# Aristotle's critique of Platonist mathematical objects: two test cases from Metaphysics M 2.

While much has been written about the Metaphysics, Books M and N (henceforth denoted by M-N) have received relatively little careful attention. This is understandable--the text of M-N is exceedingly dense, and many of its arguments are obscure. It is often difficult to see why Aristotle has chosen to make the arguments presented in these books, and to determine whether they work. For some arguments, it is even difficult to identify Aristotle's opponent. According to Julia Annas (1976), scholars generally view M-N as 'a pettifogging discussion of mystical nonsense' (1).Yet even the few scholars who give detailed analyses of the arguments in M-N dismiss many of them as hopelessly flawed and biased, and find Aristotle's critique to be riddled with mistakes and question-begging. (2) This assessment of the quality of Aristotle's critique of his predecessors (and of the Platonists in particular), is quite widespread. (3) The series of arguments in M 2 (1077a14-b11) that targets separate mathematical objects is the subject of particularly strong criticism by Annas and Ross. According to Annas, the arguments in this series 'presuppose an acceptance of Aristotle's own philosophical ideas.' Specifically, they presuppose Aristotle's view that sensible objects, rather than abstract entities, comprise what are primarily and essentially real. Moreover, these arguments 'misconceive the platonist position in some respects' (139). Two related arguments in this series (1077a14-20 and 1077a24-31) will serve as cases in point. I have chosen these two arguments because they are the objects of the most serious complaints coming from Annas and Ross, and because these very same complaints are made against other arguments in the M 2 series as well as against many other arguments in M-N. (4) In fact, the principal charges made against these arguments (that Aristotle misunderstands or misrepresents his opponents' views, and that he engages in question-begging because he presupposes his own metaphysical views) are frequently made against Aristotle's critique of Platonist positions more generally. (5) Thus if, as I argue, these charges are false for our two test case arguments, then there is good reason to think that they might also be false when they are leveled against the other arguments in this M 2 series. Furthermore, although presenting an argument for this is beyond the scope of this paper, I would suggest that these two charges are, more often than not, false when applied to Aristotle's critique of Platonist mathematical views in M-N and beyond.

The context and target of the M 2 series, and what Aristotle means by 'separate'

In M 1 of the Metaphysics, Aristotle asks about the nature of numbers and lines and 'things of this sort' ([TEXT NOT REPRODUCIBLE IN ASCII], 1076a18-19). It is clear that they are the objects of mathematics, but what exactly are these mathematical objects ([TEXT NOT REPRODUCIBLE IN ASCII], 1076a17)? In other words, what is the status of the twos and threes used in arithmetic and calculation, and what is the status of the circles and triangles used in geometry? He and his opponents agree that mathematical axioms are not true of sensible things as such, because such axioms do not apply perfectly to sensible things (N 3, 1090a35-7). For example, the rule that a tangent line will touch a circle at exactly one point is not true of any sensible line and sensible circle (see B 2, 997b35-8a4). Moreover, science must have immutable objects, but sensible things are changeable and perishable. Therefore, mathematical axioms and theorems are not unqualifiedly true of sensible things; however, both of these must be unqualifiedly true. Since (for the Greeks) to be true is to be true of something, (6) it follows that they must be true of something other than sensible things as such.

Some of Aristotle's predecessors and contemporaries conclude that mathematical objects are separate from sensible things (1076a34-5). According to Aristotle, one key problem for this view, which I will call the Platonic view, (7) is that it cannot explain 'why, if numbers are in no way present in perceptible things, their attributes apply to perceptible things.' For example, Aristotle notes that in a musical scale the attributes of number do apply to sensible things (N 3, 1090a20-5, 1090b3-5). (8) The solution, Aristotle suggests, is to adopt his own position: sensible things are the objects of the mathematical sciences, only not as sensible. What distinguishes mathematical study from the investigations of the natural scientist is that the mathematician studies sensible objects not as sensible but rather as lengths, as planes, as figures, and so on. Aristotle explains that just as there are properties peculiar to animals in virtue of their being male or female, so there are properties that are true of things as lengths or as planes. Thus, we need not posit a separate 'female' or 'male' apart from animals, nor must we posit a separate 'length' or 'plane' apart from sensible things (M 3, 1077b17-8a9). Mathematical objects are simply sensible objects treated as numbers and as geometrical objects.

In M 2, Aristotle aims to refute all alternative conceptions of the objects of mathematics, in order to leave his own position as the only remaining possibility. After first making short work of the 'fanciful' view that mathematical objects are substances that exist in sensible things (1076a38-b11), he gives a series of arguments against the Platonist view that mathematical objects are separate from sensible things (1076b11-7b11).

Before we proceed, it should be noted that in these arguments (and in certain other contexts), Aristotle uses the word 'separate' ([TEXT NOT REPRODUCIBLE IN ASCII]) in a rather technical sense. Here 'separate' means (at least) 'distinct from' and 'not present in'; at the same time, it also means quite a bit more. When Aristotle uses 'separate' in this technical way, he is connecting it with the concept of priority in ousia ([TEXT NOT REPRODUCIBLE IN ASCII]). Certainly, the precise nature of separation and priority in ousia, and of their relationship to one another, is controversial. (9) I take it that separateness and priority in ousia are mutually entailing, and that the following two conditions are jointly sufficient (and likely individually necessary) (10) not just for the priority in ousia of X to Y, but also for the separateness of X from Y:

(i) X can 'be without' Y and

(ii) Y cannot 'be without' X. (11)

We may call this 'strong separateness', to be contrasted below with 'weak separateness', according to which (i) alone is sufficient for separateness (though not for priority in ousia). (12)

In order to see how 'strong' and 'weak' separateness differ, we should begin by analyzing the concept of priority in ousia. In Metaphysics [DELTA] 11 Aristotle explains that something is prior in ousia to another if it can be without the other thing ([TEXT NOT REPRODUCIBLE IN ASCII]), but the other cannot be without it. He explicitly attributes this view not just to the Platonists, but to Plato himself (1019a2-4). It is widely understood that the capacity to 'be without' another is ontological independence from the other, and the inability to 'be without' another is ontological dependence on the other. (13) Thus, we can say that X is prior in ousia to Y if both:

(i) X is ontologically independent of Y and

(ii) Y is ontologically dependent upon X. (14)

Based on this view, (i) and (ii) are jointly sufficient for priority in ousia, and each is likely a necessary condition for such priority.

As noted above, some take separateness to be a weaker concept than priority in ousia. They maintain that:

(a) X is separate from Y if (i) X is ontologically independent of X (15) and

(b) X is prior in ousia to Y if both (i) X is ontologically independent of Y and (ii) Y is ontologically dependent upon X. (16)

On this view, while (i) and (ii) are jointly sufficient for priority in ousia (and each is likely a necessary condition for such priority), (i) alone is sufficient for separateness. (17) If this 'weak' view of separateness is correct, this should be reflected in the relationship between priority in ousia and separateness. The stronger concept--priority in ousia--should imply separation if, as is likely, (i) is a necessary condition for priority in ousia. But there are no grounds for having the weaker concept separation imply priority in ousia, since on this view priority in ousia requires something more than separation.

It turns out that the M 2 critique of Platonic mathematical objects is an ideal piece of textual evidence for Aristotle's view of the relationship between separation and priority in ousia, because this critique utilises both of these concepts. In the course of the critique, it emerges that, as both the 'weak' and 'strong' views of separateness suggest, priority in ousia implies separateness. At 1077b2-3, Aristotle explains that something is prior in ousia to another if it surpasses the other in being separate ([TEXT NOT REPRODUCIBLE IN ASCII]). He then appears to make separateness a necessary condition for priority in ousia, by arguing that since an attribute such as 'pale' cannot be separate from the compound entity 'pale human' (its substance), it may be prior to 'pale human' in account, but not in ousia (1077b1-9).

The fact that Aristotle takes priority in ousia to imply separateness is interesting, but it does not help us decide which should be preferred between 'strong separateness' and 'weak separateness'. What does help is the fact that M 2 also offers us good reason for supposing that Aristotle takes separation to imply priority in ousia. Speaking of the Platonists' mathematical objects at 1077a17-18, Aristotle writes: 'Because they exist in this way [i.e., as separate entities] they have to be prior to sensible objects.' (18) Aristotle then goes on to show that mathematical objects cannot be prior in ousia to sensible objects. Given that mathematical objects cannot be prior in ousia to sensible objects, he has shown that 'it is not possible for objects of this kind [i.e., mathematical objects] to exist in separation' (1076b11-12); this is the announced aim of the series to which this argument belongs. This argument provides strong textual evidence that for Aristotle, if X is separate from Y, then X must be prior in ousia to Y; and if X is not prior in ousia to Y, then X cannot be separate from Y. (19)

Since priority in ousia does not follow from 'weak separateness', but it does follow easily from 'strong separateness', we have good reason to prefer 'strong separateness'. Thus, when Aristotle uses the term 'separate' in M 2, I take him to mean both (i) and (ii) above.

One might, at this point, wonder why Aristotle thinks that anyone takes mathematical objects to be 'separate' from sensibles in this way. If separateness in M 2 is the strong concept I have described above, then Aristotle is not just attacking the view that there exist mathematical objects that are distinct from sensible objects. Rather, he is attacking the view that (i) mathematical objects are ontologically independent of sensibles, while (ii) sensibles are ontologically dependent upon mathematical objects. The most likely targets for Aristotle's critique are certain Platonist thinkers who attribute separate existence to 'limits and extremes'. We can see this by returning to one of the aporiai of Metaphysics B 5. There, we find Aristotle explaining that if the limits of (sensible) bodies can be without bodies, and bodies cannot be without these limits (that is, if these limits are separate from bodies in the strong sense such that conditions (i) and (ii) are fulfilled), then a body is less of a substance than are the planes and lines that define it (1002a4-8). He adds that this notion (that limits are separate from what they limit) is what motivates certain recent thinkers to believe that numbers are the first principles (1002a9-13). Aristotle seems to have in mind these same thinkers when, in [DELTA] 8, he explains that some call the limits of individual things 'substances', because when these limits are destroyed, the individual thing is destroyed as well (i.e., because condition (ii) is fulfilled). Again in N 3, Aristotle also attacks those who make 'limits and extremes' (e.g., planes, lines, and points) substances, making it clear that he objects to the separation of these limits and extremes (1090b5-13). (20) These passages suggest to me that, again, Aristotle is targeting recent Platonist thinkers who take 'limits and extremes' to have separate existence in his M 2 attack on separate mathematical objects.

The arguments at 1077a14-20 and 24-31

The two arguments we will examine are the fifth and seventh in the M 2 series that attacks separate mathematical objects. They should be taken together, because both turn on the point that what is separate ought to be prior in ousia. In both arguments, Aristotle attempts to prove that mathematical objects cannot be separate by showing that they are, in fact, posterior in ousia to sensible objects. Aristotle writes:

(v) In general, conclusions result which contradict truth and ordinary beliefs, if one takes mathematical objects to exist in this way, as separate entities. Since they exist in this way, they have to be prior to sensible magnitudes, but in fact they are posterior, since incomplete magnitude--while prior in generation--is posterior in ousia, as is the case with lifeless and living (1077a14-20, trans. Annas, slightly modified).

Here Aristotle is arguing that if one takes mathematical objects to be separate entities, then according to the Platonists' principles, they will have to be prior to sensible spatial magnitudes. Yet, in truth, the priority relation between these entities is just the opposite, because mathematical spatial magnitudes are 'incomplete', and are, therefore, prior in generation to sensible spatial magnitudes, but posterior in ousia--just as the inanimate is prior in generation but posterior in ousia to the animate.

(vii) Besides, the point is clear from the way they are generated. First length is generated, then breadth, finally depth, and then it is complete. So if what is posterior in generation is prior in ousia, body should be prior to plane and to length. It is more complete and whole in the following way also--it becomes animate. How could there be an animate line or plane? The supposition would be beyond our senses (1077a24-31, trans. Annas, slightly modified).

Here Aristotle is explaining why mathematical spatial magnitudes are incomplete and posterior in ousia relative to sensible spatial magnitudes (bodies). He says that this is clear from the way mathematical spatial magnitudes are generated. Length is generated first, then breadth, and then depth; at this point, the process of generation is 'complete'. Thus bodies, which have length, breadth and depth, are prior in ousia to mathematical entities, such as planes and lines, because they are more complete and are posterior in generation. Bodies are also more complete because they can become animate, whereas this is impossible for lines and planes.

The two arguments prove that mathematical objects cannot be separate by a simple modus tollens: If mathematical objects are separate from sensible objects, then they ought to be prior in ousia to sensible objects. Mathematical objects cannot be prior in ousia to sensible objects, because mathematical planes, lines, and points are less complete than sensible bodies. Therefore, mathematical objects cannot be separate from sensible bodies.

I find these arguments to be quite successful against the targeted view. They are also indicative of the strength that typifies Aristotle's criticisms of Platonist metaphysics in books M-N. However, others find these arguments to be seriously flawed. In the next section, I consider the strongest objections to these arguments.

Four major criticisms of the arguments

I will consider the following four criticisms of the arguments at 1077a14-20 and 24-31: (1) Annas objects that, in the course of his arguments, Aristotle attributes to the Platonists the view that sensible things lack being, when in fact the Platonists only maintain that they are unsuitable as objects of mathematics. (2) She also argues that Aristotle confuses mathematical solids with sensible bodies. (3) Both Annas and Ross charge Aristotle with equating two different kinds of generation, namely, mathematical construction and natural generation. (4) Finally, Annas objects that Aristotle presupposes his own view that the complete is prior in ousia to the incomplete.

(1) Let us begin with the first charge. Annas objects that Aristotle 'misses the point of platonism rather strikingly' by arguing that sensible bodies are prior in being to mathematical objects (144). That is, the Platonists do not wish to say that sensible objects lack being (or ousia), but rather only that they cannot be the objects of mathematics (146). They never argue that mathematical objects are prior in ousia to sensible objects, but rather only that ideal mathematical objects are necessary for mathematics, because no sensible object can be a perfect instantiation of a mathematical notion. Thus, ideal numbers and spatial magnitudes have to exist in order to 'guarantee the truth of theorems, not to be exalted entities' (144). According to Annas, Aristotle either misunderstands his opponents, or he is arguing against others who exaggerate what the Platonists say about sensible objects (144).

There appears to be some support for Annas' objection. Aristotle does say that one of his aims in this series of arguments is to show that sensible objects are prior in ousia to mathematical objects. At the end of the series, he summarizes the following accomplishments: (1) demonstrating that mathematical objects are not substantial beings more than bodies are, (2) that they are not ontologically prior to sensible objects, and (3) that mathematical objects cannot have separate existence (1077b12-14). It is clear that Aristotle intends this last point to be a critique of what he believes to be the Platonic view.

However, it remains unclear whether or not he believes that Platonists hold positions opposed to his first two points. If he explicitly attributes to his opponents the views that mathematical objects are substances more than bodies are, and that they are prior in ousia to sensible objects, then he may have misunderstood the Platonic view of mathematical objects in the way that Annas suggests. This would also mean that arguments (v) and (vii) are directed against a theory that is dubiously Platonic.

However, Aristotle cannot have simply misunderstood the Platonists, because he later makes the very point that Annas claims he misses: 'those who make number separate assume that it exists and is separate because the axioms would not be true of sensible things, while mathematical statements are true' (N 3 1090a35-b1, trans. Ross).

Aristotle may or may not think that certain Platonists claim that mathematical objects are prior in ousia to sensible objects. However, his purpose in the entire section from 1076b11-7b14 is not to counter this view; rather, to establish that mathematical objects cannot exist as separate substantial beings. It is against this view that Aristotle's critiques are directed. The relative priority of mathematical and sensible objects is merely one of the difficult consequences of this view. It is not a position that Aristotle explicitly attributes to the Platonists. (21)

The way by which he introduces the idea that mathematical objects are prior to sensible objects makes this quite clear: 'Because they exist in this way they have to be prior to sensible magnitudes' (my emphasis). This means that if mathematical objects can exist separately from sensible things, but the latter cannot exist without the planes, lines, and points that define them, then according to the Platonists' own assumptions, it is clear that mathematical objects are prior in ousia to sensible objects. (22) This neither exalts ideal numbers, nor does it imply that sensible objects do not really exist. The point is simply that something that is separate is prior in ousia to something that is not. Moreover, if mathematical entities are eternal, then it would seem they must be prior in ousia to perishable, sensible entities. (23) In other words, because the Platonists wish to say that mathematical entities are separate and imperishable, then ex hypothesi, if such beings exist, they will have to be prior in ousia to sensible objects.

Thus, it only appears that Aristotle has misunderstood the Platonists if we assume that arguments (v) and (vii) are really directed against the view that sensible objects lack being or ousia. In fact, Aristotle's aim in arguments (v) and (vii) is to show that mathematical objects cannot be separate. He proceeds by arguing that they have insufficient being for the separate existence that the Platonists believe them to have. Clearly, if they are posterior in ousia to them, then they cannot be separate from sensible things. Thus, it is against the Platonic position that mathematical objects have separate existence--and not against their view of sensible objects--that arguments (v) and (vii) are directed. This is, after all, what Aristotle had announced as the topic of discussion at 1076b11-12, when he began this series of arguments (0[TEXT NOT REPRODUCIBLE IN ASCII]). (24)

Once Aristotle has shown that the separability of mathematical objects implies that they are prior in ousia to sensible objects, he is now able to show why this is problematic. In (v) and (vii), he explains that since mathematical objects such as lines and planes are incomplete magnitudes, they must be prior in generation, but posterior in ousia to complete, sensible objects. He gives a quick analogy at the end of (v) in order to clarify his point: lines and planes ('incomplete magnitude') are prior in generation, but posterior in being to sensible objects in the same way that the lifeless or inanimate is prior in generation, but posterior in being to the living or animate being

As noted above, when Aristotle says that the complete is prior in ousia to the incomplete, he means that the complete is ontologically independent of the incomplete, and the incomplete is ontologically dependent on the complete. In [DELTA] 11, he tells us that Plato himself held that those things 'which can be without other things, while other things cannot be without them' are prior in ousia (and nature, 1019a2-4). The idea that the complete is prior in ousia to the incomplete is repeated in (vii), in which Aristotle states that 'what is posterior in generation is prior in being', and that 'what is posterior in generation' is 'more complete and whole' (1077a26-9).

An important question remains: In what sense are mathematical objects 'incomplete', and in what sense are sensible objects 'complete?' Aristotle clarifies this in argument (vii), where he gives two reasons for attributing completeness to sensible objects and incompleteness to mathematical objects. At 1077a24-8, he argues that bodies (three-dimensional objects) are posterior in generation to planes, and planes to lines. What is prior in generation is less complete and posterior in ousia. (25) Therefore, the lines and planes from which a body is generated are 'incomplete' relative to the fully three-dimensional object, just as the seed from which a human is generated is 'incomplete' relative to the fully-formed human being. (26)

He makes this same point at Metaphysics 989a15-17, where he says that that which is 'developed and combined' ([TEXT NOT REPRODUCIBLE IN ASCII]) is posterior in generation, but prior in ousia (relative to the undeveloped). At Physics 261a13-14, he explains that that which is prior in the order of generation is 'imperfect' ([TEXT NOT REPRODUCIBLE IN ASCII]) relative to that which is prior in the order of nature (or ousia). At 1077a28-31 he states that bodies can be animate, thus providing another reason why they are more complete than mathematical objects such as lines and planes. Thus, planes and lines cannot be animate, because they are less complete than bodies.

(2) This claim that bodies are posterior in generation to lines and planes is the source of the second and third objections listed above. As for the second objection: Annas argues that sensible objects are not composed of lines and planes in the same way that mathematical objects are. While one would be justified in stating that lines and planes are prior in generation (and hence posterior in ousia) to mathematical solids, one cannot claim that they are prior in generation (and posterior in ousia) to sensible bodies. She concludes that the Platonists would never say that sensible bodies consist of planes in the way that mathematical solids do, and that Aristotle has, therefore, confused mathematical solids with sensible bodies (145, 146). (27)

This objection rests on the assumption that because it is true that sensible bodies are different from mathematical solids, the Platonists would not treat them like mathematical solids and, hence, would not say that they are composed of lines and planes in the way the former are. Yet in a familiar passage from the Timaeus, Plato has Timaeus present a view according to which sensible bodies are generated from solid figures that, in turn, are generated from planes. At 53c4-4b5, he writes that the elementary bodies ([TEXT NOT REPRODUCIBLE IN ASCII]) that include fire, earth, air, and water are generated from certain kinds of triangles. These triangles are the 'originating principles of fire and of the other bodies', since bodies have depth, depth is bounded by surface, and every surface bounded by straight lines is composed of triangles. At 54d3-5c6, Plato shows how the five 'solid forms' (the regular solids) are built up from triangles. In this passage, he uses 'solid' (e.g., 'the first solid form', [TEXT NOT REPRODUCIBLE IN ASCII], 55a2-3) and 'body' (e.g., 'the second body', [TEXT NOT REPRODUCIBLE IN ASCII], 55a7) interchangeably. He then explains that the four elementary bodies (fire, earth, air and water) each have one of the first four structures just described (the fifth is the structure of the whole universe). He further explains that earth is a cube, fire a tetrahedron, air an octahedron, and water an icosahedron (55d6-6b6). When these elementary bodies are bulked together, they become perceptible (56b7-c3). Here, we can see that at least one Platonist (and a rather important one at that) has described a process of generation that moves from planes to solid figures to elementary bodies to sensible bodies.

As is well documented in the literature, however, it is not clear how seriously committed Plato is to this view, since he has Timaeus refer to it sometimes simply as a 'likely story' ([TEXT NOT REPRODUCIBLE IN ASCII], 29d2), and sometimes has him specify that what he says is 'not only likely, but correct' ([TEXT NOT REPRODUCIBLE IN ASCII], 56b4). Regardless of Plato's intention in this text, his discussion clearly and explicitly involves the construction of sensible bodies from planes.

Thus, it is likely that when Aristotle discusses the generation of sensible bodies from lines and planes in our two M 2 arguments, he has the Timaeus in mind. This is further supported by the fact that he mentions this dialogue by name in De Caelo III 1, in the course of another critique of the view that sensible bodies can be generated from planes. There, he explains that if (as some maintain) bodies are composed of planes, then clearly planes must be composed of lines and lines of points. If sensible bodies are so composed, then what applies to points, lines, and planes will have to apply to sensible bodies as well. This causes many problems. Sensible bodies become indivisible if lines are indivisible, and weightless if points are weightless. This passage is relevant to the discussion of our two M 2 arguments for yet another reason--one which indicates that when Aristotle blurs the line between the mathematical and the sensible and then transfers mathematical problems to the sensible realm in the course of his critique, this is not because he is confused about the distinction between the two realms. It may not even be because he believes that most (or any) of the Platonists are confused in this way. Rather, this is simply a consequence of the view that sensible objects consist of planes. (28) As such, it is very unlikely that Aristotle simply made a mistake when he says that planes and lines are prior in generation to sensible bodies.

Whether or not Aristotle does, in fact, recognize that sensible bodies cannot be constructed from lines and planes in the way that mathematical solids can, we might still ask why he uses this false premise in his argument. This is especially puzzling because just a few lines later, he uses the fact that 'nothing is seen to be capable of being put together out of planes or points' (1077a34-5) in order to show that mathematical objects are not substances. Obviously he himself cannot believe both that sensible bodies are generated from lines and planes (1077a24-8) and that points, planes (and lines) cannot generate anything (1077a34-5).

It will be useful here to make a brief point about the method Aristotle employs in the arguments we are considering. When he critiques the view of a particular opponent (either an individual or a group of individuals who hold a particular set of principles), he may engage in the special kind of dialectical argument that he describes in Topics VIII 5. At 159b25-7, he explains that 'if the view laid down be one that is not generally accepted (endoxon) or rejected (adoxon), but only by the answerer, then the standard whereby the latter must judge what is generally accepted or not, and must grant or refuse to grant the point asked, is himself' (trans. Pickard-Cambridge).

In his discussion of this passage, Robert Bolton (1990) explains that when the questioner's target is 'the position of some particular individual who is to be examined from his own point of view,' then just as in all dialectical discussion, both answerer and questioner are building their arguments from endoxa (received or accepted opinions). (29) What makes this a special case is that this kind of dialectical discussion is rooted in the principles that are accepted as endoxa by this particular answerer, which may or may not be endoxa without qualification. This means that, in this kind of dialectical argumentation, the questioner might need to argue from premises that he himself does not hold to be true. Aristotle seems to have this kind of dialectical discussion in mind in Topics VIII 11, where he explains:

Sometimes, also when a false proposition is put forward, it has to be demolished by means of false propositions: for it is possible for a given man to believe what is not the fact more firmly than the truth. Accordingly, if the argument be made to depend on something that he holds, it will be easier to persuade or help him (161a24-33, trans. Pickard-Cambridge).

Aristotle is describing a situation, in which a questioner makes an argument for the sake of testing rather than for the sake of instruction, that is, when his aim is only to examine and refute a particular opponent's view, rather than determine what the correct view might be. In such cases, the questioner need not argue from what he himself believes to be true, but rather only from what his opponent believes to be true--which may turn out to be false. Moreover, although a dialectical argument should generally be considered fallacious if any of its premises are false, it seems that in this kind of situation, the questioner may legitimately argue from the answerer's own false premises.

In the M 2 series to which our two arguments belong, Aristotle finds himself in just this situation. He is performing the role of questioner, and is examining the position of a particular group from that group's own point of view. (30) His purpose is to test and refute a view that competes with his own ideas about the status of mathematical objects. His aim turns instructive in M 3, where he explains what he takes to be the correct view. Thus when, in our two arguments, Aristotle says that lines and planes are prior in generation to sensible objects, we should not take him to be assuming this to be true. Rather, in order to show that their view is ultimately untenable, he is simply starting his argument from a faulty premise that either some of his opponents might reasonably be thought to accept, or that is a consequence of a view that they accept. His opponents could claim that generations, such as the one described in the Timaeus, are not to be taken seriously, and reject the premise that sensible objects can be generated from planes. However, again, I do not see evidence here that Aristotle himself has failed to understand the difference between mathematical solids and sensible bodies.

(3) This brings us to the third objection. Annas and Ross both claim that argument (vii) conflates two different kinds of generation, namely, mathematical construction and natural generation. The first kind of generation is what the geometer brings about when, e.g., she constructs a plane figure by drawing lines. The second is a developing series in nature, in which the incomplete (seed) becomes complete (man, tree). Does argument (vii) really turn on the equivocation of these two kinds of [TEXT NOT REPRODUCIBLE IN ASCII]? It certainly seems that way. The statement 'what is prior in generation is posterior in ousia' is only true of generations that begin with something that is undeveloped and incomplete, and proceed in stages towards the fully developed and complete entity. This means that it is only true of natural generations, because natural generations are teleological processes, and this teleological context is what grounds the priority relations between the various developmental stages of a natural generation (e. g., seed and man). This is not the kind of generation involved in the drawing of geometrical objects. Yet I do not agree with Ross that this equivocation of two kinds of [TEXT NOT REPRODUCIBLE IN ASCII] 'deprives the argument of whatever value it might otherwise have possessed' (414).

Here again, Aristotle is arguing from a false premise that his opponents could reasonably be expected to maintain, or that is a consequence of a view his opponents maintain. Aristotle himself does not believe that natural, sensible objects come to be from points, lines, and planes. However, if this is assumed to be true--and this is the premise from which he argues--then it is clear that we are no longer in the realm of geometrical drawings. What we have is a natural generation of sorts. It is a mistake to claim that the lines a geometrician draws are posterior in ousia to the planes he draws, because these drawn lines are not prior in 'natural' generation to drawn planes. However, if natural, sensible objects come from points, lines and planes, then this is a natural generation; thus, it is no mistake to conclude that what is prior in generation is posterior in ousia, just as with all such generations.

At this point, it might be objected that even the idea that earlier stages in a natural generation are less complete and less perfect than later stages is an Aristotelian assumption. This would mean that Aristotle assumes--without proof--that what is prior in a natural generation is posterior in ousia. However, here again, we should consider Aristotle's methodology before assessing the quality of his arguments. There is general agreement that one of Aristotle's preferred dialectical strategies, when he aims to support one of his own views, is to show that it resolves certain aporiai while remaining consistent with key endoxa. (31) In the two arguments we are considering, Aristotle employs a correlative tactic (one well suited for critiquing an opponent's view); here, he argues that the Platonists' position cannot be correct because it conflicts with key endoxa. At the start of the first of the two arguments, he writes that if (as his opponents do) one posits separate mathematical objects, then 'conclusions result which contradict truth and ordinary beliefs' ([TEXT NOT REPRODUCIBLE IN ASCII]). (32) Here, Aristotle signals to readers that in what follows, he will be trying to show that separate mathematical objects are impossible because they conflict with certain truths and customary beliefs. This means that we should not expect him to be relying on beliefs that are proprietary to his own system, but rather on widely held, customary beliefs. In responding to the third and fourth objections to our two M 2 arguments, I place particular emphasis upon these expectations in order to show that Aristotle's arguments here do not presuppose his own views.

Here is how these expectations play out. When Aristotle states that the incomplete is prior in generation but posterior in ousia to the complete, we should expect that this is an 'ordinary belief', and not something proprietary to Aristotle's system. (33) In addition, it is true that for Aristotle's contemporaries, it would have been simple common sense to think that developing series in nature move from the lifeless to the living, and that the earlier stages are incomplete relative to the fully developed being. (34) For example, in the series from earth to seed to boy to man, the earth, seed, and boy are all incomplete relative to the man, and the man is the completion or perfection of the series earth-seed-boy-man. This widespread, intuitive view (that in natural series, the whole is more complete and perfect than its earlier stages) would not have required an acceptance or even an understanding of Aristotle's other views. (35) Thus, while it is true for Aristotle that the reason the man is more complete and perfect relative to the boy involves his notions of form and final cause, nevertheless, the simple fact that the man is more complete and perfect than the boy is an ordinary belief. Moreover, it is difficult to find anything objectionable in the further claim that if something is more complete and perfect than another, then it is prior in ousia to the other.

One might still object that Aristotle's arguments only show that sensible objects are prior in ousia to planes and lines, and not that they are prior in this way to all mathematical objects. That is, they prove that bodies are prior to lines and planes, but since both sensible objects and mathematical solids are bodies, perhaps sensible objects are not prior in ousia to mathematical solids. However, when Aristotle gives the second criterion for completeness at 1077a28-9, namely, the capacity to be animate, he does so in order to prevent this identification of solids with sensible objects. Although both sensible objects and mathematical solids are bodies, only the former can be animate. Mathematical solids cannot be animate because, among other things, ideal mathematical objects are supposed to be unchanging and unmoving.

(4) This naturally raises the following question: Does Aristotle presuppose his own views when he claims that the capacity to become animate is a criterion of completeness? This is the source of the fourth objection to our two M 2 arguments. (36) The putative presupposition in this case is the characterization of animate beings as most complete and, hence, prior in ousia to all others. The worry is that this characterization is dependent upon Aristotle's own view of substance, according to which living, sensible beings comprise what are actually 'real' (i.e., being applies to them in the original sense). Yet the claim that animate beings are more complete than inanimate beings follows just as easily from the abovementioned 'ordinary belief about developing series in nature. As we have just seen, in such series the earlier stages are less complete than the later stages. What we should now notice is that in these series, living things develop from lifeless things (e.g., man from seed from earth). If a lifeless or inanimate thing is the earliest stage of the series, and a living or animate thing is the final stage, then it follows that animate beings are more complete than inanimate things (and also that something that has the capacity to become animate is more complete than something that does not). (37) Here again, although Aristotle's own principles do produce the same result as 'ordinary beliefs', I see no pressing reason for assuming that he relies on his own principles, rather than on ordinary beliefs in order to support the claim that the animate is more complete than (and hence prior in ousia to) the inanimate.

We have seen that, taken together, the arguments at 1077a14-20 and 1077a24-31 help make a strong case against the view that the objects of mathematics are separate. Two important criticisms of these arguments made by scholars, that Aristotle misunderstands his opponents and engages in question-begging arguments, are incorrect. The charges against these arguments are made against several others in the M 2 series to which they belong, and to many other arguments in Books M-N. Since it turns out that our two much-maligned arguments escape these charges, and are, in fact, quite serious and acute, we have good reason to suspect that the same applies to many other arguments in M-N.

DOI 10.1515/apeiron-2012-0012

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Emily Katz: Department of Philosophy, Michigan State University, East Lansing, MI 48824, ekatz@msu.edu

(1) I am very grateful to Ron Polansky and Jim Lennox for reading and providing comments on the earliest version of this work, and to Charlotte Witt and Christiana Olfert for their detailed comments on a later version prepared for the first biennial workshop of the Mentoring Project for Pre-Tenure Women Faculty in Philosophy (University of Massachusetts at Amherst, June 2011). I would also like to thank the workshop participants, Marie Jayasekera and Melissa Frankel, and the workshop organizers, Louise Antony and Ann Cudd. Likewise, I am grateful to the organizers of and participants in the 35th Annual Workshop in Ancient Philosophy (University of Texas at Austin, March 2012), for providing me with the opportunity to present this work and some very useful feedback.

(2) Most noteworthy of these are Julia Annas and David Ross. John Cleary (1988) does not consider the arguments in detail, but recognizes that in some of them, Aristotle tries to show that the Platonists' view conflicts with 'commonsense assumptions about the nature of reality' (86).

(3) For example, when assessing Aristotle's critique of the Platonists' mathematical views, Leonardo Taran (1981) claims that Aristotle is so convinced of his own conception of numbers that he mistakenly imposes it on Plato, thus failing to understand Plato's actual view (15-18). Of this same critique, Leon Robin (1908/1963) complains that Aristotle demonstrates a total lack of understanding of the principles of the theory he attacks (427-428). A. E. Taylor (1907/1959) proposes that Aristotle misunderstands due to his 'thoroughly non-mathematical cast of mind' (37). Speaking more generally of Aristotle's critique of his predecessors, Harold Cherniss (1945) claims that Aristotle misunderstands his predecessors, because he is blinded by his conviction that all of them tried to express the very ideas he himself holds (348, 356). In a similar vein, J.B. McDiarmid (1953, 1959) complains that Aristotle is so anxious to read his own ideas into those of his predecessors that he often fails to grasp their views.

(4) See, e.g., Paul Natorp (1903/2004) 384 (against the first argument in the M 2 series) as well as Annas (1976) 141 (the first argument) and 143 (the third argument). Against the fifth to ninth arguments, Annas writes: 'The weakness of these arguments lies partly in the fact that so many of Aristotle's own ideas are presupposed that sometimes the questions are begged by the terms f the debate; but partly also to the fact that Aristotle seems to misidentify his opposition' (144). Claims that Aristotle misunderstands his opponents and presupposes his own views can also be found in commentators' analyses of the arguments in M 6-9. See, e.g., Natorp (1903/2004) 392; Robin (1908/1963) 432-441; Ross (1951) 180 and (1924, Vol. 2) 427; Cherniss (1944) 513-517 and (1945) 34-37; I. M. Crombie (1963) 448; J.N. Findlay (1974) 436, 444-9 and 473; Taran (1981) 15-19, and Ian Mueller (1984) 257, n. 33. These same complaints are also made against Aristotle's critique of his opponents' first principles (M 8-10 and Book N). See, e.g., Ross (1951) 204 and (1924 Vol. 1) 446, Crombie (1963) 448, R. E. Allen (1970) 32-33.

(5) See note 3 above.

(6) For a discussion of this point, see Burnyeat (1987) 232.

(7) This is not to say that this is necessarily Plato's view. As many have noted, Plato does draw a distinction between sensible numbers and figures and 'the numbers themselves' or 'the square as such' and 'the diagonal as such' (see, e.g., Republic 510e3-11a1, 524c13-6a7, Theaetetus 195e1-6b2, Philebus 56d4-7a4). Furthermore, Aristotle tells us that 'Plato posited two kinds of substance--the Forms and the objects of mathematics--as well as a third kind, viz. the substance of sensible bodies' (Metaphysics Z 2,1028b19-21, trans. Ross). However, it is not clear whether or not Plato himself thinks that mathematical objects are separate from sensible objects. For this reason, I call the view Aristotle critiques here 'Platonic', rather than 'Plato's view'. Aristotle's target is discussed in more detail below.

(8) He helpfully contrasts this view with the Pythagorean view. The Pythagoreans see that the attributes of numbers do, in fact, apply to sensible things, but they go too far by making sensible things consist of numbers. Sensible things cannot be made up of numbers, because objects with weight (sensible things) cannot be produced from weightless things (numbers). Thus, the Pythagoreans--like the Platonists--are partly right and partly wrong.

(9) In the rest of my discussion of separateness, I will not continue to add 'in the technical sense', though this should be understood.

(10) While it is true that the fact that (i) and (ii) are jointly sufficient for something does not entail that either (i) or (ii) is necessary for that something, there do not seem to be any additional conditions for separateness and priority in ousia. If (i) and (ii) are the only conditions that, together, can make something separate and prior in ousia, then they should be individually necessary for separateness and priority in ousia.

(11) See Irwin (1977), 154, who also defines separation in this way.

(12) I call one view 'weak separateness' and the other 'strong separateness', because one view makes separateness weaker than priority in ousia, while the other does not.

(13) However, different scholars take 'ontological independence' to mean quite different things. Many take it to be the capacity for independent existence (see, e.g., Fine (1984), Witt (1989, 1994), and Makin (2003)). Cleary (1988) seems to share this view, but he also argues that 'the independence of things must be defined in terms of their completeness' (87). More recently, Michail Peramatzis (2008) and Phil Corkum (2008) have argued convincingly that ontological independence cannot be capacity for independent existence. Peramatzis claims that 'A is ontologically prior to B if and only if A can be what it is independently of B being what it is, while the converse is not the case' (189). Corkum argues that ontological independence is 'having claim to the ontological status of a being' independently of being either (or both) 'said of' or 'present in' another thing (as subject) (77).

(14) This is not entirely uncontroversial. See, e.g., Panayides 1999, who argues that Aristotle does not always define priority in ousia in terms of ontological independence.

(15) The view that separation terminology in Aristotle (when it is unqualified) has to do with some kind of ontological independence, rather than with spatial separation or numerical distinctness, is quite widely accepted. See, e.g., Irwin (1977) 154; Annas (1976) 136-137; Polansky (1983) 62; Fine (1984) 33, 35-36; Cleary (1988) 87; Witt (1994) 216-217 note 2, (1989) 51-52; and Corkum (2008) 66-71. See Morrison (1985a,b) for the dissenting view that separation is rather numerical distinctness.

(16) See Fine (1984), 33, 35-38, and especially 36 note 19.

(17) See Fine (1984) 38 and (1985) 159, and Corkum (2008) 69 and note 3. Corkum stops short of claiming that (i) is a necessary condition for priority in ousia, although he suggests that it might very well be when he claims that the priority of individual substances implies that they are separate from universal substances. ('Aristotle clearly holds in the Categories that individual substances are prior to, and so separate from, universal substances' (70, my emphasis).)

(18) At 1077a19, it is clear that he means priority in ousia.

(19) The upshot of the M 2 evidence is that priority in ousia and separateness are mutually entailing; see also Cleary (1988) 87, who argues that Aristotle takes priority in ousia to be a consequence of separation. I will not argue that the two terms are, therefore identical, though this seems to be a possible interpretation of the evidence. Ronald Polansky (1983) argues that 'the prior means nearly the same as the separable. While both the prior and the separable are equivocals, their primary senses converge.' For passages that support the claim that priority in ousia 'links up with the separable', he points to Metaphysics 1077b2-3,1035b22-24, and 1038b27-29 (62).

(20) While Aristotle may agree with these thinkers (and, or including, Plato) regarding the criteria for separateness, he believes that planes are ontologically dependent on bodies, and that at least condition (i) is not fulfilled. See Cleary (1988) 46 for a useful discussion of the B 5 passage.

(21) In [DELTA] 8, Aristotle tells us that 'some say' mathematical objects (he mentions planes, lines, and numbers) are substances. Even here, however, he does not say these thinkers claim that mathematical objects are prior in ousia to sensible things, or that sensible things somehow lack ousia. He does say that the mathematical objects he has mentioned limit all things, and that without them nothing exists. This seems to me to be primarily a claim about separability, and only by implication a claim about priority in ousia. In fact even in B 5, where he does speak about relative substantiality, he does not actually attribute to anyone the view that bodies are less likely to be substances than planes (nor does he say there that sensible bodies lack being). Instead, he makes a claim about the implications of taking limits to be separate. He begins with the consequent of the implication: '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.' He then gives us the basis for this speculation (the antecedent of the implication): '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' (1002a4-7). Notice that, here, he never attributes the consequent to anyone, although he does maintain that it follows naturally from the view that the limits of an entity are separable from that entity. In short, Aristotle rejects the antecedent, but accepts the implication. We should also note that what Aristotle says here is just as much about the priority of lines over planes and points over lines as it is about the priority of all mathematical objects over sensible 'bodies'. He says nothing here about whether or not mathematical solids are prior to sensible bodies. Instead, he confines himself to drawing out the consequences of his opponents' view that the limits of bodies are separable ('lines and points are substances more than bodies' 1002a15-16). Perhaps because it is not clear from what his opponents say how mathematical solids would be separable from sensible bodies, Aristotle says nothing here about the status of mathematical solids.

(22) See also Cleary, 1988, 85-88.

(23) See 0.8, 1050b6-7.

(24) In support of my claim that separation is the real issue in Aristotle's critique of Platonic mathematical objects, we may note that one of Aristotle's basic critical points, when he is addressing Platonic views, is that certain objects that Platonists make separate cannot in fact be separate. See, e.g., Metaphysics A 9 and M 4, in which he argues against separate forms.

(25) Of course, this is only true of natural generations, such as the development of an adult human being from a seed. This means that Aristotle conflates natural generation with mathematical construction when he argues here that bodies are prior in ousia to planes and lines, because they are prior in 'generation'. However, I argue below (in the discussion of the third objection) that, when we consider the likely target of Aristotle's critique, it is not unreasonable for him to conflate these two kinds of generation here.

(26) See for example Metaphysics 1050a4-7.

(27) Annas maintains that this same confusion of sensible bodies with mathematical solids also weakens the eighth and ninth arguments in the M 2 series (146-147).

(28) For a discussion of this issue as it appears in De Caelo III1, see also Cleary (1985), 30-32.

(29) Endoxa are opinions that are 'accepted by everyone or by the majority or by the philosophers --by all, or by the majority, or by the most notable and illustrious of them' (Topics I 1, 100b21-3, trans. Pickard-Cambridge, slightly modified). Later in Book I (110,104a8-10), Aristotle adds that even if an opinion is held by a wise and notable person, it will nevertheless be unsuitable as a dialectical premise if it conflicts with what people generally believe (that is, if it is [TEXT NOT REPRODUCIBLE IN ASCII]). This suggests that, in the realm of dialectical argument, endoxa that are more widely held carry more weight than endoxa that are held (only) by the wise. This explains why Aristotle can use 'ordinary beliefs' to refute the Platonic view of separate mathematical objects, even though both are endoxa (the former because they are widely held, and the latter because it is held by wise and notable people).

(30) For another interesting example of this kind of argument, see the first argument in the M 2 series (1076b12-33). In this reductio, Aristotle uses his opponents' premises (which he himself does not hold) in order to produce an absurd accumulation of ideal objects.

(31) See, e.g., Nicomachean Ethics VII 1 (1145b2-7), where Aristotle writes: 'As in other cases, we must set out what appears true about our subjects, and, having first raised the problems, thus display, if we can, all the views people hold (endoxa) about these ways of being affected, and if not, the larger part of them, and the most authoritative; for if one can both resolve the difficult issues about a subject and leave people's views (endoxa) on it undisturbed, it will have been clarified well enough' (trans. Rowe).

(32) I take it that by 'ordinary beliefs' here, Aristotle means something like the 'common things' (koina) he discusses in Sophistical Refutations 11; such 'common things' are known by everyone --even by non-scientists. For a discussion of koina as it is used in Sophistical Refutations 11 (and certain passages of the Rhetoric and Topics), and a strong argument that Aristotle means 'known by everyone' (rather than 'common to all subject matters'), see Bolton (1990), 214-17.

(33) Annas notes this as well (144).

(34) For discussions of the Greek view that life can be generated spontaneously ([TEXT NOT REPRODUCIBLE IN ASCII]) from lifeless matter, see W. K. C. Guthrie (1957) 39-42, D. M. Balme (1962) 96-104, and James Lennox (1982).

(35) At the beginning of De Caelo, Aristotle gives another reason for thinking that bodies are more perfect and complete than planes, lines and points. This second reason is also independent of his view that sensible substance is what is primarily real and does not necessarily rely on a developing series that begins with points and ends with bodies. He explains that among magnitudes, body alone is complete because it alone extends in every direction, and not just in one or two. Since the body is determined by all three dimensions, it is an 'all', i.e., complete. Aristotle explains that 'every', 'all', and 'complete' ([TEXT NOT REPRODUCIBLE IN ASCII]) really have the same meaning. He also argues that the number three is complete, because the Pythagoreans thought that it determined the universe, and because it is the first number to which the term 'all' is applied. He concludes that the magnitude determined by all three dimensions (i.e., body) is most complete and perfect ([TEXT NOT REPRODUCIBLE IN ASCII]), because it has being in all the ways possible for a magnitude.

(36) See Annas, 144.

(37) This raises a question about whether or not the earth has the capacity to become animate in the same way that the seed does (in which case the seed might not be 'more complete' than the earth). As Aristotle explains in 0 7, although the earth does become a seed, and the seed becomes a man (which is animate), strictly speaking, it is only the seed (and not the earth) that has the capacity to become animate.

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Article Type: | Critical essay |

Date: | Jan 1, 2013 |

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