SORITES ISSN 1135-1349
Issue #17 -- October 2006. Pp. 31-48
Sparse Parts
Copyright © by Kristie Miller and SORITES
Sparse Parts
Kristie MillerFoot note 2_1

1 Introduction

It is sometimes said that four dimensionalists are guilty of ontological profligacy. They admit into their ontology crazy objects such as that composed of your cat on Tuesday and my dog on Wednesday. It is thus at least implied that four dimensionalism is incompatible with the view that ontology is sparse, that is, the view that only some arrangements of basic particulars compose composite concrete objects.Foot note 3_1 This incompatibilty, however, has as yet not definitively been shown to be the case. I argue that four dimensionalism in its most common variety, perdurantism, is indeed incompatible with the view that ontology is sparse, but that this need not provide reason to reject four dimensionalism. For this incompatibility is merely a specific instance of a more general problem that a sparse ontology view faces with respect to parthood, namely the problem that we frequently quantify over non-existent parts.

Suppose we agree about the distribution of basic particulars,Foot note 3_2 and we ask what, if anything, is composed of those particulars: that is, we ask which objects exist. One natural response to this question is to hold that only some ways of arranging particulars composes an object, and some other ways do not. Call this the view that composition is restricted.Foot note 3_3 There are various different accounts of just which ways of arranging particulars will result in some object being composed.Foot note 3_4 In general however, these accounts attempt to preserve our intuitions that there is, for instance, no object composed of myself and George Bush, or of a tennis racket and your left foot. The view that composition is restricted therefore results in what we might call a sparse ontology, for it countenances the existence of far fewer objects than, for instance, the view of mereological universalism according to which every way of arranging particulars composes some object.Foot note 3_5 I will thus refer to the view that ontology is sparse in this way, as sparsism, and to proponents of this view as sparsists.

Sparsism, then, is not simply the view that some subset of all the objects that exist are special in some way. It is not the view that there are two distinct kinds of objects, natural objects and gerrymandered objects, and that while gerrymandered objects are mere mereological fusions, there is some special composition relation that holds between and only between particulars and the natural objects they compose. Sparsists are not those who see their project as providing an account of the necessary and sufficient conditions for something to count as a natural object. That view, whatever we call it, is perfectly consistent with both mereological universalism and with four dimensionalism.

Four dimensionalism is a theory of persistence. It holds that persisting objects are temporally extended: they have not only spatial dimensions, but also a temporal dimension. The most common version of four dimensionalism is perdurantism, according to which objects persist by perduring, that is, by being the mereological sum of temporal parts.Foot note 3_6 It is arguably the case that most perdurantists are mereological universalists,Foot note 3_7 and prima facie there seems be a tension between perdurantism and sparsism. Although it seems coherent to hold that only some ways of arranging particulars over time composes persisting objects, and that those objects persist by perduring, this appears to be an odd combination of views. For at least intuitively, temporal parts do not seem to be the sorts of objects that the sparsist is likely to countenance in her ontology.

Why so? Well there is much debate about whether there are any non-arbitrary, informative criteria that determine when composition occurs and when it does not.Foot note 3_8 Let us call whatever these criteria are, the composition criteria. In this paper I will assume that there are composition criteria, that is, I will assume that composition is not a brute relation. Further, I will assume that whatever these criteria are, they will preserve most of our core intuitions about which objects exist and which do not. For surely the central motivating force behind the sparsist position is that we are in general right about what exists and what does not, and about which sorts of things exist and which do not.

Sparsist accounts are founded on the idea that an arrangement of particulars composes some object just if that arrangement exemplifies some property, where this property supervenes on the complex causal relations of the particulars. So for instance the sparsist might hold that particulars compose an object just if they are continuous, or if they form a functional unit, or if they form a unit that is a member of a certain sortal or natural kindFoot note 3_9 or if their collective activity constitutes a life.Foot note 3_10 The idea is that the behaviour of particulars is such that there exists an integrated, functional unit with global properties that supervene on those particulars.

Whatever the composition criteria are, they explain why the borders of objects are where they are, and thus explain how it is that we are able confidently to pick out objects. Indeed, I take it that the sparsist takes as primary datum the fact that objects have discernible, non-arbitrary borders.Foot note 3_11 For presumably the idea is not that we begin with the intuition that there are certain functional or complex causal properties in the world, and then go about marking out the borders of the things that exemplify those properties, finally to exclaim «ah! so that's an object.» Rather, we begin with intuitions about where the borders of objects lie, we develop some general ideas about the features of these borders, and from there we attempt to construct an account that explains why objects lie within, and only within, those borders. So any sparsist account should respect this core idea that objects have what I will call natural borders.

This is not so say, of course, that all such borders are natural in the sense that they are carved out by nature. Some borders, such as those of natural kinds, will be natural in the sense that they are borders recognised by the physical sciences. But if particulars ever compose artefacts, then the borders of these objects are not carved by nature. Still, the borders of such objects are not merely arbitrary: there is some genuine difference between that which lies to one side of the border, and that which lies to the other. Broadly speaking then, we will say that a border is natural to the extent that either it is carved by nature, or it is non-arbitrary. Now of course, it is sometimes the case that the concept of non-arbitrariness is analysed in terms of the notion of being carved out by nature. It might be that for some, what it is for there to exist a non-arbitrary border, is for there to exist a border carved by nature. Clearly this is not the analysis of non-arbitrary that I wish to embrace. Nor is it the analysis that most sparsists would want to adopt, since it would mean that a great number of arrangements of particulars that we take to have natural borders and thus to compose objects, in fact fail to do so. It would turn out that only natural kinds exist. So a more robust sparsism requires that there be some account of a non-arbitrary border that does not make recourse to the idea of being carved by nature. I cannot provide such an account here. Indeed, providing such an account lies at the very heart of the sparsist project, and the difficulty of this task is one of the major stumbling blocks for sparsism. Here I assume that there is some such account, and that at least the general idea of a non-arbitrary border is sufficiently intuitively clear.

The idea that objects have natural borders lies, then, at the very core of sparsism, and thus is a notion that the sparsist should take seriously. It is a notion, however, that seems to be inconsistent with perdurantism. For perdurantists are typically committed to the idea that for any perduring object O and arbitrary temporal interval T during which O exists, there is some temporal part of O that exists during and only during T. Though the temporal extent of any temporal part is held to be an essential property of that part, the temporal borders of temporal parts are purely arbitrary. Temporal parts do not have natural borders, and are thus precisely the sorts of objects that the sparsist refuses to admit into her ontology.

Indeed, one of the reasons some three dimensionalists reject four dimensionalism is because of the apparent arbitrariness of the borders of temporal parts: they do not see how it is that at the moment one object ceases to exist, another comes into existence that is qualitatively identical to the previous object at the moment of its cessation. All they see is a unitary persisting object.Foot note 3_12 And this is precisely because they see only one natural temporal border, not a series of such borders that mark out the borders of the various temporal parts.

In the next section I begin by outlining the difficulties for a perdurantist version of sparsism, and then move on to consider a number of ways these difficulties might be met. I consider Storrs McCall's sparsist perdurantism, and argue that he is faced with a dilemma. If temporal parts are mere abstractions then they cannot do the metaphysical work proposed for them. If they are not abstractions then they appear to have non-natural borders and thus are inconsistent with sparsism. Then I consider a revised perdurantism which is consistent with sparsism, but which faces almost insurmountable metaphysical difficulties. Finally I examine sparsism itself, and conclude that it faces some difficulties of its own which may provide reason to prefer perdurantism.

2 The Problem

Perdurantism is the thesis that objects persist by perduring: by being composed of temporal parts. Roughly speaking, temporal parts as they are widely construed, are objects that exist during and only during a particular temporal instant or interval, and which during that instant or interval wholly overlap the perduring object of which they are a part.Foot note 3_13 So if I perdure, then a temporal part of me is some object that exists during and only during interval T, which during T has my spatial dimensions, and which is part of me simpliciter. More formally, following Ted Sider we will define both an instantaneous and an extended temporal part as follows:

x is an instantaneous temporal part of y at instant t=df 1) x is part of y. 2) x exists at, but only at t. 3) x overlaps every part of y that exists at t.

An extended temporal part of x during T is an object that exists at all and only times in T, is part of x at every time during T and at every moment in T overlaps everything that is part of x at that moment.Foot note 3_14

So consider some persisting object O. Perdurantists hold that for every temporal instant t at which O exists, there is some instantaneous object that exists at that time, which overlaps O at that time and which is part of O. So too for any arbitrary temporal duration T during which O exists, they hold that there is some object that exists only during T, which overlaps O during T and which is part of O. That is, the perdurantist subscribes to what van Inwagen calls the doctrine of arbitrary temporal parts:

DATP: for every persisting object P, if I is the interval of time occupied by P and sub-I is any occupiable sub-interval of I whatever, there exists a persisting object that occupies the interval sub-I and which, for every moment t that falls within sub-I, has at t exactly the same momentary properties that P has.Foot note 3_15

I think that most perdurantists are committed to this doctrine, though it is prima facie plausible that they need not be. They will almost certainly, however, want to be committed to a related doctrine. For the heart of the perdurantist thesis is that objects persist by being composed of parts at times. If an object O persists through interval T, then at each time at which O exists, some part of O must exist at that time. This does not imply that for any interval of time T during which O exists, there is some object that exists only during T and overlaps O during T. But it does imply what we will call the doctrine of instantaneous temporal parts (DITP):

DITP: for every persisting object P, if I is the interval of time occupied by P and t is any occupiable instant of I whatever, there exists an instantaneous object that occupies t and that at t has the same momentary properties that P has at t.

Perdurantists will want to adopt DITP. For suppose the perdurantist held that there exist no instantaneous temporal parts. Rather, persisting objects are composed of extended temporal parts with natural borders. Consider the example of a member of the genus Lepidoptera. On this view such an organism is composed of four extended temporal parts: an egg temporal part, a caterpillar temporal part, a pupa temporal part and a butterfly temporal part. The Lepidoptera thus perdures, but it is composed only of parts with natural borders. There are two obvious worries about this proposal. First, in the case of many persisting objects, there do not appear to be any candidates to be temporal parts with natural borders. Given our definition of temporal part then, these objects then cannot perdure. Second, in many cases the temporal borders of such temporal parts would be excessively vague: consider the example of a person with stages of childhood, adolescence and so forth. While sparsists may embrace ontological vagueness, it is difficult to see how an object could be composed of parts where the indeterminacy of the borders ranges over a number of years. Even putting these worries aside, however, few perdurantists would embrace such an account. For a rejection of DITP is a rejection of pure perdurantism in favour of some perdurantist-endurantist hybrid.

To see this, suppose we grant that a member of the Lepidoptera genus perdures by being composed of extended temporal parts with natural borders. Then how do these extended temporal parts persist? These temporal parts persist by perduring just if they are composed of temporal parts, and so too for the temporal parts of their temporal parts and so forth down the line. We get perdurantism «all the way down» so to speak, just if persisting objects are ultimately composed of instantaneous objects. If we reject DITP then we are forced to hold that persisting object perdure in virtue of being composed of objects that do not themselves perdure.Foot note 3_16 While it might be argued that such a hybrid view is plausible, it is so only if the temporal parts that endure are not themselves composite persisting objects: if Lepidoptera perdure then surely so do caterpillars and butterflies!

Even if perdurantists were willing to accept this peculiar hybrid view as a trade-off for retaining their sparsist intuitions, they would surely baulk at the loss of virtually all of the theoretical elegance of perdurantism. Perdurantists hold that if all properties are disguised relations to times as endurantists maintain, then there are no truly intrinsic properties, for no object ever exemplifies any property simpliciter. The perdurantist account allows that persisting objects exemplify properties at times in virtue of being composed of temporal parts that exemplify those properties simpliciter, and this is the sense in which properties are intrinsic.

But a temporal part exemplifies a property simpliciter only if the entire temporal duration of that part exemplifies the property. Temporal part P is red simpliciter only if P is red at all times at which it exists. Suppose persisting object O is rapidly changing colour from being all red to all blue to all red again. If O's having the property of being red is to be an intrinsic property as understood by the perdurantist, there must be some part of O that exists only during the short period in which O is red. Thus for every momentary intrinsic property that O exemplifies, there must be some instantaneous temporal part of O that exemplifies that property simpliciter. If O were composed only of extended temporal parts, then there would be properties that O exemplified which were not properties of any of O's parts simpliciter and which therefore would not be intrinsic in the relevant sense. Thus if the perdurantist is to retain the apparatus with which to explain how persisting objects exemplify intrinsic properties at times, she must at least subscribe to DITP.

Prima facie though, both DATP and DITP are problematic doctrines for the sparsist, since they seem to entail that arrangements of particulars can compose objects with wholly arbitrary temporal borders. So it seems that the very essence of perdurantism is in conflict with the core of sparsism. Is there any way to resolve this conflict? In the next section I consider a putative reconciliation of sparsism and perdurantism proposed by Storrs McCall, and argue that in fact it is no reconciliation at all. I then move on to consider two other proposals. In the first we take instantaneous objects to be basic and then ask ourselves how those objects need to be arranged in order to compose some persisting object. In the second I broaden the definition of temporal part, and argue that a reconfigured perdurantism can accommodate sparsist intuitions. Unfortunately this version of perdurantism is unsuccessful in its own terms, as a metaphysics of persistence.

3 Sparse Perdurantism

3.1 McCalls' Solution

One defender of a sparse ontology who embraces perdurantism is Storrs McCall. He suggests that rather than regarding persisting objects as being composed of more basic stages united by some unity relation, we should instead think of the four dimensional object as basic, and the stages as derivative «abstractions».Foot note 3_17 On this view, four dimensional volumes have a natural shape associated with a sortal or natural kind: they are not made of arbitrary portions of spacetime. These natural four dimensional volumes can then be divided into temporal stages or parts, just as the earth can be divided into spatial parts by meridian lines.

It is not clear in exactly what sense McCall means to count temporal parts as abstractions. The comparison to meridian lines suggests that he means to take a sort of anti-realist view of temporal parts. Just as we can imagine dividing up the earth in many different ways, corresponding to the different places we might draw meridian lines, so too we can divide up four dimensional objects in many different ways, each corresponding to one way of drawing the temporal border of a temporal part. If this is what is mean by «abstraction» however, it simply will not do. Perdurantism is the view that persisting objects have the properties they do in virtue of having temporal parts that exemplify those properties. Those temporal parts have to be real parts, not mere abstractions: no abstraction is red, only objects are red.

The other possibility is that McCall simply means by «abstraction», taking a top-down view of composition, that is, abstracting away from the whole four dimensional object to determine which temporal parts that object has. This too is problematic. For McCall's sparsism tells us that only those four dimensional volumes that have natural borders contain objects. Then four dimensional volumes contain temporal parts just if those volumes have natural borders. Since most of the temporal parts countenanced by perdurantists do not have natural temporal borders, by McCall's own composition criteria, those temporal parts do not exist.Foot note 3_18 It is not sufficient simply to say that we can use our intuitions about which particulars compose objects to determine which four dimensional objects exist, and then use a top-down process to maintain that in addition, all of the volumes contained within that four dimensional volume contain some object: a temporal part. Whatever our composition criteria are, all and only those arrangements of particulars that meet these criteria compose an object; there is no distinction to be drawn between four dimensional objects and temporal parts such that the former but not the latter need meet these criteria.

This does not necessarily mean, however, that there is no way to alter McCall's proposal so that it allows a reconciliation of sparsism with perdurantism. It is to that possibility that we next turn.

3.2 DATP and DITP

Suppose as sparsists we had the following intuition: if objects are in constant flux at the micro-level, then an arrangement of particulars that composes an object at some time t, can never be identical to an arrangement of particulars that composes an object at t*. This is a fairly standard perdurantist intuition according to which the only sense in which something that exists now is identical with something that exists at some other time, is the sense in which both of those things are parts of the same perduring object. If we accept this intuition, then we are faced with two questions: which arrangements of particulars at a time compose some instantaneous object, and which combinations of instantaneous objects compose some persisting object.

It's easy to see how this second question might be answered if we first grant that every arrangement of particulars at a time composes some instantaneous object. We could maintain that instantaneous objects compose some persisting object just if they are related in a particular way, namely, if they are causally connected in a smooth and continuous manner such that the existence of an instantaneous object at one time causes the existence of an instantaneous object at the next time. That is, a series of instantaneous objects compose some persisting object O just if they form a nice smooth four dimensional volume. This would rule out punctuate objects and other odd gerrymandered objects, for there would be no instantaneous objects causally connected in the requisite way. Since not every arrangement of instantaneous objects would compose some persisting object, we would preserve the sparsist intuition that ontology is sparse, yet it would still be true that objects persist by perduring.

Further, since the perdurantist need not be committed to DATP, she can hold that there exist only instantaneous objects and the persisting objects that they compose. Just as some object that exists at a time is composed of certain basic particulars at that time, so too an object that exists over time, is composed at each of those times of basic instantaneous objects. There are no extended temporal parts whose temporal borders are oddly arbitrary: there is no object that wholly overlaps my dog and exists for only ten minutes on Tuesday. So the appearance of a plethora of objects with arbitrary temporal borders is removed. For just as the spatial borders of a mereological simple are natural, so too the temporal borders of an instantaneous object are natural.

All well and good. The difficulty lies in conceding, as I did, that every arrangement of particulars at a time composes some instantaneous object. I do not think that many sparsists will be happy with this concession. Sparsists will not, I think, want to allow that there is some object that exists only at t, and which is composed of my dog at t, Jupiter at t, and your pillow case at t. While this object might have a natural temporal border, it certainly does not have a natural spatial border. This is not to say that this position is a hopeless one. Perhaps there are sparsists who hold that there is something special about persisting objects, such that sparsism-over-time is a more important doctrine than sparsism-at-a-time. Perhaps such a person would be willing to concede that there exist odd instantaneous objects, but no odd persisting objects. But I am not entirely sure what would motivate such a position. Why should we think that the «glue» that holds objects together over time is fundamentally different from the «glue» that holds them together at a time?

This latter question is particularly pertinent given that we are talking about a sparist version of perdurantism. For consider, the endurantist explicitly holds the view that the manner in which objects persist through time is radically different to the manner in which they extend through space. Objects extend through space by having spatial parts at spatial locations, while they persist through time by being wholly present at each time at which they exist. Perdurantists, however, construe persistence through time as analogous to extension through space, with objects persisting by having parts -- temporal parts -- at temporal locations. Given this, it is not clear what would motivate the claim that across-time sparseness is radically different to at-a-time sparseness. If there is such a case to be made, it is at least imcumbent on the perdurantist sparsist to make that case.

So while this is perhaps one way to reconcile perdurantism with sparsism, it is not a wholly attractive way, and nor, I imagine, is it an option that will find favour with many sparsists. But if the sparsist rejects the assumption that every instantaneous object exists, then we need to determine which instantaneous objects exist and why. Clearly the best proposal would be to hold that there exist the various everyday persisting objects of our ontology, and the instantaneous temporal parts of those objects. Adopting McCall's top-down approach then, it might be thought that this is precisely what we can accomplish. As sparsists, we feel confident that dogs exist. Given that dogs exist, we can conclude that the instantaneous temporal parts of dogs exist. Thus dogs perdure in virtue of being composed of temporal parts, and those temporal parts have natural borders in that they are temporally basic: they are instantaneous. There are two problems for this view. First, we do not want sparsism to be simply the view that particulars compose some object just if we think they do, that is, just if that is what our intuitions tell us. Sparsism is supposed to be the view that there are some informative composition criteria. On many criteria, composition involves complex causal relations between the composing particulars (whether these be causal relations constituting a life, or constituting a natural kind or sortal). Whatever these criteria amount to, they must apply equally to all objects. But in general, instantaneous objects do not meet the usual sorts of sparsist criteria for composition. Instantaneous objects are probably not members of sortals or natural kinds, for the underlying properties that constitute those kinds are properties of persisting objects. So too no instantaneous object exemplifies the property of having a life.

So the top-down solution does not seem hopeful: for it is plausible only if we think that the composition criteria for persisting objects is different to that for instantaneous objects. Perhaps so. But some account of the composition criteria for instantaneous objects would need to be forthcoming, and this criteria could not simply be that particulars compose some instantaneous object if that object is a temporal part of a persisting object. For that is not to provide composition criteria, it is just to state which objects exist and which do not. Until such criteria are forthcoming then, the top-down view is not at all compelling.

Moreover, the view is faced with an additional problem. On this view, there do not exist any extended temporal parts. Now consider some person P, who has some blue experience. P has the property of having a blue quale. Now suppose that blue experiences are not ever experiences of instantaneous objects.Foot note 3_19 Having a blue experience, however, is surely an intrinsic property of P. Unfortunately though, it is not a property of any temporal part of P, since P has no extended temporal parts. Strangely then, although having a blue experience would have been an intrinsic property of P if that experience had been instantaneous, since P would have had a temporal part that had that property simpliciter, as things stand, P has no such temporal part, and thus does not have that property simpliciter. So too for any «temporally extended» property. So this version of perdurantism is stuck with saying that some apparently intrinsic properties are really disguised relations to times or some such. This is even more alarming than straight endurantism, since it turns out that some apparently intrinsic properties are indeed intrinsic in virtue of being properties of temporal parts, and some other apparently intrinsic properties are not intrinsic, in virtue of failing to be properties of any temporal part.

So where does that leave us? If the perdurantist is committed to DITP, and if that doctrine is inconsistent with sparsism, then are we forced to conclude that sparsism and perdurantism are inconsistent? Before we make such a declaration, we should first consider whether there is some other way to salvage a perdurantist sparsism. In the next section we will briefly consider whether it is perdurantism that ought to be jettisoned in favour of some other version of four dimensionalism. Though this suggestion will be rejected, it does lead to the idea that we should alter the definition of temporal part. This alteration yields a version of perdurantism that is acceptable to the sparsist, and is, I will argue, the best reconciliation of perdurantism with sparsism. But while it is the best reconciliation, ultimately it too must be rejected on the grounds that it simply does not have the theoretical apparatus to explain the phenomena that an account of persistence must explain.

5 Temporal Extension without Temporal Parts?

We might think that the best way to combine sparsism with four dimensionalism is by rejecting perdurantism. This would need to involve more than, for instance, adopting Sider's stage view which accepts the same ontology as perdurantism but merely disagrees about which objects in that ontology ought to be the referents of our terms. Rather, it would require the radical view that objects can be temporally extended and thus four dimensional, and yet have no proper temporal parts. Call such an object a temporal simple. This view has recently been defended by Parsons.Foot note 3_20 Parsons does not suppose that composite persisting objects such as dogs and trees could be temporal simples, and indeed it is hard to see why a view that countenanced this possibility would be preferable to three dimensionalism. For it would no longer be possible to use the apparatus of temporal parts to explain change over time or to provide an account of temporary intrinsics. If anything, it would seem to render to nature of persisting objects all the more mysterious.

The idea of temporal simples does, however, suggest another possibility. For consider what happens when we attempt to «construct» a composite four dimensional object that lacks proper temporal parts. For the perdurantist, a four dimensional object is composed of various instantaneous objects which are themselves composed of particulars at a time. But suppose that we think of a some object not as composed of instantaneous objects at times, but simply as composed of particulars at times. That is, some persisting object O is composed of particulars S at t, P at t, Q at t, S at t1, P at t1, R at t1 etc. Then have we just described a four dimensional object with no proper temporal parts, or merely a persisting three dimensional object?

If we think that the particular S at t and S at t1 is the very same particular S, viewed at different times, then O is simply an enduring three dimensional object. If S at t and S at t1 are strictly identical, then S endures, and so too with all of the other particulars. Then O is a composite object composed of enduring particulars S, T, R, Q etc. Since by definition there exists no object that wholly overlaps O at a time and is part of O, that is, by definition O has no proper temporal parts, O must itself be an enduring three dimensional object.

On the other hand, if we think that S at t and S at t1 are distinct particulars, then we think that there exists S-at-t and S-at-t1. O is thus composed of S-at-t and S-at-t1: it has these particulars as parts simpliciter. Thus there is an important sense in which O deserves to be called four dimensional despite the fact that it is not composed of temporal parts as I defined them earlier. As I defined a temporal part, x is a temporal part of y at t only if x wholly overlaps y at t. Call this a maximal temporal part. O has no maximal temporal parts. However, O is composed of four dimensional particulars. Each particular has temporal parts: S exists at different times, and it does so in virtue of being composed of instantaneous temporal parts S-at-t, S-at-t1 and so forth.

If an object O perdures just if O persists by being the mereological fusion of maximal temporal parts, then the object just described does not perdure. It has been suggested, however, that we ought to broaden our definition of temporal part to include non-maximal parts.Foot note 3_21 Let us then define a non-maximal temporal part as follows:

x is an instantaneous non-maximal temporal part of y at t just if 1) x is part of y 2) x exists at, and only at t and 3) x is wholly overlapped at t by some part of y that exists at t.

x is an extended non-maximal temporal part of y during interval T just if 1) x exists at all and only times in T, 2) x is part of y 3) x is wholly overlapped by some part of y at all times during T.

We can then say that an object O perdures just if O persists by being the mereological fusion of maximal or non-maximal temporal parts. Then the object O described above will perdure in this sense, since it is composed of non-maximal temporal parts.

The critical question then, is whether the sparsist should think that S at t and S at t1 are distinct particulars or are strictly identical. There is a case to be made for each view. On the one hand, the sparsist thinks that we have a single unitary object just where we have natural borders. She certainly thinks that S-at-t would be a distinct particular in a world W where there is nothing S-like at temporal instants that abut t. For in that world, S-at-t clearly has natural temporal borders. In the actual world though, sparsist intuitions might steer one towards holding that there is no object S-at-t, for that putative object has no natural borders in virtue of being temporally abutted by other S-like particulars. Rather, we have one particular, S, and to claim that in addition to S, there exist various instantaneous objects that compose S, would be to claim that there exist objects with arbitrary borders.

There is something appealing about the intuition that something has a natural temporal border only if it is not temporally abutted but like particular or particulars. After all, part of the intuition that the putative temporal part of me that exists only for ten minutes today has arbitrary temporal borders, is that it is temporally abutted by person-like objects. If that same putative object existed in some world W and was not abutted by anything person-like, we would be happy to concede that it has natural temporal borders in that world. Recall however, that in the previous section we rejected the idea that any arrangement of particulars can compose some instantaneous object, on the grounds that many such objects would have arbitrary spatial borders. We noted though, that just as the spatial borders of a mereological simple are natural, so too the temporal borders of an instantaneous object are natural: these objects are temporally fundamental. So the most basic particular is one that is mereologically simple, and temporally fundamental,Foot note 3_22 and the borders of this most basic particular are natural. If the sparsist accepts this, then she can proceed to hold that some of these instantaneous simples are causally related such that they compose perduring basic particulars such as S. For the persisting object S is mereologically simple, but not temporally fundamental: it is composed of the most basic particulars S-at-t, S-at-t1 and so forth. We can then say that composite objects are composed of perduring particulars like S, and thus ultimately of mereologically simple instantaneous objects. These composite objects therefore have no maximal temporal parts, but only non-maximal parts with natural borders.

So the sparsist requirement is fulfilled: there exist no objects that have wholly arbitrary borders. Yet this view also seems to afford the perdurantist all of the metaphysical apparatus needed to explain change and temporary intrinsics. Though we cannot say that object O is red at t in virtue of having some maximal temporal part O-at-t that is red simpliciter, we can say that O is red at t in virtue of each of its non-maximal temporal parts S-at-t, R-at-t and so forth being red simpliciter. Problem solved; sparsism and perdurantism reconciled.

Not so fast. This reconciliation too is problematic. For plausibly, macro-properties such as being red, being cold, being conscious, being a person and so forth, are not exemplified by instantaneous objects. Plausibly, they are not exemplified by any instantaneous maximal temporal part, but they are certainly not exemplified by any instantaneous non-maximal basic particular. Consider some object O that is red at t. We cannot, in fact, say that O is red at t in virtue of ever basic particular that composes O at t, being red at t. For S-at-t is not red. S-at-t is too small to be red. What is red at t, is the totality of the arrangement of the particulars at t. This arrangement, however, does not compose any object, for there is no maximal temporal part of O at t. So there is nothing that exists at t and is red simpliciter, and we find ourselves faced again with the problem of temporary intrinsics. So too, on this view there exists no extended maximal temporal parts, and thus no such parts of persons. Since consciousness almost certainly supervenes on some temporally extended temporal part, here too there is a problem. Not only is it not the case that any most basic particular S-at-t is conscious, but no arrangement of these most basic particulars at a time has the property of being conscious: only some arrangement of these particulars over time could exemplify that property, but no such arrangement composes any object. So only entire four dimensional persons are conscious.

Of course, a proponent of this view might maintain that there is no real problem here. After all, this is precisely the sort of difficulty certain sorts of eliminativists, (such as eliminativists about beliefs or eliminativists about composite objects) face each day when our everyday language quantifies over non-existent objects.Foot note 3_23 For eliminativists, talk that quantifies over non-existent dogs, for instance, is true just if there is some paraphrase in which it is true that there are particulars arranged in a dog-wise way, or some such. So too, we might think, the sparse perdurantist can maintain that O is red at t just if there is some O-wise arrangement of particulars at t that is red simpliciter.

But this is all rather tortuous. For those who (plausibly) think that an arrangement of particulars exemplifies some macro-property such as being red just if that arrangement composes some macro-object, the eliminativist solution will be no solution at all. Either the arrangement of particulars at t composes some object, namely O-at-t which is a maximal temporal part of O and is red simpliciter, or the arrangement does not compose any object at t and there is no object that is red simpliciter at t. There is no middle ground, and thus no way to have one's sparsist cake and eat it too. So while there might be those who are sympathetic to the eliminativist strategy and thus willing to accept this combination of sparsism and perdurantism, they must surely be in a minority.

There is of course one other way that this version of sparsist perdurantism could deal with maco-properties, and that is by adopting an endurantist analysis of properties. There are two possibilities available, indexicalism,Foot note 3_24 according to which properties are temporally relativised, and adverbialismFoot note 3_25 according to which the having of properties is temporally relativised. For the perdurantist «O» refers to the whole four dimensional object. So if O is red at t and blue at t1, following indexicalism we can say that O has the properties of being is red-at-t and blue-at-t1. Or following adverbialism we can say O has the properties of being red tly and blue t1ly. So although there is no temporal part of O that is red or that is blue, we can still attribute these properties to O.

This is by far the best solution, and this is certainly the best combination of sparsism and perdurantism. It is consistent and workable, but it has the cost of jettisoning much of the theoretical apparatus that motivated perdurantism to begin with. Many perdurantists such as Lewis are moved to embrace perdurantism because they believe that it best deals with the problem of change, in that allows that temporary intrinsics are not disguised relations to times.Foot note 3_26 In adopting an endurantist analysis of properties, however, this version of perdurantism is forced to concede that O does not have the property of being red simpliciter, but rather has a property that is temporally relativised in one way or another. The extent to which this sparsist perdurantism is acceptable then, will depend on how repugnant one finds the endurantist analysis of properties. That is, it will depend on just how strong one's intuitions are that intrinsic properties are not relations to times. The dilemma is that the more one is attracted to the endurantist analysis, the less reason one has to prefer perdurantism in the first place, and the more repugnant one finds the analysis, the less one will be happy to accept this version of perdurantism. So while I cannot rule out this version of sparsism perdurantism as being coherent, it does require the loss of much of the theoretical elegance of traditional perdurantism, and might well not be a view that many perdurantists, even those of a sparsist bent, will be happy to embrace.

6 The Brutality of Composition

A final possibility. Suppose the sparsist relinquishes the idea that there are any non-arbitrary informative criteria of composition. She might, for instance, follow Markosian in holding that composition is brute.Foot note 3_27 Either arrangements of particulars compose an object or they do not, and there is no further story to tell. If composition is brute, then both sparsist and perdurantist intuitions can straightforwardly be reconciled. Some arrangements of particulars compose objects that appear to have wholly arbitrary temporal borders, namely temporal parts. Other arrangements of particulars fail to compose objects that would have had arbitrary borders had they existed. But there just is no reason why the former objects exist and the latter do not, for there is no principled reason why some arrangements of particulars compose objects and others do not. Indeed, there can be no such reason.

Perhaps there are good reasons to think that composition is brute. Still, this move seems wholly unsatisfactory when applied to the case at hand. The proposition that composition is brute is arrived at by considering the cases where we take composition to occur and those where we do not, and arguing that there is no informative criteria that would pick out only the former and not the latter as being instances of composition. It is by considering examples of composition in the real world that we are lead to adopt sparsism. So it had better turn out that most of our intuitions about when composition occurs are right, or the very motivation for adopting sparsism in the first place will evaporate. It would, therefore, be disingenuous to claim that even though our sparsist intuitions tell us that temporal parts do not exist, we can maintain that they do exist, and that no explanation for this inconsistency need be forthcoming since composition is brute. On these grounds I could argue that there is some object that is composed of my dog and Lincoln's foot, and that there is no object composed of your dog and Jefferson's foot. Why? Well who knows, composition is brute!

7 A General Problem

The difficulty inherent in trying to reconcile sparsism and perdurantism is that sparsism just does not afford a sufficient number of parts to meet perdurantist requirements. Because the version of perdurantism discussed in section 5 eschews the existence of maximal temporal parts, there exists nothing that is me-now. That is, there exists no object that is wholly present now and which exemplifies all of my momentary properties now. So too although there is some object that is my heart, and which is a proper part of me, my heart is a four dimensional object which is part of a four dimensional person. Since my heart is not composed of maximal temporal parts, there is no object that is wholly present now and is part of me-now. Neither endurantism nor traditional perdurantism has this odd consequence. For the endurantist thinks that both me and my heart are now wholly present, and thus that my wholly present heart is now part of wholly present me. So there is a straightforward sense in which my whole heart is part of me now. Perdurantists, on the other hand, although they disagree that my whole heart is now part of me, do agree that there is some object that is wholly present now and is part of me now, namely the maximal temporal part of my heart-now. The time-slice of me-now has that entire object as a part. So neither are forced to conclude that there is no sense in which my heart now is not a part of me at all.

This counterintuitive consequence of this version of perdurantism might lead one to conclude that sparsism and perdurantism simply are not to be reconciled, and thus that one must be abandoned. Since sparsist intuitions have a firm grip on us, there are those who will jettison perdurantism in favour of these intuitions. But is this the correct response? In fact when we consider sparsism stripped of any perdurantist overtones, we find that we are faced with an analogous difficulty. Indeed, it turns out that the counter-intuitive consequences of sparsist perdurantism is just a specific version of a general problem faced by sparsism alone.

Consider a time honoured paradox we find in the literature on persistence: Tibbles the unfortunate cat.Foot note 3_28 According to the paradox, there exists an object, Tibbles the cat, and some proper part of Tibbles, call it Tib, which includes all of Tibbles but for her tail. The paradox arises after Tibbles has her tail amputated, and we are asked to consider what we should say about the relation between Tib and Tibbles post-amputation. But what is this object Tib of which I write? At time t, prior to the amputation, Tib is a proper part of Tibbles. Both three and four dimensionalists alike accept this, though of course they gloss it differently. In general, the existence of Tib has been granted without comment by all sides of the debate, including three dimensionalists who are frequently sparsists.Foot note 3_29 The exception here is van Inwagen who rejects the existence of Tib precisely because its existence is necessary in order to generate the paradox.Foot note 3_30 van Inwagen goes further than merely denying the existence of Tib, he rejects the spatial analogue of the doctrine of arbitrary temporal parts, the doctrine of arbitrary undetached parts (DAUP):

DAUP: For every material object M if R is a region of space occupied by M at time t and if sub-R is any occupiable sub-region of R whatever, there exists a material object that occupies the region sub-R at t.Foot note 3_31

Tibbles-type paradoxes do not require that one reject DAUP. But should the sparsist accept DAUP? I think not. Whatever the composition criteria turn out to be, it is difficult to see how Tib and putative objects like it, could be seen as meeting those criteria. For suppose we follow van Inwagen in holding that particulars compose something just in case their arrangement constitutes a life. Does the Tib-wise arrangement of particulars compose something that constitutes a life? Well in one sense it does. Since Tibbles can survive the loss of her tail, Tib, if it exists, must be a sufficient supervenience base for life. But I think this misses the point. For it is clear that the tail-wise arrangement of particulars, though they are not sufficient in and of themselves to constitute a life, are part of a life, namely the Tibbles life. There is but one life there (with the exception of any cellular organisms that are floating around of course - but that is beside the point). There is not Tibbles' life, and then Tib's life, one life which includes the tail, and the other which does not. But if there is only one life, and that life if the life of Tibbles, then we cannot conclude that the Tib-wise arrangement of particulars composes anything.

Similarly, consider the view that particulars compose some object just if their arrangement constitutes something whose underlying explanatory properties mark it out as a member of a natural kind or sortal. Then we can see why the Tibbles-wise arrangement of particulars composes something: because that arrangement constitutes something that is a cat, and a cat is a member of a natural kind. But is there any natural kind or sortal of which Tib, if it exists, is a member? I cannot see that there is. Tib certainly would not be a member of a natural kind, nor does it seem plausible to think that it is a member of a sortal. If this is so, then we should conclude that the Tib-wise arrangement of particulars do not compose anything.

None of this should come as a surprise, since Tib, if it existed, would flout the sparsist intuition that objects have natural borders. For we arbitrarily defined Tib as that thing which includes all of Tibbles but her tail. Sparsists should conclude that Tib does not exist. And if Tib does not exist, then Tib is not a proper part of Tibbles. So when the sparsist says that Tib is a proper part of Tibbles she says something that should be, by her lights, false.

The case of Tibbles is by no means an isolated one. Discussion of the relation of objects to their proper parts often involves talk of proper parts that by the lights of the sparsist, do not exist. Consider the case where we talk of a statue that has a little chunk of clay removed from it. Three dimensionalists typically want to say that the aggregate of clay «Clay» that constitutes the statue, «Statue» prior to the removal of the chunk, cannot survive this removal for Clay persists only if it is mereologically constant. On the other hand, they want to say that Statue does survive the removal of the chunk, for its persistence conditions are such that it can survive such an event. This story only makes sense, however, if the chunk of clay is a proper part of the statue, and again, I see no reason to suppose that it is if one accepts sparsism.

It turns out then, that if the sparsist is right, we frequently quantify over non-existent objects. We talk of the part of Tibbles that is the small hunk of flesh on her right thigh. We talk of the small piece of clay that fell off the statue. We talk of a scoop of ice cream removed from the tub. We talk of the enamel chip that fell off the plate. And in all of these cases we will often talk of the flesh, the clay, the chip and so forth as being proper parts of the objects in question. But if sparsism is true, then I submit that we would be wrong to do so. For I can see no basis at all on which to say that these are objects at all.

If sparsism is true, then sometimes we quantify over the non-existent. So it is not merely the sparsist perdurantist who is forced to embrace the eliminativistic paraphrasing of everyday language. Just as there is no maximal temporal part of O that is red simpliciter, so too there is no chunk of clay is part of O at t. When our everyday talk appears to be quantifying over proper parts that do not exist, we are really quantifying over certain arrangements of particulars. Thus although strictly speaking Tib does not exist, there does exist a Tib-wise arrangement of particulars. And we can truly say that none of the particulars that are arranged Tib-wise, are arranged tail-wise. If this is so, then when I say that Tib is not part of the tail, what I say is true. Similarly, when I say that Tib is part of Tibbles, what I say is true just if there is some Tib-wise arrangement of particulars such that each of those particulars is part of Tibbles.

Again though, this relies on it being coherent to talk of the macro-properties of arrangements of particulars despite the fact that there is no object that exemplifies those properties. Perhaps this is not an insurmountable problem for the sparsist. What is shows, however, is that the difficulties inherent in reconciling sparsism and perdurantism are merely specific instances of difficulties that the pure sparsist faces with the notion of parthood. Not only does the sparsist not have enough parts to meet perdurantist requirements, it doesn't have enough parts to meet everyday folk requirements. It is usually held to be a major virtue of sparsist views that they take the middle ontological road between eliminativism and universalism, and thus avoid not only the plethora of odd objects posited by universalism, but also the need for paraphrasing ordinary language required by eliminativism. But sparsism does not avoid all such paraphrasing, and does not preserve all of our ontological intuitions, for it entails that Tib and others of its ilk do not exist.

8 Conclusion

Sparsism is inconsistent with any plausible version of perdurantism. Should we then reject sparsism in favour of perdurantism, or perdurantism in favour of sparsism? The answer to this question involves a delicate balancing act of weighing up the virtues of each. Perdurantism has much theoretical elegance to recommend it. Sparsism has much intuitive appeal. But not as much intuitive appeal as we might have thought, since by the lights of sparsism we quantify over non-existent objects quite frequently. Perhaps this difficulty provides sufficient weight to tip the scales in favour of perdurantism over sparsism. Or perhaps sparsist intuitions are so resilient that some will prefer to adopt an eliminativist-type strategy in dealing with non-existent objects such as Tib. The point is that some sacrifice will need to be made: it is not possible to reap the theoretical elegance of perdurantism and also the ontological parsimony of sparsism. Something has to give.


Kristie Miller
Department of Philosophy
University of Queensland
Brisbane Australia
<kristie_miller [at]>

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