both cases considering Forms that go beyond the general correlation between Forms and pluralities of sensible things that share a qualification, but he does so from different perspectives. When Aristotle suggests in A.9 that there may be more Forms than pluralities of sensibles, while not thinking that this has much effect on Equality, he may have in mind only Forms that the Platonists themselves could wish to countenance, since in this chapter he is adopting an internal perspective towards the Academy. I conjecture that these Forms may be Forms instantiated exclusively by pluralities of Forms. It is true that these Forms do not receive special discussion in Plato’s middle dialogues, but Aristotle may be thinking in A.9 that Plato would wish to take into account such Forms. If so, then there would be some Forms that do not correspond to pluralities of sensible things, so that there would be more Forms than there are pluralities of sensible things. But since it is not clear, for Aristotle, that Plato wants to take these Forms into account, and there may be very few of them in any case, their consideration does not much affect Equality.50 However, when in M.4, where Aristotle assumes an external perspective on Plato’s views, he states his claim about the relative number of Forms and pluralities of sensible things, he now considers Forms that go beyond this general correlation, and are unwanted consequences of Plato’s own claims. These include the infinite Forms generated by the Third Man, an argument which, from an initial plurality of sensible things that are F, generates successive infinite Forms of F-ness out of the consideration of pluralities that have both sensible F things and Forms of F-ness as members. Consequently, he now rejects Equality and affirms that the Forms are, ‘so to say’ (πῶς εἰπεῖν) (i.e. considering unwanted Forms), more in number than the pluralities of sensible things that share a qualification.

Finally, there is the issue of how to interpret the final phrase of the Razor: ‘both over these things and over the eternal things’ (καὶ ἐπὶ τοῖσδε καὶ ἐπὶ τοῖς ἀϊδίοις), A.9, 990b8 and M.4, 1079a4. There are two options, depending on how we take the term ‘the eternal things’. Alexander, In Metaph. 77.24-7 Hayduck and Ross 1924, 191 take the term to refer to heavenly sensible bodies, while Cherniss 1944, 199 n. 117 thinks it refers to Forms. I think the latter is preferable.51 In this case, Aristotle would be saying that there is a one over (ἐπί) many—not just over (ἐπί) many sensible things but also over (ἐπί) many Forms.52 This seems a substantive philosophical claim. And, according to my interpretation, such a claim would have a definite role in the argument. It would hint towards the reasons that, I have conjectured, are behind Aristotle’s shifting attitude towards Equality in A.9 and M.4. For such reasons involve precisely cases of pluralities that have Forms as members. And we will see in Section 5 below how this claim may also be relevant for Countability. Furthermore, this would give us a more uniform interpretation of the term ‘these things’, for in this case it would mean ‘pluralities of sensible things that share a qualification’ everywhere in the Razor. By contrast, taking ‘the eternal things’ in this final phrase to refer to celestial sensible things makes the term somewhat ambiguous. For it is clear that previously the term ‘these things (here)’ referred to pluralities of all sensible things, not only to pluralities of sublunary sensible things, since Plato does not posit Forms to explain just the latter. But in this final phrase, the term would need to have this restricted meaning.

Although this would be awkward, it is perhaps not a decisive objection in itself, however, since Aristotle may be just making clear that he does not mean that the One Over Many argument applies only to pluralities of corruptible sensible things, but to pluralities that include also eternal heavenly sensible things. The main difficulty is rather that it is unclear what the philosophical role of this claim would be. There are contexts where it is indeed philosophically relevant to point out that Forms explain both corruptible sensible things and eternal sensible things. For example, in Metaphysics K, 1060a27-36, Aristotle wonders whether the principle (ἀρχή) of corruptible sensible things and of eternal sensible things is the same or different. (According to him, if it is the same, it is hard to explain why some things are eternal and others are not.) But this question does not seem to be relevant here.53 So, for these reasons, it is preferable to take ‘the eternal things’ to refer to Forms.

Now that we have established the advantages of the Pluralities interpretation over its rivals, both textually and in accounting for Explanation and Equality, we will turn to the Countability analogy.

5 The Countability Analogy

At the heart of the Razor there is an analogy, Countability, between a certain case of counting things and postulating Forms, which suggests that the latter involves, like the former, an unreasonable multiplication of entities. We will see now that the analogy involves a critique of the Forms that is worthy of consideration. We will see, as well, that the analogy is very naturally explained in terms of pluralities of F things, which further confirms the Pluralities interpretation of the Razor.

Analogies are useful means of reasoning. They are widely employed not only in philosophy but also in science. For instance, a famous use of analogy in science is that of Galileo’s discovery of mountains on the Moon. Galileo observed that the moving dark patterns on the Moon that were visible through his telescope behaved like the shadows of mountains on Earth. He concluded, on this basis, that there are mountains on the Moon. In analogies used in argumentative contexts, it is common to distinguish the source case and the target case.54 The analogy appeals to the source case in order to illuminate in some way the target case. In an analogy, the source case has certain positive features and certain negative ones. The positive features are those in which both the source case and the target case agree, the basis of the similarity supposed to be involved in the analogy. The negative features are those in which the two cases disagree. From the claim that the source case and the target case share a certain positive feature P, the analogy intends to derive the claim that they share another positive feature Q. The analogy is useful insofar as the similarities involved in the positive features of the source case provide useful and illuminating insights into the target case, so that the dissimilarities may be tolerated as harmless. In the case of Galileo, for example, the source case is the Earth, and the target case is the Moon. The basic positive feature P is the interaction between sunlight and darkness and the inferred Q positive feature is that of having mountains.

In the Countability analogy, the source case is the counting case, and the target case is the introduction of Forms. As I have argued, the latter should be understood as the case of introducing a Form of F-ness to explain the plurality of F things, a Form of G-ness to explain the plurality of G things, and so on for every plurality of sensible things that share a qualification. The counting case has to be formulated, similarly, in terms of a plurality of F things, and I think this is not a coincidence. It corroborates the fact that the Pluralities interpretation is the more natural interpretation of the Razor. In the counting case, someone who wants to count a certain plurality of F things (say apples) thinks that she is unable to do it because the F things (the apples) are too few, so she adds another F thing (another apple) to the original plurality of F things (apples) in order to be able to count. At first sight, it is puzzling that Aristotle takes the two cases to be similar. That is why Frede 2012 thinks that the analogy is a mere joke. It should be acknowledged that there is something funny in the comparison of the two cases. I take it that this is because the source case seems so unreasonable that it is amusing to think that the target case may be similar to it. And there is where the joke contained in the Razor lies. But to think that the Razor is just a joke is a mistake. We will see that positing Forms and introducing more F things to count F things are similar in interesting ways, so that it is fair that Aristotle points to that similarity; and, by doing so, he suggests an appropriate and useful criticism of Plato. Thus, we should still be able to laugh at Aristotle’s joke while recognizing at the same time that, besides the joke, the Razor contains two interesting and fair objections against Plato’s Forms.55

I take it that the counting case has two main features that can plausibly be seen as the positive features which motivate Aristotle’s introduction of Countability, and which make this, and the Razor, appropriate as a general criticism of Plato’s Forms. The first one can be called Complication. It is plausible to say that what the counting person is doing is complicating the original task she wanted to perform by multiplying or accumulating F things on which to perform that task. That is, by accumulating F things she has now more F things to count. For example, she needs to count not merely n apples, as before, but n + 1. The second positive feature can be called Superfluity. The accumulation of F things in the counting case is superfluous. Even if it did not complicate things, it would still do nothing to solve the task of counting. Thus, the idea is that, for these reasons, it is a mistake to think that the original F things are too few to be counted so that we need to introduce one more F thing. It is plausible to hold that the target case shares these two features and leads to a similar conclusion.

According to Complication, the target case is similar to the source case because the introduction of Forms is also an instance of a case where someone who wants to perform a certain task complicates it by multiplying or accumulating things on which to perform the task. That is, the Platonists want to explain ‘these things here’, each relevant plurality of F things; but it is, according to the analogy, as if they thought that they are unable to do this because the F things are too few. For they end up multiplying F things, adding another F thing, the Form of F-ness, in order to be able to explain them. The criticism seems appropriate because one of the characteristic features of Plato’s Forms is the feature of ‘self-predication’, according to which the Form of F-ness is F: for example, the Form of Beauty is beautiful.56 It is a feature of Plato’s doctrine of Forms that is known to be a potential source of trouble. It is one of the premises of the argument that leads to the Third Man. However, the two objections, although related, are different. The Third Man points to an infinite regress in the postulation of Forms. The Razor, through Complication, does not involve a regress. It points out that it seems unreasonable to add another F thing in order to explain the many F things simply because that seems prima facie like a complication of the task of explaining the F things. By doing this, the objection goes, you do not need just to explain the original plurality of F things that you wanted to explain, but now you also have to explain one more F thing, the Form of F-ness. Behind this thought seems to be the assumption that the being F of any F thing is something that has to be explained. That is, the ultimate task at hand is to explain why (unrestrictedly) all Fs are F, why anything is F, what makes any F thing F, and not just what makes sensible F things F.57 So that when someone gives a response to this question in terms of an explanans that is itself F, that is bound to leave a bad residue: we will lack an explanation of why the Form of F-ness is F.

This seems a reasonable assumption, for something very similar seems to be going on in the counting case. Take the case of someone who is asked to count the apples on the table in order to calculate their total price (each apple costs $1.18). So, he approaches the table and says: ‘Well I cannot do this, the apples are too few, I have to add one more apple,’ and so he puts one more apple on the table. A natural reaction to this would be to tell him: ‘Hold on, you are just adding one more apple: you have now more apples to count’—and it is reasonable to say that he cannot in this way complete the task of counting. Similarly, the explanatory task was to explain the F things. By adding one more F thing, the Form of F-ness, you cannot complete the task, for you have one additional F thing to explain. Therefore, I take it, the first objection involved in the Razor is the following: an explanation of why a plurality of things is F given in terms of an explanans that is itself F cannot provide a fully general response to the question of why anything is F. That manoeuvre only invites the question of why the explanans is F.58

This is a fair point to make and it is a kind of objection that has often reappeared in later philosophy. For example, one horn of what has been called ‘Blackburn’s dilemma’ concerning a certain explanation of necessity seems to be an instance of this sort of objection. According to Blackburn, an explanation of why P is necessary in terms of an explanans that is itself necessary ultimately only shifts the question, for it leaves ‘a bad residual “must”’ (1986, 53). The objection, however, may perhaps not be decisive. There are ways of dealing with it that may seem at least prima facie viable.59 Plato, in particular, may have some options for responding to it. The following two have been attributed to Plato in order to allow him to meet the Third Man, but we can use them also to deal with the present objection. The first one is, of course, to reject commitment to self-predication, that is, to reject the assumption that the Form of F-ness is itself really (predicatively) F.60 The other is to accept self-predication, but claim that the fact that the Form of F-ness is F does not require further explanation, for the Form of F-ness, unlike the many Fs, is not derivatively F, and only what is derivatively F requires explanation.61 Of these, the latter may be more plausible, since Plato’s arguments for Forms may actually commit him to some form of self-predication.62 However, it is worth emphasizing that Plato never explicitly adopted either of these two strategies to respond to the related objection of the Third Man. So Aristotle is being fair in suggesting that the introduction of Forms may be affected by Complication.

Countability does not only suggest that the accumulation of F things complicates the explanation of why the many F things are F. It also suggests Superfluity, that it is superfluous to introduce one more F thing, the Form of F-ness, to explain the F things, as it is superfluous to introduce one more F thing to count the F things. Introducing it does no work towards solving the task at hand. Aristotle is gesturing with this claim to one of his usual objections against Plato’s Forms: postulating a separate Form of F-ness is inadequate because it is of no help towards the explanatory work Forms are supposed to fulfil.63 And the Razor may point towards something that supports the idea that Forms are not fit for the task of formal explanation. In the counting case, adding one apple to count the original plurality of apples is inadequate because the added apple is not fit for the counting task: it is just one more apple to be counted, not a tool that can help you count. So, Aristotle may be suggesting, adding one Form of F-ness to explain the plurality of F things is similarly inadequate because the Form of F-ness is not fit for the explaining task: it is just one more F thing to be explained, not a tool that can help you explain things. Aristotle may be suggesting this because he may be adopting a literal interpretation of the self-predication of Forms, according to which the Form of F-ness is F because it is just another particular instance of F-ness (albeit a perfect one). That is, the Form of Equality, for example, would be, in this interpretation, equal in the same way as particular equal things, so it would need to be equal to something. It would not be strange if Aristotle were assuming this interpretation in our passage, since he seems to adopt it in other places.64 One may, certainly, attempt to meet Superfluity by rejecting the literal interpretation of self-predication and attributing to Plato an interpretation that allows Forms to be suitable tools for explaining the being F of F things. For example, Fine 1993, 62-3 argues that Plato’s own interpretation of self-predication is not literal: the Form of F-ness is F in the sense that it is what makes F things F. If that were correct, the fact that the Form of F-ness is F would not preclude its role as a tool for formal explanation, since it would be F precisely in virtue of being an explanatory property. But, again, it should be emphasized that Plato never makes explicit what interpretation of self-predication he accepts. So, Aristotle is being fair in suggesting that the introduction of Forms may be affected by Superfluity.

In this way, although at first it seemed just a joke to say that the counting case and the introduction of Forms are analogous, we have seen that they are, after all, sufficiently similar. It seems plausible to hold, on the basis of certain characteristics of Plato’s doctrine of Forms and Aristotle’s view of it, that both cases share the positive features of Complication and Superfluity, so that the general conclusion of the argument seems at least prima facie justified: just as it is unreasonable to add another F thing to facilitate counting an original group of F things, it is unreasonable to add another F thing, the Form of F-ness, to explain the many F things. Neither manoeuvre can solve the original task at hand, since each just adds more F things on which to perform its task, and it is, furthermore, superfluous to do that. So, according to Aristotle, we should not multiply F things either to count things or to provide a formal explanation of them. I hope then to have shown that the Countability analogy, and hence the Razor, is worthy of consideration, and that adopting the Pluralities interpretation gives a natural and perspicuous reading of it and of the other two main elements of the argument, Explanation and Equality.65

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  • Van Inwagen, P. (1994), ‘Composition as Identity’, Philosophical Perspectives 8: 207-20.

  • Vlastos, G. (1956), ‘Postscript to the Third Man: A Reply to Mr. Geach’, Philosophical Review 65: 83-94.

  • Vlastos, G. (1969), ‘Plato’s ‘Third Man’ Argument (Parm. 132a1-b2): Text and Logic’, Philosophical Quarterly 19: 289-301.

  • Vlastos, G. (1974), ‘A Note on ‘Pauline Predications’ in Plato’, Phronesis 19: 95-101.

  • Waterfield, R. (1993) (tr.), Plato. Republic. Oxford.

  • White, F. C. (1977), ‘The ‘Many’ in Republic 475a-480a’, Canadian Journal of Philosophy 7: 291-306.

  • White, F. C. (1978), ‘J. Gosling on τὰ πολλὰ καλά’, Phronesis 23: 127-32.

1

The text is that of Primavesi 2012b. All translations from the Greek in this paper are mine.

2

I assume that the target of the criticism is Plato’s theory of Forms as it appears in the ‘middle-period’ dialogues. This is in general accepted concerning the first part of A.9 (990a33-991b9); see Frede 2012, 265-6. I use ‘the Platonists’ and ‘Plato’ interchangeably.

3

However, in A.9 Aristotle seems to evaluate the role of Forms as causes of change and even says that, according to the Phaedo, Forms are causes of existence and generation (991a8-11, b3-9). I think there is no inconsistency here. Aristotle is just considering whether Forms could serve another theoretical role, since, according to him, they fail as formal causes.

4

‘These things here’ are: ‘choses individuelles’, Robin 1908, 121 n. 150; ‘individual things’, Ross 1924, 121 and Cherniss 1944, 199 n. 118; ‘individual sensible things on earth and in the heavens’, Frede 2012, 352.

5

See Robin 1908, 122 n. 150.

6

Other authors have not been, in general, aware of this difficulty. Robin (see last note) is an exception.

7

E.g. Phd. 78d-e, 100b-101e; Symp. 201a-202a; Parm. 132a.

8

This interpretation is also supported by Rep. 6, 507b1-6, where Plato appeals again to Forms, introducing them globally in relation to pluralities of things. Plato does not use there the term ‘the many things’, but he uses a phrase that similarly designates all those things in relation to which he posits Forms: ‘with respect to all the things that we then posited as many.’

9

For example, to cash out τὰ πολλά, Adam 1902 uses ‘group’, Waterfield 1993 uses ‘plurality’, Shorey 1930 uses ‘multiplicities’ and Grube uses ‘many things’ and ‘manys’ (in Cooper 1997, 1200).

10

See e.g. Adam 1902, 387 and Armstrong 1978, 65.

11

Fine 1993 appeals to Crat. 387bd, 388bc for this. Sharma 2006 defends a similar view and criticizes Smith’s 1917 interpretation.

12

See Sedley 2013, 123-4 for a summary of Smith’s reasons and of responses to it.

13

Sedley appeals to Phd. 107d6-8, Crat. 385d2-3 and Rep. 4, 438a7-b2.

14

Fine 1993, 113 thinks that Aristotle holds the restricted traditional interpretation, but that in his objections to the One Over Many argument in the Peri ideōn he does not grant Plato distinctions that he does not clearly formulate, so he levels the objections as if the traditional interpretation were correct.

15

In Russell’s words, ‘a collection as one’ or ‘a collection as many’. See Russell 1903, 513-14. See also Moltmann 2016, 93, to whom I owe the reference to Russell.

16

Many contemporary logicians tend to assume this view.

17

This alternative can be adopted in two different ways, depending on whether we take a whole to be different from its parts or identical to them, as we will see later. Linguistic semanticists, such as Moltmann 1997, tend to adopt the former option. Baxter 1998 and Lewis 1986 and 1991 defend the latter.

18

See Rayo 2006, 225. Other defenders of the plurality as many view include Boolos 1984 and 1985; Simons 2016; Moltmann 2016.

19

Authors who discuss Plato’s Third Man Argument often take the many F things as the set of F things. See, for example, Geach 1956; Vlastos 1956 and 1969; Cohen 1971.

20

See Simons 2016, 60.

21

As we saw, for Plato, each plurality is one of the many pluralities of things that share a qualification. Nevertheless, it is consistent to assume a view of pluralities as many and think that they are ‘one’ among many. For the latter attribution of ‘one’ does not carry ontological commitment to the existence of pluralities as single entities, it is just a way of saying that pluralities may be structured and divided into lower-level pluralities. For example, the plurality of my students this semester is the higher-level plurality of the plurality of students a, b, c who attend my course A and the plurality of students b, e, f, g who attend my course B. The plurality of a, b, c is ‘one’ member of this higher-level plurality. But this claim involves no commitment to the existence of a plurality as a single entity, just commitment to the existence of a, b, c. See Moltmann 2016, 95.

22

See also Parm. 153a2-4 and 158c4.

23

See Ferreirós 1999, pp. xvii-xxi and Kanamori 2003, 273-4.

24

So Kanamori says: ‘It can be justifiably argued that Boole had invented the empty set’ (274).

25

One of the main motivations for those who accept the plurality-as-many view is to take it as the basis of an ontologically innocent surrogate of set theory. See Simons 2016.

26

Unrestricted Composition is a principle of classical twentieth-century mereology. Someone might suppose that, even if Plato accepted this version of the plurality-as-one view, he would not be committed to Unrestricted Composition. For Plato talks about the many beautiful things, the many just things, but would he be willing to accept that any conjunction of things is a ‘many’? For example, would he be willing to accept that this table, this dog, this beautiful, and Mount Olympus are a ‘many’? I think he would: for what could he say to the question of how many things we have here? He would have to say that they are four things—and so that they are many.

27

As McDaniel 2008 points out, Composition as Identity is more a slogan than a well-defined thesis. D. Lewis 1991, for example, stops short of affirming that composition is just the relation of identity, while Baxter 1998 accepts this. This version of the plurality-as-one view is different from the plurality-as-many view because the former accepts concerning any plurality of things a, b, c that there exists one thing, a whole, that is constituted by a, b, c. That is, it accepts the truth of: (∃x) x is constituted by a, b, c. It is a further thesis that, somehow, such ontological commitment can be had for free once you accept the commitment to a, b, c. On the contrary, the plurality-as-many view does not accept the truth of ‘(∃x) x is constituted by a, b, c’, it is just committed to the existence of a, b, c.

28

See also D. Lewis 1991, 83-84.

29

See Van Inwagen 1994 and McDaniel 2008.

30

Tht. 203-6, and Parm. 127d-130a, 131a-c and 137c-145c.

31

Parm. 146b2-5, 157b7-c8; Soph. 261d-262e; Phil. 16c5 ff. and 23c4 ff.; Tim. 29d-32d, 53c-57d.

32

Harte also interprets the notion of plurality in Plato according to the plurality-as-many view (2002, 27, 61). The Parmenides deals with several puzzles about One and Many. And, as Harte 2002, 61 notes, although it focuses on the One, there are some places where Plato problematizes the notion of plurality (πλῆθος)—in particular 157b7-c8, 158b5-c7, 164c7-d4. The main source of the puzzles seems to be the notion of ‘bare plurality’, a plurality to which we cannot apply the notion of One at all (Harte 2002, 127, 136-7). The issues raised by these passages are complex and deserve careful attention. All I can say here is that the puzzles reflect a tension between Socrates’ commitment that there must be a One that is not many and a Many that is not one (129d2-130a2), and the conception of plurality as many. For, according to the latter, the notion of ‘bare plurality’ is indeed incoherent, since you still apply the notion of One (ontologically) to a plurality in the sense that you apply it to each of the many that are the plurality (if it is a first-level plurality; if it is a higher-level plurality you apply the notion of One ontologically only to its ur-elements).

33

The authenticity of Hippias Major is not undisputed: it has been questioned often since the nineteenth century. Most contemporary scholars accept it as genuine, although some, like Kahn 1985, doubt it. However, despite this controversy, the dialogue has been subject to serious study since, as Cooper 1997, 899 says, ‘its philosophical content seems genuinely Platonic’ even if it might not be by Plato.

34

In light of our previous discussion, we can perhaps say that some of the defenders, and critics, of the Types interpretation make an assumption similar to that of Fine and Lewis: they take the many with respect to which Plato posits a Form to be an abstract thing over and above the many.

35

See Boolos 1984.

36

And with Countability, as we will see in Section 5 below.

37

Although Irwin 1977, 10 accepts Gosling’s interpretation for this passage too.

38

Fine 1993, 267 n. 20 and Harte 2010, 100-2 accept this. Harte argues that drawing the particular-universal distinction was not part of Plato’s purpose in introducing Forms. Another interesting case that she mentions is that of certain things that are metaphysically particular but which Plato never puts on either side of the contrast between Forms and the many things—such as individual souls and Gods.

39

As, perhaps, Gosling 1960 and Irwin 1977 believe.

40

Those who reject Gosling’s interpretation may hold this view, such as White 1977 and 1978.

41

There is another worry. If the plurality of F things is nothing over and above the F things, how can there be higher-level pluralities? It seems that a second-level plurality, which is what I claim Plato’s τὰ πολλά are, would have nothing out of which it could be constructed, no things out of which it can be a many. The topic of higher-level pluralities within the plurality-as-many view is controversial; some accept them (Rayo 2006 and Simons 2016) and some reject them (D. Lewis 1991 and McKay 2006). However, it is not clear why there should be an ontological controversy here. Higher-level pluralities seem to involve just more complex ways of organizing things, and so no further ontological commitments. As Linnebo 2004 puts it: ‘the second-level plurality based on Cheerios organized as oo oo oo should be no more ontologically problematic than the first-level plurality based on the same objects organized as oooooo, although the former has an additional level of structure or articulation.’ But if you have troubles concerning higher-level pluralities, you can take τὰ πολλά just as several first-level pluralities that do not comprise one further plurality. This is enough for my purposes, since we all clearly can talk about several pluralities at once.

42

Moltmann 2016, 116-17 talks about cases that involve counting pluralities.

43

We would need a final clause that says that there are no other (sensible) zz such that the zz have an instance of the feature F that Platonic Forms are supposed to explain. (What F is depends on the interpretation of the One Over Many principle one prefers: see Section 3 above.) The case of Forms that may not correspond to pluralities of sensibles will be discussed later, when we discuss Aristotle’s qualification ‘approximately’.

44

Alexander, In Metaph. 77.10-17 Hayduck is an exception. However, although he sees that this passage contains an explanation of Equality, he is not aware of its importance for the overall interpretation of the Razor, in particular for establishing what Aristotle means by ‘these things here’.

45

Aristotle uses ‘homonymous’ here not in his own sense of ‘having only the same name but different nature’ (Cat. 1a1-12), but in Plato’s literal sense of ‘having the same name’. See Fine 1993, 318 n. 10.

46

There is a controversy concerning lines 990b6-8, καθ’ ἕκαστον γὰρ ὁµώνυµόν τι ἐστί· καὶ παρὰ τὰς οὐσίας τῶν τε ἄλλων ἐστὶν ἐπὶ πολλῶν ἕν. I follow the text of Primavesi 2012b, which follows the α family of manuscripts of the Metaphysics (see Primavesi 2012a, 421-4). Ross 1924 accepts pretty much the same text, although with some differences in punctuation and order of words. Primavesi argues decisively against the main alternative reading, Jaeger 1957, who, following Bekker and Bonitz, transposes the phrase καὶ παρὰ τὰς οὐσίας to the end of the passage, to read: καθ’ ἕκαστον γὰρ ὁµώνυµόν τι ἔστι τῶν τε ἄλλων ὧν ἔστιν ἓν ἐπὶ πολλῶν, καὶ παρὰ τὰς οὐσίας. Primavesi’s main arguments are: (a) Alexander quotes the passage three times, one as in the text of the α family and the other two with minor variants; and (b) the parallel passage of M.4 repeats almost verbatim the text of the α family.

47

Plato may not postulate Forms under each Aristotelian category of being. Aristotle’s point may be that Plato postulates one Form in the case of ‘each’ plurality of things that share a real qualification, whatever these qualifications are for Plato.

48

Primavesi seems to interpret the passage similarly, since, according to him, the second and third part mean: ‘For to each set of individual things there answers an abstract item of the same name. [That is to say:] also apart from the substances and in the case of the other things there is One over many’ (2012a, 422, my emphasis). However, he does not say anything about the relevance of this passage for the interpretation of the Razor. Furthermore, his talk of ‘set’ may suggest that he is adopting a view akin to the Types interpretation. Cf. also Ross’s summary of the version of the Razor in M.4 (1924, ii, 419).

49

Types, in Annas’ interpretation.

50

Aristotle says in A.9 (991a29-31) that ‘the Forms are paradigms not only of sensibles but of themselves, for example, the genus, as genera of the Forms’ (and he takes this to be problematic, since the same thing would be both paradigm and image). Thus, Aristotle entertains elsewhere the thought that there may be Forms of Forms.

51

Cherniss supports his interpretation by saying that ‘Aristotle frequently asserts that the Ideas differ from the sensibles only in being eternal’ (ibid.). Cf. Metaph. B, 997b6-12; Z, 1028b19-20, 1040b30-34; K, 1060a16-18. The K passage is somewhat similar to the Razor, although it is not clear whether the term ‘equal’ is used there in a numerical or a qualitative sense.

52

As Cherniss (ibid.) points out.

53

Cf. also Metaph. 991a9-10, where the distinction between corruptible and eternal sensibles seems relevant because Aristotle has been talking about partaking of the eternity of the Form and about the corruptible dyads and the eternal dyads.

54

See Hesse 1966, 58-9; Bartha 2010, 15; Norton unpublished, 7-9.

55

Aristotle assumes that all instances of the counting case are unreasonable. But this is not true. Take the case of someone who is counting some dots, which are arranged in a certain pattern. It may be that the pattern is incomplete and there is one dot missing. If she adds the missing dot she may see the pattern more clearly and count the dots more easily. (I owe this point to Axel Barceló.) However, Aristotle is right in that the counting case is, in general, unreasonable.

56

See e.g. Phd. 74a-c, 100c4-6; Symp. 210e6-211a5; Parm. 129b1-c3.

57

On this requirement, see Fine 1993, 199-201. Thus, Plato cannot get off the hook by saying that he does not think that the sensible F things are too few: there are so many of them that there is no need to add one more. For, even if his project may be initially described as that of explaining what makes sensible F things F, the kind of criticism involved in the Razor points out that the most important and general project is to explain what makes any F thing F. And, as I claimed in Section 4 above, it is plausible to think that this idea is suggested by the final phrase of the Razor: ‘both over these things and over the eternal things’ (990b8). Thus, it is fair to claim that the Platonists act as if they thought that the F things are too few, for they add another F thing.

58

That the Razor involves something like the objection I call ‘Complication’ has been hinted by other authors. Thus, Ross 1924, 187 summarizes the Razor in the following way: ‘It supposes Ideas to exist in order to explain sensibles, but in doing this it merely doubles the number of things to be explained.’ Cherniss 1944, 198 similarly, says: ‘the list of objections to the ideas is headed by the charge that those who posit the ideas in attempting to ascertain the causes of phenomena have merely duplicated the objects to be explained.’ And Annas 1976, 155 says: ‘The Academy are accused of trying to solve conceptual problems about physical objects by the introduction of Forms; but since they treat Forms as just another kind of object the problems are not solved but aggravated.’ However, they do not explain what the objection exactly is or how it is supposed to work.

59

Hale 2002, for example, has argued against both horns of Blackburn’s dilemma. Yet, some have thought that Hale’s response is unsatisfactory and that there are ways of resurrecting Blackburn’s dilemma: see Hanks 2007.

60

See Allen 1960 and Vlastos 1974.

61

See Fine 1993, 228-31.

62

See Fine’s analysis of the Argument from Relatives in Aristotle’s Peri ideōn (1993, 143-4).

63

See Metaph. A, 991a8-23.

64

Cf. Metaph. Z.16, 1040b30-4, where Aristotle says that Forms are just like the sensible particulars, for the Platonists just add the term ‘itself’ (‘αὐτό’). Cf. also Metaph. B, 997b6-12. F. Lewis 1991, 34 n. 31 seems to interpret the Razor in this way. See also Cherniss 1944, 196 n. 115.

65

I would like to thank the audiences of the talks where different versions of this paper were presented. In particular, I would like to thank Martín Barbosa, Juan Pablo Bermúdez, Marcelo Boeri, Daniel Drucker, Martin Glazier, Jim Hankinson, André Laks, Raymundo Morado, Carlos Romero, Pedro Stepanenko, Alessandro Torza and Steve White. Most especially I would like to thank Axel Barceló, for discussing in numerous occasions the topic of pluralities with me, Matthew Matherne, for his very valuable feedback as my commentator in the 2017 UT Austin-UNAM Philosophy Workshop, and, above all, Ricardo Salles, for his many generous and insightful comments, both written and oral, that helped me to improve this paper in so many ways. I am also much indebted to an anonymous referee for this journal for several useful comments and suggestions.

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