Mathematical Substances in Aristotle’s Metaphysics B.5: Aporia 12 Revisited

1 Introduction

In Metaphysics B, Aristotle discusses two sets of views that hypostatize mathematical entities: one in B.2’s aporia 5 and the other in B.5’s aporia 12.^[1] He attributes the first set, which posit “so-called intermediates” (998a7), to some of the proponents of Platonic forms (995b16–17).^[2] These views about intermediates have attracted considerable interpretative attention.^[3] Aristotle is vague about the proponents of the second set of views, which hypostatize mathematical entities that are thought to limit sensible things. These views have received relatively little consideration in their own right.^[4] This may be because aporia 12 appears to be a reprise of the aporia 5 question about the substancehood of intermediates, only this time with a different kind of mathematical entity (or perhaps with the same entity under a different description). Thus for example, John Cleary argues that aporiai 5 and 12 are, respectively, “general and particular forms of the same problem; i. e. whether or not the objects of all the sciences are independent substances” (1995, 209 f.). It is typically thought that in aporia 12 as in aporia 5, Aristotle’s primary concern is simply to consider whether a certain kind of mathematical entity is a substance. This is how Ian Mueller characterizes it: aporia 12 “has the form of a complete disjunction: are surfaces, lines, etc. substances, or aren’t they?” (191).^[5] So construed, the aporia is not a terribly pressing puzzle, coming as it does after aporia 5. It is only to avoid accumulating yet another superfluous substance that Aristotle must refute the views motivating aporia 12.

The primary aim of this paper is to show that aporia 12 is more significant than its treatment in the literature suggests, as the B.5 views challenge Aristotle’s metaphysical project in at least one way that aporia 5 does not: they threaten the distinction between substance and attribute. If left unresolved, this challenge seriously undermines the Aristotelian system. This understanding of aporia 12 is informed by another thesis: that the B.5 views are directly addressed and refuted in M.2. My analysis of the M.2 treatment of these views shows that aporia 12 is not simply a puzzle about whether a kind of mathematical entity is a substance; it is also a puzzle about the ontological status of mathematicals relative to sensibles.

There is an immediate problem for my interpretation. Commentators typically emphasize the ways in which aporia 5 is related to M.2, and either under-examine or deny aporia 12’s links to M.2 – and for apparently good reason. The B.5 views seem to be incompatible with most of M.2, as the former locate mathematical entities in sensible substances, while much of the latter targets ‘separate’ mathematicals. I address this difficulty by showing that the inherence at issue in B.5 is in fact compatible with the separation at issue in M.2 (Section 3), and by establishing a robust connection between aporia 12 and M.2 (Section 4).

2 Aporia 12

Aporia 12 is introduced in B.1, where Aristotle asks “(a) whether numbers and lines and figures are a kind of substance or not, and if they are substances (b) whether they are separate from sensible things or present in them” (996a12–15).^[6] The separation at issue in (b) is typically thought to be local separation. Local separation is a spatial relation expressed by χωρίς and its cognates. A is locally separate from B just in case

no part of A is at the same location as any part of B.^[7]

Local separation is symmetric: if A is locally separate from B, then B is locally separate from A, and conversely. For example, Aristotle and Theophrastus are not at the same location: Aristotle is locally separate from Theophrastus and Theophrastus is locally separate from Aristotle.

In its B.5 development, Aristotle frames aporia 12 as a question connected with aporia 11, and asks whether numbers, bodies, planes, and points are substances or not (1001b26–8).^[8] He rules out several other plausible candidates for substancehood before arriving at the commonly held view that body is substance – a view he attributes to earlier thinkers.^[9] He then explains that according to certain more recent thinkers, points, lines, and planes are more substantial than bodies:

a body is surely less of a substance than a surface, and a surface less than a line, and a line less than a unit and a point. For a body is bounded by these; and they are thought to be capable of existing without body, but a body cannot exist without these. This is why […] the more recent and those who were held to be wiser thought numbers were the first principles (1002a4–12).^[10]

As bodies are most of all thought to be substances (1001b32–1002a4), it follows that these mathematical entities are substances. Aristotle adds that for the same reason, certain thinkers take numbers to be substances.^[11] The remainder of B.5 is a series of objections that problematize this set of views (henceforth ‘the B.5 views’).^[12]

The mathematical entities hypostatized by these B.5 thinkers are locally inherent in sensible bodies. They are the limits of sensible bodies, and as such they are, at least in some sense, at the same location as the bodies they limit. And on any reading of Aristotle’s B.5 objections, these thinkers take mathematical objects to be located where the bodies they limit are located.^[13] For example, at 1002a20–5 Aristotle objects:

no sort of shape is present in the solid more than any other; so that if the Hermes is not in the stone, neither is the half of the cube in the cube as something determinate; therefore the surface is not in it either; for if any sort of surface were in it, the surface which marks off the half of the cube would be in it too. And the same account applies to the line and to the point and the unit (emphasis mine).

For Aristotle, to be locally inherent is to be at the same location, either in whole or in part. That is:

A is locally inherent in B just in case some part of A and some part of B are at the same location.

For example, Aristotle and his soul are at the same location, as are Plato and his beard.^[14] Perhaps this sounds strange: isn’t local inherence an asymmetric relation? How can two things be in one another? According to Aristotle’s account at Physics 4.3, the primary sense of ἐν (in) is to be contained in another, “as something is in a vessel, and generally in a place” (210a24). This is an asymmetric relation. But Aristotle’s notion of local inherence is broader than this: there are several senses in which one thing can be said to be ‘in’ another (ποσαχῶς ἄλλο ἐν ἄλλῳ λέγεται: 210a14–15). The first two senses are: “as a finger is in a hand, and generally a part in a whole”, and “as a whole is in its parts; for there is no whole over and above the parts” (210a16–18). Both are local relations. And as the parts can be located in their whole and the whole can be located in its parts, two things can be locally inherent in one another (though not in precisely the same way). Thus Aristotelian local inherence should not be understood narrowly as being contained in, but rather in the broader sense captured by the formulation I have suggested above.

It is worth noting that it is appropriate to speak of local separation and inherence here, even though mathematical objects are not typically thought to be space-occupying entities. This is because the views Aristotle is targeting in B.5 take the limits of sensible things to be mathematical objects. That is, they do not distinguish between, e. g. the surfaces and edges of a table (which are sensible and space-occupying) and mathematical planes and lines (which are neither).^[15] When Aristotle introduces aporia 12 in B.1, he asks about the ontological status of numbers (οἱ ἀριθμοί), lengths (τὰ μήκη), figures (τὰ σχήματα) and points (αἱ στιγμαί). And when in B.5 he explains why some have given these objects a privileged ontological status, he uses the words body (τὸ σῶμα), visible surface (ἡ ἐπιφάνεια), line (ἡ γραμμή), unit (ἡ μονάς), and point (ἡ στιγμή). Some of these terms clearly refer to sensible entities, some clearly refer to mathematical entities, and some are ambiguous. Thus as Aristotle represents it, the B.5 thinkers reason from the relationship between sensible bodies and their sensible limits to the relationship between sensible bodies and mathematical objects.^[16] As the views Aristotle is targeting do not make a sharp distinction between the mathematical and the sensible, spatial language is appropriate here.^[17]

This brings us to the matter of the scope of aporia 12. As we have seen, in the initial description of the puzzle in B.1, Aristotle writes that there is a problem concerning (a) whether mathematical objects are substances and – if they are substances – (b) whether they are separate from or in sensible things (996a13–15). As the B.5 development of the aporia targets a view according to which mathematical objects are inherent in bodies, there appears to be a discrepancy between how the aporia is introduced and how it is developed: in B.5 Aristotle develops (a), the question of substantiality, but fails to address (b), the question of separation. Some commentators have concluded that aporia 12 is limited to (a). For example, Madigan claims that in the B.5 discussion of aporia 12, Aristotle “assumes dialectically” that mathematical objects are in sensible things (b) and debates only the question whether or not they are substances (a) (1999, 29).^[18] On Mueller’s reading, aporia 12 is concerned exclusively with the question whether or not planes, lines, etc. are substances (a); he denies that it is also concerned with whether they are separate from or inherent in sensible things (b) (2009, 190 f., 204 f.).^[19]

If this is right, then aporia 12 is only concerned with whether or not we should admit a few more items into our ontology under the category of substance. At most, then, it is another iteration of the aporia 5 question about intermediates, this time considering mathematical objects not qua object of mathematical knowledge but rather qua limits or divisions of sensible bodies. The worry driving aporia 12 would then be that allowing for the existence of such mathematical substances results in a cluttered ontology. As this is the pervading view of the aporia’s scope and importance, it is perhaps not surprising that it is also generally thought that this is one of the Book B aporiai given no direct and systematic treatment elsewhere in the Metaphysics. As Madigan notes, “[c]ommentators tend to refer in a global way to M and N as resolving aporia 12” (1999, 128).

I argue that aporia 12 is directly addressed in M.2, where Aristotle again considers a variety of views that hypostatize mathematical objects.^[20] Specifically, I show that

1. the arguments of M.2, 1077a14–b11, directly address the B.5 views’ argument for the substantiality of mathematical objects, and

2. M.2 as a whole refutes all of the B.5 views’ commitments.

The M.2 treatment of aporia 12 very clearly targets separate mathematical substances. Hence when we take account of M.2, the scope of the aporia is clearer and the apparent discrepancy between its B.1 and B.5 formulations disappears: the aporia concerns both hypostatization (a) and separation (b). We may then better appreciate the aporia’s significance for Aristotle.

But there seem to be good grounds for denying all of these claims, and indeed for denying that there is any substantial connection between M.2 and aporia 12. M.2 contains two main argument series: the first (1076a38–b11) is directed against the view that mathematical objects are distinct substances located in sensible things (IST); the second (1076b11–1077b11) targets the view that mathematical objects are separate from sensible things (SST). The separation at issue in SST has typically been understood to be local.^[21] But as we have just seen, the views targeted in B.5 take mathematical objects to lack local separation from sensible bodies: they are locally inherent.

This apparent incompatibility between the B.5 views and M.2’s SST seems to rule out the possibility that both texts target the same views, so that there cannot be a direct connection between the arguments of M.2, 1077a14–b11, and the B.5 argument for the substantiality of mathematical objects. Further, IST would be the only M.2 view compatible with the B.5 views. But Aristotle’s M.2 treatment of IST is cursory (a mere 13 lines); the bulk of the chapter (76 lines) is devoted to a treatment of SST. If the B.5 views and M.2’s SST are incompatible, then M.2 as a whole does not refute the B.5 views.

Thus, e. g. Mueller 2009 argues that since in the M.2 discussion of SST “the criterion of substantiality is the capacity to exist separately”, while in B.5 Aristotle is concerned with the possible substantiality of entities that are located in sensible things, the separability criterion “plays no role in aporia 12” (204); hence, Aristotle was unlikely to have been “conscious of a direct connection” between M.2 and B.5 (190).^[22]

This would be the correct conclusion if M.2’s SST presupposed mathematical objects that are located apart from sensible things. However, Aristotle uses χωρίς and its cognates to express several kinds of separation; and I show (Section 3) that the only kinds of separation explicitly at issue in the M.2 arguments targeting SST are distinctness and ontological separation. If being located together with something is compatible with being distinct from it and being ontologically separate from it, then the B.5 views and M.2’s SST are compatible, and we need not deny that the B.5 views are a key target of M.2. Further, there are strong textual reasons for affirming the connection between B.5 and M.2. I show (Section 4) that the B.5 views presuppose the ontological separation and priority of mathematical objects, and that this is the key premise of their argument for hypostatization. I find the very same premise targeted at M.2, 1077a14–b11. Finally, I show that as the B.5 views make mathematical objects (1) substances that (2) share a location with and (3) are ontologically separate from and prior to sensible objects, and each of these three commitments is targeted and refuted in M.2, the arguments of M.2 refute the B.5 views.

Before continuing, I will note that my claim that M.2 as a whole refutes the views that motivate aporia 12 should not be taken to imply that M.2 addresses only this aporia. There are clear connections between M.2 and the B.2 discussion of aporia 5: in the M.2 series targeting SST Aristotle mentions the “discussion of problems” and gives an argument that mirrors part of his B.2 discussion of the first of two views about intermediates (1076b39–1077a9); and in the M.2 series targeting IST, he refers the reader to the “discussion of problems”, and it is clear that the reference is to the B.2 discussion of the second view about mathematical intermediates.^[23]

These connections between M.2 and Book B’s aporia 5 have been well acknowledged; it is clear that aporia 5 is being addressed in M.2.^[24] What has either been denied or insufficiently explored is the connection between M.2 and B.5’s aporia 12. In his discussion of B.5, Ross observes briefly that the question whether mathematical objects are substances is discussed in several chapters of M and N, including M.2, but he does not explore the connection between the B.5 views and the second M.2 argument series (1953, vol. 1, 223, 247). Madigan 1999 suggests that there are two connections between Aristotle’s arguments in M.2 and B.5 (128 f.), but his final assessment is that “M and N do not directly address aporia 12’s argument for the substantiality of units, points, etc.” (129) – that is, he denies my claim (I), above.^[25] And as I have already noted, Mueller denies that there is any direct connection between M.2 and aporia 12. This is the connection I aim to establish in Sections 3 and 4.

3 Separation in M.2’s SST and Inherence in the B.5 Views

In order to show that aporia 12 is addressed by M.2 as a whole, I will need to show that the local inherence presupposed by the B.5 views is compatible with the two kinds of separation that, I argue, are presupposed by M.2’s SST: distinctness and ontological separation. This is the task of the present section.

In M.1, Aristotle writes that mathematical objects, if they exist at all, must either exist in sensible things (IST), separately from sensible things (SST) or they exist in some other way (OE).

The first M.2 argument series targets a version of IST: mathematical objects exist as distinct substances that are located in sensible things. Aristotle’s first objection to M.2’s IST is that “it is impossible for two solids to be in the same place at the same time” (emphasis mine; δύο ἅμα στερεὰ εἶναι ἀδύνατον: 1076b1). This presupposes that the targeted view takes the mathematical object and the sensible thing in which it inheres to be two, i. e. to be distinct entities.

The second M.2 argument series begins with the claim that it is not possible for mathematical objects to be separate (οὐδὲ κεχωρισμένας γ’ εἶναι φύσεις τοιαύτας δυνατόν: 1076b13). Aristotle appears to be giving an exhaustive list of the possible modes of existence of mathematical objects in M.1, to be ruling out two of them in M.2 (IST at 1076a38–1076b11 and SST at 1076b11–1077b14), and then to be defending the remaining possibility (OE) in M.3. The fact that in M.1 he writes that mathematical objects must (ἀνάγκη) either exist IST, SST, or OE (1076a32–35) contributes to this impression. So does his claim at the end of M.2 that since (ἐπεί) they cannot be separate from sensible things and they cannot be in sensible things, it is clear that (φανερὸν ὅτι) mathematical objects “do not exist without qualification” (οὐχ ἁπλῶς ἔστιν), i. e. as substances (1077b14–16).^[26]

If this impression is correct, so that the versions of IST and SST refuted in M.2 jointly exhaust the ways in which mathematical objects could exist as substances, then M.2’s IST must be the only possible version of IST, and the inherence at issue in IST must be directly opposed to the separation Aristotle objects to in M.2’s SST. As the inherence of M.2’s IST is clearly local (as I explain below), the separation in M.2’s SST would have to be local.

But while M.1’s IST and SST are stated very generally, so that they exhaust the possible modes of existence for mathematical substances, M.2 does not proceed by setting out two jointly exhaustive possibilities and then refuting them one after the other. Instead, M.2 refutes just one of at least two versions of IST, and two distinct versions of SST. M.2 is a refutation not of logical possibilities, but rather of actually held positions or premises, which do not exhaust the logical possibilities.

The details of the text make this clear. As Ross notes, in M.6 (1080a37–b4) Aristotle distinguishes the version of IST refuted in M.2 (“the way we discussed at first”) from the Pythagorean version of IST (1953, 411 f.).^[27] The version targeted in M.2 takes mathematical objects (possibly including numbers) to be distinct substances existing within (or at the same location as) sensible bodies. By contrast, the Pythagorean version of IST (on Aristotle’s interpretation) takes sensible things to be made up of numbers (ἐξ ἀριθμῶν εἶναι συγκείμενα: M.8, 1083b8–13).^[28] On this Pythagorean view, numbers are not distinct from the sensible things in which they exist. Rather, they are the substance of sensible things (see, e. g. A.5, 987a13–19). This third mode of existence for mathematical substances is neither addressed nor ruled out in M.2. It is, of course, one way of existing as a substance in sensible objects – that is, it is a version of IST, though not the version targeted in M.2.^[29] Hence it should be understood to be included in the first item of the M.1 list. The fact that it is not even addressed in M.2 indicates that M.2 is not a systematic refutation of all logically possible modes of existence for mathematical substances.

Further, the inherence at issue in M.2’s IST is not directly opposed to the separation at issue in M.2’s SST. The inherence in M.2’s IST is local, i. e. it is sharing a location. This is clear from the fact that a key problem with this view, for Aristotle, is that it becomes impossible to divide bodies, because the mathematical objects within them are indivisible (1076b4–11). Aristotle also objects that this view has two distinct substances occupying the same place.^[30] But as I will argue, the separation to which Aristotle objects in M.2 is not local, and so M.2’s SST is not directly opposed to M.2’s IST. In fact there are two subsets of arguments in the SST series, each targeting a different version of SST: SST₁ (targeted from 1076b12–1077a14) and SST₂ (targeted from 1077a14–b11).^[31] None of the M.2 objections to either SST₁ or SST₂ hinges on or even mentions place or location.

I begin with SST₁. All four of the M.2 arguments targeting SST₁ proceed by creating problems for the view that mathematical objects are substances distinct from sensible things. The arguments all show that SST₁ results in an embarrassing accumulation of objects. Specifically, they show that SST₁ results in many distinct kinds of geometrical objects (1076b12–36) and numbers (1076b36–9), as well as distinct objects for the mixed mathematical sciences (1076b39–1077a10),^[32] and distinct objects for the general (or universal) mathematical propositions (1077a9–14).^[33] Distinctness is the operative notion of separation (κεχωρισμένος) in this subset of arguments.^[34] For A and B to be distinct from one another is simply for A and B to be two (in the relevant sense).

There is ample evidence that Aristotle does use χωρίς and its cognates to express distinctness. For example, in a Metaphysics A.5 discussion of his predecessors’ views, he writes: “Down to the Italian school, then, and apart from (χωρίς) it, philosophers have treated these subjects rather obscurely” (987a9–11). And in On Generation and Corruption 1.1, he writes that we must inquire “whether alteration has the same nature as coming-to-be, or whether to these different names there correspond two separate processes with distinct natures” (πότερον τὴν αὐτὴν ὑποληπτέον φύσιν εἶναι ἀλλοιώσεως καὶ γενέσεως, ἢ χωρίς, ὥσπερ διώρισται καὶ τοῖς ὀνόμασιν: 314a4–6).

By claiming that distinctness is the operative notion of separation in 1076b12–1077a14, I do not mean to say that proponents of M.2’s SST₁ would deny that mathematical objects are located apart from sensible things. It is unclear from the M.2 discussion where they would locate mathematical objects.^[35] Given the reference to the “discussion of difficulties” and the parallel argumentation at 1076b39–1077a9 (see Section 2), the proponents of M.2’s SST₁ are likely the proponents of the B.2 version of SST. Nevertheless, issues of local separation play no role in the M.2 objections to SST₁: what Aristotle is objecting to in his critique of SST₁ is not the location of mathematical objects, but rather their status as substances distinct from sensible things.

In the second subset of arguments targeting SST, which begins at 1077a14, Aristotle’s focus shifts.^[36] He now targets the ontological separation of mathematical objects. There is a general (though not universal) consensus that in certain contexts, Aristotle uses unqualified separation terminology (χωρίς, χωριστός, κεχωρισμένoς, etc.) to mean not local separation or numerical distinctness, but rather ontological separation.^[37] Ontological separation is a matter of the capacity to be without (εἶναι ἐνδέχεται ἄνευ), i. e. ontological independence: A is ontologically separate from B just in case A is ontologically independent of B.^[38] This is a non-symmetric independence relation. There are also certain contexts in which ontological separation is best understood as an asymmetric independence relation. In such contexts, A is ontologically separate from B just in case:

A is ontologically independent of B

and

B is ontologically dependent on A.^[39]

That ontological separation is at issue in M.2, 1077a14–b11, is clear: three of the arguments in the series conclude that mathematical objects are not prior in substance, or ontologically prior, to sensible objects (1077a14–20, 24–31, 1077a36–b11).^[40] Ontological priority follows neither from distinctness nor from local separation, both of which are symmetric. That the ontological separation in M.2, 1077a14–b11, is asymmetric follows from Aristotle’s claim that ontological priority – an asymmetric ontological relation – is entailed by this kind of separation (1077a17–18).^[41] In fact these two relations are mutually entailing: asymmetric ontological separation is an ontological relation that grounds ontological priority, which is a ranking or ordinal relation.

As there are two different kinds of separation at issue in M.2, the targets of the first and second SST argument subsets should be distinguished: the former is the premise that mathematical objects are substances distinct from sensible things (SST₁), and the latter is the premise that mathematical objects are substances ontologically separate from sensible things (SST₂).^[42] Both of these kinds of separation are compatible with the inherence at issue in M.2’s IST, i. e. local inherence.^[43] A can be both locally inherent in B (some part of A can be at the same location as some part of B) and also ontologically separate from B. For example, the wood in a wooden casket is at the same location as the casket it is in. Yet it is also ontologically separate from the casket it is in, because it can exist even if the casket is destroyed, while the casket cannot exist if the wood is destroyed.^[44] And although the wood is locally inherent in the casket, the wood and the casket are distinct (i. e. they are in some sense two). As we have just seen, one of Aristotle’s targets in M.2 is the view that mathematical and sensible objects are distinct and yet share a location (IST). Clearly distinctness is – for some thinkers – compatible with sharing a location. It is for Aristotle as well, provided the items in question are not substances. For example, he locates several distinct capacities of the soul – the nutritive, perceptive, and motive – in the heart.^[45]

In sum, local separation plays no explicit role in Aristotle’s M.2 objections to either SST₁ or SST₂, and neither of the kinds of separation at issue in the M.2 SST series entails local separation. In fact both are compatible with local inherence. Hence M.2’s IST and SST are not mutually exclusive and jointly exhaustive logical possibilities (local inherence and local separation).

Nor must Aristotle’s use of modal language (ἀνάγκη) in the M.1 list indicate that M.2’s versions of IST and SST exhaust the logically possible ways in which mathematical objects can exist as substances. It is true that, necessarily, mathematical objects must either exist as substances IST (in one or more of the senses of ‘in’), or SST (in one or more of the senses of ‘separate’), or OE (i. e. not as substances). It does not follow from this that the versions of IST and SST targeted in M.2 exhaust all the possible ways in which mathematical objects can exist as substances either in or separately from sensible objects.^[46]

Even in M.1, where Aristotle does list all of the logically possible ways in which mathematical objects could exist as substances, he treats IST and SST not as mere logical possibilities, but rather as actually held views. He begins M.1 by stating that “we must first discuss the views of others” (1076a12–13). This sets up what follows as a survey of his predecessors’ and contemporaries’ views. Indeed, the entire investigative program outlined in M.1 is structured by opinions (δόξαι, 1076a17), what some people posit (1076a19–20), and what some people say (1076a17–18) or some others say (1076a21–2). And Aristotle is careful to note that IST and SST are actually held views when he first lists them (1076a34–5). It is within this framework that his M.2 investigation of mathematical objects takes place.

Finally, Aristotle concludes M.2 by claiming that “it has been adequately [ἱκανῶς] shown that <mathematical objects> are not substances more than bodies are, that they are not prior to sensible objects in substance, but only in definition, and that they cannot have separate existence” (1077b12–15; emphasis mine). Aristotle often uses ἱκανῶς to indicate that the matter under consideration has received enough or sufficient treatment for his purposes – though not the most precise or exhaustive treatment possible. For example at Prior Analytics 1.1, 24b14–15, after defining propositions and explaining how deductive, demonstrative, and dialectical propositions differ, he writes: “we have now said enough [ἱκανῶς] for our present purposes – we shall discuss the matter with precision [δι᾽ ἀκριβείας] later on.”^[47] Aristotle’s use of ἱκανῶς at the end of M.2 is evidence that he does not take himself to have eliminated every logically possible mode of existence for mathematical substances. He has nevertheless made a sufficient case that mathematical objects cannot exist as substances: he has refuted the most prominent views – even one that he considers “fanciful” (πλασματίας). Thus he is warranted in claiming that it is now clear (φανερόν) that either mathematical objects do not exist, or they exist as non-substances (1077b15–16), and he has earned the right to move on to his own account of mathematical objects in M.3.

Note that while I am claiming that M.2’s IST, SST₁ and SST₂ are actually held views, it is not clear whose views these might have been.^[48] As I have just argued, M.2’s version of IST is distinct from the Pythagorean version, at least according to Aristotle. Perhaps we might say that this version of IST is inspired by Pythagoreanism, without quite being Pythagorean.^[49] In any case, there is no consensus about the proponents of M.2’s IST.^[50] As for M.2’s SST (or SST₁ and SST₂, as I have argued), Ross attributes this view to Plato and Speusippus (1953, 412), while Annas suggests that Aristotle may have in mind a view held by certain people in the Academy “who were over-impressed by some things that Plato says about physical objects in the Timaeus” (1976, 144).^[51]

So much for the sense in which M.2’s IST posits inherent mathematical objects and the two senses in which SST₁ and SST₂ posit separate mathematical objects. Like M.2’s IST and both versions of SST, the B.5 views make mathematical objects substances. And like M.2’s IST, the B.5 views make mathematical objects somehow locally inherent in sensibles. As we have just seen, local inherence is compatible with both ontological separation and distinctness – the only two notions of separation operative in Aristotle’s M.2 treatments of SST₁ and SST₂. And since, as we have also seen, M.2’s SST₁ and SST₂ do not explicitly presuppose or imply the local separation of mathematical objects, there is no reason to deny the compatibility of the B.5 views and M.2’s SST.

4 The B.5 Views as Direct Targets of M.2

My aim is not merely to show that the B.5 views are compatible with M.2’s SST₁ and SST₂. It is also to show that aporia 12 is given another serious, targeted treatment in M.2 – an indication of its importance, for Aristotle. To this end, I now turn to securing the connection between B.5 and M.2. I claim that:

1. the arguments of M.2, 1077a14–b11 (i. e. the arguments targeting SST₂), directly address the B.5 thinkers’ argument for the substantiality of mathematical objects, and

2. M.2 as a whole refutes all of the B.5 views’ commitments.

I begin with (I). I have argued that the kind of separation at issue in M.2’s SST₂ is ontological and asymmetric. Its individually necessary and jointly sufficient conditions are:

A is ontologically independent of B

and

B is ontologically dependent on A

If the B.5 argument for the substantiality of mathematical objects is a direct target of the M.2 arguments against SST₂, then this asymmetric ontological separation must also be at issue in B.5.

As we have seen, in B.5 Aristotle reports that some think that the limits of bodies are even more substantial than the bodies they limit. He writes:

[A] body is surely less of a substance than a surface, and a surface less than a line, and a line less than a unit and a point. For a body is bounded by these; and they are thought to be capable of existing without body, but a body cannot exist without these (emphasis mine).^[52]

He goes on to claim that it is for this very reason that the more recent and wiser thinkers thought that numbers were first principles (1002a8–12). This idea that mathematical objects (surfaces, lines, points, and numbers) are more substantial than sensible bodies is the key premise in the B.5 thinkers’ argument for the hypostatization of mathematical objects. The argument is as follows: if mathematical objects are more substantial than sensible bodies and sensible bodies are substances, then mathematical objects must also be substances (1002a26–8). As Aristotle notes in Λ.8, “that which is prior to a substance must be a substance” (1073a36).^[53]

The relation Aristotle describes here between bodies and mathematical objects is an asymmetric ontological one. According to the B.5 thinkers, bodies are ontologically dependent on surfaces, lines, points (and numbers) because these mathematical objects are the limits of bodies, and bodies cannot exist without their limits.^[54] Aristotle appears to be referring to the same set of thinkers in Δ.8, when he reports that some take lines, planes, and numbers to be substances because nothing can exist if they do not exist (1017b17–21). As incapacity to exist without is one of the ways in which B can be ontologically dependent on A, these views make bodies (and indeed everything) ontologically dependent on mathematical entities. The B.5 views also stipulate that these mathematical objects can exist without bodies (1002a6–7). This is one of the ways in which A can be ontologically independent of B. Thus the B.5 views posit an asymmetric ontological relation between mathematical objects and bodies: mathematical objects are asymmetrically ontologically separate from, and prior to, sensible bodies. This is confirmed again at 1002a15–16, where Aristotle considers the consequences of presupposing – as the B.5 views do – that “lines and points are substances more than bodies” (emphasis mine; μᾶλλον οὐσία τὰ μήκη τῶν σωμάτων καὶ αἱ στιγμαί), and again at 1002a26–8, where he says that we will be at a loss to say what being and the substance of things are if mathematical objects are substance “more than body” (emphasis mine; τούτου δὲ μᾶλλον ταῦτα). Aristotle’s language here is strikingly similar to his language in M.2: at 1077b12, he claims that he has shown that mathematical objects are not “substances more than bodies are” (emphasis mine; οὐσίαι μᾶλλον τῶν σωμάτων). According to the B.5 views, mathematical objects are ontologically separate from and prior to sensible bodies in just the way that they are in M.2’s SST₂.

The assumption that mathematical objects are asymmetrically ontologically separate from and prior to sensible bodies is the key premise in the B.5 argument for the substantiality of mathematical objects. The aim of the M.2 arguments against SST₂ is to refute this premise, i. e. to show that mathematical objects cannot be separate and prior in this way. Thus the B.5 argument for the substantiality of mathematicals is directly addressed and refuted in the M.2 arguments targeting SST₂.

I turn now to (II): the claim that M.2 refutes all three of the B.5 views’ basic commitments. There is no question that the B.5 views hypostatize mathematical objects. It is equally clear that the mathematical objects in question – the limits or divisions of sensible bodies – are in some sense locally inherent in sensible bodies. I have just argued that these views also make mathematical objects asymmetrically ontologically separate from and prior to sensible bodies. In sum, the B.5 views’ three basic commitments are:

1. Mathematical objects are substances.

2. Mathematical objects are locally inherent in sensible bodies.

3. Mathematical objects are asymmetrically ontologically separate from and prior to sensible bodies.

M.2’s IST shares two of these commitments: it posits mathematical objects that are substances (1) and that share a location with sensible bodies (2). M.2’s SST₁ shares one of the B.5 views’ commitments: the substantiality of mathematical objects (1). M.2’s SST₂ shares two of the B.5 views’ commitments: the substantiality of mathematical objects (1) and the asymmetric ontological separation and priority of these mathematical substances (3). As M.2 refutes IST, SST₁ and SST₂, it refutes all three of the B.5 views’ commitments.

To summarize, claims (I) and (II) are both true. M.2 contains an important refutation of all three of the B.5 views’ assumptions; and as these views motivate aporia 12, and their assumptions are targeted and refuted by the M.2 arguments, there is strong evidence that aporia 12 is directly addressed in M.2.

5 The Scope and Significance of Aporia 12

In the course of establishing the connection between aporia 12 and M.2, I have argued that the B.5 views attribute asymmetric ontological separation to mathematical objects. If this is correct, then the scope of aporia 12 is broader than it is typically taken to be. This also distinguishes it from aporia 5 in ways that have not been explored.

Recall that when aporia 12 is introduced in B.1, Aristotle frames it as a puzzle about (a) whether mathematical objects are substances and (b) whether, if they are substances, they are separate from or present in sensible things (πότερον κεχωρισμέναι τῶν αἰσθητῶν ἢ ἐνυπάρχουσαι ἐν τούτοις: 996a13–15). Because the mathematical entities of B.5 are limits, and as such locally inherent, it is typically thought that aporia 12 in fact only considers (a).^[55]

However, B.5 does also consider (b) understood not as a question about the location of mathematical objects but rather as a question about their ontological status relative to sensible bodies. I have argued that in B.5 Aristotle considers whether such entities are asymmetrically ontologically separate from sensible bodies.^[56] He also considers whether such entities are “present in sensible things” (ἐνυπάρχουσαι ἐν τούτοις), when this too is understood as a question about ontological status – in this case, a question about ontological inherence. To be ontologically inherent is to depend for one’s being on something else. That is, B is ontologically inherent in A just in case:

B depends for its being (i. e. is ontologically dependent) on A.

There is good textual evidence that ontological inherence is a genuinely Aristotelian notion, and that it is distinct from (though not utterly unconnected to) his notion of local inherence. Aristotle’s uses of the preposition ἐν and the verb ἐνυπάρχω are not limited to the local senses of ‘in’ and ‘to inhere’.^[57] It has hardly gone unnoticed that in Categories 2, he introduces a technical sense of ἐν that includes local inherence as a necessary but not sufficient condition. This kind of inherence has an additional necessary condition, which is often understood as ontological dependence (1a24–5).^[58] And in the Physics 4.3 list of the senses of ‘in’, Aristotle includes “the affairs of Greece are in the King, and generally events are in their primary motive agent” (ἐν βασιλεῖ τὰ τῶν Ἑλλήνων καὶ ὅλως ἐν τῷ πρώτῳ κινητικῷ: 210a21–2).^[59] Here he uses the language of inherence to express an ontological dependence relation: it is insofar as the king is the primary motive agent of the affairs of Greece that these affairs are said to be in him. The primary motive agent is the efficient cause (Physics 2.3, 194b29–30; 2.7, 198a26). Aristotle explains that “the man who deliberated” and the father are causes in this way. Now it is true that the human affairs produced by a deliberator and the offspring produced by a father can all continue to exist after their efficient causes perish. In this sense, they ‘can be without’ their efficient causes. Nevertheless, human affairs and offspring – things that can only come to be by the action of a deliberator or father (respectively) – cannot or will not be without (οὐκ ἔσται ἄνευ) their proper efficient causes; and even in the case of things that can come to be either with or without a father or deliberator,^[60] the type has a necessary connection to what is by nature its efficient cause.^[61] These are the senses in which human affairs and offspring ‘cannot be without’, and so depend for their being, on their efficient causes. Thus to say that the affairs of Greece are in the king is to say that they depend for their being on the king. The inherence relation in this case expresses ontological dependence.^[62]

Local and ontological inherence will often be associated with one another. For example, the brown of your eyes is both ontologically and locally inherent in you; and in general, properties are both ontologically and locally inherent in substances. And the Categories 2 technical sense of ἐν has both local and ontological inherence as its necessary conditions. Yet these two kinds of inherence need not be associated with one another. Consider the relationship between bridle-making and the art of riding. Bridle-making depends on the art of riding as its final cause: bridle-making exists and is what it is only if and because its final cause – the art of riding – exists and is what it is (Nicomachean Ethics 1.1). As bridle-making exists only if and because the art of riding exists, it is ontologically inherent in the latter. Yet bridle-making is not located in the art of riding. In this and other similar cases, ontological and local inherence come apart.

The B.5 mathematical entities are ex hypothesi locally inherent in sensible bodies. But Aristotle also suggests that they must be ontologically inherent in (i. e. less substantial than and dependent on) sensible bodies. He makes this suggestion in several ways, but most clearly when he claims (i) that the mathematical entities under discussion are mere divisions of sensible bodies^[63] and (ii) that they are limits, which are generated and destroyed when bodies are divided or put into contact with other bodies (1002a32–b3).^[64] Both of these claims imply that the entities in question are ontologically inherent in sensible bodies. Aristotle shows that the very thing about points, lines, planes and numbers that has convinced some thinkers that they are more substantial than sensible things – that is, the fact that points, lines, and planes are all either limits or divisions (ἢ πέρατα ἢ διαιρέσεις) of bodies – proves that in fact they are even less substantial than sensible bodies.

The first point (i) is stated laconically, but can be unpacked with the help of a parallel passage in K.2. There Aristotle argues that it is because lines, planes, and points are divisions and limits that they cannot be “separate and independent” (χωρισταὶ καὶ καθ’ αὑτάς: 1060b2).^[65] Limits and divisions are limits and divisions of another thing, and as such they are always present in or belong to these other things (πάντα δὲ ταῦτα ἐν ἄλλοις ὑπάρχει: 1060b16–17). Something cannot be a limit or division unless it belongs to something else; thus to be a limit or division is to depend on a limited or divided thing.^[66] As ontological dependence is a consequence of being a division or limit, the B.5 claim that the entities under discussion are divisions of sensible bodies implies that such entities are ontologically inherent in sensible bodies.

The second point (ii) is given a longer discussion. Aristotle argues that if we consider how limits and divisions perish and come to be, it is clear that they cannot be substances.^[67] When a body is divided, a division (which is a new limit) comes to be instantaneously; and when the surfaces (limits) of two bodies are put together, both surfaces are destroyed instantaneously. Substances, by contrast, come to be and perish through a process – that is, over time (1002a28–b3).^[68] The purpose of the argument is to show that limits cannot be substances because of how they come to be and perish. But notice that the argument appeals to a premise that makes limits ontologically inherent in the bodies they limit. If limits cease to be when the bodies they limit come into contact and come to be when a body is divided, then they depend for their being on the bodies they limit.

Thus B.5 does consider (b) whether the mathematical entities in question are ontologically separate from or inherent in sensible bodies. The B.5 development of aporia 12 is entirely consistent with its initial framing in B.1, and we need not restrict the scope of the aporia to (a).

This interpretation of the aporia’s scope is also consistent, as it must be, with Aristotle’s statement that aporia 12 is connected with the previous aporia (1001b26). The first part of aporia 11 asks whether, as the Pythagoreans and Plato maintain, being and unity are the substance of beings (οὐσία τῶν ὄντων: B.1, 996a5–7, B.4, 1001a5–6). The question is not just whether being and unity are substances, but whether they are the substance of things, i. e. the primary cause of being of things.^[69] Whether one construes ontological dependence modally or causally, a thing is ontologically dependent on its own substance. Thus aporia 11 considers whether things are ontologically dependent on being and unity. In Book A, Aristotle suggests that the view that number is the substance of all things, or the cause of the substance of all things (987b23–5), follows from (διό) the view that unity is the substance of things (A.5, 987a18–19 with A.6, 987b22–3).^[70] Hence the aporia 12 idea that all things are dependent on number follows from the aporia 11 idea that all things are dependent on unity.^[71] The two aporiai are closely connected.

Further, aporiai 11 and 12 are similar in this respect: both consider together the questions whether certain items in others’ ontologies are substances and whether these items are the substances of things.^[72] That is, they consider together the questions of the items’ substancehood (a) and of their ontological status relative to sensible things (b). Aristotle proceeds in this way because of his predecessors’ reasoning: they have made unity and numbers substances because they have made them substances of (and so ontologically prior to) all other things.^[73]

This similarity between aporiai 11 and 12 points to an important difference between the views motivating aporia 12 and those motivating aporia 5: their distinct reasons for hypostatizing mathematicals. These two aporiai also differ by hypostatizing different kinds of mathematical entities.^[74] But if this were the sole difference between them, then aporia 12 would ask the same question as aporia 5 – “Is X a substance?” – only this time of a different sort of mathematical entity.

This is not the only difference between the two puzzles. While both sets of views hypostatize mathematical entities, Aristotle attributes to each of them quite different motives. He reports in B.2 that some make mathematical objects substances distinct from sensible things, and that they are motivated by an epistemological concern: it seems that the mathematical sciences will have no appropriate objects if there are no mathematical substances other than (distinct from) sensible bodies (997b32–998a6).^[75] Aristotle refutes this distinctness claim in his M.2 arguments targeting both IST and SST₁. But his M.2 treatment of SST₂ refutes an ontological priority and separateness claim that plays no role in his B.2 treatment of aporia 5. This latter claim is the B.5 thinkers’ reason for hypostatizing mathematical entities. On Aristotle’s account, they are motivated by the thought that the limits that bound bodies are “capable of existing without body, but a body cannot exist without these” (1002a6–8). From this they conclude that planes, lines, and points are substances more than sensible bodies (1002a15–16); and as sensible bodies are also thought to be substances, it follows that planes, lines, and points are substances. Thus the B.5 views’ reason for hypostasizing mathematicals is that they are thought to be more substantial than, or ontologically prior to, sensible bodies. Priority relations play no motivating role in the B.2, 997a34–998a18 views; yet they play a key role in both B.5 and several of the M.2 arguments.

In sum, while both the aporia 5 and the aporia 12 discussions of mathematical entities ask whether they are substances, the first considers whether our epistemological needs mandate distinct mathematical substances, while the second considers the metaphysical matter of whether mathematical objects are the ontologically basic or primary substances. Far from being a reprise of the aporia 5 question, aporia 12 asks about the ontological status of mathematical entities relative to sensible things.

Aporia 12’s concern with relative ontological status points to the distinct way in which this aporia challenges Aristotle. One of Aristotle’s central concerns in M.2 is to show that the B.5 thinkers and others have gotten the ontological relations wrong: they have made mathematical objects ontologically prior to sensible things – which is why they think that mathematical objects are substances – but in fact sensible things are ontologically prior to mathematical objects. That Aristotle subjects their position to a thorough refutation in M.2 is an indication of its importance. But what is at stake for Aristotle, such that he must refute this view?

Nothing less than one of his fundamental metaphysical distinctions: that between substance and attribute.

According to the B.5 thinkers, mathematical objects are ontologically prior to sensible things; they are therefore ontologically separate from sensible things. But if mathematical objects, which are quantities, are allowed to be ontologically separate, then there is no prima facie reason why any of the other non-substance categories (quality, action, etc.) could not be ontologically separate. Yet for Aristotle, substances alone are separate.^[76] If the ontological separation of quantities cannot be ruled out, then the distinction between substance and attribute is endangered. But Aristotle’s metaphysics does not get off the ground without it.^[77] And at least according to Aristotle, the source of many of his predecessors’ errors is confusion about precisely this distinction. As he observes in his Γ.2 discussion of the subject-matter of first philosophy, those who study the attributes peculiar to being qua being “err not by leaving the sphere of philosophy, but by forgetting that substance, of which they have no correct idea, is prior to these other things” (1004b8–10).^[78] The B.5 views, like the B.4 views about being and unity, commit this error: they make attributes substances – in fact they also make them the substance of other things. It is essential to Aristotle’s project of defining and delimiting first philosophy that such views be refuted.

6 Conclusion

I have argued that the B.5 treatment of aporia 12 concerns both (a) whether planes, lines, points, and numbers are substances and (b) whether these are ontologically separate from or inherent in sensible things. So understood, aporia 12 challenges Aristotle’s metaphysical project in several significant ways, at least one of which extends beyond the aporia 5 discussion of mathematical entities. Hence the B.5 views motivating aporia 12 merit the serious refutation they are given in M.2. In M.2, Aristotle addresses aporiai 5 and 12 alongside one another: he shows that mathematical objects cannot be substances, and that they can be neither distinct from nor ontologically separate from sensible things.

After thus showing that none of the prominent positions hypostatizing mathematical objects are tenable, Aristotle devotes M.3 to developing his own solution to these aporiai. His view of mathematical objects – which is most fully explicated in M.3, but which informs numerous statements in other books – allows him to escape the difficulties faced by the B.2 997a34–998a18 and B.5 views. Briefly, for Aristotle, mathematical objects are non-substances.^[79] They are separable from sensible objects in thought, but they are not ontologically separate. And they are in a sense located in certain sensible objects, so that our mathematical theorems turn out to be about sensible things (though not qua sensible), and epistemological concerns need not lead us to posit a whole new class of substances. In fact mathematical objects are just the quantitative attributes of certain sensible objects. Since for Aristotle only movable bodies are in places, it turns out that – strictly speaking – mathematical objects have no place.^[80],^[81]