Aristotle on Matter Kit Fine Mind, New Series, Vol. 101, No. 401. (Jan., 1992), pp. 35-57. Stable URL: http://links.jstor.org/sici?sici=0026-4423%28199201%292%3A101%3A401%3C35%3AAOM%3E2.0.CO%3B2-Q Mind is currently published by Oxford University Press.
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Aristotle on Matter KIT FINE It is my belief that there is still a great deal to be learnt from Aristotle's views on the nature of substance; and it is my aim in a series of papers, of which this is the first, to make clear what these views are and wha,t it is in them that is of value.' A peculiarity of my approach, compared to current scholarly practice, is the attempt at rigour. I have tried to provide what is in effect a formalization of Aristotle's views. I have, that is to say, attempted to make clear which of his concepts are undefined and which of his claims underived; and I have attempted to show how the remaining concepts are to be defined and the remaining claims to be derived. I can well understand a traditional scholar being suspicious of such an approach on the grounds that the various parts of Aristotle's thought are either too unclear to be capable of formalization or else are clear enough not to require it. Since the matter is not one for a priori dispute, I can only ask the scholar to put his suspicions at bay until the details of the case are examined. I then think that it will be found that the attempt at rigour provides a most valuable guide for the study of the text. I have not tried to deal with all aspects of Aristotle's thought on substance. I have concentrated on those which centre on the concepts of matter, form, part, and change; and I have neglected those which concern the related concepts of predication, function, priority and power. It is to be hoped that the investigation will be rounded out at some later time to include all of the central aspects of his work. It should also be mentioned that my treatment of the text has not been altogether scholarly. Partly this has been a matter of competence, and partly of inclination. I have been more concerned with the broad sweep of Aristotle's views than with exegetical detail; and this has led me to conjecture that he held a certain opinion, not because of direct textual evidence but because it is what his view most naturally requires. Thus the Aristotle I have presented here is much more consistent, definite and complete than the Aristotle of the texts. I This paper is based upon the first two sections of my unpublished paper "Aristotle on Substance". I should like to thank the members of a seminar I held at UCLA in the winter of 1991, and Frank Lewis in particular, for many helpful discussions on some of the topics of the paper. I am also grateful to Richard Sorabji for valuable remarks on an earlier version of the paper.
~ i n dVol. , 101.401. January 1992
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1. The concept of matter Underlying Aristotle's account of the things in the world is the doctrine of hylomorphism. This doctrine declares that things in the world are compounds of matter and form. Thus central to the view are three features: the two "ingredients", matter and form; and the manner of composition, or the compound. It is quite plausible to suppose that, for Aristotle, the notion of matter is to be understood in terms of composition: matter is what plays a certain role in the It compound; it is, as it were, the passive participant in the act of compo~ition.~ is therefore important to appreciate that the two can, in principle, be detached and that one can endorse the view that things have matter, or a constitution, without thereby supporting the idea that there is a form in virtue of which they have that constitution. In order to accommodate such a view, I have made it my aim in the present paper to discuss the concept of matter on its own; and it is only in the sequel that I have considered its relationship to the concepts of form and compound. If we compare Aristotle's notion of matter with the modem one, we find two striking differences. The first is that Aristotle has a hierarchical conception of matter; what is matter may itself have matter. Thus the matter of something need not be identified with that of which it is ultimately composed, the elementary particles or what have you. Rather, the matter of something exists at each level of its organization, ranging from the most immediate constituents at one extreme to the most basic constituents at the other. In the second place, Aristotle's conception of matter is comprehensive in its scope. It applies, not merely to physical, but also to non-physical objects; for they may have other non-physical objects as their matter3 Thus not only will the token letters constitute the matter of a token expression, the letter types will also constitute the matter of the expression type. Indeed, as Aristotle says in connection with the objects of mathematics, "there is some matter in everything which is not an essence and a bare form but a 'this"' (Metaphysics, 1037al-2). It might be thought that this issue over the nature of matter is not a real one. For Aristotle's use of the term "hule" is somewhat technical; and so it is conceivable that the apparent difference in view is to be attributed to a difference in the It is not so plausible to suppose that the notion of form is also to be understood in terms of the compound, with the form being the active participant in the act of composition. For the pure forms can hardly be understood in terms of such a role. There is the question of whether non-physical objects can have physical objects as their matter and of whether physical objects can have non-physical objects as their matter. I am inclined to think that the answer is yes in both cases: for the members of a set, which is non-physical, may be physical; and the meaning of a word is, arguably, as much an "ingredient" of the word as the underlying token. However, there is no indication that Aristotle would have allowed either of these two possibilities.
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use of terms. But the dispute is not verbal, but concerns the question of whether there is a viable conception of constitution of the sort that Aristotle supposes, one which is uniformly applicable to physical and non-physical objects alike and which is capable of hierarchical application. We may grant;-if you like, the narrow use of the term "matter". 'The question then is whether it signifies a special case of the more general concept, whether matter so understood is the ultimate matter, in the Aristotelian sense, of something physical. On this further issue, it seems to me that Aristotle's views have not been given their due. His advocacy of intelligible matter has commonly been regarded as some kind of aberration, implausible in itself and hardly in line with his nominalistic leanings; and his hierarchical conception of matter, though treated with a great deal more sympathy, has rarely been endorsed. But on both counts, it strikes me that Aristotle's views are on the right track and most in accord with our intuitions. Of course, if one regards the matter of an abstract circle as some kind of abstract goo or dough, then the idea appears faintly ridiculous. But if one considers Aristotle's standard example of a definition, then it is entirely plausible that its defining terms (e.g. "plane figure" in the case of circle) should be constitutive of it in exactly the same general way as physical matter is constitutive of something physical; both go to make up the other and nor is there any mystery, in the case of definitions, as to what the abstract matter is. There is also a clear sense in which something may be differently constituted at different levels. We have no difficulty, for example, in supposing that a word is made up of letters and the letters of strokes. Of course, the division of the word into letters may be treated as one of the many possible decompositions of the word. But the fact is that we would not normally suppose that the word was constituted by the "funny" letters obtained under an arbitrary decomposition. Indeed, in the case of certain abstract objects we are in no doubt as to their hierarchical structure. With the set { ( a ,b ) , { c ,d ) ), for example, the matter at the first level is constituted by the sets { a ,b ) and {c, d ) , and at the next level by the elements a, b, c and d4It would be absurd to regard the set { a ,c ) as a constituent of the original set, somehow on a par with the others; and it is unclear why it would not be equally absurd to regard the funny letters as constitutive of the word. But great as the virtues of Aristotle's concept of matter may be, his treatment of the concept is not without its problems. The first of these is relatively minor and has to do with the distinction between matter as many and one. In referring to the matter of something, one may be referring to some single thing which is the matter or to several such things. Aristotle is not always careful to heed this distinction and so it is not always clear when he refers to the matter of something, as with the body being the matter of a man, whether he is making a singular or plural reference. For us, it is sets which constitute the most natural example of a hierarchical structure within the abstract realm. But for Aristotle it would have been definitions, via their natural division into genus and differentia.
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In our own treatment, we shall attempt to be more sensitive to the distinction and will first give separate accounts of matter as many and as one, and only then consider the relationship between the two. The other difficulty is more serious and has to do with the. way that Aristotle's conception of substance interferes with his views on what constitutes what. For he assumes that concrete substances are not, strictly speaking, the matter of anything else; they constitute the terminus of the matter-of relation. But this leads one to ask: if the parts of a body, or the body itself, can constitute a man, then why should men not constitute a family? Why draw the line at the level of the man? The difficulty becomes even more acute if the family is compared, not to the man, but to a house; for it is then even less clear what the relevant difference might be. Now it was entirely reasonable for Aristotle to believe that there was a difference in the two cases, especially if artifacts are excluded from consideration. For there is an evident difference in the unity possessed by a man and by a family; and it is plausible that this difference in unity is to be attributed to a difference in the way that the constituents come together. The constituents of the one genuinely come together to form something which is genuinely one, while the constituents of the other come together in some looser way to form something which is only loosely one. But however reasonable it may have been for Aristotle to hold this view, it is not reasonable for us. For with the advance of science, we know that there is no special force or principle which binds together the different parts of the body and yet is not operative in the universe as a whole; and in the absence of any such force or principle, it is rather hard to see what ontological basis there could be for distinguishing between the constituency of substances and of mere heaps. Thus the idea that there is a distinctive notion of constitution, terminating in the concrete substances, is one that should be given up. However, this is not necessarily to give up the idea that there is something distinctive about the concrete substances themselves. For one can grant that something is genuinely one, without thereby granting that what makes it genuinely one is some distinctive way in which its constituents come together. In a similar fashion, it might be supposed that certain complex concepts were more of a natural unity than others. But it would be odd to explain this unity in terms of the fact that the component concepts were put together in a special way, that the definitional glue was in these cases somehow different.
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2. Linear constitution The formalization in this paper and its sequels will proceed in stages. At each stage, I present those principles that are proper to a certain 'dluster of concepts; I provide, that is to say, afragment of the whole theory. In some ways this approach is unnatural, since the full justification or even understanding of one principle may require concepts not involved in its formulation. However, by a judicious choice of concepts, it is possible to obtain a natural stratification of the theory in which the conceptual disorder is reduced to a minimum. The present paper deals with a single primitive notion. This is the notion of one object being the matter of another; and for the purposes of the present section, I take this relation to obtain when the one is the matter of the other at some level, and not necessarily at the proximate level. I shall sometimes talk of one object constituting another (a tern which I suspect corresponds better to Aristotle's "hule"), although my tendency will be to employ "matter" for the singular or linear relation and to employ "constitution" for the non-linear or plural relation. There is some question as to the intended domain of application of the theory. At its very broadest, the domain could be taken to consist of everything, both non-physical and physical; and, in this case, I shall speak of "objects" as opposed to "things". But even when the domain is restricted to things, some uncertainty remains as to how broad it should be. At the very minimum, the domain should contain the concrete substances (strictly speaking), their matter, the matter of their matter, and so on. The concrete substances will include living things, or at least those that are fully formed. Whether they should also include artifacts is a matter of dispute. But important as this issue might be elsewhere, it will not be important for us here. There are two respects in which this minimal domain might be extended. First, it might be taken to include "strays", i.e. matter which is not the matter of any substance. Examples might be the water in my bathtub or the air in my study.5 Secondly, it might also be taken to include "mere heaps", things without any genuine unity. Now an obvious example of a heap is a pile of trash. But I think that Aristotle is also forced to recognize more organized objects, such as families or nations, as heaps; for otherwise, they would have people as their matter and this goes against people being substances. Of course, this is not to deny that there might be some derivati;e sense in which a family or a nation could have people as its matter; but this would not be in a strict or proper sense. Let us call the first of these domains the central ontology, the second the extended ontology, and the third the bloated ontology. The ontology which includes absolutely everything might, by contrast, be called universal. Sometimes the difference in the choice of ontology will matter; and sometimes it will It is fairly plausible to suppose that only matter can stray. That is to say: any nonuniform thing, if not itself a strict substance, must be the (contributory) matter of a strict substance.
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not. But in order to avoid needless complexities of formulation, we may adopt the convention that the difference will be taken not to matter unless there is some indication to the contrary. It is supposed that each ontology is closed under constitutitjn, i.e. contains the matter of anything which it contains. However, the sense of constitution varies accordingly as linear or non-linear constitution is under consideration. In the latter case, the matter can be taken to be any contributory matter; but in the former case, it should be taken to be the whole matter. If, for example, we allowed a hand into the ontology, then we would have to falsely declare it a substance since it is not the matter (i.e. the whole matter) of anything else. There is one complication, pertaining to the bloated ontology, which I shall ignore. Suppose a quantity of earth is the sum of what, at a given time, is the earth in me and some stray earth. Then it is somewhat hard to know what to say about it. We do not want to say that it is the matter of something, for part of it is not; and we do not want to say that it is the matter of nothing, for part of it is. In a theory which had a place for an appropriate notion of part, the ambiguous status of such things could be properly defined. But within the present theory, we cannot distinguish them both from those things which are wholly the matter of something else and from those which are wholly not the matter of something else. It might be thought that such a problem did not arise within Aristotle's ontology, properly conceived, for there is no reason to suppose it closed under arbitrary mereological sums. But the problem can arise without any such assumption. The earth which is in Callias, for example, may later become distributed between me and a clod. Thus the persistence of matter is sufficient, even in the absence of closure, to generate the wayward cases. I suspect that there may be a way of removing the difficulty, once the doctrine of persistence is stated in an appropriate form.6 But rather than prejudge the issue, I shall simply presuppose that the wayward cases have been excluded from consideration. One of the most fundamental principles concerning the matter-of relation is the following: Axiom 1 (Foundation). There is no infinite sequence of objects x , , x2, x,,. .. such that x, is the matter of x,, x, is the matter of x2, and so on ad infinitum. This principle is very plausible and has been advanced, in one form or another, by many different thinkers throughout the history of philosophy. It provides the basis for the Fundierungsaxiom in set theory, with the matter-of relation now being understood as set-theoretic membership. For a related discussion on the impossibility of infinite regresses, see 11.5.332b6333a15, of Generation and Corruption.
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It is clear from Metaphysics 994a1-5, that Aristotle held to such a prin~iple.~ And in the ensuing passage 994a10-18, he appears to argue for the principle on the grounds that there must be a first cause ("if there is no first there is no cause at all" and, by implication, if there is an infinite regress there is no first). This is a common fallacy. From the fict that there is always a first cause, it does not follow that there is no infinite regress of causes; for any infinite regress might be preceded by a first cause. There may, however, be a more charitable way of construing his argument or, at least, the intuition which lies behind it. Aristotle seems to think that it is necessary for there to be a first cause for there to be any cause at all. But suppose we strengthen this requirement to the demand that any cause must be grounded in first causes, i.e. either itself be a first cause or be (caused) by a first cause, or be caused by such causes, and so on indefinitely. Then it will follow that there cannot be an infinite regress of causes, since none of the causes in the regress could be grounded. We thus have something like the standard justification for Foundation, viz. that it must be possible to generate all of the entities in question from the bottom Foundation has an immediate consequence for the notion of proximate matter. Let us make the following two definitions:
Definition 1. x is matter if it is the matter of something; Definition 2. x is the proximate matter of y iff x is the matter of y and there is no z such that x is the matter of z and z is the matter of y. We may then conclude: Theorem 1. Any matter is the proximate matter of ~omething.~ Given our understanding of the matter-of relation, we may also assume that it is transitive: Axiom 2 (Transitivity).If x is the matter of y and y the matter of z then x is the matter of z. An explicit statement of Aristotle's willingness to use a transitive conception of matter-of is to be found at 1044a20-23, of Metaphysics. He there writes, "And The basic idea is to say that the matter in Callias persists not as one thing x but as several x,,x2,.... The view that matter persists can then be reconciled with the view that it persists as something connected. There might be special reasons, in the case of concrete things, for denying the possibility of a first cause being,followed by a downward infinite regress of causes. For let us suppose that we link up, in the way sometimes countenanced by Aristotle, the present matter which constitutes a thing with the past matter from which it comes. The possibility of a sequence x,,x2,x3,... of the sort excluded by axiom 1 might then be taken to yield the possibility of a sequence a,, a2, a3.. . of comings to be, where a,is x, coming from x2,a2 is x2 coming from x3,and so on. It then seems reasonable to suppose that the sequence of comings to be can be so selected that a2wholly precedes a,, a3wholly precedes a2, and so on. If it is now further assumed to be true of the select sequence that each of its comings to be takes more than a fixed period of time (no matter how small), then it follows from the standard properties of the temporal continuum that no coming to be can precede all of the members of the sequence. The proof is a simple exercise on relations. Other proofs will be omitted when they are straightforward.
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there come to be several matters for the same thing, when one matter is the matter for the other; e.g. phlegm comes from the fat and from the sweet, if the fat comes from the sweet". Foundation and Transitivity have consequences for the notion of ultimate matter. It is clear that Aristotle had a conception of "first" matter. For example, in Metaphysics 1049a24-26, he writes, "And if there is a first thing, which is no longer, in reference to something else, called 'thaten', this is prime-matter". Accordingly, let us say: Definition 3. (i) x is ultimate matter iff x is matter and nothing is the matter of x (otherwise the matter x is said to be enmattered); (ii) x is the ultimate matter of y iff x is ultimate matter and is the matter of y. I have used the expression "ultimate matter" in preference to the traditional term "prime matter", since the traditional term is commonly taken to signify something with additional properties, such as being indeterminate, merely potential, and the matter of the elements. But whereas there may be a genuine question as to whether Aristotle believed in prime matter as thus characterized, there is no real question as to whether he believed in ultimate matter. The definitions of "matter" and of "enmattered are not really appropriate if strays and heaps are admitted into the ontology. For the water in my bathtub intuitively is matter but is not matter according to the definition, and a pile of trash intuitively is enmattered and yet is not enmattered according to the definition. The first of these difficulties might be removed by allowing anything with a suitable material form to be matter, and the second difficulty could be removed by introducing an analogous relation of quasi-matter between a heap and its members and allowing an object to be enmattered if its quasi-matter was enmattered. Unfortunately, we lack the conceptual resources within the present framework to make either change to the definitions. My own view is that in a proper development of the theory a general relation of constitution (covering both matter and quasi-matter) would be taken as primitive. The asymmetries in the treatment of ordinary matter, strays and heaps would not then arise. From Transitivity and Foundation, it follows that: Theorem 2. Any matter either is ultimate matter or has something as its ultimate matter. Aristotle accepts that the converse of the matter-of relation is well-founded. Axiom 3 (Reverse Foundation). There is no infinite sequence of objects x,, x,, x3,... such that x, is the matter of x,, x, is the matter of x3, and so on ad infinitum. This principle is not as reasonable as Foundation itself and has not generally been embraced by those who advocate Foundation. In the case of sets, for exam-
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ple, we may have x a member of {x], {x}a member of { {x}1, and so on ad infinitum. It therefore seems clear that Reverse Foundation should not be accepted for objects in general. It is also plausible that the assumption should not be accepted for concrete things. For there seems to be nodifficulty in principle in conceiving of one artifact being the matter, or part of the matter, of another artifact, and so on ad infinitum. And even without any departure from the physical limitations of the universe, we may imagine that the constitution of some organization prescribes that each of its committees should help constitute an oversee committee for that committee. The oversee committees would then constitute an infinite ascending regress. Aristotle's argument for the principle is stated in the paragraph beginning at 994a19 of the Metaphysics. He considers regresses x,, x,, x,, ... in which each term x,,, "comes from" its predecessor x,; and presumably it is intended that the four ways in which x can cause y should be cases in which y comes from x. These regresses are of two sorts. There are first those in which y comes from x by the destruction of x. In these cases, Aristotle supposes that the items of the regress must cycle. It is not clear why they must cycle. But even if they did cycle, we may note that this would at most establish that there cannot be an infinite regress of different things, with each term in the regress being distinct from any other term,1° not that there cannot be an infinite regress at all. Regresses of the second sort are those in which y comes from x by helping to complete x. Presumably regresses constituted by the matter-of relation are of this second sort. In these cases, Aristotle supposes that the items of the regress must have some end-point, i.e. that each thing must eventually come'to be something which does not itself come to be anything else. We thus have an argument which is dual to the one concerning regresses in the other direction. The same considerations therefore apply. Just as a first cause was not sufficient to prevent a backward regress, a final cause will not be sufficient to prevent a forward regress. But it is also true, just as before, that the argument can be mended by adding the requirement that each cause be grounded in one or more final causes. However, it is hard to see in this case why there should be a grounding from the top. For a substance is most naturally taken to be built up from its matter, not the matter from the substance. And even if it is conceded that construction could proceed in the opposite direction, it is hard to see why it should proceed in both directions. For we only rtquire that something be built once. It may be that the two forms of grounding have a different source; the one from the bottom up is required for the constitution of the thing to be intelligible; the one from the top down is required for the essence of the thing to be intelligible. Indeed, Aristotle explicitly remarks that the matter of a thing is in one sense prior to the thing and in another sense posterior. (See Metaphysics 1019a8-10, "in l o It might not even establish this. For it may be that each of the itemsx,, x2,xj,. .. reappeared after the whole sequence.However, this is a possibility that might be ruled out in the same way as in the case of backward regresses.
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potency the half line is prior to the whole line, and the part to the whole, and the matter to the concrete substance, but in complete reality these are posterior".) But if both relations of priority are well-founded, then infinite regresses of matter in either direction will be blocked.l! A good analogy for the purposes of understanding Aristotle's position is with the parts of linguistic expressions. Under an unsophisticated conception of part, one would allow a sentence S to be a part of its negation --d,-S' to be a part of 7 4, and so on, thereby creating an infinite regress. But under a more sophisticated conception of part, it would be required, in order for an expression e to be . a part of another expression f,that the linguistic role of e should be understood in terms of the linguistic role off.12 No sentence would then ever be part of another sentence; and, in general, the simple sentences (i.e. those containing no other sentences as parts) would be the counterpart to the Aristotelian substances and would provide the terminus to the role-oriented part-whole relation. There is another standpoint from which Reverse Foundation might be justified, one which does not require us to make sense of the idea that a thing is prior to its matter. For let us suppose that a special class of objects, corresponding to the Aristotelian substances, can somehow be picked out, whether or not on the basis of ontological priority. Call these objects the core. Let us further suppose that the core is subject to Reverse Foundation and that no object in the core is infinite, in the sense of containing an infinite ascending sequence of constituents. Define the extended core as the closure of the core under the matter-of relation. It then follows that the extended core will also be subject to Reverse Foundation.I3 In providing Such a justification, we do not presuppose that the matter-of relation terminates with the core. We may grant, for example, that men constitute families, that families constitute extended families, and so on. But somehow we succeed in drawing a critical line between those things which are too high in the constitutive hierarchy and those which are not; and we then confine the matter-of relation to those things which lie at or below the critical line. Thus our justification, or even the one in terms of priority, does not serve to undermine our previous criticism of Aristotle. For although the defined notion of constitution terminates with substance, the underlying relation does not. The ceiling is imposed from the outside and does not arise from within the relation itself. Reverse Foundation has consequences dual to those for Foundation.I4 I I The restriction of a well-founded relation is well-founded. So if the matter-of relation implies well-founded priority and well-founded posteriority, then both it and its converse will be well-founded. l 2 It is to such a distinction that Dummett appeals in his elucidation of Frege's dictum that only in the context of a sentence does a word stand for anything (1973, pp. 192-6). l 3 The proof depends upon assuming Upward Linearity, stated below. The general concept of the core will be discussed in more detail in a later paper. l 4 It turns out that a general duality theorem will hold with respect to the replacement of the matter-of relation with its converse. I shall not always bother to state the dual.
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The analogue of theorem 1 is: Theorem 3. Anything enmattered has something as its proximate matter. (For a statement which strongly suggests that Aristotle would endorse this result, see Metaphysics, 1044a, 15-20.) Dual to the definition of ultimate matter, we have: Definition 4. x is a substance iff x is enmattered and is not the matter of anything. As with the definitions of "matter", this definition is not appropriate if strays are admitted into the ontology. Thus the water in my bathtub or the air in my study would be substances according to the definition but would not be strict Aristotelian substances. Such strays could be excluded from substancehood by requiring that substances have a suitable substantial form. But again, we lack the conceptual means to make the required change to the definition. With the help of Reverse Foundation, we may deduce the analogue of theorem 2: Theorem 4. Any enmattered object is either a substance or is the matter of something which is a substance. Although I have postulated that the matter-of relation is transitive, I have not postulated either its asymmetry or its irreflexivity. Asymmetry follows from either Foundation or Reverse Foundation: Theorem 5. If x is the matter of y then y is not the matter of x. For if x and y were the matter of each other, then x, y, x, y,. .. would constitute an infinite sequence in both the upward and the downward direction. (Direct evidence that Aristotle accepted asymmetry is to be found at 1013b8-l l , where he writes "things can be causes of one another... not, however, in the same way".) From the above result it follows that the matter-of relation is irreflexive: Theorem 7. x is never the matter of itself. The assumptions so far stated are compatible with branching in either the downward or upward direction, i.e. with something having two distinct proximate matters or with something being the proximate matter of two distinct objects. It is unlikely that Aristotle would have countenanced either of these possibilities. Downward branching goes against his apparent presupposition that the matter of something (at any given level of analysis) is unique. (See also the previously cited passage at 1044a20-23.) Against upward branching, we may argue as follows. Suppose x branches off into y and z. In the absence of downward branching, y and z will (respectively) either be identical to or the matter of two distinct substances y' and z'; and so there would then be two (actually existing) substances with the same ultimate matter, a possibility which Aristotle would reject. Let us, therefore, lay down the following two linearity assumptions: Axiom 4 (Downward Linearity). If x and y are both the matter of z, then either x = y or one of x or y is the matter of the other. Axiom 5 (Upward Linearity). If z is the matter of both x and y then either x = y or one of x or y is the matter of the other.
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From previous results and Downward Linearity, it follows that the ultimate and proximate matter of something enmattered are unique: Theorem 8. Anything enmattered has exactly one object as its ultimate matter and as its proximate matter. Of course, the ultimate and proximate matter in a given case might be the same. The dual of this result states that for any matter (i.e. matter of something) there is exactly one object of which it is the proximate matter and exactly one substance of which it is the matter. Together, the various conditions imposed on the matter-of relation amount to the requirement that the totality of objects should constitute a set of finite linear sequences. Thus the world according to Aristotle would look like this:
with the dots representing the objects and with the lines going down from a object to its matter. There may, however, be one respect in which our account (in regard to the central or extended ontology) is incomplete. For it allows something to be isolated in the sense that it is neither the matter of anything nor has anything as its matter (in the diagram it would be represented by an isolated dot). But is this a real possibility for Aristotle? The question turns on whether the elements constitute the ultimate matter. If they do, then isolation is possible; for the elements will presumably be capable of an isolated existence. If they do not, then isolation will be impossible; for the ultimate matter will be the matter of the elements and presumably will be incapable of isolated existence. Granted that the elements possess matter, we should therefore also postulate: Axiom 6. Anything either is matter or is enmattered.
3. Nonlinear constitution In the previous section I have dealt with a linear conception of constitution, one according to which each object has at most one other object as its proximate matter. There is, however, a natural way in which many objects can be regarded as
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the proximate matter of an object. Thus we may take the proximate matter of the set { 0 , 1 ] to be comprised by its two members 0 and 1; and we may take the proximate matter of a body to be comprised by its various parts. It is my aim, in the present section, to develop an account of the non-linear conception. Aristotle is most naturally taken as subscribing to a linear conception of matter. But there is evidence that he was also willing to construe the matter-of relation in a non-linear fashion.Thus the remarks on unification in Metaphysics 1045a7-19, strongly suggest that the constituents are many, it being the role of the form to make the many into one. Or again, he seems to think that the bodily parts are individually matter for the body. Indeed, it may be surmised that often when he seems to take the matter as one he is really thinking of it as many. So the elemental matter of a mixture is really the several different elements, and perhaps even the body is only the matter of a man through its bodily parts. I therefore think that the ensuing theory is very much in the spirit of what Aristotle has to say about constitution. If only he had been more sensitive to the distinction between linear and non-linear constitution, he would have been willing to subscribe to the principles that are here laid down for the non-linear notion. There is some question as to what we should take as primitive in a theory of non-linear constitution. We have the notion of several objects x,, x,, ... constituting another object y. But there are various ways in which this notion might be understood. First, x,, x,, ... might either immediately or mediately constitute y. For example, the set { 101, { 1 ] ] is immediately constituted by ( 0 ) and { 1 ] but mediately constituted by 0 and 1. Let us agree that constitution is to be understood as immediate (for reasons which will later become apparent). Second, there is the question as to whether the order or multiplicity of the constituents x,, x,,. .. is relevant to whether the constitution relationship holds. Do we say that x,, x,-as well as x,, x,--constitute the sequence x,x2;and do we say that x,-as well as x,, x,--constitute the sequence x,x,? Let us agree, for purposes of simplicity, that order and multiplicity are to be irrelevant, even though the ability to convey certain information is thereby lost. Finally, there is the question as to the sense in which x,, x,,. .. are totally to constitute y. It is conceivable that there are different (immediate) analyses of y. That x,, x,, ... constitute y may then be taken to mean either that x,, x,, are all of the constituents which belong to some given analysis of y or that they are all of the constituents which belong to some analysis or another. Let us agree to abide by the first sense, which aft& all is more natural and "fine-tuned". Under this interpretation, it is in principle possible that x,, x,, ... and y,, y,, ... should both constitute z , where the set {x,,x,, ...I is distinct from the set {y,, y,,. .. ] . However, there is some plausibility in the supposition that the immediate constituents under any two analyses of the same object are the same. This is not necessarily to deny that there cannot be two distinct analyses, but only that the difference between them cannot rest on a difference in the constituents. It would take us too far afield to defend this supposition against all of the various objections which might be raised against it. But we may consider one such
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objection, which is to the effect that the mereological sum x,+ x, + x, has both x,,x2+ x3and x,+ x,,x, as constituents. To this objection, it may be replied thatsomething is not a constituent of the mereological sum if it is itself a mereological sum; the analysis of a mereological sum, if it has an analysis, must be in terms of mereological atoms. Granted the supposition, the distinction between the two forms of total constituency disappears. But also, it is then possible to define the total notion of constituency in terms of a partial notion. Let us say that x helps constitute y if x is among the constituents which totally constitute y (i.e. x,,x,,... constitute y, where one of x,,x2,... is x). Then we may say that x,,x2,... (totally) constitute y iff each of x,, x2,.. . helps constitute y and nothing else helps constitute y. There is one case in which this definitional scheme fails; and that is the case in which the polyadic constituency relation holds with no constituents on its left. We will want to say, for example, that.. . totally constitute the null set, where.. . is blank; but we will not want to say of anything which has no constituents that it is in this way degenerately constituted. But with this one exception, the correctness of the definition can be upheld.15 The interest of the more general polyadic notion of constituency is not to be denied. But for the purposes of simplicity, I shall take as primitive the immediate and partial notion of constituency. It might be wondered whether an alternative possibility in this case is to take the mediate partial notion of constituency as primitive, just as in the linear case. But there are in fact decisive reasons against so doing. We cannot define x immediately (helps) constitute y in the usual way as x mediately constitutes y and there is no z such that x mediately constitutes z and z mediately constitutes y. For we want to say that 0 immediately constitutes (0, ( 0 ) 1, even though it mediately constitutes {O),which mediately constitutes (0, (01 ) . Indeed, it may be proved that it is in principle impossible to give any definition of the immediate notion in terms of the mediate notion.16 We are therefore obliged to adopt the immediate notion as the primitive. We have a form of Foundation for partial constitution: In the case of concrete things, it might be argued that such a case cannot arise. For suppose that.. . constitutes x. Then granted that the spatial location of x should be included in the union of the spatial location of its constituents, it follows that x has null spatial location, which might be ruled out as impossible. l 6 More specifically, we may prove that the ancestral E- of E is not definable (even in principle) in terms of &. For let $ be the permutation on a set-theoretic universe V with urelements which "interchanges" { { a ] ,a ] and { { a ]) in any set, for a a fixed urelement of the universe. Then $ is an automorphism on the structure . But it is not an automorphism on the structure . For { { a ]] is the set whose sole member is { a ) ;but $({ {a.)])= { { a ] ,a ] is not the set whose sole member is $({a])={ a ] . T h ~ rsesult is of some independent interest for set theory and raises the question of what a theory with E- in place of E would look like.
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Axiom, 1(Foundation,). There is no infinite sequence of objects x,, x2, x,,. .. such that x2 (helps) constitute x,, x, (helps) constitute x,, and so on ad infinitum. This principle may be justified in the same way as its linear counterpart.17 However, there is an objecti'on to the principle which is peculiar to the nonlinear case. For it may be supposed that x, + x, + x, +. .. is partially constituted by x, + x, + ..., which is partially constituted by x, + ..., and so on ad infinitum. But in response to this objection, it may be again maintained that the parts of a mereological sum will not be constituents if they are themselves mereological sums. Indeed, since Aristotle himself believed that mereological sums (such as quantities of uniform matter) are indefinitely divisible and since he also believed in Foundation, it would appear that he is committed to treating the relationship between a thing and its mereological parts as different from the relationship between a thing and its material or constitutive parts. The one kind of relation holds exclusively in the horizontal dimension, as it were, without any hylomorphic descent, while the other relation holds exclusively in the vertical dimension. This point is of some importance for understanding Aristotle's views on uniform matter. For I have heard it said that flesh cannot be constituted by water and earth, let us say, since any part of flesh is flesh; and so no matter how far one goes in breaking up the flesh, one never comes across water or earth. But this is to presuppose that the material and mereological parts constitute the flesh in the same way.ls There are other objections to this form of Foundation: for example, from the theory of ill-founded sets as developed by Aczel and others, and from the possibility of the analysis of matter into atoms, elementary particles etc., proceeding indefinitely. I am not sure what to make of such cases and shall not discuss them any further. Of course, Aristotle would want to adopt the nonlinear form of Reverse Foundation as well as of Foundation: Axiom, 2 (ReverseFoundation). There is no infinite sequence of objects x,, x,, x,,. .. such that x, constitutes x,, x2 constitutes x,, and so on ad infinitum. Those of the previous results which did not depend upon the linearity assumptions can then be proved in the same way as before. It is not clear whether any further principles, within the completely general theory of constitution, should be laid down, either by Aristotle or by the neo-Arisl 7 It should be noted that we also have a spatial argument for the principle in application to concrete things, somewhat analogous to the earlier temporal argument. For it might be supposed that a violation of the principle would require that things get indefinitely small. I s Nor, in my view, is the fact that the water and earth are only potentially present in the flesh a reason for denying that they are constitutive of the flesh. Indeed, I would assume that their potential presence, whatever this comes down to, is indicative of the peculiar vertical way in which they constitute.
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totelian.19 However, it does seem reasonable to make certain other assumptions for concrete things, where these additional assumptions are justified in terms of special features of the physical realm. These assumptions are most aptly explained in terms of what one might call constituency trees. For any given object y, the constituency tree for y is defined in the obvious way. The initial node is labelled with y; its immediate descendants are then labelled with the immediate constituents of y; and so on. Thus a typical tree might look as follows:
Given the way the constituency tree has been defined, there is nothing to prevent two distinct nodes being labelled by the same object. We may think of each node of the tree with label x as an occurrence of the constituent x. Thus there is nothing to prevent an object from having several constituent occurrences. However, in the case of concrete things, it is plausible to suppose that various cases of multiple occurrence should be excluded. Let us define mediate constitution by: Definition, I . x mediately constitutes y if for some XI,x2,. .., x,, n>l, x = x, constitutes x,, x, constitutes x,,. .., and x ,,., constitutes x, = y. (Ordinary constitution may be called immediate by contrast). The first exclusionary assumption then states: Axiom, 3(i)-.No thing x both mediately and immediately constitutes the same thing.y. In terms of the constituency trees, this assumption rules out subtrees of the following form:
l 9 Foundation will be complete for the universal theory of constituency as long as for any set x of objects (or of pure sets, for that matter) there is a proper class of objects y with the property-that the set of its constituents is x. We know that there is always one such y; for we can let y be the set x itself. It is also plausible to suppose that there are other such y; we may, for example, let y be a multi-set of the objects in x. However, it is somewhat hard to know on what basis to determine the size of the class of y's.
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There is, of course, no question of this axiom (or the subsequent ones) holding for abstract objects. The set ( ( a ] , a ] , for example, has a both as an immediate and as a mediate constituent. Or to make the comparison more direct, the sentence (P&(P&Q)) has P as an immediate and mediate constituent when viewed as a type but, when viewed as a token, has the first token of P occurring only as an immediate constituent and the second token of P occurring only as a mediate constituent. It is natural to wonder what can justify the striking contrast between the concrete and abstract cases; and one answer is in terms of the underlying spatial properties of concrete things and their constituents. These are defined by: (a) Subsumption. Any immediate constituent of a thing is spatially included in it. (b) Non-inclusion. Neither of two distinct immediate constituents of a given thing is spatially included in the other.20 From these two assumptions, axiom 3(i) can then be derived. Given axiom 3(i)', immediate constitution can be defined in terms of mediate constitution in the obvious way: Theorem, I. x immediately constitutes y iff x mediately constitutes y and for no z does x mediately constitute z and z mediately constitute Thus it was the possibility of "mediate reoccurrence" which rendered the definition incorrect in the case of sets. The axiom does not rule out all cases of reoccurrence within the constituency tree of a given thing. For it allows the possibility that xl and x2 are constituents of something and that x is a constituent of them both. By a constituency path from x toy, we mean a sequence x = x,, x ,,,...,x , = y in which each xi is a constituent of xi+l,for i = 1,2,..., (n-1). Then the general exclusion of reoccurrence within a tree amounts to the claim that: 20 The proof goes as follows. Suppose that x is both a mediate and an immediate constituent of y. Then for some y', y' is an immediate constituent of y and x is a mediate constituent of y'. Now (1) y' is distinct fromx; for otherwise, x would be a mediate constituent of itself, which is contrary to Foundation. Furthermore, by Subsumption and the transitivity of the relation l for spatial inclusion, it follows that (2) xly'. But the conclusions (1) and (2) are contrary to Non-inclusion. The left-to-right direction follows immediately from the axiom. The right-to-left direction follows by definition of mediate constitution.
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Axiom, 3(i) No two constituency paths ever have the same origin x and the same destination y. The strengthened exclusionary assumption can be derived from a strengthening of Non-inclusion: (c) Disjointness (for constituents). No two immediate constituents of a thing overlap; (d) Non-nullity (for constituents). Any constituent has non-null spatial extension.,, These underlying assumptions are rather plausible for the objects of the central ontology. One possible kind of counterexample to Constituent Disjointness is illustrated by the case of the neck overlapping with the head. But we may suppose that it is indeterminate where the boundaries of the head and neck lie. Although it is possible to draw the boundaries of either so that there is overlap, it is not possible so to draw the boundaries of them both; the head must begin where the neck ends. Another possible kind of counterexample is illustrated by the case of a leg and a table top having some glue in common. But we may suppose that, properly speaking, the constituents of the table are the leg and the top without the glue and the glue itself. However, such responses are less plausible for other cases. One might maintain, for example, that two highways may both be immediate constituents of a highway system and yet overlap; and it is implausible in this case to suppose either that there is some indeterminate boundary between the two roads or that the area of intersection belongs to neither of them. All the same, there does seem to be some sort of presumption in favour of the axiom and the underlying assumptions. It does seem to be true, in a way that is hard to make precise, that any counter-example to these principles is to be regarded as an abnormal case. A special problem over Constituent Disjointness may arise for Aristotle from his account of uniform matter. For on one understanding of his view, the elements ,,The proof goes as follows. Say that a path from x to y is accompanied if there is another path from x toy. Suppose that there is an accompanied path. Let x=x,, x,,. .. xn= y be a shortest such path; and let x=x',, x',,.. . x',=y be another such path. Then x,., is distinct from x',,.,, since otherwise x,, x,, ... x,., would be a shorter accompanied path. So, by Disjointness, x,., and x',., are spatially disjoint. By Subsumption and Transitivity of 5, x = x , l x,,., and x=x', 5x'l,,.,, contrary to Non-nullity. It is worth noting that the some of the spatial assumptions so far made can also be used to justify Foundation. We must make the additional assumptions that (i) any thing with a constituent has at least two constituents and that (ii)there is a non-zero limit to the size of the spatial region occupied by any thing. With the help of Disjointness and Subsumption, we then see that any violation of Foundation would require the size of the constituents to get indefinitely small. However, it is not clear that ( i ) is a justifiable assumption; and it is certainly not justified for Aristotle, who would take the sole constituent of a man to be his body or of a quantity of earth to be its ultimate matter.
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which constitute flesh, let us say, interpenetrate and hence the spatial location of each of them will be the same as that of the flesh itself.23 But even on such an understanding, it would still be possible to save the stronger form of our exclusionary hypothesis. For the notion of spatial location played rather an abstract role in its proof. Suppose now that we substitute for the spatial location of a thing its material content, i.e. the mereological sum of the ultimate matter which constitutes it. Then the proof from the analogue of Constituent Disjointness will still go through. Even the stronger principle is not sufficient to rule out all cases of multiple occurrence; for it is still possible to have repeated occurrences across trees, rather than within a tree. Say that: Definition 2. Two objects are unconnected if they are not both mediate constituents of a common object. The remaining possibility may then be excluded by requiring that: Axiom, 3(ii) No thing is the common constituent of two unconnected things. Let us define the notion of substance in the same way as before, but using the contributory rather than the holistic concept of constitution. Assume that any enmattered object is either a substance or is a mediate constituent of a substance (something which follows from Reverse Foundation). Then our assumption can be more simply rendered as: Axiom, 3(ii)'. No two substances have any mediate constituent in common. Together, the two axioms 3(i) and 3(ii) are equivalent to: Axiom, 3 (Upward Linearity). No thing is the immediate constituent of two things. Thus this axiom serves to exclude all cases of multiple occurrence; the difference between constituent and occurrence (between label and node) disappears.Given that Aristotle accepted the above three axioms (Foundation, Reverse Foundation, and Upward Linearity), the picture of his universe which then emerges is that of a collection of trees, rather than of sticks. The substances fan out, as it were, as we move downwards, with each constituent part occupying its own unique position. To derive axiom 3 from our spatial postulates, we need to derive the case described under axiom 3(ii), since the other case has already been take care of. This can be done by extending the Constituent Disjointness assumption: (c+) (Disjointness for the Unconnected) Any two unconnected things are spatially disjoint.24 Or given Reverse Foundation, we may more simply require: 23 This view is usually associated with the Stoics. See Chapter 6 of Sorabji (1988). But for reasons which I shall not go into, I think that the view may with some plausibility be attributed to Aristotle. 24 The proof proceeds in the same way as before.
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(c') (Disjointnessfor Substances). Any two substances are spatially disjoint. These assumptions are not without their exceptions. Consider a sentence which is used ambiguously. Then it could be argued that two sentence tokens have been produced, each with its own meaning. Thus if neither of the sentence tokens were a constituent of anything else, we would have a case in which two substances were spatially coincident. Or again, in regard to "heaps", someone may be a member both of a family and a gang, for example, even though the two are unconnected. The assumptions are also problematic in their application to stray bits of matter; for it seems quite plausible to suppose of two stray bits of matter which are not the matter of anything else that one may be properly included within the other. And nor does it help, in this case, to construe content materially rather than spatially. One way out of the difficulty in this case is to deny the possibility of proper inclusion. Perhaps one only admits into the ontology those stray bits of matter which are of a piece and which are not properly included in any other stray matter which is of a piece. But it is interesting to note that another solution is open to Aristotle. For let it be assumed, compatibly with proper inclusion, that the constituents of a quantity of uniform matter interpenetrate and hence occupy the same location. Then axiomc 3(ii)' can be derived.25 But even though there are exceptions to the axioms and assumptions, I am still inclined to think that there is a presumption in favour of their truth, especially in application to the central ontology; and some account needs to be given of why this should be so. The explanation, I think, goes roughly as follows. The presumption in favour of the principles derives from the desirability that they should hold; and their desirability derives, in its turn, from the fact that a universe ordered in conformity with them is more manageable than one that is not. Thus the principles derive from our attempt to make the universe manageable. There are two relevant aspects to our having a manageable conception of the universe, one relating to the axioms (which merely concern constituency) and the other relating to the assumptions (which also concern spatial location). The second of these, which is perhaps the easier to understand, is that the things of inter2 5 Suppose that x and y are two substances with u as a common (mediate) constituent. We know, from the previous reasoning, that x and y cannot both be substances in the strict sense. So let us suppose that x is a substance in the broad sense, i.e. a quantity of uniform matter. Then u must have the same location as x and hence the location of y must at least include that of x. But then y cannot be a substance in the strict sense and hence must also be a quantity of uniform matter. S o y must have the same location as x; and since they are both substances, they must be the same. On the other hand, without interpenetration (and with the possibility of proper inclusion), it hard to see how the axiom is to be defended. For suppose that x is a substantial quantity of uniform matter and that u is a constituent of x with a smaller spatial location. Let y be the part of x with the spatial location of u.Then presumably y is also a substance with x as a constituent.
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est to us should be readily discernible. Now things are discerned in terms of their boundaries; and it is in general hard to see how the boundaries are to be discerned if the things overlap. Thus the various disjointness assumptions can be seen as part of a format which will facilitate the discernment of boiindaries. The special constituency axioms follow from the spatial assumptions and so can be indirectly motivated in the same way. But they also have a more direct motivation in terms of our ability to track the consequences of change. For the purposes of prediction and control, we would like to limit the changes which are conceptually dependent upon a given change. In particular, we would like to limit those changes which are induced by the replacement of one constituent by another. Now two obvious requirements to impose in this regard are that, in replacing an immediate constituent, one does not affect any other immediate constituent or any unconnected thing. But it is exactly the constituency axioms which guarantee that this requirement is met. For example, if we had u an immediate constituent of y and z , and y and z immediate constituents of x, then in replacing y with the result y' of replacing u in y with u', we would also be effecting a change in z.
4. Linearization I want to consider the question of how, and to what extent, the two accounts of constitution might be reconciled. Let us call the notion of constitution from the previous section partial and the notion from the section before that singular. Then one possibility is to deny that there are any partial, as opposed to singular, constituents. Anything which we take to be a partial constituent is in reality a part of a singular constituent. Consider, for example, the sequence of letters "cat". Then one naturally takes its constituents to be the letters "c", "a" and "t". But on the proposed view, the (single) constituent of the expression is in fact some collective of the three symbols. There are various things wrong with this view. First, it is arbitrary what we take the collective to be; it could be a set, a mereological sum or something else altogether. Thus the account of the constituents of an object seems arbitrary in a way in which it should not. Second, the manner in which a complex such as a sequence is formed from its constituents should not depend upon the internal structure of those constituents; we should not have to reach inside the constituents, as it were. Finally, there are, as we shall see, certain technical difficulties which arise from the possibility of constituents themselves having constituents. The view is also not Aristotle's. For as we have already noted, there is reason to suppose that Aristotle would have been willing to countenance a plurality of immediate constituents. A second possibility is to treat the singular relation of constitution as the restriction of the partial relation to those cases in which there is no competing constituent. Thus we may say that 0 is a constituent of 10) but not that it is a con-
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stituent of ( 0 , 11; and Aristotle might say that the body was a constituent of the man but not that anything was a constituent of the body. Certainly, such a proposal would deliver the right formal properties for linear constitution. But it would not deliver the right meaning. We would be forced to sap, for example, that the body was the ultimate matter of the man, since there was no singular constituent for the body. The final, and most plausible, possibility is to take the notion of singular constitution to be the result of linearizing the notion of partial constitution. The tree, which is used to represent the constituency structure of a given thing, is somehow converted into a branch. The topic of linearizing a structure is one which can be treated with mathematical precision. A linearization may be identified with some kind of homomorphism or partition on the structure. Various linearization procedures can then be considered and played off against various conditions that one might want them to satisfy. A very familiar linearization, for example, is given by the notion of rank in set theory; and it can be characterized as the most coarse-grained linearization which is subject to the condition that x ~ -+ y f(x)
