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ARISTOTLE AND THE IDENTIFICATION OF FORMS AND IDEAL NUMBERS IN PLATO

Published online by Cambridge University Press:  12 December 2025

Rareş Ilie Marinescu
Rareş Ilie Marinescu*
Affiliation:
University of Toronto
*
r.marinescu@utoronto.ca
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Abstract

By offering a fresh reading of several partially overlooked passages from Aristotle’s Metaphysics Μ and Ν, this article argues that the identification of Forms and ideal numbers in Plato is not presented as Aristotle’s own reconstruction. Instead, Aristotle sets forth what he takes to be Plato’s views. This reading enhances not only our understanding of the Academic debates with which Aristotle engaged but also his status as a historian of philosophy.

Keywords

PlatoAristotletheory of formsmetaphysicsOld Academymathematics

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Research Article
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The Classical Quarterly , Volume 75 , Issue 1 , May 2025 , pp. 195 - 208
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Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (https://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
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© The Author(s), 2025. Published by Cambridge University Press on behalf of The Classical Association

INTRODUCTION

One of the most puzzling aspects of Aristotle’s account of Plato’s philosophy is his identification of Forms and ideal numbers. This has been largely ignored in recent decades, although it plays a major role in the last two books of the Metaphysics.Footnote 1 This supposed identification is for various reasons problematic. Above all, it is unclear whether all Forms are numbers or whether we need to distinguish between different types of Forms—some numbers, some not.Footnote 2 Given what we know about Aristotle’s way of dealing with Plato and his predecessors more generally, it seems from the outset questionable whether Aristotle presents us with a consistent theory or faithful representation of Plato’s views.

More fundamentally still, scholars debate whether this identification goes back to Aristotle himself. What can be termed the traditional—but, at least in the Anglophone world, no longer dominant—view regards Aristotle’s account as a representation of Plato’s (later) thought.Footnote 3 However, since at least Teichmüller and Shorey, scholars have questioned this supposition and instead maintained that the identification is based on a reconstruction by Aristotle whereby he draws a logical conclusion that is not genuinely Platonic.Footnote 4 As Cherniss puts it in his influential work Aristotle’s Criticism of Plato and the Early Academy (at 198):

the apparent unity of the account in Α, chap. 6 is an unhistorical fiction; the connection of the ideas and numbers there has nothing to do with the theory of ideas as such, and the influence on the theory there imputed to the Pythagoreans, inseparable from this connection as it is, is shown by Aristotle himself to be at best a reconstruction based upon a later conception of the nature of the ideas.

On this reading, Aristotle brings together two different theories—one about the nature of universals, another about numerological speculation and the origin of numbers—which were either not connected at all by Plato or were connected only superficially and tentatively.

Since the latter position has recently gained some traction through Annas’s influential commentary on Metaph. Μ and Ν and Steel’s discussion of Metaph. Α 6, I offer in the following a defence of the traditional view by offering a fresh, in-depth reading of some—partly overlooked—passages from Metaph. Μ–Ν. I argue that, based on the way in which Aristotle presents the identification, it seems very unlikely that it is just his reconstruction—without denying that Aristotle obviously presents us with a picture that suits his agenda. Instead, he sets forth what he takes to be Plato’s final views on the Forms—if not entirely faithfully and consistently. According to Aristotle, Plato initially developed the theory of Forms on its own and then connected it to the theory of ideal numbers. My main interest lies here both in discussing Aristotle’s views on Plato, as we find them in his directly transmitted work, and in analysing the language Aristotle uses to present his most important predecessor and interlocutor. I do not claim that Aristotle’s presentation is the historically accurate one—this scholars of Plato (and the Early Academy) may decide. Instead, I maintain that there is no reason to believe that anything in Aristotle’s presentation of the Form-number identification suggests it is his reconstruction. I thus attempt a rehabilitation of Aristotle as a doxographical source, since the view that he essentially made up and attributed such a bizarre theory as the identity of Forms and ideal numbers to his teacher of twenty years did great damage to, if not completely undermined, his trustworthiness as a doxographer in certain scholarly circles.Footnote 5 The focus throughout will be on passages which seem to suggest that Forms are numbers—and not vice versa.Footnote 6 The latter would indeed be far less controversial.Footnote 7 I begin by discussing the two main reports on (the development of) Plato’s philosophy in Μ 4 and Α 6, before dealing with a few central passages on this identification divided into dialectical-style arguments and matter-of-fact statements.

THE DEVELOPMENT OF THE THEORY OF FORMS (METAPH. Μ 4)

Aristotle himself casts the connection of the Forms with numbers as a later development when he discusses Plato’s account of the theory of Forms and its genealogy in Μ 4.1078b9–32:

περὶ δὲ τῶν ἰδεῶν πρῶτον αὐτὴν τὴν κατὰ τὴν ἰδέαν δόξαν ἐπισκεπτέον, μηθὲν συνάπτοντας πρὸς τὴν τῶν ἀριθμῶν ϕύσιν, ἀλλ’ ὡς ὑπέλαβον ἐξ ἀρχῆς οἱ πρῶτοι τὰς ἰδέας ϕήσαντες εἶναι.

But concerning the Forms, we must first examine the theory of Forms itself, without associating it with the nature of the numbers, but rather as those who first posited the existence of Forms understood it initially. (Metaph. Μ 4.1078b9–12)

Although this passage is often taken as an indication that the connection of Forms with numbers is a development of the late(r) Plato,Footnote 8 the wording is actually quite ambiguous, as I would like to show.Footnote 9 Take, for instance, the term referring to the connection itself, συνάπτοντας. The phrase μηθὲν συνάπτοντας πρὸς τὴν τῶν ἀριθμῶν ϕύσιν implies as a subject ἡμᾶς, since the verbal adjective ἐπισκεπτέον implies ἡμῖν.Footnote 10 This leaves open as to who exactly brought the theory of Forms and that of numbers together. In fact, one could even take it—as the pronoun in the first-person plural suggests—that Aristotle himself is making this connection. Yet ἐξ ἀρχῆς in the next line clearly implies that some development took place, and it is not just Aristotle who makes the connection. Rather, he here refers to how the Forms were initially (ἐξ ἀρχῆς) conceived by those who first posited their existence, that is, Plato. This alone does not imply that Plato himself developed his theory in this way, as the Forms’ association with numbers could be a later addition by other Academics.Footnote 11 In other words, the phrase ἐξ ἀρχῆς logically could here indicate either an external development, as just sketched, or an internal one. According to the latter, we would have to distinguish between (at least) two stages in Plato’s thought: in the first stage, the Forms were considered on their own, while, in a later stage, they were attached to the nature of the numbers. Forms in general seem to be subsequently connected to numbers and not just a subset, which fits Aristotle’s broad and oft-repeated claim that ‘Forms are numbers’. At any rate, the association is clearly a later one, and Aristotle regards it as rather contingent since he believes that one can—reasonably—consider the Forms on their own. Most importantly, in order to settle the question of who developed the theory of Forms in such a way, we need to consider some other crucial passages.

METAPHYSICS Α

While in the doxography of Plato in Μ 4 the theory of Forms and its background are treated on their own, in the more famous report on Plato’s philosophy in A 6 we get a different picture. A cursory overview of the chapter shows that the Pythagoreans—completely absent in Μ 4—play a prominent role, while the contribution of Socrates is downplayed—in contrast to Μ 4. As will be seen, this is due to Aristotle’s different intentions in these passages.Footnote 12 Unlike in Μ 4, Forms and ideal numbers are here directly brought together:

ἐπεὶ δ’ αἴτια τὰ εἴδη τοῖς ἄλλοις, τἀκείνων στοιχεῖα πάντων ᾠήθη τῶν ὄντων εἶναι στοιχεῖα. ὡς μὲν οὖν ὕλην τὸ μέγα καὶ τὸ μικρὸν εἶναι ἀρχάς, ὡς δ’ οὐσίαν τὸ ἕν· ἐξ ἐκείνων γὰρ κατὰ μέθεξιν τοῦ ἑνὸς τὰ εἴδη εἶναι [τοὺς ἀριθμούς]. τὸ μέντοι γε ἓν οὐσίαν εἶναι, καὶ μὴ ἕτερόν τι ὂν λέγεσθαι ἕν, παραπλησίως τοῖς Πυθαγορείοις ἔλεγε, καὶ τὸ τοὺς ἀριθμοὺς αἰτίους εἶναι τοῖς ἄλλοις τῆς οὐσίας ὡσαύτως ἐκείνοις·

Since the Forms are causes of other things, he thought that their elements were the elements of all beings. As matter, the great and the small were principles; as essence, the One; for from the former [that is, the great and the small], by participation in the One, are the Forms [MSS: are the Forms, that is, the numbers]. But, in saying that the One is essence and not another thing that is said to be one, he agreed with the Pythagoreans; and in saying that the numbers are causes of the essence of other things, he likewise agreed with them. (Α 6.987b18–25)

The passage in question poses serious textual problems.Footnote 13 Most scholars are not happy with accepting the transmitted τὰ εἴδη εἶναι τοὺς ἀριθμούς, and delete either τὰ εἴδη (Gillespie, Ross) or τοὺς ἀριθμούς (Susemihl, Christ, Jaeger and Primavesi). However, since at least Alexander, there have been propositions of keeping both. The Peripatetic takes τοὺς ἀριθμούς as an apposition to τὰ εἴδη (In Metaph. 53.9–11), in which he is followed by Bonitz, Robin, Stenzel and, more recently, Crubellier (n. 1), 305.Footnote 14 A variation of this position is Ueberweg’s τὰ εἴδη εἶναι ὡς ἀριθμούς. Lastly, we have Asclepius’ solution of inserting a καί: τὰ εἴδη εἶναι <καὶ> τοὺς ἀριθμούς (endorsed by Isnardi Parente). The choice between the different variations is often ideologically motivated, as the editors choose the version which fits more with their understanding of Plato or rather with their understanding of Aristotle’s understanding of Plato.Footnote 15

It is dangerous to eliminate either term. Besides the textual issues—the oldest manuscripts as well as our ancient witnesses, Alexander and Asclepius, clearly have a text with both terms—there is a more philosophical concern: the passage starts with the Forms and ends with the ideal numbers, attributing to both the characterization as causes of all other things (αἴτια τὰ εἴδη τοῖς ἄλλοις; τοὺς ἀριθμοὺς αἰτίους εἶναι τοῖς ἄλλοις τῆς οὐσίας).Footnote 16 To guarantee the logical continuity of the passages one should be reluctant to delete either.Footnote 17 If we keep both, we get the result that Aristotle assimilates the two and treats them as if they occupy the same level on the ontological hierarchy.Footnote 18 How exactly they are related to each other is left open.

How do we explain the discrepancy between the two accounts, Μ 4 and Α 6? The most plausible explanation seems to be to assume that in Α 6 Aristotle presents the final version of Plato’s theory where Forms and numbers were associated, while in Μ 4 he focusses on its initial (ἐξ ἀρχῆς) formulation by Plato.Footnote 19 Only later, under Pythagorean influence, as Aristotle stresses (987b23–4), did Plato associate the Forms with numbers.Footnote 20 To this innovation, one might add the One and the Indefinite Dyad which are absent from Μ 4 as causes of the Forms. It comes as no surprise that he goes on after the passage quoted above to discuss the differences between Plato and the Pythagoreans (987b25–988a1). This means that Aristotle takes a developmentalist stance on Plato’s philosophy. Aristotle alludes to a development also at 987a32–b1, where he refers to doctrines that Plato held from his youth (ἐκ νέου) and also at a later time (ὕστερον), namely that sensibles are in a flux and that knowledge of them is impossible. Such a statement can be read as implying that Plato did change his mind on some issues and that such a doctrinal continuity is missing in other areas.Footnote 21 In Μ 4 Aristotle presents an earlier and incidentally more simplified version of Plato’s philosophy. This accords with the methodological procedure he sets out for Μ–Ν in Μ 1, where he states that he will offer only a restricted account of three views. Μ–Ν concern eternal, unmoved and separate substances (Μ 1.1076a11), while A has a broader scope, looking into Aristotle’s predecessors’ first principles and their causality. What is important for my current purpose is that both Α 6 and Μ 4 suggest that Plato himself associated—if not straight out identified (as we will see in other passages)—Forms with numbers.

ONLY ARISTOTLE’S RECONSTRUCTION?

After Shorey, several important scholars have cast serious doubt on Aristotle’s presentation of Plato’s views. In consequence, the identification of Forms and numbers has been seen as a reconstruction by Aristotle, whereby he only draws a logical conclusion that is not genuinely Platonic. Apart from the apparent absence of such an identification in the Platonic corpus, the main reason for this is that a fair amount of references to the theory are presented in the form of a hypothesis whereby the protasis includes or is the claim ‘if Forms are (not) numbers’ and the apodosis usually follows with an absurd consequence (Α 9.991b9–21, Μ 7.1081a12–17, Μ 8.1083a17–20, Ν 4.1091b26–30).Footnote 22 As evidence for this position, the following passage (which is the first use of the hypothesis in Μ–Ν) is often cited:Footnote 23

εἰ δὲ μὴ εἰσὶν ἀριθμοὶ αἱ ἰδέαι, οὐδ’ ὅλως οἷόν τε αὐτὰς εἶναι (ἐκ τίνων γὰρ ἔσονται ἀρχῶν αἱ ἰδέαι; ὁ γὰρ ἀριθμός ἐστιν ἐκ τοῦ ἑνὸς καὶ τῆς δυάδος τῆς ἀορίστου, καὶ αἱ ἀρχαὶ καὶ τὰ στοιχεῖα λέγονται τοῦ ἀριθμοῦ εἶναι, τάξαι τε οὔτε προτέρας ἐνδέχεται τῶν ἀριθμῶν αὐτὰς οὔθ’ ὑστέρας)·

But if Forms are not numbers, they cannot exist at all. (For from what principles will Forms be? For number is from the One and the Indefinite Dyad, and the principles and elements are said to be of number; but it is not possible to rank Forms either before or after numbers.) (Μ 7.1081a12–17)

Cherniss believes that the negative hypothetical clause indicates that the identification is an Aristotelian reconstruction. According to Cherniss, Aristotle here does not pick up someone else’s assertion but makes it himself, that is, he is proving the statement that Forms must be numbers according to Plato: ‘If your opponent asserts a thesis which you undertake to refute, you do not begin by proving that he asserts it’ (Cherniss [n. 4 (1945)], 59). If the principles of numbers are the One and the Indefinite Dyad, which are generally the first principles, they must also be the principles of Forms since these are neither prior nor posterior to numbers. Therefore, Plato must conclude that Forms are numbers—otherwise his theory is illogical. But, so Cherniss, Plato himself did not conclude this—rather we have here merely an Aristotelian reconstruction whereby Aristotle takes over some Platonic premises and arrives at a non-Platonic conclusion (that is, the identity of Forms and numbers).

Yet Cherniss misses the context of the passage which, I believe, makes clear that the point in question is not Aristotle’s reconstruction. 1081a12–17 is part of a larger discussion (Μ 6—Μ 8.1083b23) about the nature of units of numbers that are here understood in the Platonist sense as ‘separate substances and primary causes of beings’ (Μ 6.1080a14 οὐσίας αὐτοὺς εἶναι χωριστὰς καὶ τῶν ὄντων αἰτίας πρώτας).Footnote 24 Aristotle asks whether (i) none of the units (μονάδες) of numbers are combinable (συμβληταί) with each other or (ii) all or (iii) some (Μ 6.1080a17–35).Footnote 25 In other words, are all the units of the number three, conceived in this way, combinable with those of the number four? ‘Combinable’ implies here being able to undergo operations such as addition and subtraction—as is the case with units of mathematical numbers, which, through such processes, can form different mathematical numbers.Footnote 26 Aristotle first goes through the option that all units of the number are combinable (ii):

If all the units are combinable (συμβληταί) and undifferentiated (ἀδιάϕοροι), mathematical number results and only this one kind, and Forms cannot be numbers. (What sort of number will the Man itself (αὐτὸς ἄνθρωπος) be, or Animal or any other Form? For there is one Form of each thing, for example one of the Man itself and another of the Animal itself; but these similar (ὅμοιοι) and undifferentiated numbers (ἀδιάϕοροι) are unlimited, so that this particular three (ἥδε ἡ τριάς) cannot be the Man itself any more than any other.) (Μ 7.1081a5–12)

Now if all the units are combinable with each other (both within the same number and with units of other numbers), then the resulting numbers cannot be ideal numbers but are mathematical numbers. Yet this makes it impossible to identify Forms with numbers (a7). Why can Forms not be mathematical numbers? Because ‘these similar and undifferentiated numbers are unlimited’ (a10–11), that is, there are infinite instances of any given mathematical number and they do not have any singularity (one instance of three cannot be distinguished from another instance of three). But if these numbers lack singularity they cannot be identified with any Form, since ‘this particular three cannot be the Man itself any more than any other [that is, particular three]’ (a11–12). Forms are singular, whereas mathematical numbers are not.Footnote 27

It is in this context that Aristotle concludes that Forms are not numbers and where our passage a12–17 picks up. Aristotle intends to infer from the impossibility of the Forms’ identification with numbers the impossibility of the Forms’ existence, since Forms derive from the same principles as numbers and have the same ontological status (as also clarified in Α 6). The assumption in a13–17 is thus not his own but rather presented as acquired from Plato to demonstrate the absurdity of an identification, if indeed Forms are said to be identical with numbers that have combinable units, that is, are mathematical.Footnote 28 Aristotle’s argument is strengthened by accepting that he takes over the identification thesis.

The two other references to the identification thesis in Aristotle’s discussion of the combinability of numbers also suggest that Aristotle takes the thesis as given. One is formulated as a conclusion of an argument (Μ 7.1082b23–8) and the other as a hypothesis (Μ 8.1083a17–20). Let us start with the former:

Nor will the Ideas be numbers. Those who claim that the units differ (διαϕόρους τὰς μονάδας) speak correctly, if indeed (εἴπερ) Ideas will exist, as was said earlier: for the Form is unique (ἕν), but if the units are undifferentiated (ἀδιάϕοροι) the twos and threes will be undifferentiated also.

This passage is part of Aristotle’s discussion of option (iii) (Μ 7.1081b35–Μ 8.1083a17), that is, the units within the same number are combinable with (and, thus, undifferentiated from) each other, but they are not combinable with the units of a different number from whose units they differ (Μ 7.1081b35–7). Here Aristotle argues that, if the units of a number are undifferentiated from each other, the resulting numbers will be also undifferentiated. Now, given that Forms are numbers, it would mean that Forms consist of undifferentiated units and are thus not unique (26–8). But this conclusion is absurd. That is why Aristotle prefers the view that, if Forms are numbers, then the units of these numbers must differ from the units of other numbers (24–5): that is, the Three itself consists of ‘ddd’ and the Four itself of ‘eeee’, to follow Crubellier’s useful illustration.Footnote 29 Important here is the expression ‘if indeed Ideas will exist, as was said earlier’ (25–6), which I take to refer to the discussion at 1081a5–17 where Aristotle argued that Forms would cease to exist if they were not numbers.Footnote 30 This shows that both discussions, 1081a5–17 and 1082b23–8, are connected and take the assumption that Forms are numbers as a given.

The second reference is formulated as a hypothesis and appears at the end of his discussion of the combinability of units:

It is clear then that, if indeed the Ideas are numbers, all units can be neither combinable (συμβλητάς) nor non-combinable (ἀσυμβλήτους) with one another in either way. (Μ 8.1083a17–20)

This hypothesis is not ‘hypothetical’ but rather serves as a conclusion to Aristotle’s discussion and repeats the background assumption that—as I have argued—he simply took over. Just as making units non-combinable (either within the same number or with units of another number) led to problems in identifying Forms with numbers, so does making units combinable, as he concludes here. At any rate, Μ 7.1082b23–8 and Μ 8.1083a17–20 cannot be explained away as talking, in fact, about Form numbers—Aristotle clearly talks about Forms themselves and their unicity, which needs to be maintained if they are to be identified with numbers. Neither can the identification be regarded, as Annas (n. 1), 66 states, ‘as a scornful shorthand way of referring to Plato’s idea about Forms and numbers in general’ or as the oversight of a later editor of the text.

Aristotle’s procedure becomes clearer if we consider other hypothetical formulations and, more generally, his procedure in book Α on which the doxographical account of Μ and its method of engaging with his predecessors are based.Footnote 31 There, we encounter the only occurrence of the (positive) hypothesis: ‘Furthermore, if indeed (εἴπερ) the Forms are numbers, how will they [that is, numbers] be causes (αἴτιοι)?’ (991b9–10).Footnote 32 The discussion goes up to 992a10 and must be connected to Aristotle’s earlier and more general query about the causal contribution of Forms (991a8–10). 991b–10 picks up this topic and further develops it under the assumption that Forms are numbers.Footnote 33 And if so, he asks, how do these numbers cause sensible things? Thus, Aristotle here introduces this objection, without having argued previously for why Forms should be numbers—he had only touched on this supposed identification in Α 6 (see above). As Crubellier (n. 1), 305 has stressed by pointing also to the use of εἴπερ, the latter is simply accepted as a fact and a starting point for his objection which again emphasizes that he simply takes it as a given. Now, if we have a cursory look at other hypothetical formulations in Α 9, we can see that the protasis is usually an unquestionably Platonic supposition. For instance, at Α 9.990b28–9 he states that, based on the theory of Forms, Forms must be only of substances if they are participated in (εἰ ἔστι μεθεκτὰ τὰ εἴδη)—the latter being clearly a Platonic supposition, as Aristotle himself claims at Α 6.987b9–10. Indeed, the whole passage 990b22–991a8 is instructive, as Aristotle builds up on the assumption that Forms exist (990b22–3 κατὰ μὲν τὴν ὑπόληψιν καθ’ ἣν εἶναί ϕαμεν τὰς ἰδέας) and continues his argument.Footnote 34 The same goes also for the Platonic/Platonist arguments that he lists in favour of the Forms at 990b8–22: if anything, it is not the premises of these arguments that are Aristotelian but rather the unwanted conclusions. Before he lists the arguments, Aristotle emphasizes that the problem is with what follows—or, indeed, does not follow—from them (990b10–11).Footnote 35 Taking over an accepted statement and drawing absurd consequences from it is a procedure that Aristotle employs in his treatment of other predecessors in Α and Μ–Ν. In fact, he announces this type of aporetic examination in the transitional passage Α 7.988b20–1, stating that he will discuss the possible difficulties (τὰς ἐνδεχομένας ἀπορίας) regarding their views on the principles. Aristotle’s goal is thus to construe possible aporiai that arise from the principles that earlier thinkers have accepted. This accounts for the hypothetical structure we encounter not just in his discussion of Plato and the Academics but also of other thinkers.

For instance, in his treatment of Empedocles he claims that ‘the same is true if someone assumes (τίθησιν) more of these, as Empedocles says the matter is four bodies’ (Α 8.989a19–21). Just as in the case of the monists (Α 8.988b22–989a18), Aristotle intends to show how a pluralist like Empedocles runs into the same problems, for example failure to account for incorporeal things and essences. Similarly, the Pythagoreans are described as unable to account for motion ‘if limit and unlimited and odd and even are the only things assumed (ὑποκειμένων)’ (Α 8.990a8–10).Footnote 36 Both of these statements seem to be based on positions that these philosophers held—or, at least, in the challenging case of the Pythagoreans, positions that cannot be reasonably excluded from having been held.

There are doxographical accounts in Aristotle that are formulated differently and include suppositions which the thinkers addressed clearly did not make. In contrast to the former case (that is, using accepted suppositions), there is a certain reluctance in constructing a hypothetical position for his interlocutors, which is—at least sometimes—expressed tentatively by an optative governed by an indefinite third-person singular. Of Anaxagoras, he says:

if one were to suppose (εἴ τις ὑπολάβοι) that he said there were two elements, the supposition would accord thoroughly with a view that he himself did not articulate, but he would have necessarily followed those who stated this view. (Α 8.989a30–3)

Aristotle imputes a view to Anaxagoras which the latter did not hold explicitly (οὐ διήρθρωσεν), but which would follow from what he said, namely that both νοῦς and the mixture could be regarded as elements, that is, as certain causes. This type of procedure shows a certain sensitivity in Aristotle’s interpretation of what the actual text of Anaxagoras states and what can be reasonably inferred from it. The fact that Aristotle feels the need to comment on his method shows that he wants to defend his interpretation against possible objections. A similar approach is found in the discussion of Empedocles (Α 4.985a4–7).Footnote 37 Such sensitivity and reluctance are not, however, encountered in his discussion of Plato’s identification of Forms and numbers—which again suggests that he is not the one coming up with it.Footnote 38 But Aristotle does clarify when he goes beyond Plato: for instance, when he associates the Form of man with the number three or Threeness, he introduces the claim with οἷον εἰ (Μ 8.1084a14).Footnote 39

What can we conclude from this? First, Aristotle is fond of formulating his objections in the form of dialectical-style arguments which set up and test a hypothesis. While in these arguments the apodosis may well be a conclusion that Aristotle himself draws—indeed, this is precisely his aim: to show what unfortunate consequences the theory leads to—the content of the protasis is often original (or at least regarded as such by Aristotle). His whole emphasis is on the apodosis and not on the protasis. In other words, his account focusses primarily not on the (uncontroversial) if but on the then—an argumentative structure that helps him to convince the reader of absurd consequences that would follow from a certain philosopher’s assumptions. Thus, an assumption formulated as a hypothesis does not need to be hypothetical—just as Plato really believed that there are Forms and Empedocles that four elements exist. Second, in assessing whether the argument’s premises are based on Aristotelian assumptions we need to consider context and language. As we have seen in the case of Μ 7 and Α 9, the context does not allow us to conclude that the hypothesis ‘Forms are numbers’ is Aristotle’s own supposition. Additionally, the type of conditional he sets up helps assess the provenance of the content of the protasis: his use of the optative in the protasis suggests that he himself comes up with it.

OTHER PASSAGES

Besides these hypothetical formulations, several passages refer to the identification in a matter-of-fact tone, which can be contrasted with the previous type of statements. These make the case for the ‘reconstructionists’ much more difficult. Two of these, which are often ignored by those who believe that Aristotle himself made the identification, could not be clearer that Aristotle regarded Plato as identifying Forms with numbers:

  1. 1. ὁ δὲ πρῶτος θέμενος [sc. Plato] τὰ εἴδη εἶναι καὶ ἀριθμοὺς τὰ εἴδη καὶ τὰ μαθηματικὰ εἶναι εὐλόγως ἐχώρισεν

  2. And the first who claimed that the Forms exist, that Forms are numbers and that mathematicals exist, reasonably separated [them]. (Μ 9.1086a11–13)Footnote 40

and

  1. 2. οἱ μὲν οὖν τιθέμενοι τὰς ἰδέας εἶναι, καὶ ἀριθμοὺς αὐτὰς εἶναι, <τῷ> κατὰ τὴν ἔκθεσιν ἑκάστου παρὰ τὰ πολλὰ λαμβάνειν [τὸ] ἕν τι ἕκαστον πειρῶνταί γε λέγειν πως διὰ τί ἔστιν, οὐ μὴν ἀλλὰ ἐπεὶ οὔτε ἀναγκαῖα οὔτε δυνατὰ ταῦτα, οὐδὲ τὸν ἀριθμὸν διά γε ταῦτα εἶναι λεκτέον·

  2. Those who posit that the Forms exist and that they are numbers are trying at least to say, by taking a certain one according to their method of setting each one out over the many, why each one exists. However, since these considerations are neither necessary nor possible, one should not claim that the number exists because of them, at least. (Ν 3.1090a16–20)

It seems unreasonable to assume that doxographies worded like this—in contrast to dialectical-style statements such as ‘if the Forms are numbers, then …’—are also just based on logical inferences from Aristotle.

First, let us consider the context of these passages. In Μ 9.1085b34–1086a21, Aristotle presents the doctrines of Speusippus, Xenocrates and Plato on number before pointing towards the reason (αἴτιον) for their contradictions: they base their theories on false assumptions and principles (1086a15–16 αἱ ὑποθέσεις καὶ αἱ ἀρχαὶ ψευδεῖς). The appearance of the term ὑπόθεσις here not only explains Aristotle’s widespread use of the formulation ‘if Forms are numbers’ but also emphasizes right after he presents Plato’s views that he takes this to be his master’s assumption (and the same goes for the other Academics). Here, the absurdity of attributing the identification thesis to Aristotle becomes obvious: it would mean that Aristotle includes in his portrayal of Plato this thesis—which he has made up himself—only to criticize Plato for basing his account of numbers on false hypotheses, including one Plato had never subscribed to. This would make Aristotle’s presentation more than just disingenuous. Ν 3.1090a16–20 is part of a section dealing with reasons for the existence of numbers (Ν 2.1090a2–4).Footnote 41 Its meaning is straightforward: proponents of the Forms—who assert that these are numbers—explain the Forms’ existence by positing one term over many, that is, the universal cat over the individual cats. Since these types of arguments are insufficient, so Aristotle, the existence of numbers cannot be based on them.Footnote 42

Second, the statements are presented as a set of doctrines: Plato (and/or other philosophers) claims or posits (θέμενος/τιθέμενοι) that (1) Forms exist and that (2) they are numbers, as well as, in the case of Μ 9, that (3) mathematicals are separate from Forms, that is, ideal numbers. This type of presentation makes it unlikely that Aristotle here incorporates a reconstruction, if one compares it to other doxographical accounts where the term τίθημι appears. We find plenty of such cases in Μ–Ν, for example Xenocrates identifying Forms with mathematical numbers and not seeing that he does away with the mathematical number if he posits (θήσεται) these principles (Μ 9.1086a5–11) or a reference to τῷ μὲν γὰρ ἰδέας τιθεμένῳ (Ν 2.1090a4). The same goes for Book A: Aristotle mentions, for instance, Anaximenes and Diogenes who posit (τιθέασι) air as principle (3.984a5–7) and those who posit (τιθέμενοι) Forms as causes (9.990a34–b1).Footnote 43 These are all matter-of-fact statements and cannot be considered merely hypothetical reconstructions. This does not exclude that there is an Aristotelian interpretation in the background—but it is part of the general caveat when faced with his doxographies. External evidence of these philosophers (in so far as it is independent of Aristotle) suggests that they did assume those principles.Footnote 44

There are various passages besides Μ 9.1086a11–13 and Ν 3.1090a16–20 (and Μ 7.1082b23–8, discussed above) which simply presuppose that Forms and numbers are in a certain way identical without elucidating that this is an assumption by Aristotle himself:

  1. 3. ταῦτα [sc. lines, planes and solids] γὰρ οὔτε εἴδη οἷόν τε εἶναι (οὐ γάρ εἰσιν ἀριθμοί)

  2. for these can neither be Forms (for they are not numbers) (Α 9.992b15–16)

  3. 4. ἡ μὲν γὰρ περὶ τὰς ἰδέας ὑπόληψις οὐδεμίαν ἔχει σκέψιν ἰδίαν (ἀριθμοὺς γὰρ λέγουσι τὰς ἰδέας οἱ λέγοντες ἰδέας, περὶ δὲ τῶν ἀριθμῶν ὁτὲ μὲν ὡς περὶ ἀπείρων λέγουσιν ὁτὲ δὲ ὡς μέχρι τῆς δεκάδος ὡρισμένων· δι’ ἣν δ’ αἰτίαν τοσοῦτον τὸ πλῆθος τῶν ἀριθμῶν, οὐδὲν λέγεται μετὰ σπουδῆς ἀποδεικτικῆς)·

  4. For the theory of Ideas has no specific investigation [of this question]; for those who claim that there are Ideas claim that the Ideas are numbers, but they speak of the numbers sometimes as infinite and sometimes as limited by the number 10; but as to the reason why this is the quantity of the numbers, nothing is said with demonstrative rigour. (Λ 8.1073a17–22)Footnote 45

  5. 5. οἱ μὲν οὖν ἀμϕοτέρους ϕασὶν εἶναι τοὺς ἀριθμούς, τὸν μὲν ἔχοντα τὸ πρότερον καὶ ὕστερον τὰς ἰδέας, τὸν δὲ μαθηματικὸν παρὰ τὰς ἰδέας καὶ τὰ αἰσθητά, καὶ χωριστοὺς ἀμϕοτέρους τῶν αἰσθητῶν·

  6. Some claim that both kinds of numbers exist, the one with a before and after being the Forms, and the other mathematical being distinct from both the Forms and the perceptible things, but both being separate from the perceptible things. (Μ 6.1080b11–14)

The context here is crucial.Footnote 46 Annas (n. 1), 64 correctly states that not all passages where we seem to get an identification carry the same weight. But she misses the point when she thinks that those which only briefly touch upon the issue—for example Α 9.992b15–16 (3), Λ 8.1073a17–22 (4) and Ν 3.1090a16–20 (2)—are to be devalued. Similarly, she objects to the weight of Μ 6.1080b11–14 (5) and Μ 9.1086a11–13 (1), as they are primarily concerned with distinguishing ideal numbers and mathematical numbers. Passage (5) is clearly concerned with ideal numbers and not number Forms, as the article τὰς ἰδέας makes clear, that is, one type of numbers, namely those which have an anterior and a posterior, are the Forms.Footnote 47 If Aristotle wanted to refer to Forms of numbers—as he does multiple times, in fact—he would have left the predicative ‘Forms’ without an article (Μ 8.1084a7–8, Ν 2.1090a5, Ν 4.1092a8). Far from being insignificant, these passages may be the most important pieces of evidence because, by casually inserting an identification of Forms and numbers, Aristotle presupposes that his reader is aware of it and accepts it. In other words, Aristotle does not have to argue for this position or present it as the conclusion of his own reconstruction. This becomes evident in a passage like Λ 8.1073a17–22 (4), where the context has little to do with ideal numbers; in fact, their relation is only discussed here in Book Λ.

CONCLUSION

Aristotle presents Plato’s identification of Forms and numbers as a development of his original theory of Forms. Perhaps under the influence of the Pythagoreans and by introducing the One and Indefinite Dyad as ultimate principles of reality, Plato introduces this identification. The considerable number of references to this identification in Aristotle, the diversity of contexts in which it appears and the manner in which he presents it make it, as I argued, unlikely that Aristotle himself invented it. It is simply implausible that in all the listed references Aristotle expects the reader (or listener) to recognize the identity thesis as his reconstruction of Plato, particularly if it is neither presented as such nor otherwise discussed in the context. This point is reinforced when considering the audience of Aristotle.Footnote 48 While precise statements concerning the date and audience of Α and Μ–Ν are impossible to make, one can reasonably assume that—given his exclusive engagement with the Academic theories of Plato, Xenocrates and Speusippus in Μ–Ν—a significant proportion of his addressees are members of the Academy who would have been well versed in these debates. If Plato (or, for that matter, Xenocrates or Speusippus) never admitted that Forms are somehow identical with ideal numbers, Aristotle’s entire argumentative strategy would be rendered ineffective and unpersuasive. For the Academics presumably had some knowledge of Plato’s theories which were still in circulation, given that Aristotle reacted to them so vigorously (just as Xenocrates and Speusippus did).

Now the problem is that this identification does not seem to occur anywhere in the dialogues but rather appears to be based on Plato’s unwritten doctrines.Footnote 49 Nevertheless, efforts have been made to locate it within Plato’s writings. For example, Sayre (n. 21) believes that of the four kinds mentioned in the ontological passage of the Philebus the limit(ed) should be identified with the Forms and, more importantly, that it should be characterized by imposing measure on things. The Forms are numbers in so far as they are measures.Footnote 50 In contrast, Burnyeat (n. 3) argues for the importance of the Republic in understanding Aristotle’s Metaph. Μ–Ν, especially Μ 6–9. According to Burnyeat’s interpretation of the line and cave analogies, Plato posits Forms of mathematical numbers, which are the highest Forms and explanatory of all other Forms. Unfortunately, Burnyeat only points to these highest Forms for understanding Aristotle’s phrase ‘Forms are numbers’ without spelling out what it would actually imply (Burnyeat [n. 3], 235 n. 57).Footnote 51

Leaving aside the success of such interpretations, one should be wary of a certain type of exegetical method that tries to brand everything in Aristotle’s doxography of Plato that is not found in the dialogues—but could not be presented in more explicit terms by Aristotle—as a mere ‘reconstruction’, ‘misinterpretation’, ‘inference’, etc. These mental acrobatics might be, in some instances, warranted but cannot be assumed as a general interpretative strategy when dealing with Aristotle’s views of Plato. A strong focus on Aristotle’s misrepresentations and misinterpretations risks missing the point of his discussion. Aristotle treats his predecessors neither for their own sake to provide an objective history of philosophy nor (only) to present himself as the pinnacle of a certain solution. In the case of the Metaphysics, his discussion serves a productive function in establishing his own project of wisdom.Footnote 52 Consequently, he starts his discussion in Μ–Ν by asserting that ‘first, we must investigate the opinions of the others, so that, if they are mistaken about something, we may not be guilty of the same mistakes’ (Μ 1.1076a12–14). An advance can only be made if Aristotle approaches his predecessors with a certain intellectual honesty—otherwise, it would be impossible to learn anything from them. This becomes clear not only in Α 3–6, where Aristotle identifies the principles of his predecessors with his four causes, but also in Α 8–10, where his critical examination of their views helps him make progress in his own project.Footnote 53 To reduce Aristotle’s position on Plato to pure polemics—to which an identification of Forms and numbers amounts, as Aristotle believes it is absurd anyway—risks misunderstanding how he engages with his predecessors and how this, in turn, helps us understand his metaphysical project.

Footnotes

*

I thank Christian Pfeiffer, Léa Derome and the anonymous referees of this journal for their helpful comments on earlier drafts of this paper.

References

1 Aristotle’s Metaphysics is cited after W.D. Ross, Aristotle’s Metaphysics. Volumes I–II (Oxford, 1924), except for Book Α for which I use the edition of O. Primavesi, ‘Aristotle, Metaphysics Α. A new critical edition with introduction’, in C. Steel (ed.), Aristotle’s Metaphysics Alpha. Symposium Aristotelicum (Oxford, 2012), 385–516. All translations are mine. For a distinction between the views of Plato, Speusippus and Xenocrates on Forms and numbers in Metaph. Μ–Ν, cf. e.g. Μ 1.1076a19–22, Μ 6.1080b4–30, Μ 8.1083a20–b8, Μ 9.1086a2–13. Cf. also Ζ 2.1028b18–27, Λ 1.1069a34–6. The communis opinio on these passages results in the following picture: Speusippus denies the existence of Forms and accepts only mathematical numbers, while Xenocrates identifies both Forms/ideal numbers and mathematical numbers (thereby, in fact, getting rid of mathematical numbers; cf. Μ 9.1086a5–11). The following passages, which are concentrated in Metaph. Α 9, Μ 6–9 and Ν 2–4, suggest an identification of Forms and ideal numbers in Plato: Α 9.991b9–21, Α 9.991b21–7, Α 9.992b13–17, Λ 8.1073a17–22, Μ 6.1080b11–14, Μ 7.1081a5–17, Μ 7.1082a28–b1, Μ 7.1082b23–8, Μ 8.1083a17–20, Μ 8.1083a21–2, Μ 8.1083b34–5, Μ 8.1084a7–10, Μ 8.1084a12–29, Μ 9.1086a11–13, Ν 2.1090a2–7, Ν 3.1090a16–20, Ν 4.1091b26–30, Ν 4.1092a5–8 and Ν 6.1093b21–4. On Α 6.987b18–25, see below. The only reference outside of Metaphysics is De an. 1.2.404b24–5 which has received different attributions; cf. J. Opsomer, ‘The Platonic soul, from the Early Academy to the first century CE’, in B. Inwood and J. Warren (edd.), Body and Soul in Hellenistic Philosophy (Cambridge, 2020), 171–98, at 174–5. Eth. Eud. 1.8.1218a15–32 has also been attributed to Plato by J. Brunschwig, ‘EE I 8 1218a 15–32 et le ΠEΡΙ ΤΑΓΑΘΟΥ’, in P. Moraux and D. Harlfinger (edd.), Untersuchungen zur Eudemischen Ethik. Symposium Aristotelicum V (Berlin, 1971), 197–224. For different lists of passages, cf. L. Robin, La théorie platonicienne des idées et des nombres d’après Aristote (Paris, 1908), 267–8 n. 254; J. Annas, Aristotle’s Metaphysics. Books Μ and Ν (Oxford, 1976), 64 n. 78; M. Crubellier, ‘The doctrine of forms under critique – part II (Metaphysics Α9, 991b9–993a10)’, in C. Steel (ed.), Aristotle’s Metaphysics Alpha. Symposium Aristotelicum (Oxford, 2012), 297–334, at 303 n. 14.

2 See the overview of different solutions in T.A. Szlezák, ‘Die Lückenhaftigkeit der akademischen Prinzipientheorie’, in A. Graeser (ed.), Mathematics and Metaphysics in Aristotle = Mathematik und Metaphysik bei Aristoteles (Bern and Stuttgart, 1987), 45–67, at 58.

3 Cf. Ross (n. 1), 2.421 and W.D. Ross, Plato’s Theory of Ideas (Oxford, 1951), 15, 162; H.J. Krämer, Arete bei Platon und Aristoteles. Zum Wesen und zur Geschichte der platonischen Ontologie (Heidelberg, 1959), 35; K. Gaiser, Platons ungeschriebene Lehre (Stuttgart, 1968), 294; I. Mueller, ‘Aristotle’s approach to the problem of principles in Metaphysics Μ and Ν’, in A. Graeser (ed.), Mathematics and Metaphysics in Aristotle = Mathematik und Metaphysik bei Aristoteles (Bern and Stuttgart, 1987), 241–59, at 243 n. 7; E. Berti, ‘Il Filebo e le dottrine non scritte di Platone’, in E. Berti (ed.), Nuovi studi aristotelici. Vol. II: Fisica, antropologia e metafisica (Brescia, 2007), 539–51, at 550; Crubellier (n. 1), 301–3 (the thesis is not an ‘unqualified’ identification for him); D. Lefebvre, ‘Aristote, lecteur de Platon’, in M. Dixsaut, A. Castel-Bouchouchi and G. Kévorkian (edd.), Lectures de Platon (Paris, 2013), 291–320, at 309–10; M. Rashed, ‘Plato’s five worlds hypothesis (Ti. 55cd), mathematics and universals’, in R. Chiaradonna and G. Galluzzo (edd.), Universals in Ancient Philosophy (Pisa, 2013), 87–112, at 94, 100–1; S. Menn, The Aim and the Argument of Aristotle’s Metaphysics. online: https://www.philosophie.hu-berlin.de/de/lehrbereiche/antike/mitarbeiter/menn/contents (accessed on 18 November 2024), 1β1, 8. Possibly also M.F. Burnyeat, ‘Platonism and mathematics. A prelude to discussion’, in A. Graeser (ed.), Mathematics and Metaphysics in Aristotle = Mathematik und Metaphysik bei Aristoteles (Bern and Stuttgart, 1987), 213–40, at 235 with n. 57 and M. Isnardi Parente, Testimonia Platonica. Per una raccolta dei passi principali della tradizione indiretta riguardante i λεγόμενα ἄγραϕα δόγματα. Le testimonianze di Aristotele (Rome, 1997), 428–9. The view goes back to the influential treatise of F.A. Trendelenburg, Platonis de ideis et numeris doctrina ex Aristotele illustrata (Leipzig, 1826), to which P. Shorey, De Platonis idearum doctrina atque mentis humanae notionibus commentatio (Munich, 1884) reacts.

4 For this view, cf. G. Teichmüller, Literarische Fehden im vierten Jahrhundert vor Chr. Vol. 1 (Breslau, 1881), 229–30 note; Shorey (n. 3), 31–9 and P. Shorey, The Unity of Plato’s Thought (Chicago, 1903), 82; H. Cherniss, Aristotle’s Criticism of Plato and the Academy. Vol. 1 (Baltimore, 1944), 194–5, 198 and H. Cherniss, The Riddle of the Early Academy (Berkeley, 1945), 58–9, 60; Annas (n. 1), 62–72, especially 67–8. L. Tarán, Speusippus of Athens. A Critical Study with a Collection of the Related Texts and Commentary (Leiden, 1981), 18 n. 88 holds that this view is likely and C. Steel, ‘Plato as seen by Aristotle (Metaphysics Α6)’, in C. Steel (ed.), Aristotle’s Metaphysics Alpha. Symposium Aristotelicum (Oxford, 2012), 167–200, at 189–91 is sympathetic. Robin (n. 1) takes a slightly different view. He argues that the theory of numbers is a later development (453) but believes that Aristotle’s account of its relationship to the theory of Forms is imprecise (268–9, 454)—in both resembling E. Zeller, Die Philosophie der Griechen in ihrer geschichtlichen Entwicklung dargestellt. Vol. 2 (Leipzig, 1922), 679–86, 750–60, 946–51. By focussing on a fragment of Theophrastus (Metaph. 6b11–15), Robin concludes that they are not identical and that ideal numbers are prior to ideas (458). For other earlier scholarship on this issue, see the Foreword in Cherniss (this note [1944]) and M. Isnardi Parente, E. Zeller – R. Mondolfo. La filosofia dei Greci nel sviluppo storico. Parte II, volume III/2 (Firenze, 1974), 734–8. Already Syrianus accuses Aristotle of using his own hypotheses (Commentary on Aristotle’s Metaphysics 80.21–2 ἰδίαις δὲ ὑποθέσεσι) in criticizing Plato in Μ–Ν.

5 See R.I. Marinescu, ‘Aristotle on Platonic efficient causes. A rehabilitation’, Elenchos 45 (2024), 203–28, which includes a bibliography on Aristotle as doxographer.

6 The eleven passages I deal with are (besides Α 6 and Μ 4) Α 9.991b9–21, Α 9.992b15–16, Λ 8.1073a17–22, Μ 6.1080b11–14, Μ 7.1081a5–17, Μ 7.1082b23–8, Μ 8.1083a17–20, Μ 8.1084a12–29, Μ 9.1086a11–13, Ν 3.1090a16–20 and Ν 4.1091b26–30. This list differs somewhat from Szlezák (n. 2), 58 n. 33 and Crubellier (n. 1), 303 n. 14.

7 Plato mentions Forms of numbers in Phd. 101b–d. On this, cf. Cherniss (n. 4 [1945]), 33–7. See also Resp. 525c–526b; Phlb. 56d–57a.

8 Ross (n. 3), 154 n. 3; H.J. Krämer, Der Ursprung der Geistesmetaphysik (Amsterdam, 1967), 45 n. 80. Robin (n. 1), 11 is unclear.

9 Ross (n. 1), 2.420–1 has shown conclusively that οἱ πρῶτοι τὰς ἰδέας ϕήσαντες cannot refer to Socrates (as J. Burnet, Greek Philosophy. Part 1. Thales to Plato [London, 1928], 157 claims, followed by L. Gerson, From Plato to Platonism [Ithaca, NY, 2013], 98–9 n. 4) but rather to Plato.

10 This phrase echoes Μ 1.1076a23 (which refers to mathematicals).

11 As Steel (n. 4), 189 correctly notes.

12 On this, cf. Menn (n. 3), 1β1, 7 and 1γ1.

13 For a more detailed overview, cf. Steel (n. 4), 186–7.

14 Perhaps also by Menn (n. 3), 1β1, 8.

15 On the textual issues, cf. Isnardi Parente (n. 3), 407, 412 and Isnardi Parente (n. 4), 107–9 n. 94.

16 Cf. Ν 5.1092b8–9.

17 Cf. Cherniss (n. 4 [1945]), 180–2 n. 104; M. Vogel, Platons Philosophie der Zahlen (Baden-Baden, 2024), 19.

18 Steel (n. 4), 186 believes the latter is unclear, but the expression τοῖς ἄλλοις emphasizes it. In the same way, it is used at Α 6.988a10–11; cf. Marinescu (n. 5).

19 I take Aristotle’s reference to Speusippus’ rejection of both the Forms conceived on their own (ἁπλῶς) and as some numbers (ἀριθμούς τινας) (Μ 8.1083a22) as possibly referring to both stages in Plato’s theory of Forms, although the thesis ‘Forms as numbers’ applies also to Xenocrates (cf. Tarán [n. 4], 312).

20 This proximity to the Pythagoreans and their possible influence are also suggested by Aristotle’s On the Good fr. 2 Ross. Cf. M. Crubellier, ‘Platon, les nombres et Aristote’, in E. Barbin and J.-P. Le Goff (edd.), La mémoire des nombres (Caen, 1997), 81–100, at 90–1.

21 Cf. K. Sayre, Plato’s Late Ontology (Princeton, NJ, 1983), 10.

22 I will not discuss the last passage here, as the same conclusions drawn from the other references apply to it. Similarly, one can find the claim ‘if numbers are Forms’ (Μ 7.1082a28–b1, Μ 8.1084a7–10, Ν 2.1090a2–7), which, as mentioned in my introduction, implies a less controversial doctrine and is thus of no concern here.

23 Cf. Cherniss (n. 4 [1944]), 180–2 n. 104 and Cherniss (n. 4 [1945]), 58–9; Annas (n. 1), 67–8; Steel (n. 4), 191.

24 This is an important restriction (pace Ross [n. 1], 2.426); cf. Isnardi Parente (n. 3), 442.

25 Aristotle takes Plato’s ideal numbers to be (iii): the units, for instance, of Threeness are combinable with each other but not with the units of, say, Fourness (cf. M. Crubellier, Les livres “Mu” et “Nu” de la Métaphysique d’Aristote : traduction et commentaire [Lille, 1994], 259). This view is usually rejected as an authentic interpretation of Plato’s doctrine; cf. I. Mueller, ‘On some Academic theories of mathematical objects’, JHS 106 (1986), 111–20, at 111–15. On Aristotle’s classification here, cf. L. Tarán, ‘Aristotle’s classification of number in Metaphysics Μ 6, l080a15–37’, GRBS 19 (1978), 83–90.

26 For a different understanding of συμβλητόν as ‘comparable’, cf. Crubellier (n. 25), 223–6 and Menn (n. 3), 1γ3, 49.

27 Cf. Μ 7.1082b23–8; Ν 1.1087a29–b18.

28 On the role of mathematicals, which Aristotle regards as intermediate between Forms (and, thus, ideal numbers) and sensibles in Plato, cf. especially Metaph. Α 6.987b14–18 and Β 6.1002b14–16. For a discussion, cf. J. Annas, ‘On the “Intermediates”’, AGPh 57 (1975), 146–66, who concludes that Aristotle’s reports are not based on Plato’s dialogues.

29 This position—namely that the units of a number differ from those of another number—brings its own set of issues according to Aristotle. As he pointed out in the previous part (b19–23), it would follow that ideal numbers are not comparable, e.g. the ideal three would not be greater than the ideal two.

30 Cf. Szlezák (n. 2), 52–3 n. 15; Crubellier (n. 25), 275.

31 Cf. Crubellier (n. 1), 299–301; D. Frede, ‘The doctrine of Forms under critique – Part I (Metaphysics Α 9, 990a33–991b9)’, in C. Steel (ed.), Aristotle’s Metaphysics Alpha. Symposium Aristotelicum (Oxford, 2012), 265–96, at 266, 269–70; Menn (n. 3), 1γ3, 2–3. Cf. e.g. the reference to an earlier discussion of Forms at Μ 1.1076a28–9.

32 Cf. also Ν 5.1092b8–25; Ν 6.1093a1–19. Annas (n. 1), 65–6 believes that this statement does not amount to an identification but rather deals with Forms of numbers or Academic theories of principles more generally. As evidence for her view, she takes αἴτιοι as referring to numbers, which is obviously true. Annas does not discuss why this would mean that only Forms of numbers are treated.

33 Cf. Metaph. Λ 10.1075b27–8.

34 For similar language, cf. Α 9.991b4–5; Λ 8.1073a17–18; Μ 7.1082b32–3; Μ 9.1086a15–16 (see below); Ν 2.1090a4–7. Additionally, cf. DC 1.9.278a16–17. This ‘hypothetical’ language goes back to Plato: cf. Phd. 100b3–9 and Prm. 128e–135c (especially 134e9–135a2).

35 This does not mean that every supposition is non-Aristotelian; see e.g. 991a2–3. Here, it is important to note that it is not the starting point of Aristotle’s discussion but located in the middle of his argument.

36 The idea here is that, at least for one group of the Pythagoreans, the elements of numbers, that is, the odd (limited) and the even (unlimited), are the elements of all things (Α 5.986a1–2, a17–19).

37 Cf. also Arist. Ph. 1.3.186a32–4 on Parmenides.

38 On the presence of this exegetical sensitivity in Μ–Ν, cf. Szlezák (n. 2), 66.

39 When he assumes later (1084a25) that the Form of man is Twoness, we must conclude that this is yet another example. This indicates that the theory in the background was either not fully developed or not completely understood by Aristotle. There is no reason to take this Aristotelian example as evidence for the Aristotelian provenance of the identification thesis itself. Pace Annas (n. 1), 179.

40 Annas does not discuss this passage. Cherniss (n. 4 [1944]) mentions it in different contexts at 480 and 483 but fails to refer to it in his subsequent work (n. 4 [1945]). Steel (n. 4), 188 thinks that this passage and Ν 3.1090a16–17 could be based on Aristotle’s conclusions.

41 The attribution of 1090a16–20 is uncertain, but, given the almost identical description in Μ 9.1086a11–12 (where Plato is addressed), Plato can be seen as implied here. Pace Tarán (n. 4), 320 n. 121.

42 Already Tarán (n. 4), 320 n. 121 expressed his amazement at Annas (n. 1), 208, who denies that an identification of Forms and numbers is meant here by claiming that only ‘some forms are numbers’, which contradicts the text.

43 Cf. also Α 3.984b20–2; 5.987a7; 7.988a35–b1; 8.988b22–3; 8.989b25–6.

44 Cf., for instance, on Anaximenes 138, 140–1 KRS and on Diogenes 598 KRS.

45 Connected to this passage is Μ 8.1084a12–17, which, although not referring explicitly to the identification thesis, implies it. If numbers run only to ten and Forms are numbers, ‘Forms will soon run out’ (ταχὺ ἐπιλείψει τὰ εἴδη)—because each Form would have to be assigned to a number and there could not be more than nine Forms presumably. Regardless of whether this and other absurd consequences drawn here (1084a18–29) are warranted, some form of identification is clearly in the background. Cf. also Arist. Ph. 3.6.206b27–33.

46 Worth mentioning is also Aristotle’s characterization of Speusippus (among others) as not identifying Forms and numbers (Μ 6.1080b27–8; M 8.1083a21–2), which in turn implies that some do identify Forms and numbers.

47 Pace Annas (n. 1), 164.

48 On this, cf. Crubellier (n. 1), 303.

49 Cf. Crubellier (n. 1), 303 and Isnardi Parente (n. 3), 413.

50 On the problematic understanding of numbers as measures, cf. Vogel (n. 17), 48–9.

51 For other (failed) attempts, cf. Cherniss (n. 4 [1944]), xiii.

52 Cf. Marinescu (n. 5).

53 On this, cf. O. Primavesi, ‘Second thoughts on some Presocratics (Metaphysics Α 8, 989a18–990a32)’, in C. Steel (ed.), Aristotle’s Metaphysics Alpha. Symposium Aristotelicum (Oxford, 2012), 225–63, at especially 226–7.