COMMENT ON BRIAN LEFTOW'S
"AQUINAS ON DIMENSIONS AND INDIVIDUATION"

Like certain of their seventeenth-century predecessors, many contemporary philosophers simply assume that any sort of Aristotelian or Scholastic form/matter ontology has been effectively undermined by recent advances in physics. I myself have come to believe that this assumption, in both its seventeenth-century and twentieth-century versions, is false and that a broadly Aristotelian ontology of form/matter and substance/accident is fundamentally rightheaded--though (luckily) I do not have time to argue for either of these claims here. I mention this only to forestall the facile judgment that Brian Leftow's paper, for all of its philosophical rigor and resourcefulness, is of merely antiquarian interest.

My comments on Leftow's paper are divided into two parts: first, I will provide some background by noting a disagreement among late medieval Thomists about what role St. Thomas assigned to materia signata quantitate or designated matter; second, I will raise some problems for Leftow's attempt to fill out the theory of individuation found in q. 4, a. 2 of St. Thomas's commentary on Boethius's De Trinitate.

1. Three questions about individuation

Following Francisco Suarez in Disputationes Metaphysicae V, we can mark off three distinct metaphysical questions concerning individuation, each of which was taken by at least some late medieval Thomist to be the question St. Thomas meant to answer by his appeal to designated matter:1

(A) The individuality question: What is the intrinsic principle by virtue of which a thing of a given species, say the species aardvark, is this aardvark or numerically one aardvark or an individual aardvark? That is, what constitutes it as something which is, as the Latin term individuum suggests, indivisible into things of the same species?2

(B) The distinctness question: Given a pair of individuals of the same species, what is the intrinsic principle by virtue of which this one is distinct from that one? (Note that distinctness is different from individuality or numerical unity, since distinctness, unlike individuality, is a relation that presupposes the existence of at least two individuals.)3

(C) The plurality question: What makes numerical plurality within a given species possible? That is, what is the metaphysical ground for the possibility that there should exist more than one individual of a given species? (This question is obviously different from the individuality question, since it is conceivable that an individual should belong to a species that cannot be multiplied into many individuals; this, of course, is just what St. Thomas himself believes to be true of the angelic species. The plurality question also differs from the distinctness question, even though they are intimately related; for the distinctness question has a place only on the assumption that there is a plurality of individuals within a given species.)

Leftow assumes that on St. Thomas's view designated matter is the principle of individuality for corporeal substances, and there is indeed ample textual support for this assumption.4 However, in the face of strong objections to the claim that designated matter is the principle of individuality, some late medieval Thomists retreated to the view that, according to St. Thomas, primary matter taken by itself is the principle of individuality, whereas matter as quantified is the principle only of distinctness and not of individuality. This position is supported by passages such as the following:

"Just as diversity of matter and form taken absolutely makes for diversity in genus or in species, so this form and this matter make for numerical diversity."5

"It is by these indefinite dimensions that matter is made to be this designated matter ... and it is in this way that numerical diversity within the same species is caused by the matter."6

"It is only because of quantity that this matter is divided from that matter; and so matter as subject to dimensions is understood to be the principle of this diversity."7

Still other Thomists took St. Thomas's invocation of quantified matter to be directed toward the plurality question and not toward either of the other two questions. They, too, could appeal to passages such as the three just quoted, and, in addition, they could point out that when, in q. 2, a. 4, St. Thomas takes up Boethius's disputed claim that accidents individuate, he introduces the objections to that claim as follows: "It seems that the variety of accidents cannot be a cause of numerical plurality,"8 by which he means numerical plurality within the same species.

My own tentative view, for what it is worth, is that (i) if designated matter has any role at all to play here, it is only as a principle of plurality, but that (ii) it is better to claim that the principle of plurality for material substances is just primary matter, conceived of as itself an individual (though not an individual substance) having by its nature an intrinsic aptitude or capacity for being extended in three dimensions. But be that as it may, anyone who has read and struggled with discussions of designated matter in St. Thomas and his followers will undoubtedly appreciate Suarez's remark that "what the term ['matter designated by quantity'] signifies is so obscure that in explicating this topic the defenders of the position in question disagree among themselves to an astonishing degree (mirum ad modum)."9

2. Some Problems with Leftow's View

Nonetheless, Leftow does a fine job of laying out clearly the distinction between determinate (or, better, definite) dimensions (DD) and indefinite determinate extension (IDE), and of wrestling with the difficulties attendant upon the attempt to turn matter as the subject of IDE into the metaphysical principle of individuality. I do believe, however, that he should have refrained from calling IDE an accident, even a "special" accident, since (i) accidents are by their nature apt to inhere in substances, and primary matter, which is the subject of IDE, is not by itself a substance, and since (ii) what Leftow calls a "case of IDE" is, as he acknowledges at least implicitly, either an abstraction that cannot have being on its own or else just identical with a case of DD. Of course, as Leftow well realizes, this is just the problem. How are we to think of matter's being the subject of IDE prior to (i.e., metaphysically prior to) its being formed or structured into a substance with definite dimensions? What is it for matter to have indefinite dimensions metaphysically prior to its having definite length, breadth, and width?

Leftow tries to help us out here. We can, he tells us, think of a case of IDE "as a function that takes primary matter as its argument and generates as a value a distinct clump of 'pure quantified matter'--in effect, a distinct clump of primary matter, in a place discrete from all other clumps" (p. 8). This may or may not be what St. Thomas had in mind, but I find it problematic for at least two reasons.

The first is this: Is it all of primary matter that is the subject of a given case of IDE? The answer, it seems to me, can only be negative. There are, after all, many cases of IDE, corresponding to the multiplicity of actual material substances, and in generating a material substance the various relevant agents must, it seems, be effecting dimensions in this matter, as opposed to that matter. But if this is so, then it seems that (i) distinct parcels of primary matter are the subjects of various cases of IDE, and thus that (ii) even prior to being extended in three dimensions, each such parcel is a this with an intrinsic capacity for being the subject of dimensions. It follows that, pace St. Thomas, this matter is a this metaphysically prior to its being extended by any sort of dimensive quantity, whether definite or indefinite. (In fact, if, like Suarez, we think of primary matter as antecedently divided in this way, then we can easily reinterpret talk about IDE as talk about the intrinsic general capacity that parcels of primary matter have for receiving DD.)

Second, how can a case of IDE yield "a distinct clump of primary matter, in a place discrete from all other clumps" (p. 8)? Here we have a dilemma. If a case of IDE is distinct from any case of DD, then a case of IDE as such results only in indefinite dimensions. But is a clump with indefinite dimensions a clump?10 On the other hand, if every clump has definite dimensions because, as Leftow puts it, IDE and DD "are really identical" (p. 11) and "must arrive and exist together" (ibid.), then how is it legitimate to distinguish the roles of DD and IDE from one another? To his credit, Leftow delineates this problem clearly, but, unlike him, I have no confidence at all that there is a plausible way out.

(I note in passing that these same considerations militate against Leftow's suggestion, at the end of the paper, that IDE is a forma corporeitatis. As far as I know, in the medieval debates over the postulation of a distinct forma corporeitatis in animals, it is assumed by all sides that such a form (i) would be capable of yielding a complete substance and (ii) would give definite dimensions to the matter it structured.)11

My final reservation centers on Leftow's skillful attempt, in section 5, to answer the "thorny question" of how we might interpret St. Thomas's claim that cases of IDE "of themselves have a certain nature of individuation according to a determinate place (situs)." (Actually, situs is better translated as 'position' than as 'place'.) Leftow suggests that we conceive of IDE as being identical with the attribute of occupying some unique set of points or other, where a unique set of points defines a unique spatial location:

The per se individuality of geometrical objects can be viewed as a consequence of their nature as a set of points. As per se individual ... points can serve as a source of individuality for other things which are per accidens individual--e.g., clumps of matter. I suggest, then, that the attribute of IDE = the attribute of occupying some unique set of points or other ... We can represent Aquinas's point in calling such a clump self-individuated in this way: when we have spoken of it in its own proper nature, as the matter which occupies such-and-such spatial points, we have ipso facto distinguished it from any other clump of pure quantified matter" (p. 10).

My worry is this: If we take IDE to be a principle of individuality (as opposed to a principle of distinctness or plurality) and if occupying some unique set of points thus constitutes a bodily substance as an individual, then it seems to follow that it is absolutely or metaphysically impossible that two distinct bodily substances should occupy exactly the same set of points (or the same place) at the same time.

However, in q. 4, art. 3 of the De Trinitate commentary St. Thomas explicitly claims that while it is naturally impossible for two bodies to occupy the same place, it is nonetheless not absolutely impossible for two bodies to occupy the same place, though this can happen only miraculously. Now I will be the first to acknowledge that it is not easy to see how St. Thomas can admit the possibility of such a miracle, given his claim that "it is by the nature of matter subject to dimensions that more than one body is prohibited from being in the same place."12 For it seems to follow that it is no more possible for two bodies to occupy exactly the same position than it is for two geometrical objects to occupy exactly the same position--and St. Thomas unambiguously claims that the latter is indeed absolutely impossible. Nevertheless, to the extent that I understand it, here is his argument for the metaphysical possibility that two bodies should occupy the same place:13

"Clumps of corporeal matter are constituted as distinct from one another only by virtue of having distinct dimensions; and dimensions, considered abstractly as geometrical shapes with length, breadth and width, are constituted as distinct from one another only by virtue of having positions that are distinct from one another. (This is why it is absolutely impossible for more than one three-dimensional geometrical object to occupy exactly the same position.) So two clumps of matter are constituted as distinct from one another only because their dimensions, and hence their positions, are distinct from one another. The distinctness of their positions is, then, an intrinsic natural cause of the distinctness of the two clumps; and since a material body is constituted as a distinct body only because its matter, which is one of its constitutive principles, is antecedently distinct, it follows further that a material body cannot, at least in the order of natural causes, occupy the same place as another body.14

"Notice, however, that once two material bodies, call them A and B, have already been constituted as distinct bodies, the definite dimensions inhering in A (call them DD₁) and the definite dimensions inhering in B (call them DD₂) are distinct from one another not only, as in the case of geometrical objects, by virtue of occupying distinct positions but also by virtue of inhering in distinct subjects, viz., A and B--even though, as we saw above, the occupation of distinct positions was the original intrinsic cause of the distinctness of DD₁ from DD₂. It is this extra ground for distinctness, viz., inherence in distinct subjects, that underlies the metaphysical possibility that A along with DD₁ should remain distinct from B along with DD₂ even if the original cause of their distinctness, viz., the distinctness of their respective positions, is absent. Now, as noted above, this possibility lies beyond the nature and power of the bodies themselves to actualize, since in the order of nature distinctness of material subject itself presupposes distinctness of position as its cause. But, says St. Thomas, 'God, who is the first cause of all things, is able to conserve an effect in being without its proximate [secondary] causes; hence ... he is able to conserve the distinctness of the corporeal matter and of the dimensions in it in the absence of a diversity of position.'"15

This difficult, but ingenious, line of reasoning raises three questions that I will end my comments with. First of all, is the argument a good one for establishing the metaphysical possibility that two bodies should occupy exactly the same place? Second, are the premises of the argument consistent with the claim that designated matter is the principle of individuality (as opposed to the principle of distinctness or plurality)? Finally, an exegetical question: Can Leftow's reconstruction of St. Thomas's theory of IDE accommodate the absolute possibility that two distinct material bodies, along with their distinct inherent matters, should occupy exactly the same place?

1. I do not mean to imply that these three questions exhaust the metaphysical problem of individuation. Most significantly, there remains the question of what makes an individual the same individual despite changes it might have undergone through time.

2. Notice that things signified by mass nouns (as well certain lower animals such as worms) require a slightly more sophisticated formulation of this principle of indivisibility. Suarez considers the objection that this volume of, say, water is an individual and yet is divisible into parts each of which is water. His reply is that this water is nonetheless an individual, since it is not divisible into parts each of which is this water.

3. Of course, it may still be that whatever it is within a thing that constitutes it as an individual is sufficient to make it distinct from other individuals of the same species, so that the only difference between the individuality and distinctness of a given individual is, as it were, an extrinsic factor, viz., the mere existence of other individuals of the same species. This is, in fact, Suarez's position.

4. For instance, in q. 4, a. 2 of the commentary on the De Trinitate St. Thomas says: "The parts of an individual are this matter and this form ... A form becomes a this by being received in matter ... A form is individuated by being received in matter only insofar as it is received in this matter, which is distinct and determined to the here and now; but matter is divisible only through quantity" (Bruno Decker, ed., Sancti Thomae de Aquino Expositio Super Librum Boethii De Trinitate (Leiden: E.J. Brill, 1959), p. 142, line 26, and p. 143, lines 2-5). Later, in replying to the second objection he says: "Since dimensions are accidents, they cannot of themselves be the principle of unity for an individual substance; instead, it is matter as subject to such dimensions that is understood to be the principle of such unity" (op. cit., p. 144, lines 10-13).

5. op. cit., p. 142, lines 18-20.

6. op. cit., p. 143, lines 20-21.

7. op. cit., p. 145, lines 1-3

8. op. cit., p. 137, lines 22-23.

9. Disputationes Metaphysicae V, § 8.

10. Recall, by the way, that St. Thomas makes the move from definite dimensions to IDE in part because a given substance can survive changes in its definite dimensions. But this seems irrelevant if what we are concerned with is the individuality question rather than questions about identity through time.

11. The most compelling argument for the postulation of a forma corporeitatis distinct from the sentient soul in an animal is that it allows us to say that the corporeal accidents, quantitative and qualitative, of an animal that has just died are the very same accidents had immediately beforehand by the living animal. But this clearly presupposes that the forma corporeitatis is a principle that makes a parcel of matter into a bodily substance having definite dimensions. See, e.g., Ockham, Quodlibeta Septem II, q. 11.

12. op. cit., p. 150, lines 27-28.

13. The following reconstruction is based upon the commentary on the De Trinitate commentary, as well as Quaestiones Quodlibetales I, q. 10, a. 1-2; Summa Theologiae III, q. 54, a. 1, and Supplement, q. 83, a. 1-3.

14. According to the account of natural necessity which I laid out in "The Necessity of Nature," Midwest Studies in Philosophy 11 (1986): 215-242, a material body has a strong deterministic natural repugnance to the property of occupying the same place as a body distinct from itself.

15. Quaestiones Quodlibetales, q. 10, a. 2, resp.