Could Extended Objects Be Made Out of Simple Parts

Could Extended Objects Be Made Out of Simple Parts? An Argument for "Atomless Gunk" Dean W. Zimmerman Philosophy and Phenomenological Research, Vol. 56, No. 1. (Mar., 1996), pp. 1-29. Stable URL: http://links.jstor.org/sici?sici=0031-8205%28199603%2956%3A1%3C1%3ACEOBMO%3E2.0.CO%3B2-X Philosophy and Phenomenological Research is currently published by International Phenomenological Society.

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Philosophy and Phenomenological Research Vol. LVI, No. 1, March 1996

Could Extended Objects Be Made Out of Simple Parts? An Argument for "Atomless Gunk" DEAN W. ZIMMERMAN

University of Notre D a m e

1. Introduction Let us say that an extended object is "composed wholly (or entirely) of simples" just in case it is an aggregate of absolutely unextended parts spread throughout an extended region-that is, just in case there is a set S such that: (a) every member is a point-sized part of the object, and (b) for every x, x is part of the object if and only if it has a part in common with some member of S. Could a truly extended substance be composed entirely of unextended ("simple") parts? Reflection upon the fact that it must be at least possible for extended objects to touch one another suggests that the answer to this question is: No. (For the purposes of this paper, it will be convenient to use "extended" throughout to mean "three-dimensionally, spatially extended", and "unextended" to mean "dimensionless" or "having the dimension of a geometrical point".) Although Zeno's mathematical paradoxes of plurality were long thought to raise insurmountable difficulties for the supposition that an extended thing could be composed of unextended simple parts, Adolf Griinbaum has shown that these paradoxes are significantly defused by Cantor's discovery of the distinction between denumerably and non-denumerably infinite numbers.' If Griinbaum is right, the traditional reasons for doubting the consistency of "conceiving of an extended continuum as an aggregate of unextended elements" have been laid to rest,2 and the question of this paper is ripe for reconsideration. Numerous commonplaces of the modern analytic geometry of physical space will be assumed in the arguments to follow. Perhaps this is a mistake; Cf. Adolf Griinbaum, Philosophical Problems of Space and Time (New York: Alfred A.

Knopf, 1963), ch. 6.

Griinbaum, Modern Science and Zeno's Paradoxes (Middletown, Connecticut: Wesleyan

University Press, 1967), 115.

COULD EXTENDED OBJECTS BE MADE OUT OF SIMPLE PARTS?

1

perhaps extended spatial regions do not really consist of sets of continuummany points. But if space were not made out of points, spatially extended objects could not be made out of point-sized simples. Where would the simples be if there were no point-sized locations for them to occupy? By adopting the analytical, punctual approach to the geometry of physical space, I merely afford the view I intend to refute a running start. These assumptions from analytic geometry are inconsistent with the hypothesis that space is "discretew-i.e., that extended spatial regions consist of finitely many minimal "spatial atoms". Some theories of discrete spacethose according to which the ultimate space-atoms are themselves extendedare, I believe, patently absurda3The only version of discrete space that need be taken seriously is that according to which the space atoms are truly unextended. Perhaps such a theory is not as preposterous as it sound^.^ In any case, if my conclusions are right, one could only make extended objects out of simples if space were discrete.

2. The concept of an extended material object The question in my title is intended to be a peculiarly metaphysical one; it is not, "Are there any extended bodies, and if there are do any happen to be composed of simples?", but rather, "If there were such things as extended physical bodies, could they be composed entirely of simples?" I begin with an examination of the concept of a three-dimensionally (spatially) extended substance. What properties would something have to have in order for it to count as an "extended body" in my use of the phrase? It must be a three-dimensional, spatially-located thing (not a mere point-sized atom, or even a cloud of disconnected point-sized atoms, each at some distance from all the others) which excludes similar things from the locations it occupies. If something is, in this sense, an extended body, then (1) it fills a precise three-dimensional region of space (and therefore has a certain size and shape; and, if it persists for any length of time, must therefore be at rest or in motion with respect to other space-occupiers), and (2) it is in some manner impenetrable with respect to other classes of space-occupiers. These more or less exhaust Locke's account of the "primary qualities of matter"." An extended entity which lacks precise boundaries-because, for instance, there is an extended region R surrounding it which is such that it is neither true nor false that any subregion of R is completely filled by a part of the en-

"f.

Cf. my arguments in Zimmerman, "Indivisible Parts and Extended Objects: Some Philosophical Episodes from Topology's Prehistory", Monist 79 (January 1996), $2. Peter Forrest develops a discrete space theory of this sort in "Is Space-Time Discrete or Continuous?-An Empirical Question", Synthese 103 (1995): 327-54. Anthony Quinton's discussion of Locke in "Matter and Space", Mind 73 (1964): 332-52, esp. 341-42.

2 DEAN W. ZIMMERMAN

tity-will not satisfy the concept of a solid extended substance as I am elucidating it here. Unlike, say, fields of force or perceived patches of color which may have fuzzy edges or gradually fade away, an extended object cannot indeterminately fill a region. The sort of impenetrability which is a part of the concept of a solid, extended object must be understood not as a power to resist the pressure of other objects, but as an essential inability to share the same spatial location with them. Many contemporary philosophers insist upon a very strong impenetrability thesis: if one extended object exactly fills a region, then there cannot possibly be another completely discrete object filling that same region." David Sanford argues, however, that the concept of a physical object does not really require that such objects be impenetrable with respect to every other physical object, but only with respect to at least some other class of physical objeck7 If a space-occupying thing is penetrable by anything and everything, then there is no sense in which it still qualifies as a material body or physical object. But a set of moveable space-occupiers that necessarily exclude one another ought to qualify as material bodies, even if there is some other class of bodies which can pass right through them. The most noteworthy fact about this impenetrability debate, however, is that all parties agree that some kind of "logical impenetrability" must be possessed by a thing if it is to count as an extended material substance. And Sanford's weak impenetrability is sufficient for my purposes here. To simplify matters, let it be understood that, in the sequel, whenever two objects are said to come into contact with one another, they are of the same "impenetrability kind". Although Newton and his contemporaries believed that the physical world consisted entirely of extended physical objects in the sense just e~plicated,~ times have changed. Many scientists and philosophers now hold that, if we take contemporary physics seriously, we can only conclude that the traditional concept of extended substance has no appli~ation.~ Some philosophers, "f.

Quinton, "Matter and Space", 341-42; and Richard Swinbume, Space and Time (London: Macmillan, 1968), 15-17. Cf. also David Wiggins, "On Being in the Same Place at the Same Time", Philosophical Review, Vol. 77 (1968), pp. 90-92; Frederick Doepke, "In Defence of Locke's Principle: a Reply to Peter M. Simons", Mind, Vol. 95 (1986), pp. 238-41; and Peter Simons, Parts: A Study in Ontology (Oxford: Clarendon Press, 1987), pp. 221-28. Cf. also John Locke, An Essay Concerning Human Understanding, Bk. 2, Ch. 27, 881-2. Cf. David Sanford, "Locke, Leibniz, and Wiggins on Being in the Same Place at the Same Time", Philosophical Review 79 (1970): 75-82. Cf. Sir Isaac Newton's Mathematical Principles of Natural Philosophy and His System of the World, vol. 1, trans. Andrew Motte, trans. revised by Florian Cajori (Berkeley: University of California Press, 1962), xxvi (Cotes's preface to 2nd ed.); and Newton's Opticks (New York: Dover, 1952), 400. It is worth noting that even those scientists and philosophers who come to bury, not praise, extended substance bear witness to the general contours and familiarity of the concept. The alleged disappearance of precise boundaries and impenetrability at the

'

COULD EXTENDED OBJECTS BE MADE OUT OF SIMPLE PARTS?3

including Pierre Bayle, have advanced an even harsher critique of extended substance: not only are there in fact no extended substances, but there could not be any; the very concept of extended substance harbors contradictions, like the concept of a round square."' Considering the familiarity of this notion, such a result would be nearly as surprising as Berkeley's conclusion that there can be no unperceived objects, Zeno's that there can be no motion, or McTaggart's that there can be no time. Only if every attempt to understand extension and contact ended in paradox would we have cause to conclude, with Bayle, that extended bodies "can only exist i d e ~ l l y . " ~ ~ And herein lies the real significance of my project. Extended substances are more like gold mountains than round squares. Perhaps there aren't any, but there could be. And if the argument of this paper is correct, the possibility of extended objects implies the possibility of objects consisting of "atomless gunk"-objects that are infinitely divisible and not wholly composed of partless parts. But, as I have shown elsewhere, the bare possibility of atomless gunk has far-reaching consequences for the metap'hysicsof living things and artifacts.I2 -

lo

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subatomic level is cited as the strongest argument for the "dematerialization" of matter. (Cf., for example: Arthur Eddington, The Nature of the Physical World (Ann Arbor, Michigan: University of Michigan Press, 1958; first published, 1928), xiii-xiv; MiliE b p e k , The Philosophicul Impact of Contemporary Physics (Princeton: D Van Nostrand Co., Inc., 1961), 244-329; N. R. Hanson, "The Dematerialization of Matter", in The Concept of Matter, ed. Eman McMullin (Notre Dame, Indiana: University of Notre Dame Press, 1963), 549-61; and Erwin Schrodinger, "What is an Elementary Particle?", in Space, Time, and the New Mathematics, ed. Robert W . Marks (New York: Bantam Books, 1964), 100-1 15.) Such attempted "dematerializations" show at least this much: we know what it would be like for something to be an extended material substance, and we are surprised to be told that nothing is really like this. Thus the very project of demonstrating that the notion of extended substance is outmoded in the light of contemporary science presupposes that we do have a notion of physical substance which is not so plastic as to fit harmoniously with "whatever science says". And so the question whether extended bodies could be composed of simple parts would remain a perfectly meaningful one even if there were conclusive empirical evidence to show that there are no such things. According to Bayle, "If extension existed, it would not be possible for its parts to touch one another, and it would be impossible for them not to penetrate one another. Are not these very evident contradictions enclosed in the existence of extension?" (Bayle, Historical and Critical Dictionary: Selections, trans. Richard H. Popkin (Indianapolis: Bobbs-Merrill, 1965), 364). There is considerable doubt about how seriously Bayle took this and many other of the paradoxical conclusions drawn in his dictionary. Cf. Richard H. Popkins's introduction to Bayle, xxiv. Bayle, 363. Cf. also Franz Brentano, Philosophical Investigations on Space, Time and the Continuum, ed. Stephan Korner and Roderick M. Chisholm, trans. Barry Smith (London: Croom Helm, 1988), 39-44 and 138-49. In "Theories of Masses and Problems of Constitution" (Philosophical Review 104 (1995): 53-110) I argue that, if there could be objects made of atomless gunk, then (unless we are prepared to deny the absoluteness of identity, or accept a metaphysics of temporal parts) living things and other objects that can be constituted by different masses of matter at different times must be construed as either (i) a kind of process or

4 DEAN W.ZIMMERMAN

3. The "atomless gunk argument" In order to facilitate discussion of questions about the alleged point-sized parts of extended objects, let us adopt an extremely latitudinarian definition of a "region of space", according to which any set of spatial points counts as a region.13If an "open sphere" about a spatial point p is a region whose members consist of just those points less than some distance or other r from p, then a "boundary point of a region A" is a point such that every open sphere about it includes points in A and points not in A. A region is said to be "closed" if and only if it includes all of its boundary points, "open" if and only if it includes none of its boundary points, and "partially open" just in case it includes only some of its boundary points. We call the set of all the points in a region that are not boundary points the "interior" of the region. Likewise, the union of a region and its boundary points is called the "closure" of that region. If an extended object were really made up entirely of point-sized parts, it would have to fill its three-dimensional location by having an unextended part at every spatial point in the region. Since every spatial region is either closed, open, or partially open, it follows that an extended object made wholly of simples would also have to be either "closed", "open", or "partially openv-depending on whether the simples of which it is made occupy a closed, open, or partially open set of points. The notions of closure and openness for objects which are important for present purposes have more to do with the presence or absence of boundary elements than with the topological properties of the regions occupied. Roughly, an extended object should count as "closed" if and only if it has an outermost "skin" of simple parts-i.e., if and only if it has a simple part for each point in the boundary of the region it occupies,-"open" if and only if it lacks such a skin of atomic parts altogether-i.e., if and only if it has no parts located just in the boundary of the region it occupies,-and "partially open" if and only if it has a "skin" on some parts of its surface but not oth-

l3

event, or (ii) logical constructions out of more basic entities. It must be noted, however, that, by limiting the scope of the present enquiry to the question whether extended material objects in the traditional sense could be composed entirely of simple parts, I forfeit any claim to reach conclusions about the nature of "substantial fields", or extended objects conceived as having fields among their literal parts. As long as "substantial fields" are interpreted as being at least partially penetrable by any kind of thing or as having vague boundaries, my conclusions can throw no a priori light upon the question of whether fields have simple parts, or whether the forces in such fields should be regarded as having values for point-sized locations. The familiar topological definitions of "region", "boundary point", "open", "closed", "interior", "closure", and "connected" given here are advocated in a similar context by Richard Cartwright. Cf. Cartwright's "Scattered Objects", reprinted in his Philosophical Essays (Cambridge, Massachusetts: MIT Press, 1987), 171-86.

ers. These intuitive characterizations can be improved upon once we have the notion of an object's being adjacent to a region. (Dl) Object x is adjacent to region R =df the region exactly filled by x has no points in common with R , and the union of the two regions is a connected region (Roughly, disconnected regions are those separated by at least a point that is included in neither region, while connected regions must not even be that far apart. According to (D4), then, if an object occupies an open region, it can only be adjacent to regions which are-at least at the adjoining locationclosed, and vice versa.) (D2) x is a closed object =df x is a spatially located object; and for every y such that y is a part of x adjacent to a region which is not filled by a part of x , there is a set A of simple parts of y such that each member of A is adjacent both to regions filled by parts of x and regions not filled by any part of x (D3) x is an open object =df x is a spatially located object; x has proper parts; and there is no set of simple parts of x such that each member is adjacent both to regions filled by parts of x and regions not filled by any part of x (D4) x is a partially open object =df x is a spatially located object; x has proper parts; and x is neither open nor closed Notice that, if we assume (as we surely should) that every simple occupies exactly one spatial point, then an object is closed in the above sense if and only if it exactly fills a closed region. Given the further (controversial) assumption that three-dimensionally extended objects are composed entirely of simple parts, then an object is open in the above sense if and only if it exactly fills an open region. The truth of this further assumption is, of course, the main question at issue here. If the assumption is eventually shown to be false, the way is cleared for a metaphysics of extended objects (such as the "Whiteheadian" view discussed below) according to which such objects lack a final skin of simple parts-and so qualify as open according to (D3)-but still cannot properly be said to fill just open (and not closed) regions. Herein lies the importance of defining closure and openness for objects in terms of the presence or absence of exterior simples, and not by direct appeal to the topological closure or openness of the regions occupied. Every (finite) extended object must be either closed, open, or partially open. Broadly speaking, then, there are really only a few alternative metaphysics of extended objects to choose from. The most obvious three are: (i) necessarily, every extended object is closed; (ii) necessarily, every extended 6 DEAN W.

ZIMMERMAN

object is open; (iii) necessarily, every extended object is partially open. In addition to these three, there are numerous "mixed-bag" metaphysics of extended objects-metaphysical theses about the nature of extended objects implying the denial of each of the above three alternatives. For instance, one might hold that, as a matter of necessity, objects only come in the open and closed varieties, and that it is possible for there to be objects of both sorts; or that, necessarily, extended objects are all either open or partially open, and that it is possible for there to be objects of both sorts. Although there are clearly many varieties of mixed-bag metaphysics that could be formulated, the only ones which require separate consideration are those which allow for the possibility of coexisting objects of two or more kinds: open and closed objects, open and partially open objects, closed and partially open objects, and all three kinds together.14 The "atomless gunk argument" shows that, if extended objects were wholly constituted by simples, then they would not be so constituted-and that, consequently, extended objects cannot be wholly made up of simples. Here is the argument's overall structure: (1)

If extended objects could be composed entirely of simples, then one of the following "metaphysics of extended objects" would have to be correct: (i) necessarily, every extended object is closed; (ii) necessarily, every extended object is open; (iii) necessarily, every extended object is partially open; (iv) some mixed-bag metaphysics is true.

(2) Alternatives (iii) and (iv) are impossible.

(3)

If alternative (ii) were true, then extended objects could not be composed entirely of simples.

(4)

If alternative (i) were true, then extended objects could not be composed entirely of simples.

(5) So if extended objects could be composed entirely of simples, then they could not be composed entirely of simples.

Therefore, extended objects could not be composed entirely of simples.

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Mixed-bag metaphysics which do not allow objects of more than one of the three kinds to coexist will be ignored, since what I have to say about alternatives (i), (ii), and (iii) will apply to them. For instance, if open and closed objects are possible, but cannot coexist in any world, then the arguments I give pertaining to open-objects metaphysics suffice to show that in those possible worlds which contain open objects, those objects cannot be composed entirely of simples; and mutatis mutandis for the possible worlds which contain only closed objects. COULD EXTENDED OBJECTS BE MADE OUT OF SIMPLE PARTS?7

Thus, if there are extended objects, they must have extended parts which (a) are infinitely divisible (since they are extended), but (b) cannot be identified without remainder with a sum of unextended parts. Consequently, if there are extended objects, then they have at least some parts that satisfy David Lewis's description of a piece of "atomless gunk": "[aln individual whose parts all have further proper parts".15

4. Four constraints on any adequate metaphysics of extended objects The terms used in premise (1) have just been defined, and its truth is obvious. Premises (2), (3), and (4) will be defended in turn. My arguments for these premises will be found to turn upon the following four assumptions, (A)-(D): (A) Every extended object has a left and right half which are discrete and are themselves extended objects. This assumption, which-given the relativity of "left" and "right"-implies both infinite divisibility and "the doctrine of arbitrary undetached parts", will be contested by some.I6However I cannot but think that such objectors are using the word "part" in a special sense to mean something like "organically unified part" or "naturally demarcated part". After all, however arbitrarily the division may fall between the right and left half of an object, it cannot be denied that the object has a right and left half. If the object didn't completely fill up the left half of the region it occupies, then it wouldn't be occupying the whole of that region after all. And since the left-half region does not contain the whole object, it must be exactly filled by only a part of it. Three more presuppositions will be needed to support the atomless gunk argument; all have to do with the nature of contact between extended objects. The problem of contact is an ancient one. Although Aristotle was surely right that "[tlhings are said to be in contact when their extremities are together",17 it is not so obvious what talk of "the extremities" (or boundaries) of extended objects is really about, nor is it obvious what it is for "extremities" to be "together". A satisfactory metaphysical theory about the nature of the boundaries of extended objects (i.e., of their surfaces, edges, and corners) must provide an account of what "boundary-talk" means, including an explanation of the relationship holding between two extended objects just in case l5

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David Lewis, Parts of Classes (Oxford: Basil Blackwell, 1991), 20. Peter van Inwagen, "The Doctrine of Arbitrary Undetached Parts", Pacific Philosophical Quarterly 62 (1981): 123-37. Aristotle, Physics, bk. 5, ch. 2 (trans. R. P. Hardie and R. K. Gaye in The Basic W o r h of Aristotle, ed. Richard McKeon (New York: Random House, 1941); the quoted passage is on 306); cf. also bk. 6, ch. 1.

8 DEAN W. ZIMMERMAN

"their extremities are together". Assumptions (B), (C), and (D) below represent constraints upon any adequate metaphysical "fleshing-out" of the contact relation. A theory of contact must respect these principles, on pain of changing the subject: i.e., if a metaphysics implies that extended objects cannot be related in a manner that meets these standards for being the contactrelation, then it is a metaphysics on which contact and, I shall argue, extended objects themselves are impossible. An initial constraint on any adequate theory of the nature of contact is this: it should not imply that some pairs of extended objects which are in contact are closer to one another than other such pairs. The notion of contact is essentially that of a limit on how close things can get without either interpenetration or the sharing of parts. This fact is even hallowed in popular song: the answer to the rhetorical musical question, "Can I hold you closer to me and not feel you going through me?" is meant to be obvious: "It's impossible.. .." If two people are in contact, they cannot get any closer together without interpenetration. Thus the following principle should come out true on any acceptable metaphysical theory of contact: )

If two objects are in contact, then it is impossible for two distinct non-overlapping objects to be closer together than the two objects in question.

This conception of contact as an absolute limit of closeness is disparaged by Boscovich. He calls it "mathematical contact" and claims that our everyday use of the word "contact" to describe relations between observable objects does not connote this species of unsurpassable closeness, but rather a relation of "physical contactv-that degree of proximity "in which the distance is too small to affect our senses, and the repulsive force is great enough to prevent closer approach being induced by the forces we are con~idering."'~ Thus Boscovich's "physical contact" is a purely relative notion, varying with the amount of force we are interested in applying in particular cases. But if all contact among physical objects were "mere physical contact" and not an absolute limit of proximity, then, as Boscovich is well aware, every part of a composite thing would be separated from every other part, and there would be no solid extended objects. Thus a metaphysics of extended objects that rejects (B) is not a metaphysics of extended objects at all. I also assume what might be called the thesis of "the metaphysical possibility of contact". In essence, the thesis asserts that extended objects are not kept from touching one another just because of the shapes of their surfaces or the structure of their outermost parts. Ix

Roger Joseph Boscovich, A Theory of Natural Philosophy, trans. J. M. Child (Cambridge, Massachusetts: M.I.T. Press, 1966; translation of an edition first published in 1763), 57. COULD EXTENDED OBJECTS BE MADE OUT OF SIMPLE PARTS?9

(C) For any two possible shapes of a part of the surface of an extended object, it is possible for there to be extended objects having surfaces with those shapes that are in contact.

I intend "shape" to be understood here as referring in the broadest way possible to the configuration or structure of an extended object's exterior parts. Thus not only does the surface of a cube differ in shape from that of a sphere, but the surface of an open cube differs in shape from that of a closed cubeeven if their sides are of the same length. Perhaps there are extended objects which, due to repulsive forces among their essential properties, are necessarily untouchable. Still, it would not be the shape or configuration of parts on the surfaces of such objects that drive other extended objects away. According to (C), there could be extended objects lacking such forces but with precisely the same shape which were in contact. Powerful arguments in favor of (C) will emerge from the discussion of Bolzano's views about the nature of surfaces and contact below; so I reserve its further defense until then. A final assumption about contact displays the close connections between contact and extension. Since the left and right hemispheres of a solid globe are a paradigm case of objects in contact, a metaphysics of extended objects must allow that objects which are truly in contact can get as close to one another as the two halves of a sphere. More generally, every undetached proper part of an unscattered object must be in contact with the remainder of the object: (D) For every extended object x that fills a connected region, if extended objects y and z are discrete proper parts of x such that every

part of x is a part of the sum of y and z, then y and z are in contact. (D) is essentially Aristotle's point that "if there is continuity there is necessarily contact.. .."Iy Or, in Bayle's words, "An extended substance that might exist should necessarily allow for the immediate contact of its part^."^" Recall that, according to (B), contact is that relation which holds between two discrete extended things when they are as close as they can possibly get. To deny (D) would be to deny that, for example, a pair of discrete extended parts that compose a sphere are as close to one another as they could be. But if there is a gap-however small-between every pair of discrete extended parts of a sphere, then the sphere is nothing but gaps through and through! So if (D) were false because the internal parts of extended objects could not be in contact, then there would be no extended objects. -

Iy 20

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Adstotle, Physics, bk. 5, ch. 3 (McKeon, 307). Bayle, 363.

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5. The problems of contact in "partially-open" and "mixed-bag" worlds According to premise (2) of the atomless gunk argument, both of the following metaphysics of extended objects are unacceptable: a theory on which such objects are all partially open, and a mixed-bag metaphysics which would allow for various combinations of closed, open and partially open objects. The premise is justified by the fact that neither sort of view can measure up to the standards set in the previous section for an adequate metaphysics of extended objects. We begin with the problems facing a theory that countenances partially open objects, but both partially-open and mixed-bag metaphysics fail for the same basic reason-a reason which is suggested by Franz Brentano in the following passage taken from his discussion of the origin of our concept of a continuum:

. ..[G]eometry teaches that a line that is halved is halved in a single point. The line & therefore in the point b . And further, that one is able to lay the one half over the other, for example in such a way that cb would come down on ba, the point c coinciding with the point b , the other end coinciding with the point a . According to the doctrine [that continua are aggregates of infinitely numerous point-sized elements], in contrast, the divisions of the line would not occur in points, but in some absurd way behind a point and before all others of which however none would stand closest to the cut. One of the two lines into which the line would be split upon division would therefore have an end point, but the other no beginning point. This inference has been quite correctly drawn by Bolzano, who was led thereby to his monstrous doctrine that there would exist bodies with and without surfaces, the one class containing just so many as the other, because contact would be possible only between a body with a surface and another without. He ought, rather, to have had his attention drawn by such consequences to the fact that the whole conception of the line and of other continua as sets of points runs counter to the concept of contact and thereby abolishes precisely what makes up the essence of the c o n t i n ~ u m . ~ ' The atomless gunk argument as a whole can be seen as an attempt to make good on the last sentence in this passage; while the present section explores in more detail the features of Bolzano's view which Brentano finds so "monstrous". Let us begin with the supposition that all objects are partially open. In that case, two objects reach the greatest degree of closeness when an open surface of one is pushed up against a closed surface of the other, yielding a continuous stretch of simple parts between the two with not a single point-sized region unoccupied by some part of one or the other object. Consider what happens when, instead, two closed surfaces meet. Since space is a continuum, there are a non-denumerably infinite number of points between any two spatial points. If we set aside for the moment the possibility of discrete objects penetrating one another by having simples located at the very same points, it 21

Brentano, Space, Time and the Continuum, 146-47 ("Nativistic, empiricist and anoetistic theories of our presentation of space", $8). COULD EXTENDED OBJECTS BE MADE OUT OF SUlPLE PARTS?11

follows that however hard you push two closed surfaces together, there will always be a three-dimensional region separating the simples on their skins. Thus the open faces of a partially open perfect cube will be able to get smack up against a closed surface, while the closed faces will only be able to approach or hover over it. Since, as stated in (B), contact is that degree of proximity which cannot be surpassed, it follows that a closed surface may touch an open surface, but not another closed surface. Clearly, similar paradoxes confront all of the mixed-bag alternatives as well, such as Bolzano's mixedbag metaphysics according to which some objects are closed and some open.22 To see the absurdity of these results, consider a Bolzano-world which contains three closed cubes and one open cube, all having the same measure and mass. Take the open cube and one of the closed cubes, set them ten feet apart on a smooth surface, and push them toward one another with sufficient force so that they come into contact in two seconds. Now perform the same operation with the pair of closed cubes. Where are the two closed cubes after two seconds have elapsed? They cannot be touching. In fact, there must be some finite distance between their surfaces, Was their progress toward one another slower than that of the other pair, or did it stop sooner? In either case, we seem forced to attribute repulsive powers of some kind to the cubes-an ability each cube has to "let the other know" that it has a skin of simples so that, if both the approaching surfaces are closed, the bodies can make sure to slow down or stop. Unless we ascribe repulsive forces to closed surfaces, forces possessed necessarily by surfaces of that shape, Bolzano's world becomes one in which a certain class of objects are unaccountably deferential to one another-always just managing to step out of one another's way-while they bang heedlessly into the members of another class of objects. Surely repulsive forces would have to be posited to explain such behavior. Situations analogous to that of the pairs of cubes just described are possible not only in Bolzano's world, but in every other mixed-bag or partially-open world as well. But why should a certain shape of extended object be necessarily such that objects of that shape possess special repulsive powers? Is it not at least possible for there to be completely inert substances of any shape and size you like? The inappropriateness of ascribing forces or powers to objects just because they have certain shapes provides the promised further vindication of assumption (C) above. If some configurations of the parts of the surface of an object were such that no objects having those configurations could be brought into contact-i.e., if (C) were false,-we would find ourselves forced to posit repulsive powers which must be possessed by all substances having those shapes. Surely it would seem to be a contingent matter whether a certain 22

Cf. Bernard Bolzano, Paradoxes of the Infinite, trans. Dr. Fr. Prihonsky (London: Routledge and Kegan Paul, 1950; first published, 1851). 167-69 (166-67).

12 DEAN W. ZIMMERMAN

shape always comes with a repulsive power. Such necessary associations could, of course, be postulated by the metaphysician without fear of empirical refutation-for the activity of the force could always be supposed to kick in at distances just smaller than whatever degree of imprecision there might be in current methods of measurement. It would be a suspicious usurpation of physics by metaphysics, however, to insulate this seemingly contingent question from the sorts of empirical verification more appropriate to it. Such a move should only be taken for weighty reasons; each particular temptation to postulate a surprising necessity should be resisted. Thus (C) would seem to be at least a methodologically sensible working assumption, to be given up only as a last resort. Thus partially-open and mixed-bag metaphysics seem unable to provide a coherent account of contact. Recall, however, that at a crucial point in the above argument I set to one side the possibility of contact occurring between closed objects by means of their having distinct but coincident skins of simples. The defender of a mixed-bag or partially-open metaphysics could try to put this possibility to use in an account of contact along the following lines. The contact-relation comes in three slightly different varieties, she might insist, depending upon what kinds of surfaces are touching: (a) an open surface touches a closed surface when there are no empty points between them, (b) two open surfaces are in contact when there is only a less-than-three-dimensional empty region between them, and (c) two closed surfaces are in contact when portions of their two skins of simples are in the same place. If the exterior simples on a closed surface can really coincide in the way postulated by (c), then closed surfaces need not always keep their distance from one another, and the above argument against mixed-bag and partially-open theories all but evaporates. So, in order to support premise (2), I must also demonstrate the unacceptability of this trebly-disjunctive analysis of contact. Combining a mixed-bag or partially-open metaphysics with the proposed three-way account of contact leads to a number of difficulties. First of all, the proposed threefold definition of "contact" sins against (B), the assumption that contact is an unsurpassable limit of proximity. If having distinct simples in the same place does not count as overlap but rather as contact, then surely two open objects with empty points between them are not as close to one another as are two closed objects with coincident outer boundaries. If the two open objects could "grow" skins of simples, would they not thereby come to be in closer contact? Their surfaces, which were separated by an empty region, would move together to become coincident. If two open objects can be "in contact" without being as close together as two closed objects that are in contact, then the relationship between the open objects was not really one of contact after all.

COULD EXTENDED OBJECTS BE MADE OUT OF SIMPLE PARTS?13

Secondly, the view is undermined by some of the same problems which will be found to afflict the closed-objects metaphysics discussed in section 7. For instance, the possibility of coincident simples seems inconsistent with the impenetrability of extended objects-unless one admits that objects are not wholly composed of simples. In addition to these problems, the threefold disjunctive analysis is rendered implausible by the following very general objection which cuts against every partially-open and mixed-bag metaphysics: all such theories introduce a great deal of complexity where none is needed. Notice that the supposed difference between otherwise similarly-shaped open and closed surfaces could not be detectable in any way. Compare a pair of spheres each having diameter d, one open and the other closed, with a similar pair differing only in that both are open. When either pair of spheres is so aligned that the centers of the spheres are separated by the distance d, the spheres in that pair will be in contact according to the threefold analysis. In the case of the two open spheres, however, there must be an empty point-sized region where they meet; whereas, in the other case, this point is filled by a simple part of the closed sphere. Surely this could not be an empirically discriminable difference! Since an open region and its closure do not differ in measure, an open sphere would not "expand" in any measurable way if a layer of simples were spread over it so as to give it a closed surface. The dimensions of all four spheres are exactly the same, and therefore no comparison of their sizes could reveal which spheres are open, which closed, or which pairs touch in the one manner, which in the other. Whether or not a surface has a final "skin" of simples cannot be simply a matter for stipulation if objects are really made out of simples. Therefore the partially-open and mixed-bag metaphysicians must recognize real differences here, but of a highly occult and seemingly undetectable sort. Although there is nothing inherently wrong with positing necessarily undetectable differences for a priori reasons, surely a metaphysical theory is simpler-and therefore preferable-to the extent that it introduces fewer such differences. On this score, the "Whiteheadian" and "Brentanian" metaphysics to be set forth below (cf. sections 6 and 9) both come out far ahead. Where the threefold analysis of contact must recognize three distinct but indiscernible modes of contact, these metaphysics posit only one; where the partially-open and mixed-bag metaphysics have to recognize unmeasurable differences between objects of the same size, the Whiteheadian and Brentanian metaphysics see no distinctions. As long as I succeed in showing that the Brentanian and Whiteheadian metaphysics are as simple and unobjectionable as I have claimed, considerations of simplicity add substantially to the evidence against the trebly-disjunctive account of contact. They show that, even if partially-open and

14 DEAN W.ZIMMERMAN

mixed-bag metaphysics did not face all of the problems adduced earlier in this section, there would still be a good reason to look elsewhere for the best metaphysics of extended objects.

6. Problems of contact among open objects In defense of premise (3), I shall argue that an open-objects metaphysics cannot fit into a larger metaphysics of extended objects which satisfies assumptions (A)-(D) as long as extended objects are taken to be wholly composed of simples. If two objects are open and composed entirely of simples, then they must occupy open regions; and two discrete open regions are necessarily separated by at least a point. The only real candidate for the contact-relation in an openobjects metaphysics is, therefore, obvious enough: if all objects are open, then two non-overlapping extended objects have attained an absolute limit of proximity (short of three-dimensional interpenetration) just in case they are separated by no more than an empty zero-, one-, or two-dimensional region. An open-objects account of contact must, then, look something like this: (D5) x is in contact with y =df there is a line which (a) contains no two-dimensional segments falling entirely outside of regions occupied by either x or y, and (b) which contains no point in a region occupied by both x and y Combining an open-objects metaphysics with the thesis that extended objects are composed of simples leads to numerous difficulties. For one thing, such a combination entails that the contact relation is instantiated in two ways, one of which puts the relata closer together than the other. Even when two open objects are in contact, they remain "separated" from one another by an empty point, line, or plane; but a simple part inside a solid object is in even closer proximity to the extended parts of the object which surround it. There are no empty regions, however small, between each interior simple and the rest of the object in which it is embedded. If there were, the object would not be an extended body after all, but a mere "cloud" of separated, point-sized atoms. But then doesn't this account of contact violate assumption (B)? Some pairs of objects which satisfy (D5) (e.g., two open spheres separated by a point) are not as close together as others (e.g., the simple at the center of one of the spheres and the sum of the sphere's remaining parts). So (D5) cannot, it would seem, be an analysis of "contact" after all. But perhaps the appeal to (B) is out of place in the present context. Contact between extended objects must certainly be an absolute limit of closeness unsurpassable by any other pairs of extended objects; but if extended objects

really had unextended parts, then the species of contact relating unextended parts to their extended wholes would no doubt be of a wholly different sort.23 Nonetheless, the suggestion that all extended objects are open but also composed of simple parts remains utterly incredible. Consider a solid open sphere which, on the present view, consists of a collection of simple parts filling every plane and line running through its interior. Don't the simples filling the plane that separates the sphere's left and right halves form an object-a very thin film cutting the sphere in half? But then why is it impossible for the left hemisphere and this thin film (which are, after all, right up against one another) to form a partially closed object? The two-dimensional layer of simples is there inside the sphere, but-since all extended objects are supposed to be open-the (open) left hemisphere and the inner film do not now constitute a partially closed part of the sphere. Anyone sympathetic to the doctrine of arbitrary undetached parts will find this denial hard to swallow. Take the arbitrariness away and the result becomes even more implausible. Suppose all the simples in the (open) right hemisphere lack some property which the rest of the sphere's simples have. How could one deny that the sphere has a natural division into two parts, the right hemisphere and the partially closed remainder? Recognizing partially closed objects puts a much greater strain on the open-objects analysis of the contact relation. Now there are two species of contact even among pairs of extended objects. The left (partially closed) and right (wholly open) hemispheres just described must count as in contactthey satisfy (D5), and no extended objects could get closer. But two open hemispheres separated by an empty plane qualify as in contact by (D5), too. Suarez, as we shall see, was forced to recognize such distinct modes of contact in his closed-objects m e t a p h y s i c ~ But . ~ ~ now we surely have a situation in which (B) is violated: the two open objects are supposed to be in contact, but there are non-overlapping extended objects with less of a gap between them. An even more difficult question is: what happens to extended objects when they are broken up? For example, what happens to the disc-shaped film of simples inside the sphere when the hemispheres are separated? Could the film of simples be supposed to adhere to either (open) hemisphere? Surely not, since a free-standing hemisphere with a skin of simples on its circular face would certainly constitute an extended object with a partially closed surface. But then where do these inner simples go? Do they wink out of existence or float off by themselves in a cloud? Are they "absorbed" into one of the hemi23

24

Suarez was somewhat disconcerted by this result. Cf., his Disputationes Metaphysicae, disp. 40, sect. 5, $96 and 67. Cf. also Aristotle, Physics, bk. 6, ch. I, for Aristotle's related argument about the impossibility of contact between "indivisibles". Cf. Suarez, disp. 40, sect. 5, $ 9 5 8 4 6 .

16 DEAN W. ZIMMERMAN

spheres? And how do the thin films of simples come into the picture when two (open) extended objects are brought together to form a larger extended object? Or is breaking up an irreversible process? Clearly, anyone brave enough to hold this view faces a barrage of difficult questions, each of which admits of any number of equally good-because equally silly-answers. Although no one has ever actually advocated the sort of open-objects metaphysics I am criticizing here, we shall see that Suarez's closed-objects metaphysics is confronted by many analogous questions. Examination of his views will illustrate in detail how hard it is to answer them sensibly. Just as the sense data theory was dealt a blow from which it has never recovered by simple questions like, "What color is the back side of a red sense datum?', "Where is this red sense datum located?", and, "When did that red sense datum pass away and this pink one take its place?"; so the outlandish nature of all the possible answers to the above questions about the simple parts of extended objects suggests that there is something wrong with a theory that requires that they have answers. In summary, it is clear that an open-objects metaphysics, when conjoined with the thesis that extended objects are wholly composed of simples, faces several serious objections. (i) It goes against the spirit of the doctrine of arbitrary undetached parts, (ii) it posits two modes of contact between extended objects, the one of which is a more immediate sort of contact than the other, (iii) it faces numerous puzzling questions which seem to have no plausible answers but which, if the theory is true, must have answers. The cumulative force of these considerations is enough, I believe, to substantiate premise (3) of the atomless gunk argument. There are, however, open-objects metaphysics which do not suffer from these three defects. Consider, for instance, the thesis that, necessarily, every extended object is open and made entirely of atomless gunk-there can be, in other words, no point-sized parts and no closed or even partially open parts anywhere in an extended object. This view may be justly described as "Whiteheadian". Incorporating (D5) within Whiteheadianism yields a contactrelation that satisfies all three of my constraints: (B) no distinct non-overlapping extended objects could be closer together than objects so related (since Whiteheadianism regards closed and partially open objects as impossible, no two discrete extended objects could possibly have parts filling a less-thanthree-dimensional region between them), (C) two objects of any shape can stand in this relation, and (D) the relation must hold between the left and right halves of every extended thing (i.e., the interior parts of things are all open objects, too). Whiteheadianism is pretty clearly the only feasible open-objects metaphysics. Adding simple parts to the interior of an open object made of atomless gunk would merely reintroduce most of the problems discussed in this

COULD EXTENDED OBJECTS BE MADE OUT OF SIMPLE PARTS?17

section. Furthermore, if extended objects can be in contact without exterior skins of simples, as an open-objects metaphysics must allow, then surely simples are not needed in order for the interior parts of an object to be in contact. Interior simples on top of atomless gunk in an open-objects metaphysics do nothing but cause trouble: they have no theoretical role to play, and they confront us with absurd and unanswerable questions. Not too surprisingly, Whiteheadianism is the only form of open-objects metaphysics anyone has ever espoused. It may, in its heyday, have come closer to the status of philosophical orthodoxy than has any other metaphysics of extended objects. Its advocates include Ockham, Descartes, Tarski, is an extremely important "fellow traveller", due to and Q u i n t ~ nWhitehead .~~ his famous "method of extensive abstracti~n"~~-a"method" that was immediately adopted by Russell, Nicod, and Broad.27 Whitehead's method identifies points, lines, and planes with infinitely converging "abstractive sets" of nested extended regions, so that space may be regarded as devoid of less-than-three-dimensional regions. Recently, highly sophisticated "regionbased topologies" have been developed which avoid certain problems found in earlier attempts to carry out a Whiteheadian construction of points.28If, as seems likely, the program of constructing spatial points from three-dimensional regions can be carried out satisfactorily, the availability of the White-

25

.f"'

27

2R

Cf. William of Ockham, Quodlibetal Questions, first quodlibet, question 9, reply to arguments 1 and 2; Descartes, Principles of Philosophy, part 2, $20; Tarski, "Foundations of the Geometry of Solids" in his Logic, Semantics, Metamathematics, trans. J. H . Woodger (Oxford: Clarendon Press, 1956), 24-29; and Anthony Quinton, "Matter and Space". Whitehead, The Orgunisation of Thought, Educational and Scientific (London: Williams and Norgate, 1917). ch. 7. I cannot tell whether Whitehead's own peculiar event-ontology has a use for the concept of extended substance which we are exploring. But be that as it may, Whitehead's treatment of space obviously does yield the result that, if there were extended objects, then they would have no simple parts--there being, on Whitehead's view, no simple regions in which to put them. Cf. Russell, The Analysis of Matter (London: George Allen and Unwin, 1927), ch. 28; Nicod, Geometry and Induction (Berkeley: University of California Press, 1970), 2132; and Broad, Scientific Thought (Paterson, New Jersey: Littlefield, Adarns and Co., 1959; first published, 1923). ch. 1. Bowman L. Clarke offers an elegant Whiteheadian constmction based on the fundamental concept "connected" ("Individuals and Points", Notre Dame Journal cg Formal Logic 26 (1985): 61-75); but Peter Simons has shown that Clarke's theory has a very unintuitive consequence: namely, that there are regions with all the same proper subregions but which are nonetheless distinct (Simons, Parts: A Study in Ontology (Oxford: Clarendon Press, 1987). 98). More recent constructions which do not, as far as I can see, succumb to Simons's objection may be found in Peter Forrest, "From Ontology to Topology in the Theory of Regions", and Peter Roeper, "Topology as a Theory of Regions", both in Monist 79 (January 1996). Cf. also G. Gerla and R. Volpe, "Geometry Without Points", American Mathematical Monthly 29 (1985): 707-11; and G, Gerla, "Pointless Metric Spaces", Journal of Symbolic Logic 55 (1990): 207-19.

headian approach suggests that one may deny that objects have simple parts without thereby jeopardizing any physical or metrical facts.2y I have not the space to mount a full defense of Whiteheadianism here, but shall assume that it can withstand critical scrutiny at least as well as the Brentanian closed-objects metaphysics described below. Elsewhere, I have responded to some of the more obvious objection^.^"

7. Are all extended objects closed and made of simples? At this point, only premise (4) remains undefended: it remains to be shown that, if all extended objects were closed, then they could not be entirely composed of simple parts. As we saw in the discussion of Bolzano's metaphysics, two closed objects could be in contact only if simple parts in their boundaries were in the very same place. Could such a relation really constitute touching? Is it not rather interpenetration (if the simple parts remain distinct though coincident) or the sharing of parts (if the simples do not remain distinct)? Many philosophers have held that objects in contact have coincident external simple parts.31Two "skins" may be in the same place, they say, but this does not constitute interpenetration. Real interpenetration only occurs if three-dimensional parts coincide. In this section, I raise two problems which suggest that contact by boundary overlap is inconsistent with the supposition of complete decomposability into simple parts. First, the penetrability of the boundaries of closed objects sits uneasily with the impenetrability of their interiors. Second, if extended objects were really just sums of simples filling closed regions, then any closed region full of material simples should constitute an extended object. But this could not be the case. Whatever one might think about the permeability of zero-, one-, and twodimensional boundaries, it is clear that no extended material body could be so 2y

3" 31

It is worth noting that some feel there are aspects of contemporary physics which support the view that space or space-time contains no point-sized parts: John Earman has suggested that, in order to make General Relativity compatible with the possibility of determinism, space-time points should be treated as constructed entities; and Peter Forrest has offered an argument for the same conclusion based on the probabilistic nature of quantum physics. Cf. Earman, World Enough and Space-Time (Cambridge, Massachusetts: MIT, 1989), ch. 9; and Forrest, "From Ontology to Topology in the Theory of Regions". Zimmerman, "Indivisible Parts and Extended Objects: Some Philosophical Episodes from Topology's Prehistory", 31V. Cf. Suarez, disp. 40, sect. 5; W. K. Clifford, The Common Sense of the Exact Sciences ed. Karl Pearson, newly ed. James R. Newman (New York: Dover Publications, 1955; first published, 1885), 43-47; Leibniz, Philosophical Pctpers ctnd Letters, 2nd ed., trans. and ed. by Leroy E. Loemker (Dordrecht: D. Reidel Publishing Company, 1969). 141; Franz Brentano, "On what is continuous", Space, Time and the Continuum, 1-38; David Sanford, "Volume and Solidity", Australasian Journal of Philosophy 45 (1967): 329-40; and R. M. Chisholm, "Boundaries", ch. 8 of his On Metaphysics (Minneapolis: University of Minnesota Press, 1989), 83-89. COULD EXTENDED OBJECTS BE MADE OUT OF SIMPLE PARTS? 19

intangible that any space-occupying thing of any sort whatsoever could pass through it. The "logical impenetrability" of material objects prohibits at least this much; to be so completely penetrable is to no longer be an extended material thing. But if all the simples on the exterior of an object can overlap with other simples, why is it not possible for just any simple to overlap with any other? Take two closed cubes, for example, which move continuously toward one another until they touch by having faces which occupy the same plane. What prevents their motion from continuing, so that the cubes pass right through one another? Since each cube is the mereological sum of a set of two-dimensional objects lying in planes parallel to the coincident surfaces, and since these internal two-dimensional objects differ from the coincident surfaces only in being inside the cubes, why can they not coincide with two-dimensional parts of other objects as well? If extended objects are made up out of simples, and distinct simples can be in the same place at the same time, then does it not follow that any two distinct extended objects could be in the same place at the same time? It will not do to suppose that there is something different about the simples inside an extended object which make them unfit to coincide with other simples. No property they possess could render them unfit for coincidence since, according to the sort of closed-objects metaphysics being considered in this section, they already are coincident with other simples. If all extended objects are closed-including those that constitute the proper parts of extended objects-and all contact occurs in virtue of overlapping simples, then it follows that all the boundaries inside an extended object contain overlapping simples. The plane in the middle of a cube, for instance, contains simples forming the boundaries of both the left and right halves of the cube. The two halves are closed objects which touch along that plane, and the two sets of coincident simples would "come apart" if the cube were split. In fact, it is easy enough to see that every point in the cube or on its surface is occupied by at least a denumerable infrnity of simple parts! Even the outermost point of one of the cube's corners is converged upon by an infinite number of discrete pyramid-shaped parts of the cube, each with its own distinct simple part coinciding with those of all the others at the cube's "tip". Adding even a denumerably infinite number of simples to a spot that already contains as many does not increase the number of points at that location. So it cannot be that one cube keeps the other from passing through it because its simples are already packed together too densely to admit the other cube's simples into its space. The two cubes could be superimposed point for point without increasing the number of simples within their boundaries! Once it is admitted that the simple parts of things can overlap, the closedobjects metaphysician who thinks extended objects are made of simples will have difficulty maintaining that for every extended object, there are similar extended objects with which it cannot possibly coincide. It seems plausible to suppose that, if it is impossible for physical objects of some kind to stand in a certain intrinsic relationship, then a situation exactly like the impossible 20 DEAN W. ZIMMERMAN

one in every intrinsic respect must also be impossible. The superposition of two precisely similar objects of the same impenetrability-kind is impossible. But the sum of the two superimposed objects would not seem to differ in any intrinsic respect from one of the objects standing alone. Each is an array of simples filling an extended region in exactly the same way with the same number of simple parts having the same intrinsic properties. So the superposition cannot be impossible after all. One might try to account for the impossibility by ascribing repelling powers to simples in the interior of a closed object.32It would be a mistake to postulate that the interior simples necessarily keep all other simples away, since they already overlap with other simples. To stipulate instead that they necessarily repel every simple that belongs to another extended object would force one to say that each extended object is made out of a physically distinct kind of stuff. So the postulated powers must be supposed to repel only other surface simples. The positing of essential repulsive powers associated with objects of a certain shape (in this case, with open interiors) is the sort of a priori closure of a seemingly contingent matter to which I objected above. But even if the proposal were not objectionable for this reason, it would still leave it quite mysterious why two extended objects could not come into existence in the same location, or jump discontinuously into the same spot. These forms of interpenetration are just as impossible, but would involve no changes in the numbers or kinds of simples located in the boundaries and interiors of the objects involved. At the very least, these problems of impenetrability show that it would be passing strange if all the ultimate parts of an extended substance could share their locations with as many parts of the same kind as you like, even though the substance itself could not share its location with another substance exactly like it. But everything about this view is strange. Consider, for instance, the odd manner in which the simple parts of an extended thing are supposed to fill the region it occupies. It would be most natural to construe an extended object made entirely of simples as an aggregate of simple parts arranged so that there is one simple at each spatial point in the region the object fills. If this were the case, God could create an extended object simply by filling a space with one simple for every point in the region. But once the assumption is made that all extended objects are closed, this simple picture of how extended objects are constituted by aggregates of point-sized parts must be rejected. The metaphysics that emerges in its place is baroque and incredible. According to the new picture, once God has created one simple at every point, His work is barely begun! He must create another simple at each point in the region, then another, and another .... Only after creating an infinite 32

The suggestion was made by Richard Miller. COULD EXTENDED OBJECTS BE MADE OUT OF SIMPLE PARTS?21

number of simples for every point is the work of filling the region complete. No wonder God rested on the seventh day! Surely, if an extended whole were nothing over and above a sum of unextended parts filling an extended region of space, there would be no need for God to keep adding parts once every point in the region had a simple in it. To bring the problem into focus, consider a set A containing continuum-many simple parts of the sort that are supposed to be able to compose extended objects. It seems clear that if an extended object were simply a sum of pointsized, unextended simple parts appropriately arranged so as to fill an extended region, then this set of simples would necessarily compose an extended object if every point in an extended region R were occupied by exactly one of its members. But this thesis leads to contradiction when combined with the supposition that all objects are closed. By assumption (D) above, each of the undetached extended parts of an object-such as its left or right half-is in contact with the rest of the object. Furthermore, if all objects are closed, then all contact involves overlap of simples. Thus one who holds a closed-objects metaphysics must admit that every extended object has a left and right half which touch one another by having boundaries that are in the same place. In particular, then, if there were an extended object composed of just the elements in A, then its left and right halves would have to be in contact. They cannot be, however, since A has exactly one simple for every point in R. Let us call this the "overlapping-simples argument", and make its structure a bit more explicit. 1.

If extended objects are identical with mereological sums of simple parts, then there must be an extended object composed of A.

2.

There is an extended object composed of A. (Assumption for conditional proof)

3.

For any extended object x , there is a left and right half of x with coincident simple parts. (A theorem of the closed-objects metaphysics of extension and contact, given (A), the doctrine of arbitrary undetached parts)

4.

The extended object composed of A has a left and right half with coincident simple parts. (From 3 by universal instantiation)

5.

So if there were an extended object composed of A, then A would have coincident simple parts. (Conditional proof from 2)

6 . But A does not have coincident simple parts. (By the description of A) 7 . So there is no extended object composed of A. (5 and 6 by modus tollens)

22 DEAN W. ZIMMERMAN

Therefore, extended objects are not identical with mereological sums of simple parts. (1 and 7 by modus tollens) Thus, on the assumption of a closed-objects metaphysics in which contact is a matter of coincident boundaries, extended objects cannot be entirely composed of simples.

8. Objections to the overlapping-simples argument from Suarez The success of the overlapping-simples argument depends entirely upon the plausibility of premises 1 and 3. Premise 3 is justified by the fact that, if all extended objects were closed, then this would go for all the undetached internal parts of extended objects as well-and so they, too, could only touch one another by having coincident simples in their boundaries. There is, so far as I can see, only one way for a closed-objects metaphysician to deny the latter assumption. She may (i) agree that, for instance, the left and right halves of a sphere are closed and in contact by virtue of their each having simple parts in the same plane, but (ii) deny that there need always be two surfaces in this plane. The internal faces of the two hemispheres must both be terminated within the sphere by a final skin of simple parts, since all extended things are closed; but this skin may in some cases be a two-dimensional object shared by both hemispheres. A theory along these lines was advocated by Suarez. On his view, all extended objects are closed, but there are two,distinct modes of touching: some extended objects touch by being continuous with one another-i.e., by sharing a zero-, one-, or two-dimensional boundary part in common,-and some touch by being contiguous with one another-i.e., by having distinct but coincident boundary parts.33If Suarez's closed-objects metaphysics were correct, then premise 3 of the overlapping-simples argument could be false. The premise depends upon the assumption that in a closed-objects metaphysics all contact must involve overlapping simples. But extended parts inside an object, if they are merely continuous with one another, touch by sharing a single terminating surface, edge, or simple in common (§§21-25). Thus, in Suarez's account, there is no need to posit an infinite number of simples at every point in the region occupied by an extended object. Interestingly, Suarez agrees that extended objects are not wholly composed of simples. Since he also believes that contact requires simple parts, he concludes that, although extended objects have simple parts on their surfaces and at every point inside their boundaries, they nonetheless are not composed of such parts alone. Suarez holds that every extended body is made out of two 33

Suarez, disp. 40, sect. 5, 521 (parenthetical references pertain to the paragraphs of this section).

sorts of constituents, each playing a different role: (1) there are the two-, one-, and zero-dimensional elements which are responsible both for the contact among all of a body's (infinitely numerous) internal extended parts and also for its aptitude for coming into contact with other bodies through the coincidence of their external boundaries; and (2) there are also threedimensionally extended parts, which are responsible for a body's ability to fill a region of space (8835-36). Since the latter parts could not be made out of the former, every extended object is, on Suarez's view, composed in part of atomless gunk. For my purposes, the fact that Suarez himself believed in atomless gunk is of little consequence. After Cantor, Suarez's Zeno-inspired Aristotelian reasons for rejecting the view that extended bodies are made up out of simples appear to be groundless. The important question is whether his closed-objects metaphysics is intrinsically incompatible with the supposition that extended objects are made entirely of simples. If it is, then the only closed-objects metaphysics which can escape the overlapping-simples argument is tenable just in case all extended objects contain some atomless gunk. Conjoining Suarez's disjunctive analysis of contact with the thesis that extended objects are closed and made out of simple parts leads to intolerable and absurd consequences. The conjunction of these views would imply both: that every breakage produces enough new entities, created out of nothing, to fill the universe; and that there are pairs of extended objects which cannot touch by means of either continuity or contiguity. And there is a third, more general objection to Suarez's view: he must either deny that certain seemingly reversible natural processes are in fact reversible, or else posit deeply mysterious changes which, when they occur, do so for no reason at all and could not possibly be detected. Although Suarez's exploration of the metaphysics of extension and contact is perhaps the most thorough and subtle that has ever been carried out, there are many problems with the view he finally advocates. For one thing, all of the difficulties for open-objects metaphysics discussed above have analogues here. In particular, there is the problem of where the common terminus goes when a continuous object is broken in half. Since all objects are closed, the breakage produces two halves each of which must have its own terminating surface. But, before the break, both parts were terminated by one and the same surface. So who gets custody? Since Suarez can think of no principled reason to give the original internal surface to one half rather than the other, he supposes that it is destroyed and that two new terminating surfaces are created ($56). Clearly, a new terminus must materialize somehow on at least one of the two halves. For all his ingenuity, Suarez has a hard time making this result palatable even to himself. Since it is "not philosophical in this situation to posit cre-

24

DEAN W. ZIMMERMAN

ation or annihilation" of simples and surfaces, he suggests that the new simples arise from out of the extended parts through some sort of "resultancy"; or, alternatively, that their appearance is due to the activity of God by which He conserves matter and, since their production would then be dependent upon a pre-existing action (God's), it is a kind of "quasi-creation (con-creatio), not to be called creation ..." (4456-57). Just why the production of simples by means of "resultancy" or "quasi-creation" is not creation, simpliciter, is not successfully made out; and Suarez is clearly dissatisfied with both his solutions. But the situation is much worse for someone who thinks extended objects are made of simples. Even though the set of simples created when an extended object fractures are arranged so as to form a very thin layer, there are so many of them that they cannot be regarded as insignificant. Given the fact that there are as many points on a line or in a plane as there are in any threedimensional region, it follows that every breakage produces enough simples out of nothing to fiI1 the entire universe! If extended objects were just sums of simples filling space, we could provide ourselves with the materials to make an object of any size we like simply by breaking an extended object, "harvesting" the simples that pop into existence, and then distributing them appropriately. Secondly, the conjunction of Suarez's theory and the thesis that all objects are made of simples seems to imply that continuous bodies have extended parts which cannot possibly touch other extended parts, a result incompatible with assumption (C).A continuous sphere has closed parts imbedded within it. But shouldn't the set of simples filling just the (open) interior of a (closed) internal part itself constitute an open extended part of the sphere? If extended objects were just sums of simples filling extended regions, why wouldn't this sum of simples filling an extended (albeit open) region constitute an object? If the rest of the object were very carefully peeled away, that set of simples would continue to fill its open region, but with a brand new skin of simples somehow forming around it. How could one deny that, in this case, there is a part of the object which was first imbedded in the sphere and then disclosed wearing a new skin? Or suppose that the outer skin of an object is green, but its interior entirely red. Would not such an object consist of a green and a red part-a closed green skin, and an open red interior? Of course if all objects are closed, the answer must be "No". But there would seem to be no justification for refusing to confer the honorific title "object" upon a collection of simples merely because the region they fill is an open one. Once this is admitted, however, the Suarezian account of contact is in jeopardy. Suppose that an open region inside a sphere is filled with the sort of simples that make red objects, while the simples constituting the closed part of the sphere which surrounds the region are of a different sort. The red part of the sphere is an open object "extrinsically terminated" (as

COULD EXTENDED OBJECTS BE MADE OUT OF SIMPLE PARTS?

25

Suarez would say) by the surface of the non-red surrounding part (cf. @5866). This state of affairs runs afoul of assumptions (B) and (C), according to which contact is an absolute limit which objects of any two shapes can attain. Closed objects in contact fit together more intimately than the open red part and the closed surrounding part of the sphere ever can, so the the latter pair cannot really be touching. Even on Suarez's own disjunctive analysis of contact, the red and non-red parts of the sphere do not touch. They are neither continuous nor contiguous with one another-since they share no simple parts in common, and have no coincident simple parts, either. The open inner part is suspended just beyond the boundary of the enclosing object, and no amount of squeezing could bring the two into contact without altering the shape of at least one of them. Notice that, if Suarez is right and every extended object is made of a threedimensional portion of atomless gunk with the simples merely thrown in for good measure, then neither of the above problems arises. The infinity of simples which results ex nthilo from a fracture would not be enough by itself to make an extended object of any size-not without the addition of a threedimensional portion of atomless gunk, at any rate. Nor would an open set of simples like those in the red interior part of the sphere ever by themselves constitute an extended thing: they are not the real fillers of that subregion; the sphere must have a part made of atomless gunk to do that. So Suarez's peculiar closed-objects theory of contact may be plausible if it is conjoined with the thesis that extended objects are made, in part, of atomless gunk. Finally, with or without atomless gunk, Suarez's metaphysics has highly paradoxical implications. Should not the processes of breaking something in half and then putting the two halves back together at least sometimes be simple mirror images of one another? According to Suarez, two continuous halves, when separated, become two closed objects. The two pieces can thereafter only be contiguous with one another-unless we are to suppose that there is some mechanism whereby one of the newly created skins may be destroyed, absorbed, or scraped off, allowing the contiguous halves to become continuous once more. Can Suarez legitimately introduce a process of division-in-reverse which produces continuous objects out of contiguous ones, or must he admit that, on his view, division is an irreversible process? To accept the irreversibility of breakage would be to adopt a kind of "HumptyDumpty theory": once a continuous object falls and "cracks", the broken parts can never be put back together again to form a continuous whole. To avoid the Humpty-Dumpty theory, Suarez must suppose that contiguous bodies sometimes become continuous. However, since the annihilation and creation of simples which is said to accompany breakage is posited for purely a priori, metaphysical reasons, it would be inappropriate for Suarez to imagine that there is some empirical mark which accompanies the reversal of

26 DEAN W. ZUIMERMAN

this process. After all, the passage from boundary-sharing to the possession of distinct boundaries is one which must occur no matter what goings-on could possibly be observed at the split; so the transition back to boundarysharing must also be consistent with any empirical findings. Thus the difference between continuous bodies and contiguous bodies becomes in principle indiscernible. In order to retain the reversibility of breakage, Suarez must say that whenever two objects are brought into contact they are either continuous or contiguous, but we cannot possibly tell which. But then any object, no matter how it has been put together or taken apart, may easily be supposed contiguous all the way through. By Suarez's own lights, therefore, it is quite possible that all contact is a matter of coincident simples, and that there are no merely continuous objects. So why suppose that there really are two distinct species of contact? Suarez's distinction between continuity and contiguity is both empirically ungrounded and (more importantly) metaphysically disastrous. It does no work and leads to the strange consequencesjust described: infinities of simples created at every turn, and parts that cannot get closer together but yet are not quite touching. Surely Suarez is trying to sell us a difference without a difference here, and one which we do best to eschew. There are, then, good reasons to think both that: (a) Suarez's closed-objects metaphysics leads to absurdities unless one follows him in denying that extended objects are made up entirely of simple parts; and, in any case, (b) the sort of closed-objects metaphysics that is vulnerable to the overlapping-simples argument is preferable to Suarez's theory.

9. The Brentanian closed-objects metaphysics We have seen in the last two sections that the only theories of contact available to the friends of closed objects result in metaphysics of extension according to which extended objects must contain atomless gunk. Thus premise (4) of the atomless gunk argument is justified. Of course my criticisms do not impugn every sort of closed-objects metaphysics. Consider Franz Brentano's views: every extended body, including every extended proper part of a body, is "closed" by a skin of simples. All contact, whether among the internal proper parts of a continuous object or between distinct objects, is a matter of coincident but distinct simples. Extended objects, although they contain simple parts as boundaries, are not composed entirely of simples. Rather, the simples filling an object's zero-, one-, and two-dimensional (internal and external) boundaries are dependent entities, things that could not exist except as boundaries of the truly extended parts of the 34

Cf. Brentano, Space, Time, and the Continuum, 10-12, 146-48, and xvi-xvii (the editors' introduction); and Brentano, The Theory of Categories, trans. Roderick M . COULD EXTENDED OBJECTS BE MADE OUT OF SIh4PLE PARTS?

27

Like Suarez, then, Brentano believes that extended objects are wholes composed of two radically different kinds of parts: (1) extended parts, which, though infinitely divisible into extended parts within extended parts, ad injinitum, cannot be "infinitely divided" (even "in thought") into a set of simple parts; and (2) non-extended (zero-, one-, and two-dimensional) parts which are necessarily present at or along the inner and outer boundaries of every extended part. Recognition of atomless gunk frees Brentano's metaphysics from problems of impenetrability and the paradoxes emphasized in the overlapping-simples argument. Here's how: First off, admitting both atomless gunk and simples successfully lays to rest the worry that an extended object composed only of simples could not be naturally jealous of its space. If extended objects have extended parts that are fundamentally different in kind from their boundaries, not being themselves made up of penetrable simples, then the air of mystery surrounding their impenetrability vanishes. A pair of superimposed cubes would naturally differ intrinsically from a cube standing alone. The space filled by the pair of cubes contains two cube-sized pieces of atomless gunk; the space filled by a single cube does not. Furthermore, the overlapping-simples argument is obviously no threat, since the Brentanian accepts its conclusion. He can likewise dismiss the absurd picture of God having to fill a region over and over again with an infinite number of simples for every point in order to create an extended object. Every point in the region occupied by an object must contain an infinite number of simples not because they are needed to "really" fill the space, but because they are not the real fillers of the space at all. Boundaries, unlike the truly extended parts made of atomless gunk, are dependent entities"parasites" that are automatically present wherever one finds extended "hosts". God fills the cube by creating one extended part for every spatially extended region within the cube, and the simples just come along for the ride. Similarly, there is no need to imagine extra simples coming into existence when an extended object is broken up; each of the resulting pieces already had its own boundary of simples before the breakup, coinciding comfortably with the pre-existing simples in the boundaries of the other parts. The simples always accompany their hosts; it is not possible or necessary for them to disappear or pop into existence when their hosts are brought into contact or separated. Given the general objections to Suarez's views set forth at the end of the previous section, and the results of the overlapping-simples argument, it seems safe to conclude that the only viable closed-objects metaphysics is Brentano's.

Chisholm and Norbert Guterman (The Hague: Martinus Nijhoff, 19811, 55, 128, and 157-58. Cf. also Chisholm, On Metaphysics, ch. 8.

28 DEAN W.ZIMMERMAN

10. Conclusion Every premise of the atomless gunk argument has now been defended. Each of the four possible kinds of metaphysics of extended objects has been examined, with the following results: mixed-bag and partially-open metaphysics are simply unacceptable; so either an open-objects or closed-objects theory must be correct; and in either case, extended objects could not be decomposable without remainder into simples. Thus extended objects must be madeat least in part-of atomless gunk.35

35

I am deeply indebted to Roderick Chisholm and Fred Freddoso for help over the years this paper has been incubating. Chisholm introduced me to the problems about boundaries explored here, and his patient and careful criticisms saved me from some serious blunders early on; and Freddoso spent countless hours explaining the important but little-known metaphysics of extended objects developed by Francisco Suarez. Talks based on the paper were given at Brown University, the University of Notre Dame, and the 1992 Central Division meetings of the APA. I am particularly grateful to the following people for helpful comments and criticisms, or for supplying unpublished materials: Donald L. M. Baxter, Robin Collins, Michael Detlefsen, Peter Forrest, Jaegwon Kim, Michael Kremer, Larry Lombard, Alasdair MacIntyre, Richard W. Miller, Hugh McCann, Phil Quinn, Peter Roeper, Barry Smith, Richard Swinburne, Jim Van Cleve, Eric Watkins, and an anonymous referee for this journal. Although I learned a great deal from all these people, I doubt whether I have done complete justice to some of their criticisms. I hope that the paper has been considerably improved by my trying. COULD EXTENDED OBJECTS BE MADE OUT OF SIMPLE PARTS?29

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Matter and Space Anthony Quinton Mind, New Series, Vol. 73, No. 291. (Jul., 1964), pp. 332-352. Stable URL: http://links.jstor.org/sici?sici=0026-4423%28196407%292%3A73%3A291%3C332%3AMAS%3E2.0.CO%3B2-U 6

Matter and Space Anthony Quinton Mind, New Series, Vol. 73, No. 291. (Jul., 1964), pp. 332-352. Stable URL: http://links.jstor.org/sici?sici=0026-4423%28196407%292%3A73%3A291%3C332%3AMAS%3E2.0.CO%3B2-U 6

On Being in the Same Place at the Same Time David Wiggins The Philosophical Review, Vol. 77, No. 1. (Jan., 1968), pp. 90-95. Stable URL: http://links.jstor.org/sici?sici=0031-8108%28196801%2977%3A1%3C90%3AOBITSP%3E2.0.CO%3B2-R 6

In Defence of Locke's Principle: A Reply to Peter M. Simons Frederick Doepke Mind, New Series, Vol. 95, No. 378. (Apr., 1986), pp. 238-241. Stable URL: http://links.jstor.org/sici?sici=0026-4423%28198604%292%3A95%3A378%3C238%3AIDOLPA%3E2.0.CO%3B2-1

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Locke, Leibniz, and Wiggins on Being in the Same Place at the Same Time David H. Sanford The Philosophical Review, Vol. 79, No. 1. (Jan., 1970), pp. 75-82. Stable URL: http://links.jstor.org/sici?sici=0031-8108%28197001%2979%3A1%3C75%3ALLAWOB%3E2.0.CO%3B2-8 12

Theories of Masses and Problems of Constitution Dean W. Zimmerman The Philosophical Review, Vol. 104, No. 1. (Jan., 1995), pp. 53-110. Stable URL: http://links.jstor.org/sici?sici=0031-8108%28199501%29104%3A1%3C53%3ATOMAPO%3E2.0.CO%3B2-F 25

Matter and Space Anthony Quinton Mind, New Series, Vol. 73, No. 291. (Jul., 1964), pp. 332-352. Stable URL: http://links.jstor.org/sici?sici=0026-4423%28196407%292%3A73%3A291%3C332%3AMAS%3E2.0.CO%3B2-U 28

Geometry Without Points G. Gerla; R. Volpe The American Mathematical Monthly, Vol. 92, No. 10. (Dec., 1985), pp. 707-711. Stable URL: http://links.jstor.org/sici?sici=0002-9890%28198512%2992%3A10%3C707%3AGWP%3E2.0.CO%3B2-5 28

Pointless Metric Spaces Giangiacomo Gerla The Journal of Symbolic Logic, Vol. 55, No. 1. (Mar., 1990), pp. 207-219. Stable URL: http://links.jstor.org/sici?sici=0022-4812%28199003%2955%3A1%3C207%3APMS%3E2.0.CO%3B2-5

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