Aristotle on the Purity of Forms in Metaphysics Z.10–11

3.1. The Main Body of Z.11 (1036a26–b32)

Z.11 starts out with a puzzle: ‘One wonders reasonably both what sort [of parts] are parts of the form and what sort are not, but [parts] of the composite (τοῦ συνειλημμένου)’ (1036a26–27). While Z.10 was concerned with the question whether the parts of a whole should be mentioned in its definition, Z.11 aims at distinguishing the formal from the material parts of the composite. The way Aristotle sets up that task favours the hard purist. For Aristotle claims that ‘the definition is of the universal and the form’ (1036a28–29), apparently assuming that only the formal parts of a composite should be mentioned in its definition (1036a29–31). In the same vein, Aristotle goes on to claim that the flesh and bones of a human being are not ‘parts of the form and account’ (1036b5), with the implication that only the parts of the form should be mentioned in the definition of the human being.[15]

Overall, then, the introductory section of Z.11 proceeds as predicted by the hard purist and, by the same token, spells trouble for the soft purist. But hard purism, too, begins to face problems when, in the ensuing Mathematical Passage, Aristotle objects to his opponents’ practice of defining geometrical objects:[16]

Hard purists take Aristotle still to be concerned with distinguishing the formal from the material parts of the composite (Devereux 2010: 177; Frede & Patzig 1988: 206). On their reading, Aristotle’s opponents claim that geometrical objects should be defined without reference to ‘lines and the continuous’ (1036b9–10) because they mistake them for material parts.[18] But according to Aristotle, as presented by hard purism, geometrical objects should be defined with reference to the continuous because the continuous is one of their formal parts (Frede & Patzig 1988: 201). That last claim will lead to problems for hard purism if we look beyond the MP.

At the beginning of the SYP, Aristotle describes the mistake criticized in the MP thus:

It seems that, in defining geometrical objects in terms of numbers, and without reference to the continuous, Aristotle’s opponents define geometrical objects with reference to their formal properties alone, ‘taking away the matter’ (1036b23) from the definition.[19] If that impression is not deceptive, hard purism is in trouble. For hard purists claim that the continuous is a formal part of a geometrical object. But if Aristotle also takes the continuous to be the matter of geometrical objects, hard purists must admit that some of the material parts of a geometrical object are also among its formal parts, which is the impurist view (Peramatzis 2011: 58–59; Whiting 1991: 626).

In response, Devereux (2010: 180) has suggested that Aristotle complains not that his opponents omit reference to what is the matter of geometrical objects by his own lights, but merely to what they mistake for matter. Hence, Aristotle’s point is that the continuous is a non-material formal part of geometrical objects, just as the hard purist maintains. However, if Devereux is right, according to Aristotle, the continuous should not count as the matter of geometrical objects. But there is good evidence that Aristotle does take the continuous to be the matter of geometrical objects, namely, their intelligible matter.

In Z.10–11, Aristotle explicitly analyses geometrical objects as composites of a geometrical form and intelligible matter (1036a2–12; 1036b32–37a5). He contrasts intelligible matter with perceptible matter (1036a9–12; 1037a4–5), and he seems to think of intelligible matter as something like pure spatial extension which ‘is present in perceptible things not qua perceptible’ (1036a11).[20] Indeed, elsewhere, he identifies the matter of geometrical objects with the continuous.[21] Moreover, the mention of ‘lines’ at 1036b9 is helpful. For in his summary of Z.10, Aristotle contrasted the perceptible with the mathematical right angle, treating the lines as the (intelligible) matter of the latter in analogy with the bronze of the bronze right angle (1036a20–22).

Hence, it seems that, for Aristotle himself, the continuous is the (intelligible) matter of geometrical objects. But then Devereux’s response fails. It is not just that Aristotle’s opponents mistake the continuous for matter. Rather, they omit reference to what, by Aristotle’s own lights, is matter, and they are wrong for doing so. According to Aristotle, geometrical objects should be defined with reference to their (intelligible) matter. But if all the parts of an object mentioned in its definition must be formal parts, as the hard purist has it, some of the formal parts of a geometrical object are also material parts, and hence geometrical forms must be impure.

Analogous problems arise for the hard purist interpretation of the SYP. The section at 1036b21–24 refers not only back to the MP but also forward to the SYP. Aristotle generalizes his conclusion from the MP: One should not remove (reference to) the matter from the definition of anything which is ‘this in this or these being in this state’ (1036b23–24), that is, ‘this form in this matter’ or ‘these material parts in this (formal) state’. He then relies on that general principle to confront the Platonist Socrates the Younger:[22]

Prima facie, Aristotle’s line of reasoning is as follows: Socrates the Younger is an extreme exponent of the view rejected in the MP. He wants to define not only geometrical objects but even sensible objects without reference to their matter. Hence, he treats human beings, and animals more generally, in analogy with the (mathematical) circle. Circles are intelligible composites (1036a3–4) and can be defined without reference to perceptible matter. Similarly, according to Socrates the Younger, animals can be defined without reference to perceptible matter. But, Aristotle objects, there is a disanalogy: Animals, unlike (mathematical) circles, are perceptible objects.[23] Hence, the animal ‘cannot be defined without motion, nor therefore without its parts being in a certain state’ (1036b29–30).[24]

That reading, if correct, is disastrous for the hard purist. For the hard purist claims that sensible objects should be defined with reference to their formal parts alone. Hence, if they should be defined with reference to their perceptible matter too, some of their formal parts must be material as well, and impurism is true. In order to avoid that consequence, Frede and Patzig have attempted to show that, according to Aristotle, animals should not be defined with reference to their material parts. Rather, Aristotle criticizes Socrates the Younger because his position may mislead one into thinking that forms can exist without matter. According to the hard purist, then, animals should be defined with reference to the capacities of their souls alone, such as the capacity for perception. The material bodily parts of the animal are not explicitly mentioned in its definition, but their existence is implied since they are required for the animal to have the relevant capacities (Frede 1990: 119–121; Frede & Patzig 1988: 209–210).

However, the issue of ‘taking away the matter’ (1036b23) with which both the MP and the SYP are occupied, is a puzzle ‘about definitions’ (1036b21), an instance of which has been stated as a puzzle ‘about the circle and the triangle that it is not appropriate for them to be defined by lines and the continuous (γραμμαῖς ὁρίζεσθαι καὶ τῷ συνεχεῖ)’ (1036b9–10). But surely defining an object by the continuous has to involve explicit reference to the continuous, and similarly, one would think, for other instances of the puzzle. For instance, the puzzle whether the definition of an animal should refer to its perceptible matter must be concerned with an explicit definition of the animal no less than the puzzle whether the definition of a geometrical object should refer to its intelligible matter. Plausibly, then, Aristotle claims that the animal should be explicitly defined in terms of its material parts, contra Frede and Patzig.

Reformist hard purists admit that last point but draw on the fact that, in the SYP, Aristotle discusses the relation between the whole and its functional parts. In that respect, as we will see in detail below, the SYP importantly differs from the Puzzling Passage, where Aristotle was interested in the relation between a whole and the bits of matter into which it perishes. Some hard purists, then, hold that the functional parts mentioned in the SYP, such as the hand, are not ‘material’ parts at all, properly speaking, and can be parts of the form (Chiaradonna 2014: 384–388; Devereux 2010: 183–184; Wedin 2000: 338). Yet, Aristotle speaks of the functional parts of sensible composites as their matter, as we will see in more detail later.[25] Moreover, if the hard purist admits that the functional parts of a composite are parts of the form, and merely insists that those are not ‘material’ in the right way, the distinction between hard purism and impurism is in danger of becoming merely notational.[26] Overall, therefore, while hard purism fared well as a reading of the introductory section of Z.11, it is difficult to sustain as an interpretation of the MP and the SYP.

Unlike the hard purist, the soft purist is not troubled by the MP and the SYP. For unlike the hard purist, the soft purist accepts that a composite can be defined with reference to its non-formal material parts. Hence, the material parts with reference to which composites are said to be defined in the MP and the SYP need not be formal parts, and neither geometrical forms nor forms of sensible objects turn out to be impure. Of course, the details of the soft purist reading of those passages are not trivial. As we will see in our discussion of hierarchical purism, additional work is needed to show that there can be any sort of account of a composite which mentions its material parts. But in general terms, soft purists can take the MP and the SYP to support their view: Composites should be defined with reference to some of their material parts, and the point of that conclusion is precisely that those material parts are not formal parts of the composites (Gill 1989: 133–136; Heinaman 1997; Ross 1924: 203).

That said, we should not forget that the soft purist had trouble accounting for the implication in the introductory section of Z.11 that the definition of a composite is a definition solely in terms of its form. That issue will arise more forcefully in the summary of Z.4–11 to which we are turning now.

3.2. The Summary of Z.4–11 (1037a21–b7)

At first sight, the summary of Z.4–11, or better, that part of it which is concerned with Z.10–11 (1037a21–33), supports hard purism against soft purism. However, underneath its surface, the summary will be seen to display the same structure as the Puzzling Passage in Z.10 and the main body of Z.11, and hence to be difficult for both hard and soft purists.

Hard purism rests largely on what I will call the ‘substance claim’ in the summary:

Here, Aristotle says explicitly that the only definition of a composite is in accordance with its primary substance or form, which is just the hard purist thesis. As we will see below, a later remark in the summary will qualify that claim and cast doubt on whether the summary provides straightforward evidence for hard purism. But first, let us consider how soft purists try to make sense of the substance claim.

As far as I am aware, there have been three direct soft purist responses to the substance claim: First, Ross (205) holds that only prime matter is excluded from the definition of the composite because only prime matter is indefinite. Second, Heinaman (1997: 295) argues that only the definition of the form must not refer to the matter of the composite, while the definition of the composite can, or should, refer to its matter. Third, Michael Ferejohn (1994: 315–316) has suggested that the identification of substance as ‘the immanent form (τὸ εἶδος τὸ ἐνόν)’ (1037a29) implies that not only the form but also the matter of the composite should be mentioned in its definition.

None of those responses are convincing. As for Ross, Aristotle treats matter quite generally as ‘indefinite’, not just prime matter (if Aristotle even has a notion of prime matter).[27] Hence, there is no reason to think that Aristotle’s claim here is restricted to prime matter. Heinaman’s answer, in turn, does not succeed because the substance claim explicitly concerns the definition of the composite, not just its form. Nor is Ferejohn’s suggestion any more successful. For even if forms are immanent in matter, it may be that the composite should be defined without reference to the matter in which the forms are immanent. Unsurprisingly, then, soft purism cannot accommodate the substance claim any more than other instances of the same claim found in part [A] of the Puzzling Passage or the introductory section of Z.11 (see also Z.10, 1035b31–36a2).

That said, the summary as a whole does not support hard purism either. For Aristotle follows up on the substance claim as follows: ‘But in the composite substance, such as snub nose or Kallias, the matter will be present too’ (1037a32–33). Superficially, that claim does not have any definitional import, but it appears to refer us back to a passage which does. The only other place in Z.10–11 where Kallias is mentioned (in tandem with the snub) is in the aftermath of part [B] of the Puzzling Passage. There, both the snub (1035a26) and Kallias (1035a33) are cited as examples of ‘things taken together with matter’ which should be defined with reference to their material parts. Since Aristotle makes the effort to remind us of that earlier passage, presumably the definitional claim made there is still in place (see Gill 1989: 136–137). But if that is right, even though the substance claim is in line with hard purism, the summary as a whole is not. For hard purism cannot account for the implied definitional claim at 1037a32–33 any more than for part [B] of the Puzzling Passage.

Hence, the summary, no less than the Puzzling Passage and the main body of Z.11, exhibits the very dialectic which has proven so troublesome: First, Aristotle claims that the composite should be defined with reference to its formal parts alone, but then he goes on to claim that some of its material parts should be mentioned too. Neither hard nor soft purism sits well with that dialectic. If impurism can do better, and if there are no further purist alternatives, we may have good reason to abandon purism.

4.1. The Impurist Reading of Z.10–11

Let us first return to the Puzzling Passage in Z.10. In part [A] of that passage, Aristotle claimed that composites should be defined without reference to the material parts into which they perish. For example, a human being should be defined without reference to its flesh and bones. But in part [B] of the same passage, Aristotle said that composites should be defined with reference to the material parts into which it perishes. For instance, Kallias should be defined with reference to his flesh and bones. The impurist may aim to resolve that apparent inconsistency by appeal to the material principles of the composite.[28]

Remember that some impurists distinguish between the particular bits of matter into which a composite perishes and the material features of the composite. Only the latter are parts of the form. That distinction promises an elegant solution to our puzzle: In [A], Aristotle argues that the particular bits of matter into which a composite perishes should not be mentioned in its definition. But in [B], Aristotle speaks of material parts as ‘principles’ (1035a24), and the impurist can take those to be material features of the composite rather than parts into which the composite perishes. Thus, Aristotle argues consistently that the material parts into which a composite perishes should not be mentioned in its definition, and are not parts of its form, but that the material principles or features of the composite should be mentioned in its definition and are parts of its form.[29]

The impurist takes the MP and the SYP in Z.11 to confirm the same sort of point. First, according to the impurist, Aristotle argues in the MP that the continuous is both a formal and a material part of the geometrical composite, and hence that geometrical forms are impure, as seen in our discussion of hard purism (Peramatzis 2011: 58–59; Whiting 1991: 626). More importantly, the SYP is taken to directly support impurism about the forms of sensible objects. According to Irwin (1988: 245–247) and Whiting (1986: 372; 1991: 631), Aristotle commits himself to the view that the animal should be defined with reference to its functional material parts. On Peramatzis’s version of impurism, the animal, or human being, should be defined with reference to its material features, such as ‘having fully functional hands’ (2011: 98–99; 2015: 209–210; see also Charles 2008: 15–17, discussing the implications of DA I.1). But on either view, forms of sensible objects turn out to be impure. For if the animal is defined with reference to its functional matter or material features, and it is defined exclusively with reference to its formal parts, the functional matter or material features must be formal parts of the composite.

Thus, apparently, impurism can draw on the MP and the SYP as support no less than soft purism. But unlike soft purism, impurism does not seem to have any trouble with passages like the summary of Z.4–11 where Aristotle claims that only the formal parts of a composite should be mentioned in its definition. For on impurism, it is true that the definition of a composite mentions only its formal parts, not the indefinite non-formal material parts of the composite (Peramatzis 2011: 51). Nonetheless, ultimately impurism does not work as an interpretation of Z.10–11.

4.2. Why Impurism Fails

A first problem with impurism arises from the MP. According to the impurist, Aristotle’s claim in the MP is that the lines and the continuous are both formal and material parts of the geometrical object, and hence that the geometrical form is posterior to some of the material parts of the composite, namely, to the lines and the continuous. But in the summary of Z.10, Aristotle claims that ‘the right angle without matter is posterior to the parts in the account, yet prior to the parts in the particulars’ (1036a22–23, my italics). In the case of the intelligible right angle, then, the right angle without matter, that is, the form of the right angle, is posterior to its own parts but prior to the intelligible material parts of the intelligible composite, namely, its lines (1036a21). Aristotle effectively restates the same point in Z.11 when he claims that the intelligible semi-circles ‘will not be parts of the universal circle but of the particulars’ (1037a2–3). It would be surprising if Aristotle argued for the negation of that claim in the MP, as the impurist has it.

But the main objection concerns the impurist reading of both the Puzzling Passage and the SYP. As for the Puzzling Passage, the impurist move was to distinguish between the material parts into which a composite perishes and its material principles. However, it is doubtful that such a distinction can be found in the passage. Aristotle says, ‘Some things are from the parts into which they perish as from principles (ὡς ἀρχῶν), but others are not’ (1035a24–25). It seems quite clear that the principles are not distinguished from the material parts into which composites perish but identified with them. That would also fit Aristotle’s treatment of the bits of matter from which a composite comes to be as its material principle in the Physics (see, e.g., Phys. I.7, 190b17–91a3).

If that is right, the impurist faces the same difficulty as the hard purist. Although the impurist thinks that either the functional matter or the material features of the composite should be mentioned in its definition, and hence are among the parts of its form, the impurist wants to exclude reference to the bits of matter into which the composite perishes from its definition. Just like the hard purist, therefore, the impurist has trouble accommodating [B], where Aristotle tells us that the composite is defined with reference to the material parts into which it perishes.

Peramatzis (2011: 49) recognizes that problem and argues that [B] is concerned merely with the existential dependence of composites on the material parts into which they perish. However, in [B], Aristotle is explicitly concerned with ‘accounts’ of composites, not merely with an existential dependence relation. After all, he claims that ‘the account of such parts will be in [the accounts] of some things’ (1035a22–23). The Puzzling Passage, far from supporting impurism, is in fact a problem for the view.

A similar difficulty comes about for the impurist reading of the SYP. Recall what Aristotle says in the crucial section of the SYP:

There are two impurist options here. Prima facie, Aristotle tells us that the animal should be defined with reference to ‘its parts being in a certain state’, such as the ensouled hand ‘capable of fulfilling its function’. That is, prima facie, Aristotle’s claim is that the animal should be defined with reference to its concrete functional material parts, such as the (ensouled) hand, and it is those concrete functional material parts which are parts of the form. That is the first impurist option, defended by Irwin (1988: 246).

The problem with the first impurist option is that, in Z.10, Aristotle explicitly denies that functional material parts are prior to the form or soul of an organism. Having identified the soul as the substance or form of the animal (1035b14–16), he says that ‘the parts [of the soul], either some or all of them, are prior to the composite animal, and similarly in each case, but the body and its parts are posterior to that substance’ (1035b18–21). Indeed, in the SYP itself, Aristotle seems to recall that doctrine. For ‘if [the hand] is not ensouled, it is not a part [of the human being]’ (1036b32), which suggests that the hand of a human being is posterior to its soul. Assuming that Aristotle maintains a coherent account of the relation between body and soul in Z.10–11, we should reject an interpretation of the SYP which makes it straightaway inconsistent with the doctrine stated in Z.10.

The second impurist option, advocated by Peramatzis, avoids the problem just mentioned. On that version, it is not the concrete functional material parts of a composite (nor types of such parts), which are parts of the form (Peramatzis 2011: 10), but rather material features, such as ‘having living and fully functional hands which can perform successfully their relevant function’ (Peramatzis 2011: 98). While the functional material parts of the composite may be posterior to the soul, the relevant material features are prior. The force of the SYP, then, is that the animal should be defined with reference not to its (functional) material parts but to its material features.

That impurist move is sophisticated, but perhaps too sophisticated for the SYP. For Aristotle is quite explicit that the animal cannot be defined ‘without its parts being in a certain state’ (1036b30). It is true that the state of the parts is important for the definition, and perhaps we can understand the state of the parts as a material feature of the organism such as ‘having fully functional hands’. But it is equally true that the parts themselves are important for the definition: It is the parts in some state which should be mentioned (see Devereux 2010: 180). Hence, for the brand of impurism we are considering, the SYP (much like the Puzzling Passage) shows too much, namely, that the concrete material parts (or perhaps types thereof) should be mentioned in the definition of the composite, and so, by impurist lights, are parts of the form. Overall, then, neither of our two impurist interpretations of the SYP is persuasive.

Where does that leave us? On the one hand, neither hard purism nor soft purism promises a coherent reading of Z.10–11. On the other hand, impurism cannot handle crucial passages, such as the Puzzling Passage or the SYP, either. I will argue next that hierarchical purism offers a unified interpretation of Z.10–11 which avoids the problems encountered so far. If that is right, we should take Z.10–11 to argue for the purity of forms after all.

5.1. Solving the Z.10 Puzzle

Let us return to a passage early on in Z.10, which I will call the ‘Hylomorphic Passage’:

Large parts of the passage are dedicated to explicating the thesis that ‘there is a sense in which matter, too, is said to be a part of something, and there is a sense in which it is not’ (1035a2–3). Aristotle distinguishes between a composite whole (σύνολον) and an object ‘said as form’ (1035a7). Probably, Aristotle has in mind a linguistic distinction which he draws more clearly elsewhere. In Metaphysics H.3, for example, he says, ‘We must not ignore that sometimes it is hidden whether the name signifies the composite substance or the actuality and the form’ (1043a29–31; see also Meta. Z.8, 1033b17–18). Names such as ‘statue’ can either refer to a composite object, or to the form of that object. Thus, if we use a name to refer to a composite, then matter is a part of the referent, but if we use the same name to refer to a form, matter is not a part of the referent. Bronze is a part of the composite statue, but not of the form of the statue (see Bonitz 1848–49: 333; Bostock 1994: 148; Frede & Patzig 1988: 174; Ps.-Alexander 503.16–17).

Aristotle goes on to state the principle more generally: ‘Both the form and each thing insofar as it has a form (ᾗ εἶδος ἔχει) should be called [‘something’], but the material by itself should never be called [‘something’]’ (1035a7–9).[30] By any name, we can refer either to the form of a composite or to the matter-form composite itself insofar as it has a form, but never to the matter of the composite by itself. For instance, both the soul of Socrates and the composite of body and soul (insofar as it has a soul) can be called ‘Socrates’, but never the body by itself (for the example, see Meta. Z.11, 1037a5–10).

Crucially, Aristotle’s principle leaves open another option. Even if we cannot use a name to refer to the matter of a composite by itself, we can use it to refer to the composite insofar as it has both a form and matter. Aristotle will take recourse to that option later, or so I will suggest. Equally importantly, however, only the composite insofar as it has a form is under consideration as of yet. The composite insofar as it has both a form and matter is disregarded for the time being. Hence, Aristotle can conclude—‘that is why (διό)’ (1035a9)—that only the formal parts of a composite should be mentioned in its definition, for instance, the letters in the definition of the syllable (but not the segments in the definition of the circle). After all, it is a natural thought that, if one considers a composite only insofar as it has a form, it should be defined with reference to its formal parts alone.

One might object that, in speaking of the composite insofar as it has a form, Aristotle does not mean to introduce a special way of considering the composite but merely to highlight that the composite is a definite, nameable object in virtue of its form.[31] On that point, I simply agree with the objector: In the Hylomorphic Passage, Aristotle does not draw a distinction between different ways of considering the composite. Rather, he simply assumes that composites are nameable, and definable, in virtue of their forms, without entertaining the possibility that the matter of the composite could play a role for explaining and defining what the composite is. As I see it, Aristotle’s confusing move will be precisely that he drops that background assumption later, at least in some contexts.

With the Hylomorphic Passage in mind, then, we can understand the Puzzling Passage. As noted above, there is a curious shift in terminology between parts [A] and [B] of the Puzzling Passage. In [A], Aristotle refers to the composite as a ‘σύνολον’ or ‘whole’, but in [B], he begins to call it a ‘συνειλημμένον’ or ‘thing taken together with matter’.[32] Moreover, in [A], he argues that the composite (under the description of ‘whole’) should be defined without reference to the material parts into which it perishes because they are not parts of its form. But in [B], he claims that the composite (under the description of ‘thing taken together with matter’) should be defined with reference to the material parts into which it perishes.

The terminological difference extends beyond the Puzzling Passage. In the Hylomorphic Passage, Aristotle called the composite a ‘σύνολον’ or ‘whole’, speaking of the ‘whole statue’ (‘τοῦ συνόλου ἀνδριάντος’) at 1035a6. On the other hand, in the aftermath of the Puzzling Passage, he keeps speaking of the composite as a ‘συνειλημμένον’ or ‘thing taken together with matter’ (1035a25–b3). Moreover, as discussed, Aristotle claims in the Hylomorphic Passage that only the formal parts of the composite should be mentioned in its definition. By contrast, after the Puzzling Passage, Aristotle insists that, for example, ‘the clay statue perishes into clay and the sphere into bronze and Kallias into flesh and bones’ (1035a32–33), with the implication that those composites should be defined with reference to the material parts into which they perish.

My suggestion is that the shift in terminology marks a shift in how Aristotle thinks about the composite, and that the latter shift explains his divergent claims about the definition of the composite. As long as he refers to the composite neutrally as a ‘whole’, Aristotle is interested in the composite only insofar as it has a form, and hence, in [A], he adheres to the claim from the Hylomorphic Passage that only the formal parts should be mentioned in its definition. On the other hand, as soon as Aristotle begins to call the composite a ‘thing taken together with matter’ in [B], he draws on an option absent from the Hylomorphic Passage, namely that we might use a name to refer to the composite insofar as it has both a form and matter. For example, by ‘Kallias’, we might mean the composite insofar as it has both a soul and flesh and bones. However, in that case, Kallias should be defined with reference to both his soul and the flesh and bones into which he perishes.[33]

Thus, the distinction between the ‘σύνολον’ or ‘whole’ in [A] and the ‘συνειλημμένον’ or ‘thing taken together with matter’ in [B] is not an extensional one between composites of different classes (let alone between forms and composites, as soft purists have it), but rather an intensional one between different ways in which composites can be considered, either only insofar as they have a form, or insofar as they have both a form and matter. Moreover, depending on how a composite is considered, it should be defined in different ways. Hence, on hierarchical purism, the tension in the Puzzling Passage finally disappears. If a composite is considered one way, it should be defined with reference to its formal parts alone, as we are told in [A], but if the same composite is considered another way, it should be defined with reference to its material parts as well, as we are told in [B].

None of that is to say that the terminological distinction between ‘σύνολον’ and ‘συνειλημμένον’ always tracks the distinction between the composite only insofar as it has a form and the composite insofar as it has both a form and matter.[34] Indeed, we will see shortly that, later in Z.11, ‘σύνολον’ can refer to the composite insofar as it has both a form and matter. Nonetheless, in the Puzzling Passage, the terminological distinction points at the distinction between different ways in which the composite can be considered. In other passages, different markers will serve the same purpose, as I am about to discuss.

5.2. Hierarchical Purism and Z.11

I argued in Section 3 that both the main body of Z.11 (1036a26–b32) and the relevant part of the summary of Z.4–11 (1037a21–33) have the same dialectic structure as the Puzzling Passage. I will now explain how the hierarchical purist can account for Z.11 in just the way in which she has handled the Puzzling Passage.

To begin with, the introductory section of Z.11 (1036a26–b7) is reminiscent of part [A] of the Puzzling Passage. For Aristotle recalls the example of the human being from [A], once again to argue that the flesh and bones of the human being are not parts ‘of the form and the account’ (1036b5, recalling 1035a21). A similar point applies to the ‘substance claim’ at 1037a26–29 in the summary of Z.4–11 (that is, to the claim that the only definition of a composite is in accordance with primary substance). For the substance claim occurs in the context of what is effectively a paraphrase of the contention in [A] that the material parts of the composite are not parts of the substance of the composite (1037a24–26, paraphrasing 1035a19–21). Plausibly, then, both the introductory section of Z.11 and the substance claim in the summary concern the composite only insofar as it has a form, no less than [A].

In the Puzzling Passage, Aristotle’s shift of interest from the composite considered only insofar as it has a form to the composite considered insofar as it has both a form and matter was marked verbally: In [A], Aristotle spoke simply of the ‘whole’, while in [B] he began to speak of ‘the thing taken together with matter’. At the outset of Z.11, Aristotle calls the composite the ‘thing taken together with matter’ (1036a27), which suggests that the composite insofar as it has both a form and matter might play a role in the chapter. But in the main body of Z.11, he does not explicitly refer to the composite at all. Nonetheless, there is a linguistic marker that, as Aristotle proceeds beyond the introductory section of Z.11 to the MP and the SYP, his interest shifts to the composite considered insofar as it has both a form and matter.

At the beginning of the SYP, Aristotle says, pointing both back to the objects discussed in the MP and forward to the objects to be discussed in the SYP, that they are ‘this in this or these in this state’ (1036b23–24). The paradigm example for something which is ‘this in this’ is the snub, which is concavity in flesh (or in the nose) (see, e.g., Meta. E.1, 1025b32–33; Z.5, 1030b16–20; Z.10, 1035a4–6; Phys. II.2, 194a4–7; DA III.4, 429b14). But right after the Puzzling Passage, the snub was also treated as an example of a ‘thing taken together with the matter’ (1035a26; similarly, at Meta. E.1, 1025b32). Presumably, then, when Aristotle speaks of things which are ‘this in this’ he is concerned with the composite insofar as it has both a form and matter no less than when he speaks of ‘the thing taken together with its matter’. If that is right, unlike the introductory section of Z.11, the MP and the SYP target the definition of the composite considered insofar as it has both a form and matter.

The summary of Z.4–11 calls for a similar remark. We saw that, a few lines beyond the substance claim, Aristotle says that ‘in the composite substance (ἐν τῇ συνόλῳ οὐσίᾳ) the matter will be present too’ (1037a32–33). Above I argued that Aristotle implicitly evokes the doctrine from Z.10 that composites should be defined with reference to their material parts. And even though Aristotle uses the word ‘σύνολον’ or ‘whole’ to evoke that earlier doctrine, the examples of the snub and Kallias, which were treated as ‘things taken together with matter’ in Z.10 (1035a26; 1035a33) suggest that here, unlike in the context of the substance claim, Aristotle has in mind the composite insofar as it has both a form and matter.

On those textual grounds, the hierarchical purist can extend her reading to Z.11. Both in the introductory section of Z.11 and in the context of the substance claim, Aristotle is interested in the composite only insofar as it has a form. Hence, he concludes (or assumes) that the composite should be defined with reference to its form alone. But in the MP and the SYP as well as in later parts of the summary of Z.4–11, Aristotle’s interest shifts to the composite insofar as it has both a form and matter. As in Z.10, he argues that, if we consider the composite in that way, it should be defined with reference to both its form and its matter.

One might doubt, though, whether hierarchical purism can account for Aristotle’s discussion of geometrical objects. I have argued that, in the MP, Aristotle objects to his opponents because they define geometrical objects with reference to their forms alone. Yet, on hierarchical purism, one might expect that Aristotle himself has to allow for such purely formal definitions of geometrical objects. For geometrical objects are intelligible composites, and the hierarchical purist holds that, in some contexts, composites should be defined with reference to their forms alone. But if that is right, why does Aristotle object to the purely formal definitions offered by his opponents?

In response, the hierarchical purist should urge the analogy between Aristotle’s treatment of intelligible and sensible composites. First of all, there is some evidence in Z.10–11 that, even for geometrical objects, Aristotle sometimes calls for purely formal definitions. In Z.10, he excludes mention of the segments of the circle from the definition of the circle on the grounds that they are its parts ‘as matter in which [the circle] comes to be’ (1035a12). Since he also says that the segments in question are ‘closer to the form than the bronze when roundness comes to be in bronze’ (1035a13–14), Aristotle seems to exclude mention of the intelligible material parts from the definition of the mathematical circle.[35] Similarly, in Z.11, he excludes mention of the semi-circles from the definition of the circle on the grounds that they are its intelligible material parts (1036b32–37a5). Hence, Aristotle does not seem to object to purely formal definitions of geometrical objects tout court.

Nonetheless, in the MP, Aristotle does object to such purely formal definitions, or so I have argued. The hierarchical purist can offer a resolution of the apparent conflict in line with her interpretation of Aristotle’s treatment of sensible composites. The passage from Z.10 where Aristotle excludes mention of the segments from the definition of the circle occurs in the Hylomorphic Passage which leads into part [A] of the Puzzling Passage. In those passages, on the hierarchical purist interpretation, Aristotle discusses the composite only insofar as it has a form, and that is why he concludes for geometrical objects, no less than for sensible composites, that they ought to be defined with reference to their forms alone. Presumably, the same is true of the related passage in Z.11 (1036b32–37a5), although it is notoriously controversial what its exact place in the argument structure of Z.10–11 is.[36]

By contrast, in the MP, on the reading just offered, Aristotle is interested in geometrical objects insofar as they have both a form and (intelligible) matter. Hence, in the MP, but not in the other passages mentioned above, Aristotle concludes that one should mention intelligible matter in the definition of geometrical objects. His objection to the opponents is that they remove reference to intelligible matter no matter what. But in that case, removing mention of the matter is ‘περίεργον’, that is, ‘needless trouble’, or literally, ‘overworking’ (1036b23): It is not always the case that one should define composites, including both sensible and intelligible composites, with reference to their forms alone. As we will see shortly, in some contexts, it would be a mistake to omit reference to the matter of the composite.

That said, one can put a different spin on the objection, offering what we might call the no-context objection. In the MP, Aristotle does not appear to qualify the claim that geometrical objects should not be purely formally defined. Instead, he seems to condemn his opponents outright for their definitional practice. In fact, the same problem seems to arise from Aristotle’s discussion of sensible composites. For instance, when Aristotle says that the animal should be defined with reference to its functional material parts in the SYP, his claim does not seem to be restricted to any particular context. In the summary of Z.4–11, too, Aristotle appears to claim unqualifiedly that the only definition of a composite is in accordance with primary substance. The objection to hierarchical purism, then, is that it restricts Aristotle’s claims to certain contexts, even though Aristotle himself does not explicitly restrict them in that way.

I will concede that Aristotle does not explicitly restrict his claims to any particular context. For instance, he does not come out and say that it is wrong only in some contexts to define geometrical objects purely formally. But I have argued above that there are markers which suggest implicit restrictions. On the hierarchical purist view, then, the explanation why Aristotle does not make the restrictions explicit is simply that, in the relevant passages, Aristotle is already operating within one or the other context. For example, in the SYP, Aristotle can put forward the claim that an animal should be defined with reference to its functional material parts without explicitly restricting that definitional practice to the context where one studies animals insofar as they have both a form and matter. For, in putting forward that claim, he is already operating in that context; and similarly for other passages.

If we grant the hierarchical purist all those moves, her burden becomes to elucidate what the different contexts are in which a composite should be considered and defined in one or the other way. Let us therefore turn to Aristotle’s definitional claims in more detail.

5.3. Two Definitions

On hierarchical purism, each composite has two definitions, one in terms of its formal parts alone, and one in terms of both its formal and (some of) its material parts. That consequence of hierarchical purism becomes less surprising if we compare the view with Frede’s version of hard purism. On Frede’s (1990: 113–114) view, composites are defined with reference to their forms alone in first philosophy, but in Aristotle’s natural philosophy, composites are defined with reference to both their form and their matter. Where the hierarchical purist differs from the hard purist is that he takes the definitional concerns of natural philosophy to feature in Zeta alongside the more austere definitional concerns of first philosophy.

There are some general considerations in favour of taking the definitional concerns of natural philosophy to be present in Zeta. The central question of Zeta, what primary substance is, is answered in terms of matter and form, apparently culminating in the conclusion that form is primary substance (Z.11, 1037a29). Indeed, Aristotle introduces the details of the matter-form distinction into Zeta by way of a précis of some results from the Physics in Z.7–9.[37] It is hardly shocking that he should import the definitional concerns of natural philosophy along with that technical machinery. Towards the end of Z.11, Aristotle even makes explicit that, in Z.10–11, the interests of physics have not been excluded, on the contrary, ‘in a sense, the enquiry concerning perceptible substances is a function of physics and second philosophy’ (1037a14–16).

But the main reason why we should acknowledge that, contra the hard purist, definitions in the style of natural philosophy have a place in Zeta is simply that, in Z.10–11, Aristotle prominently discusses how a composite should be defined if it is considered insofar as it has both a form and matter. For it is not the first philosopher but the physicist who is interested in the composite insofar as it has both a form and matter. In Metaphysics E.1, for example, we are told that ‘physics is about separable but not unmovable things’ (1026a13–14), and just like in the Socrates the Younger Passage (1036b29–30), Aristotle connects movability with (perceptible) matter to conclude concerning the definition given by the physicist: ‘The account of none of [the natural things] is without motion, but it always has matter’ (1026a2–3).[38]

Similarly, I submit, it is the definitional concerns of the physicist which are heeded when, in Z.10–11, Aristotle argues that the composite insofar as it has both a form and matter should be defined with reference to both its form and its matter. By contrast, when Aristotle argues that the composite insofar as it has only a form should be defined in purely formal terms, he is taking on the definitional concerns of first philosophy which abstracts away from the material features of matter-form composites.

However, the two definitions, the first-philosophical or metaphysical one and the second-philosophical or physical one, are not on equal footing. Indeed, strictly, only the metaphysical definition is a definition at all. For a definition is an account which states the essence of an object (Meta. Z.5, 1031a12), and the essence of a composite is its form (see Meta. Z.7, 1032b1–2; Z.10, 1035b14–16; Z.11, 1037a28–29). Hence, only the metaphysical definition states the essence of the composite, and only the metaphysical definition is a definition, strictly speaking. How does that outcome bear on the status of the physical definition?

In Section 1, I said that the accounts or λόγοι with which Z.10–11 is concerned are definitions. However, I also noted that Aristotle does not always use definitional language in the strictest sense, for example, in speaking of ‘defining’ organs and geometrical objects (1035b7–11; 1036b8–10), which, as non-substances, are, strictly speaking, not definable. There is some evidence that matter-form composites do not turn out to be strictly definable either if one thinks about them in the way the physicist does, namely, insofar as they have both a form and matter.

For in Metaphysics E.1, Aristotle says that ‘all natural things are said like the snub’ (1025b34–26a1), and proceeds to assign the study of such snub-like things to the physicist (1026a24–26). But in Z.5, Aristotle claims that coupled entities like the snub can be defined only in a loose sense of ‘definition’ (1031a5–11). Hence, the definitions offered by the physicist of composites in terms of both their form and matter are not definitions in the strictest sense (see Code 2010: 207–208). On the other hand, since the essence of a composite is its form, the purely formal definitions provided by the first philosopher are definitions in the strictest sense.

The two definitions of the composite, then, are hierarchically ordered: Strictly, only the metaphysical definition is a definition of the composite, but loosely, one can also speak of a physical ‘definition’ which mentions not just the form or essence of the composite. Of course, the physicist is also interested in the essence or form of the composite: ‘For the physicist needs to gain knowledge not only of matter but also of the substance in accordance with the account, and more so’ (Z.11, 1037a16–17). Still, the definitional project of the physicist aims at a ‘what-it-is’ of the composite which is broader than its essence or form: ‘It is clear how one needs to search for the what-it-is (τὸ τί ἐστι) and [how one needs to] define (ὁρίζεσθαι) [things]’ (Metaphysics E.1, 1026a4–5), namely, always with respect to their matter since the physical definition ‘always has matter’ (1026a3), as discussed above.

The hierarchical order, then, of the two definitions of the composite arises from their content. The metaphysical, purely formal definition is a definition in the strict sense because it mentions only the essence of the composite. The physical definition is, strictly speaking, not a definition because it states not the essence or form of the composite but its broader τί ἐστι or ‘what-it-is’. For Aristotle, that ‘what-it-is’ of the composite, stated in a physical definition, is not exhausted by its essence or form but comprises some of the material features of the composite, too.[39]

Nonetheless, Aristotle speaks of the physical definition in definitional terms, not merely in the neutral terms of an ‘account’ (λόγος). In Metaphysics E.1, as just seen, he says that the physicist’s search for the ‘what-it-is’ of an object is an endeavor to ‘define’ (ὁρίζεσθαι) that object (1026a4–5). Similarly, when Aristotle argues in the SYP that objects which are ‘this-in-this’ should be defined with reference to some of their material parts, he speaks of ‘defining’ (ὁρίζεσθαι) the animal (1036b29), and he sets up his discussion as a response to a puzzle ‘about definitions’ (1036b21). The crucial point, then, is not that physical definitions cannot be called ‘definitions’ but rather that those definitions are not definitions in the strict sense because they do not state the essence but rather the broader ‘what-it-is’ of the composite.

Analogous considerations apply to intelligible composites. Above, I argued that, on the view developed in Z.10–11, intelligible composites, too, can be considered either only insofar as they have a form, or insofar as they have both a form and matter. In the former case, intelligible composites should be defined purely formally, in the latter case, they should be defined with reference to their intelligible matter as well. This is not the place to enter the difficult details of the relation between first philosophy and mathematics.[40] But the picture in Z.10–11 suggests that, unlike geometry, first philosophy abstracts away from the intelligible material features of geometrical objects, just as it abstracts away from the perceptible material features of sensible objects. It is only in first philosophy, then, that one should consider and define composites just insofar as they have forms.

5.4. Matter and Definition

There is, however, the further question how there can be any definition of a composite which mentions both its form and its matter, even in a loose sense of ‘definition’. In [B] of the Puzzling Passage, Aristotle discusses the definitions of particular composites in terms of the bits of matter into which they perish. Similarly, in the SYP, Aristotle seems to suggest that particular composites should be defined with reference to their concrete functional parts, such as the hand.[41] But in Z.10, Aristotle also claims that there is no definition of particulars (1036a2–9), and the reason he supplies is that ‘matter by itself is not graspable (ἄγνωστος)’ (1036a8–9; see also Z.15, 1039b27–30). Aristotle’s point is not that there is no definition of particulars at all. Rather, he claims that particulars ‘are always said and grasped by the universal account’ (1036a8). One strategy is to consider the composite only insofar as it has a form, and hence to define it in terms of its form alone (1035b34–16a1). But how can one define the composite, if one considers it insofar as it has both a form and matter?

I think that Aristotle recognizes that problem already in Z.10, and that it prompts him to introduce his notion of ‘matter as universal’ (Z.10, 1035b30; see Gill 1989: 135–136). Aristotle contrasts the matter of universal composites, that is, species, with the concrete material parts of particulars, such as the flesh and bones of Socrates (1035b30–31; see also Z.11, 1037a6–7). The relation between the matter of the particular composite and the matter of the universal composite can be understood as a relation of universalization: If one universalizes the particular bits of flesh and bones that make up Socrates one will end up with the flesh and bones ‘as universal’ of the human species.

Thus, although one cannot define a composite with reference to the particular bits of matter which make it up, one can define it with reference to its universalized matter, that is, the relevant kind of matter. For example, one can define Kallias insofar as he has both a soul and flesh and bones with reference to soul and the material kinds flesh and bones. The same solution can be applied to the concrete functional material parts mentioned in the SYP. It is the universalized functional parts which are mentioned in the definition of a composite. Moreover, even though Aristotle does not explicitly extend his account of universal composites to geometrical objects, a similar point might hold for intelligible matter.[42] The spatial extension of a particular geometrical object cannot be mentioned in its definition, but the universalized extension, say, its dimensionality, is apt for being mentioned in the definition. There are, then, two ways of providing a universal account of particulars, either by mentioning only their forms, or by mentioning both their forms and their universalized material parts.

With respect to sensible composites insofar as they have both a form and matter, there is the further question whether their definitions mention the (universalized) material parts into which they perish or whether the definitions mention their (universalized) functional parts. We saw that Aristotle seems to claim the former in the Puzzling Passage but the latter in the SYP. As far as I can tell, Aristotle does not settle that question in Z.10–11. Still, given that Aristotle’s prime example in Z.10–11 for matter into which animals perish, namely, flesh, turns out to be functional in the biological works (see, e.g., PA II.8, 653b19–25; GA II.1, 734b25–28), and that functional parts become more important over the course of Z.10–11, it seems that, ultimately, the (universalized) functional parts should be mentioned in the definition of the sensible composite.

Despite that last point, I think we should stop short of concluding with the reformist hard purists that, since ultimately only the functional parts of a sensible composite can be mentioned in its definition, none of its properly material parts will ever be mentioned. For Aristotle clearly speaks of the functional parts of a composite as its matter. For example, in arguing that the finger is posterior to the whole human being, he says that ‘those things which are parts as matter and into which [the whole] is divided as into matter are posterior’ (1035b11–12). Similarly, he urges later that the composite animal, unlike its soul, is divided into the bodily parts ‘as into matter’ (1035b21). The functional parts of a composite are material no less than the parts into which the composite perishes.