Aristotle’s Critique of Form-Number

1 Introduction

Plato posited Form-numbers to account for ordinary numbers. Just as a Form in general represents a One over its many particular instances, a Form-number, say Three, represents the Unity of the many phenomenal threes in circulation. Thus Three was thought to impose form and structure onto the three units that happened to make up a particular three and which would otherwise lack the requisite intelligibility as three. Another closely related function of Form-numbers was to guarantee order to ordinary numbers. Judging from Aristotle’s account, Plato seems to have thought that if ordinary numbers are ordinal and participate in Form-numbers, their ordinality must be the result of Form-numbers’ ordinality. As a hypostatisation of the natural numbers, the Form-numbers were thought to exhibit ordinality independently of ordinary numbers.

Many scholars take the view that the Form-numbers exhibit ordinality per se as obvious. They claim that Aristotle not only reported on Form-numbers’ (purported) ordinality but also granted it to Plato. But there is prima facie reason to rethink this view. First of all, reporting on a particular view need not mean endorsement and the reporter may have independent reasons for resisting it. More importantly, if the Form-numbers are indeed a hypostatisation of ordinary numbers, one might well think the reverse and say that it is Form-numbers’ ordinality that is based on ordinary numbers’ ordinality.

My aim in this article is twofold. First, I want to show that for Aristotle not only can there not be ordinality among Form-numbers, but positing Form-numbers brings disorder rather than order in our usual understanding of the number-series. Second, the broader aim is to show that Aristotle’s critique of Form-number forms part of an effort to disentangle properly philosophical questions from mathematical ones. I begin in Section 2 by arguing against the abovementioned scholarly view by reconsidering Metaph. M 6.1080a14–36, where Aristotle offers his classification of ideal number. I focus on the fact that Aristotle characterises Form-number as contiguous to one another, while mathematical numbers are successive. In Section 3 I set Aristotle’s critique within the larger context of his concern to find principles commensurate to the things they are intended to account for. I suggest that Aristotle’s refusal to consider number in terms of form is part of his larger project to separate metaphysics from mathematics. In Section 4 I present three consequences Aristotle says would follow from positing the Form-numbers. This section is intended to show systematically that for Aristotle the Form-numbers positively disrupt our understanding of the order of numbers. Finally, Section 5 reinforces my argument by looking at how some of the later Platonists set about answering questions about number property, particularly primeness. I show that the consequences from Section 4 are all found in the later Platonists and prevented them from considering two as a prime.

2 Ordinality among Form-Numbers?

A commonly discussed passage for Aristotle’s views on Form-numbers is Metaph. M 6.1080a14–36. There Aristotle classifies four ways in which numbers and their units must exist if numbers are separate substances:

1) there is a first in it and a second (τὸ μὲν πρῶτόν τι αὐτοῦ τὸ δ᾽ ἐχόμενον), each being different in kind (τῷ εἴδει), and this is true of the units without qualification, and any unit is non-comparable with any unit

2) they [units] are all directly successive (εὐθὺς ἐφεξῆς), and any of them is comparable with any, as they say is the case with mathematical number

3) some units must be comparable and some not

4) Or one kind of number is like the first that was named, one like that which the mathematicians speak of, and that which we have named last must be a third kind.^[1]

Much of the discussion on this passage has focused on whether Plato held that Form-numbers contain units and conclude that Aristotle mischaracterised Plato’s views.^[2] Here I want to focus on a different aspect of the passage and show that Aristotle’s subsequent arguments can be made intelligible whether or not his monadic (unit-containing) characterisation of Form-numbers is correct. A number of scholars take Aristotle in this passage to be suggesting that Form-numbers exhibit ordinality. Ross (1924, 427), for example, equates the Form-numbers with the natural numbers and claims that “oneness”, “twoness”, and “threeness” exhibit the order of the natural numbers 1, 2, and 3. Tarán (1981, 15) cites the above passage and claims that “Aristotle refers more than once to the ‘ordinal’ aspect of the ideal numbers, explicitly stating that they stand to one another in the relation of prior to posterior”. Similarly, Annas (1976, 163) takes the Form-numbers to be in an “ordered series”, and Burnyeat (2012, 169) even states that “According to Aristotle, Form numbers, unlike mathematical numbers, exhibit serial order” (emphasis added). Cleary (1999, 438–9) not only argues that Plato’s Form-numbers are ordinal, he also suggests that “Plato’s originality lay in grasping the primacy of ordinal number over cardinal number”.

In the quoted passage, however, Aristotle seems to be refusing ordinality to Form-numbers. In particular, the views just mentioned overlook the contrast Aristotle makes between two types of relation in 1) and 2). Aristotle says that if each number differs in kind, there will be a first and one that is ἐχόμενον to it, whereas comparable units imply that numbers are ἐφεξῆς. For Aristotle, that the units are comparable is the explanation for why (διό) mathematical number is successive, while each of the Form-numbers are considered distinct and non-addible (1080a30–34). Whether the units in Form-numbers are incomparable completely or only partially would be irrelevant in this context, as long as each Form-number is considered formally distinct. In the case of Forms, being distinct in form means to be unique, or individual, and Aristotle thinks that ‘among individuals’ (ἀτόμοις) one is not prior and another posterior.^[3] The apparent claim then is that formal distinction among numbers is not sufficient to guarantee the kind of successive order exhibited by the number-series. Aristotle’s distinction is obscured, however, when τὸ ἐχόμενον is translated “a next in succession” (Annas 1976, 100) or “a successor” (Reeve 2016, 224). Ross’s “a second” in the translation quoted above also seems unsatisfactory as it assumes, unwarrantedly for Aristotle, ordinality.

There are compelling reasons to think that Aristotle is using ἐχόμενον and ἐφεξῆς deliberately. To see this, I suggest we look briefly at Aristotle’s report of the Platonist account of number generation. On Aristotle’s telling, Form-numbers were generated by an interaction between a formal principle and a material principle, the One and the Indefinite Two.^[4] While different theories about the type of interaction involved have been suggested, the Form-numbers seem to be produced when the One acts upon the Indefinite Two, or the “Great-and-Small” and limits it.^[5] Robin (1908, 446–7), for example, suggests that the One “stops” the material principle by an “act which determines and limits”, each time the expanding material principle reaches the right number.^[6] On this picture, the Great-and-Small is a sort of increasing continuum which, when the correct ‘amount’ is reached, is marked off by the One. Besides the fact that some sort of action is attributed to the One, this raises the question of what it is that differentiates one Form-number from another, if each Form-number is supposed to differ essentially. Arguably for Aristotle, one must first identify the nature of the difference between various things in order to subsequently claim that the difference is such that it constitutes a series among those things. On the view mentioned, however, the only thing that seems to differentiate one Form-number from another is the ‘magnitude’ each number acquires by the limiting act of the One. Thus what differentiates numbers would be on the side of the matter rather than the form. This is perhaps why Aristotle, in a critical remark about the Platonist conception of number, says that “they [the Platonists] make many things from the matter, whereas the form generates only once” (Metaph. A 6.988a2).

Since the Platonist position is that the Form-numbers are distinct formally, the above view would be hardly satisfactory. There must be something more intrinsic (i.e. non-material) to each Form-number that differentiates it. One might respond by saying that the Form-numbers, as simple essences, are simply unanalysable into prior elements, that “numbers are just Twoness, Threeness, and Fiveness, each being a unity which is irreducibly itself and nothing else” (Tarán 1981, 14). But as mentioned, simply saying that the numbers are unique does not specify what exactly necessitates their successive order. An alternative view locates the differentiating factor in position. According to Blyth (2000, 24 and 30), each Form-number is “differentiated essentially by position in sequence” and “defined by its discrete position in the sequence”.^[7] As a ‘progression’ of forms, each number “progresses from one to the next as distinct entities”. Thus, the Platonic numbers are held to be “essentially ordinals, not cardinals”. Blyth’s view of number (2000, 27–8) is based on the Platonic premise that if a group is irreducible to its members, it must be ontologically prior to them. The priority in question is cashed out in terms of a one-to-one mapping of contingent quantities onto a necessary sequence comprising the Form-numbers. Thus counting for Blyth requires reference to independently existing numbers by which we count (also Tarán 1981, 15). However, Aristotle himself was aware of this line of thought when he says that the Platonists puzzled over whether we count by addition or by “separate parts”. His considered view is that we use both ways and the difference between the two should not warrant the positing of separate numbers.^[8] More to the point, Blyth’s argumentation seems to assume the very point at issue. If the Form-numbers do in fact progress in a sequence, then they would occupy positions in that sequence. But the whole point is to show that they progress in a sequence, and not only the that but the “how and why”, to use Aristotle’s own phrase. What is it about the Form-numbers that causes them to be positioned in precisely that order?

The model on which Aristotle construed the ‘progression’ of the Form-numbers is what might loosely be called the points-along-a-line model. The two views mentioned above seem to more or less exemplify this model. Robin’s Form-numbers seem quite literally to be ‘stopping points’ on a continuum, and Blyth’s Form-numbers take up a ‘discrete position’ in a continuous ‘progression’. More importantly, Plato’s own statements in the Parmenides pertaining to number generation (148d–149d) seem to justify this model. In drawing out the many consequences of a One that is, Parmenides asks “whether the one touches (ἅπτεσθαι) or does not touch itself and the others” (148d).^[9] Both positive and negative answers are given, depending on the aspect of the One considered. As something “in itself” (ἐν ἑαυτῷ), the One does not touch the others but only itself; but as something also “in the others” (ἐν τοῖς ἄλλοις), it also touches them. Thus the One is considered to touch both itself and the others. However, further consideration of the notion of ‘touch’ complexifies Parmenides’s answer. Touching means that two (or more) things are “adjacent” (ἐφεξῆς) to each other; but the One is not two but one; therefore, it cannot touch itself. The same reasoning is then applied to ‘the others’ to establish that the One does not touch them either. Touch requires there being more than one thing, and “if there is only one, and not two, there could not be contact” (149c). Moreover, ‘the others’ cannot be said to be ones since they are ‘other’ than the One. It follows from this that ‘the others’ cannot be numbers (ibid.): Number requires ones other than the One that are touched by the One, which ex hypothesi is impossible.^[10] Hence, the One alone is one and there cannot be any contact between it and the others. Summing up both his answers, Parmenides claims that “the one both touches and does not touch the others and itself”.

Parmenides (or Plato) of course may simply be engaging in a dialectical exercise without committing to either view. But the takeaway for our discussion is that a model is proposed in which numbers require other ones that are touched by the One and, by implication, touch each other.^[11] Parmenides thus seems to suppose that contact between several ones is necessary for the existence of numbers. This allows us to see the significance of Aristotle’s distinction between ἐχόμενον and ἐφεξῆς. At Phys. V 3.227a10, Aristotle defines “the contiguous” (τὸ ἐχόμενον) as a “thing that is in succession and touches” (ἅπτηται).^[12] Contiguity is a specific type of succession, which means that not all things that are successive touch. Based on this distinction, Aristotle argues against those who separated the point and the unit:

Hence, if as some say points and units have an independent existence of their own (κεχωρισμένας), it is impossible for the two to be identical; for points can touch while units can only be in succession (ταῖς μὲν γὰρ ὑπάρχει τὸ ἅπτεσθαι, ταῖς δὲ μονάσιν τὸ ἐφεξῆς).^[13]

The point Aristotle wants to make against the Platonists^[14] is that the point and the unit cannot be identical. This implies that the Platonists, despite their giving separate existence to these two types of indivisibles, failed to see the difference between them. Aristotle distinguishes the point and the unit by saying that “points can touch while units can only be in succession”. However, the translation of this passage can be misleading and makes Aristotle flatly contradict what he says in the following book of Physics, that indivisibles do not touch, and so putting together points does not make a line (VI 1.231a23–24). A literal translation of ταῖς μὲν γὰρ ὑπάρχει τὸ ἅπτεσθαι would be “for in case of points, touching inheres”,^[15] which is quite different from saying “points can touch”. Based on Metaph. Δ 6, which defines the point as an indivisible quantity having position and the unit as an indivisible quantity without position, Aristotle seems to have in mind that the point is such as to be an abstraction from things capable of touching, like magnitudes. A point is what is reached after dividing magnitude, and hence is called “indivisible” (Metaph. Δ 6.1016b27–30). By contrast, units are only successive; they do not involve magnitude and so do not have position. In Categories 6 Aristotle clarifies how numbers and magnitudes differ based on the distinction between the point and the unit. The parts of a number (units) do not touch because they have no common boundary (κοινὸς ὅρος), whereas the parts of a line (segments) do have a common boundary at which they touch, namely the point (4b25 and 5a1–2). Hence “there is nothing between the numbers one and two” (Phys. V 3.227a35), but there would be something between two points in the case of a line, namely, extension.^[16]

Thus underlying Aristotle’s characterisation of Form-numbers as contiguous is the criticism that the Platonist conception of number is not sufficiently ‘formal’. Whether number is conceived as being determined by the One’s limiting act on the Great-and-Small, as taking up different positions on a continuous progression, or as involving contact between different ones, the Platonist account of number is essentially bound up with magnitudinal considerations. But the very notion of number as Form would require the Platonist to conceive number as discrete rather than continuous. Form is if anything separate and unitary, which runs counter to the idea of Forms ‘touching’ each other. Aristotle’s own view of number is based on the unit, which is up another level of abstraction from the point by lacking position, and is in that sense more ‘formal’. Because of their formality, it does not matter how the units in a given number are positioned relative to each other; it is enough for them to be counted together in one act of counting. Aristotle’s critique then would be twofold. First, the relevant successiveness in numbers, or ordinality, requires that each term of the series be discrete rather than continuous.^[17] If number is Form, this much would have been obvious. But second, Plato also wanted a generative account of number, in which the One functions not only as a unit but also as having a sort of causal efficacy. The One either ‘limits’ a material principle or ‘touches’ other ones. And since numbers were generated by the One, they must have been thought to be continuous from the One.^[18] Thus the order of Form-numbers was explained not through something inherent to the Form-numbers themselves, which as individuals lacked the relevant difference for priority and posteriority, but through the causality of the One that produced them. In taking a ‘formal’ approach to the problem, Aristotle is giving his negative answer to Parmenides’s question, “Does the One touch other ones or not?” and in so doing puts forward his own method for metaphysics.

3 Different Types of ‘One’

In the previous section we saw how Aristotle’s distinction between the discrete and the continuous bears on his view of number ordinality. The type of ‘one’ required for the number-series was different from the type of ‘one’ required for magnitudes. But Aristotle’s critique of Form-number can be viewed from the broader context of the Metaphysics, where his primary concern is another type of ‘one’ relevant for our conception of form. Here I want to briefly show that this was Aristotle’s concern and focus on the two types of ‘one’ that will be instrumental for our discussion in the following section. In distinguishing these different types of ‘one’, Aristotle will again make the point that the Platonist conception of Form was not formal enough due to its neglect of the different ways things exhibit oneness.

A number of Metaphysics passages suggest that Aristotle was critical of thinkers who in his eyes failed to see the distinctness of different types of ‘one’. A case in point is when Aristotle, in his discussion of the eleventh aporia, reports the Zenonian view that whatever lacks spatial magnitude is not a being (Metaph. B 4.1001b4–9): “that which neither when added makes a thing greater nor when subtracted makes it less, he asserts to have no being”. Since the point and the unit, as indivisibles, seemed immaterial to the thing they were added to or subtracted from, Zeno reportedly concluded that the one-itself, as an indivisible, is nothing. It followed from this that the Platonic and Pythagorean one-itself would be non-existent. In the immediate context, Aristotle defends the one-itself from Zeno’s ‘crude’ reasoning by pointing out that there is a sense of the indivisible that makes a thing more numerous (πλεῖον) and not greater in magnitude (μεῖζον). The unit, for example, is not nothing just because it does not add to magnitude. But having said that, Aristotle also distances himself from the Platonist and Pythagorean one by turning the tables on them: If the one-itself is indeed indivisible, how does magnitude, which is divisible, come from it?

Aristotle’s concern to distinguish between different types of one grows out of his concern to discover causes and principles commensurate to the ‘one’ at hand. If one type of ‘one’ differs non-trivially from another, then presumably one would have to posit two different causes or principles for each. Aristotle’s report that Speusippus was led to posit distinct ‘ones’ for numbers, for magnitudes, and for the soul in addition to the one-itself suggests that discussions about the different senses of one were already active within the Academy (Metaph. Ζ 2.1028b21–22). Aristotle’s critique of Form-numbers then is a continuation of that discussion and closely relates to his broader critique of the Platonist account of being. Aristotle contends that the Platonists adopted two different standpoints at the same time, that of mathematics and universal accounts (Metaph. M 8.1084b23–33). From a mathematical viewpoint, they treated the ‘one’ as a unit. ‘One’ was thus construed as the ἀρχή of mathematical number, and since a number is made up of units, this ‘one’ was taken to be a principle qua element.^[19] The problematic move, according to Aristotle, was when the mathematical conception of ‘one’ was generalised to cover not just numbers but beings (τὰ ὄντα) in general. That is, the Platonists “make 1 the starting-point in both ways” (1084b18). Because they were searching for universals, the Platonists are said to have “taken the predicable one also as a part in this way” (τὸ κατηγορούμενον ἓν καὶ οὕτως ὡς μέρος ἔλεγον).^[20] It is important to note the opposition being made here between two types of ‘one’. Ross’s translation of this passage seems unsatisfactory as it limits the scope of the ‘predicable one’ to the ‘mathematical one’: “they treated the unity which can be predicated of a number, as in this sense also a part of the number” (emphases added).^[21] The context however requires that Aristotle set a universal sense of ‘one’, which cannot be taken as a part or element, against the unit which is a part or element.

The general discussion (1084b2–31) shows that the two senses of ‘one’ Aristotle distinguishes are understood on the analogy of form and matter. In one way, ‘one’ is a principle as form or substance (ὡς εἶδος καὶ ἡ οὐσία), in another way as part or matter (ὡς μέρος καὶ ὡς ὕλη). For Aristotle, the universal ‘one’ cannot be a part or an element since it is predicated of all things, and elements are not predicated of things they are the elements of (1088b4). The elements of a composite are not the same as the composite itself, which is why Aristotle notes that ‘being’ and ‘one’ cannot be elements.^[22] The same point is made in terms of the notion of indivisibility. As Aristotle sees it, the Platonists posited a single ‘one’ as principle for both unit and form because both seemed to be indivisible. Aristotle’s view is that these are indeed indivisible but in different ways. The unit is indivisible in discrete quantity while the form is indivisible in the category of substance.^[23] Since Aristotle thinks the indivisible comes to be known by dividing what is divisible,^[24] our concept of the quantitatively indivisible (unit) would ultimately depend on the quantitatively divisible (extension). Further, extension or ‘intelligible matter’ for Aristotle results when we consider sensible matter qua extended.^[25] Hence to posit a single ‘one’ as principle for two different things, unit and form, would be to conceive that principle partly in material terms. Thus Aristotle charges that the Platonists’ error stemmed from making every principle an element (1092a5). Treating the universal one which is predicable of the form or substance like an element led these thinkers to “put things together out of the smallest parts” (1084b27).

So far, I have tried to show two main points: that number for Aristotle is not the same as magnitude, and so the Platonist view of number which was conceived in magnitudinal terms cannot explain number ordinality (Section 2); and that the type of ‘one’ constitutive of beings is different from the type of ‘one’ constitutive of numbers (Section 3). These two points seem to be related. Aristotle’s general criticism is that in their attempt to generate all things from a single One, the Platonists conflated different senses of ‘one’ and were lead to incorrect views about numbers and beings. In particular, we have seen that Parmenides’s One was one in two ways: as something separate (Form) which does not touch anything, and as something in other things and touching them (element). By distinguishing these two senses of one and drawing out the consequences, Aristotle wants to say that the same type of ‘one’ cannot be responsible for both numbers and beings: numbers require the mathematical one, or the unit, and beings require the ‘metaphysical’ one, or form.

4 Three Consequences of Positing the Form-Numbers

In this section I want to look at some consequences Aristotle says would follow from the Platonist view of number. Focusing on specific arguments in light of the different senses of ‘one’ distinguished in the previous sections, the aim will be to show that positing Form-numbers results in a disordering of our ordinary understanding of numbers.

5 Is Two a Prime?

While the evidence for the views Aristotle attributes to Plato and the Academy are often indirect reports given by Aristotle himself, developments in later Platonism provide us with enough material to gauge the validity of his criticisms. In this section I would like to put Aristotle’s criticisms to the test by focusing on a specific problem. One question that arises in connection to the Platonist view of number is which sort of object one has in mind when determining number properties. Plato himself thought it was arithmetic that deals with the even and odd, and “everything that pertains to number”.^[42] He distinguishes the theoretical study of numbers considered in themselves from a practical use of numbers that pertains to “visible or tangible bodies”. Arithmetical number can be “grasped only in thought” and is studied for the sake of knowledge, whereas the art of calculation is used for trading and the like (Resp. VII 525d–526a). Given that theoretical knowledge for Plato is not to be looked for in perception, this would explain Aristotle’s earlier statement that the Platonists separated the unit.^[43] Thus the object of arithmetic must be on another level than the sensible and, according to Aristotle, “they [Platonists] must set up a second kind of number (with which arithmetic deals)” (Metaph. A 9.991b27). These ‘intermediate’ numbers, as Aristotle calls them, differ from ordinary numbers by being “eternal and unchangeable” and from Form-numbers in that there are many of them (Metaph. A 6.987b14–15).

Suppose now that we want to find out which numbers are prime. In particular, one may wonder whether there are any even numbers that are prime. While the answer to this question is considered obvious today, Aristotle in several passages suggests that the question was far from settled in his own day. For example, in Topics VIII 2 (157a34–b2) Aristotle is concerned with how to object to a universal proposition that purportedly derives from particular instances. Ordinarily, denying a universal proposition involves adducing a counter-example that the said proposition fails to take account of. However, Aristotle makes an exception in cases where the only available counter-example is among the instances that have already been enumerated. An example of such a counter-example, he says, is “two is the only even number which is a prime”. The universal proposition against which this serves as a counter-example is obviously “No even number is a prime”, and the particular instances on which it is based must have included two, e.g. two, four, six.^[44]

In another passage, Aristotle says specifically why confusion might arise when it comes to deciding questions like whether two is a prime. In Metaph. Θ 10, Aristotle states that the question whether any even number is a prime is prone to error: “while we may suppose that no even number is prime, we may suppose that some are and some are not” (1052a8–9). To understand the significance of this statement, we need to consider it within the general context.^[45] The statement occurs in a discussion of truth and falsity as they pertain to necessary or contingent states of affairs. The contingent state of a white man, for example, is truthfully stated when ‘white’ and ‘man’ are combined; separating the two when in fact they are combined will mean falsity. However, there are also items that are always combined and always divided (1051b8–10). Unlike contingent statements or thoughts whose truth value depends on the time at which they occur, statements or thoughts about “incomposites” (ἀσύνθετα) are said to be always true or always false (1051b16). It is commonly accepted that what Aristotle is referring to with ‘incomposites’ are essences. While ‘white’ and ‘man’ are distinct items that can be either combined or separated, the parts of an essence are always combined, as in ‘two-footed-animal’. Given this nature of essence, one essence will always be divided from another essence. The relevant assessment of thoughts or statements about such objects then will no longer be in terms of separation or combination. Instead, truth and falsity are said to consist in contact or non-contact: one either grasps the object or fails to do so, and failure in this case simply means ignorance (1051b18–25). Crucial to my interpretation of Aristotle’s remark about the possibility of error regarding the primeness of even number is the further distinction Aristotle apparently makes among incomposites. After saying that incomposites require a different mode of truth and falsity, he says “and the same holds good regarding non-composite substances” (μὴ συνθετὰς οὐσίας). If by ‘incomposites’ Aristotle simply meant essences in general, it would be difficult to make sense of this additional remark. Rather, he seems to be making a distinction between essences that are incomposite in themselves but which always occur in a composite and essences that are substances in their own right (“And they all exist actually, not potentially”, 1051b27).

It is on the basis of this reading that I propose to interpret the (apparently) mathematical examples that follow (1052a4–10). We must first note how Aristotle begins the paragraph: “It is evident also that about unchangeable things (περὶ τῶν ἀκινήτων) there can be no error in respect of time, if we assume them to be unchangeable” (1052a4–5). The reasoning that follows proceeds in three steps. The example of the triangle is meant to show that error about mathematical objects does not occur in terms of time (1052a5–7). The example about the possible primeness of even number is meant to show in what terms there can be mistakes in mathematical reasoning (1052a8–9). The final remark that “ἀριθμῷ δὲ περὶ ἕνα not even this form of error is possible” (1052a9–10) is the key to understanding the entire chapter and whose interpretation I want to dispute.

The most relevant passage for making sense of the absence or presence of cognitive fallibility in regard to unchangeable things seems to be Metaph. Ε 1, where Aristotle distinguishes mathematical science and theological science. There he says that both mathematics and theology are about unchangeable objects, but adds that mathematics deals with objects inseparable from matter (ὡς ἐν ὕλῃ), while first philosophy deals with things that are unchangeable and separable (1026a14–16). Tying this into our discussion, mathematics and theology would both be immune to temporal error since they deal with unchangeable objects. If one thinks that the triangle is such that statements about it cannot be true at one time and not true at another, the same would apply to the object of theology. Still, Aristotle seems to be saying, that does not mean mathematical truth and truth about separate substance are on the same level. While mathematical thinking is immune to temporal error, it is not immune to another sort of error, namely, when determining whether an attribute belongs to a certain class or not (1052a7). Based on the Metaph E 1 passage, fallibility in mathematical thinking must derive from its inherent dependence on matter. Aristotle gives the incommensurability of the diagonal as an example of a composite truth (1051b19). Even if one knows the essence of the square, one may still be in doubt about whether there can be a common measure for its side and diagonal. On our interpretation, this is ultimately due to the material aspect of geometrical figures, which requires the measuring of lengths.^[46] Similarly, even if one knows what ‘prime number’ means, one may still wonder which of the numbers fall under it. The answer again depends on the material aspect of number, whose determination involves counting the units, and temporal conditions (being eternal) are considered irrelevant.^[47] For Aristotle, mathematical objects are not the type of objects which, once one grasps their essences, one either grasps the whole truth about them or not. But if something is entirely separate from matter, then arguably these sorts of doubts would be precluded from the outset. Contrary to a common reading, then, Aristotle seems to be contrasting mathematical thinking, which is unmistakable as regards time but mistakable in other respects, to thinking “about something that is one numerically” (ἀριθμῷ δὲ περὶ ἕνα), namely separate substance, where “not even this form of error is possible”.^[48] Having alluded to the possibility of separate essence, Aristotle is adding the important qualification that what he is referring to is not, as the Platonists would have it, the object of mathematics.^[49] This is the theoretical background that allows Aristotle to hold two as a prime,^[50] which marks a departure from the earlier Pythagorean view and predates Euclid’s definition of prime number as a number measurable by the unit alone.^[51]

Now it has been pointed out that the later Platonists retained the earlier Pythagorean position on this issue. As Tannery (1912, 196), Heath (1921, 84), and Klein (1968, 111) show, the later Platonists excluded two from the prime numbers, despite their knowledge of Euclid’s definition. However, that they did so is less important for our purposes than their reasoning behind it. It must first be pointed out that there is nothing in Plato’s intermediate numbers that prevents one from holding two as a prime. The intermediate numbers consist of pure, separate units and are susceptible to quantitative considerations. But Plato also thought that number properties are separate forms. In Phaedo 103e, he says there is a Form of Odd and a Form of Even, in participation of which numbers become either odd or even. Apparently, it was this aspect of number property, its separate and eternal nature, coupled with the view of two as material principle, that the Platonists latched onto when deciding which number is prime. Thus instead of looking for numbers that satisfy primeness (counting the units), they proceeded top-down and seem to have been prevented from seeing an exception. For example, Heath (1908, 285) reports that Iamblichus first posited the Odd and the Even as two genera of number. He then subdivided even number into the even-times-even, even-times-odd, and odd-times-even; and since none of these subdivisions seemed to include primeness, he excluded two from prime numbers. For Iamblichus, two was only potentially an even-times-odd number, that is, when the potentially odd (the unit) is multiplied by two. Iamblichus, however, was only following Nichomachus’s Introduction to Arithmetic, where prime number is said to be a subdivision of odd number and three is claimed to be the first prime.^[52] By contrast, Aristotle held that primeness belongs to number qua number and regarded terms like odd, even, and prime as attributes.^[53] Perhaps the most explicit formulations of the later Platonist conception of two is found in the Theology of Arithmetic. In it, the anonymous author states that two “is not number, nor even, because it is not actual”.^[54] Moreover, two is said to be “mid-point” between one and plurality. The first plurality is three, which explains why two is not a number.^[55] One might object that the author is dealing with the principle rather than the number two. However, all the usual mathematical operations are said to yield with this ‘two’. For example, the sum of two and two is said to equal their product.^[56] Interestingly, the only exception to the prevailing view seems to have been Domninus of Larissa, rival of Proclus, who according to Tannery (1912, 113) was careful to include two among the primes. Domninus’s deviation went hand in hand with his reaction against Nicomachus and his effort to return to Euclidean principles (107).

What does this brief discussion show? It certainly does not show what Plato himself thought on the issue of two’s primeness. Plato in fact seems silent on the matter.^[57] What it does show is that the three consequences that Aristotle said would follow from positing the Form-numbers are witnessed as a whole in the later Platonists. For these thinkers, two is neither a prime nor the first number because it is not a number at all. Two is considered “not actual”, that is, indeterminate, and its ambiguous position between the One and the Many renders it not-yet-number. Moreover, this two, as what is only potentially number, must be prior to the unit. On the Platonist view, what we take to be the first number after one does not follow in any evident manner, nor do any of the properties that we normally ascribe to it. But given that these are views Aristotle infers from Plato himself, and given his attempts to delimit mathematical procedures from metaphysical ones, it seems doubtful that Plato would have given a clear answer to this issue.

6 Conclusions

The initial aim was to show, based on Aristotle’s arguments, that his critique of Form-number involves disentangling metaphysical questions from mathematical ones, which Plato apparently failed to do. This was most obvious in the case of the Two, on which the Form-numbers depended as their principle. As we have seen, in the Platonist conception of the Two a numerical notion was intertwined with a metaphysical one. As a ‘multiplier’ of the One, the Indefinite Two had to be some sort of multiplicity. On the other hand, the view that the Two was only potentially number made it ambiguous whether two was the first number (if a number at all), and affected the later Platonist view concerning two as a prime. An aspect of the Indefinite Two, moreover, was seen to be common to both it and the Form-numbers in general, namely their having been construed in terms of the apeiron. Some problems followed directly from the Form-numbers themselves: two becoming prior to one and the Form-numbers becoming the units of mathematical number. It was argued that these problems stemmed from an indiscriminating use of different types of ‘one’. By distinguishing the type of ‘one’ constitutive of number from that constitutive of beings, Aristotle separates form from number and delimits their respective disciplines. The science of being treats one qua one and not qua numbers, lines, or fire (Metaph. Γ 2.1004b5–6). Moreover, it was suggested that Plato’s construal of number ordinality in magnitudinal terms was for Aristotle a manifestation of overlooking the inherent features different objects exhibit, in favor of an extensional approach centred on a single type of ‘one’.