Abstract
In this paper, I focus on three issues intertwined in current debates between endurantists and perdurantists—(i) the dimension of persisting objects, (ii) whether persisting objects have timeless, or only time-relative, parts, and (iii) whether persisting objects have proper temporal parts. I argue that one standard endurantist position on the first issue is compatible with standard perdurantist positions on parthood and temporal parts. I further argue that different accounts of persistence depend on the claims about objects’ dimensions and not on the auxiliary claims about parthood and temporal parts.
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Notes
Discussions of endurantism and perdurantism (or, three- and four-dimensionalism) are complicated by the fact that significantly different definitions of ‘three-dimensionalism’, ‘four-dimensionalism’, ‘endurantism’, and ‘perdurantism’ have been offered in the literature. For a sampling of options, compare Merricks (1999), Sider (2001), Sattig (2006), Hawley (2001), Hawthorne (2008). In addition, although some authors use the terms ‘three-dimensionalism’/’endurantism’ and ‘four-dimensionalism’/’perdurantism’ interchangeably, many prescribe distinct usages for the terms in the latter pair. Most notably, ‘perdurantism’ is most often used to pick out a specific version of four-dimensionalism—worm theory as opposed to stage theory. (See, e.g., Sider (2001). But see Parsons (2000) for different uses of ‘four-dimensionalism’ and ‘perdurantism’.)
In this paper, I use the terms ‘endurantist’, ‘three-dimensionalist’, ‘perdurantist’, and ‘four-dimensionalist’ to refer to philosophers who label themselves as such. I use the terms ‘endurantism’/’three-dimensionalism’ and ‘perdurantism’/’four-dimensionalism’ very loosely to pick out the two general sorts of theories endorsed by endurantists or perdurantists. For the most part, though, I try to avoid confusion in what follows by focusing on specific positions on (DIM), (TP), and (PART) instead of on the different theories endorsed by actual endurantists or perdurantists.
Throughout the paper, I use the term ‘object’ to refer only to material objects—to things like electrons, cells, statues, and lumps of clay (and not to immaterial things like properties, sets, spatiotemporal regions, or holes).
To keep things simple, I will from here on ignore lower-dimensional objects (surfaces, boundaries between surfaces, and so on). I think it is easy enough to see how the points made in this paper may be modified to accommodate lower-dimensional objects.
Of course, given the diversity of approaches to the two sorts of positions (see footnote 1), there are exceptions to these generalizations. For example, as one reviewer rightly points out, some endurantists and perdurantists have focused predominantly on issue (TP) and have had little, if anything, to say about issue (DIM). Still, I do not know of any perdurantist who claims that persisting objects are three-dimensional or any endurantist who claims that objects are four-dimensional. In the end, the central thesis of this paper—that two specific positions on issue (DIM) embody different accounts of persistence which, taken separately, are each compatible with standard perdurantist positions on (TP) and (PART)—does not depend on actual endurantists’ or perdurantists’ positions on issue (DIM).
I might instead have distinguished between philosophers who adopt an A theory of time (which assumes that the present is objectively distinguished from the past and the future) and philosophers who adopt a B theory of time (which denies any objective distinction between the past, present, and future). B theorists are committed to eternalism, but A theorists might deny presentism while still holding that there is some objective distinction between the present, the past, and the future. But, as a matter of fact, most A theorists (and, as far as I know, all who have entered into three- and four-dimensionalism debates) are presentists. Moreover, the sorts of relations I introduce later in the paper assume only eternalism and thus might be adopted even by eternalists who are A theorists. For these reasons, I do not see the need for taking the (somewhat exotic) combination of A theory and eternalism into special consideration.
Still, there is an important difference between the presentist’s non-time-relative parthood relation and the perdurantist’s atemporal parthood relation. The perdurantist generally treats parthood as a relation that cannot change over time. For the presentist, by contrast, the extension of the binary parthood relation changes over time (and facts about parthood are, in this sense, time-relative).
Mereological essentialism is the position that objects cannot gain and lose parts over time. See Chisholm (1973) for an endorsement of mereological essentialism. In the interest of simplicity, I will throughout the remainder of this paper ignore mereological essentialism. In other words, I will operate under the highly plausible assumption that some persisting objects gain and lose parts.
Hawley defines endurance as follows: “an object endures if and only if (i) it exists at more than one moment and (ii) statements about what parts the object has must be made relative to some time or other”. For Hawley, endurantists are committed at least to the claim that ordinary objects endure in this sense (2001, p. 27).
Of course, things are more complicated in relativistic spacetimes where shapes and distances might be taken to vary with reference frames.
Note that the exact location relation used in this paper (and in Hudson (2001), Sattig (2006), Gilmore (2006), and Balashov (2008)) is not the same as the relation called ‘exact location’ in Parsons (2007). Parsons introduces the term ‘exact location’ for a relation that is supposed to hold between an object x and a region r just in case (i) every region outside of r is completely free of x and (ii) no subregion of r is completely free of x (2007, p. 203). Unlike the exact location relation used in this paper, Parsons’ relation plausibly holds between an object and at most one region. Parsons explicitly makes this assumption and points out that what he calls the ‘exact location’ of an object is what Gilmore calls instead the ‘path’ of the object (Parsons, 2007, p. 220). (I introduce Gilmore’s path function later in this section.)
Parsons raises doubts about the intelligibility of an exact location relation that does not restrict each object to at most one exact location (2007, p. 219). As I will briefly indicate in the conclusion of this paper, once we shift the discussion to relativistic spacetimes, I have some sympathy with Parsons’ expectation that proponents of ‘multiple exact location’ owe us a better explanation of their relation than they have so far supplied. (Gilmore is also sympathetic with this sort of concern. See Gilmore (2006).) But I do not, like Parsons, think that the case against multiple exact location is already decided. Moreover, in the context of a Galilean spacetime, I think that the sort of multiple location assumed in some representations of endurantism—amounting roughly to my (3D)—is unproblematic since here the endurantist does offer a clear account of where objects are exactly located within their spatiotemporal paths.
Notice that we need not assume any sense in which an object has a shape, size, or dimension in a way that is independent of the geometrical properties of the regions at which it is located. In fact, if we, say, take the appropriate notion of three-dimensionality for material objects to be
\( \begin{aligned} {\text{object}}\;x\;{\text{is}}\;{\text{three-dimensional}} = & \; _{\text{def}} x\;{\text{occupies some spatiotemporal region and all of the spatiotemporal regions}} \\ {\text{that }}x\;{\text{occupies are three-dimensional}} \\ \end{aligned} \)
then it follows immediately that all three-dimensional objects occupy only three-dimensional regions.
In addition, Gilmore (2006), Sattig (2006) and Balashov (2008) consider relativistic reformulations of (3D) and (4D). See also van Inwagen (1990b) where accounts in the spirit of (3D) and (4D) are contrasted with a stage theoretic take on spatiotemporal location. But van Inwagen leaves open the possibility that, on an endurantist account, an object might occupy both three-dimensional subregions of timeslices and the four-dimensional sum of these regions.
In Barker and Dowe (2005), something like (3D) is presented as the default option for endurantists who are eternalists. Barker and Dowe use a strong assumption about how objects are located in spacetime—namely, that location is ‘additive’ in the sense that x is located at multiple spatiotemporal regions r i only if x is also located at their union ∪r i —to argue that endurance is paradoxical. Like Bittner and Donnelly (2004), Gilmore (2006), Sattig (2006), Balashov (2008), and Gibson and Pooley (2006), I do not assume that location is additive.
The purpose of disjunct (ii) is to take into account the view that parthood is metaphysically basic and so not reducible to location relations. Nonetheless, it may still be the case that atemporal parthood behaves (almost) exactly like a defined relation R in the sense that the parthood relation and R apply to the same pairs of objects (except perhaps in unusual cases). If so, then, despite their difference, the availability of R to both 3D-ers and 4D-ers is surely at least a good prima facie reason for thinking (in the absence of considerations to the contrary) that the closely matching parthood relation is also available to both 3D-ers and 4D-ers. I address this point in Sect. 3.
The term ‘temporal segment’ is borrowed from Gilmore ( 2006, p. 206): “x is a temporal segment of y iff: (i) x’s path is a ‘temporal slice’ or ‘temporal chunk’ of y’s path and (ii) x and y ‘share all of their matter’ within x’s path.” I tend to think that the Gilmore’s second condition is redundant. It seems to me that if x and y are material objects and x’s path is included in y’s path, it should follow immediately that x and y share all of their matter within x’s path. However, this point is not relevant to the discussion that follows.
More precisely, Parsons uses an atemporal parthood relation to define ‘wholly located at’ as: x is wholly located at t = def all of x’s parts are weakly located at t (2007, p. 212) (If I understand correctly, when restricted to objects and times, Parsons’ weakly located at relation roughly matches my present at relation.) Parsons then defines: x endures throughout = def x persists and, for any time t such that x is weakly located at t, x is wholly located at t. Note that the right hand side of Parsons’ definition is equivalent to: x persists and, for any time t such that x is weakly located at t and any part y of x, y is weakly located at t. In the terminology of this paper, Parsons’ definition states that x endures throughout just in case x persists and all (atemporal) parts of x are present whenever x is present.
See also Thomas Sattig’s argument that a proponent of a position along the lines of our (3D) cannot endorse the claim that persisting objects have (proper) temporal parts (2006, p. 55). I admit that I do not entirely understand how Sattig’s argument is supposed to work. But it explicitly relies on the assumption that if y is multiply located, as are the 3D-er’s persisting objects, and if x is any part of y simpliciter, then “x occupies many instantaneous regions just as y does, or else x occupies no spacetime region at all”. It follows immediately that, on Sattig’s view, no object which persists in the (3D)-sense has any instantaneous object as a part. Sattig also thinks that (3D) rules out extended (proper) temporal parts.
Why? The chain was sitting on a shelf at the bicycle shop for several weeks before it was installed on my bike. Thus, the beginning stretch of the chain’s path is not included in my bike’s path. It follows that the chain is not path included in the bike. And it is exactly for this reason that perdurantists deny that the chain is—in the atemporal sense—part of my bike.
More precisely, for any set A ≠ ∅, <⊆, ℘(A)/{ϕ}> is a model of classical mereology, where parthood is interpreted as the subset relation ⊆ over ℘(A)/{ϕ} (the set of non-empty subsets of A).
The reflexivity of the path inclusion relation follows immediately from the reflexivity of the subregion relation. Given either (3D) or (4D), any object’s path is a subregion of itself. The transitivity of the path inclusion relation follows immediately from the transitivity of the subregion relation. Given either (3D) or (4D), if x’s path is a subregion of y’s path and y’s path is a subregion of z’s path, then x’s path is a subregion of z’s path.
In particular, I see no reason why 3D-ers cannot adopt the modal counterpart treatment of the statue/lump scenario that is espoused by perdurantists like Lewis. On this solution, exactly one object has the path of the statue/lump, but there are different kinds of counterpart relations linking it to objects in other possible worlds. A ‘statue counterpart relation’ links the actual object to possible objects, all of which are statues of roughly the same shape as the actual object, but which need not be made entirely of clay. A ‘lump counterpart relation’ links the actual object to possible objects all of which are lumps of clay but which need not be statues. See Lewis (1986).
The relevant relation here is that which holds between any four objects when they are all hinged to this particular plank.
For another example of an expansive enduratist ontology, see Thomson (1998). Thomson suggests that endurantists might take a liberal stance on the existence of indefinitely many varieties of sums. For example, in addition to the ‘path-sums’ considered above, Thomson suggests that for any set S of objects which includes a member that is at some time red, there is an object which is present only when at least one member of S is red and is composed of members of S whenever it is present (1998, p. 167). Following a variation on Thomson’s suggestion, a 3D-er might hold that there is an object which (i) is present only when all of the sticks are attached to our plank and (ii) is comprised of the sticks whenever it is present. Such an object would be path included in the table but discrete from the plank.
Notice also that, like the 4D-er, the 3D-er might accept that persisting objects have arbitrarily many proper temporal segments without holding that distinct objects share a spatiotemporal path. Thus, if he wishes, the temporal segment friendly 3D-er can avoid one strain of the supervenience problem for coincidents—namely, the problem of how objects that share all of their non-modal properties can differ in their modal properties (see, e.g., Bennett 2004). An object and one of its proper temporal segments do not share all of their non-modal properties since the former is present at some time when the latter is not.
See also the interesting suggestion in Sidelle (2002, p. 129) and Bennett (2004, p. 355) that by embracing a maximally full ontology, the pro-coincidence three-dimensionalist might also avoid having to provide supervenience bases for the different persistence conditions of temporarily coinciding objects (like a lump of clay and a statue which the lump temporarily forms). The idea is that if all possible ‘ways of persisting’ are instantiated wherever there is any object, then we do not seem to need something in the immediate situation which grounds the persistence conditions of each object under consideration. As Sidelle puts it, “since wherever you have an object, every possible criterion of identity/method of tracing is instantiated, there is no special problem saying why, for each one, it is instantiated, and coincidence follows trivially” (2002, p. 129).
See Heller (1984, pp. 327–328): “a temporal part of O is the material content of a temporal sub-region of R”, where O is an object and R is the unique spatiotemporal region at which O is located. Heller defines a “temporal sub-region of R” as “a spatiotemporal sub-region that shares all of R’s spatial boundaries within that sub-region’s temporal boundaries” (1984, p. 328). As far as I can tell, the only difference between my temporal segment relation and Heller’s temporal parts relation is that the latter applies only to objects with unique spatiotemporal locations (thus excluding the 3D-er’s persisting objects) while the former applies to objects with any number of spatiotemporal locations.
Here is an extreme example of what this sort of case might look like. Suppose o is a persisting object that is located within each timeslice of its career at a spherical region. (I am assuming (3D). But the example can easily be reworked with the same results for assumption (4D).) Suppose further that o has as proper parts only instantaneous objects, each of which is located at a cubical region within one of o’s locations and no two of which are present at the same time. Then, given definition (*), each of these cubical objects is a temporal part of o. In this case, each of o’s temporal parts is not only spatially smaller than o, it also has a noticeably different shape than does o.
Even if the parthood relation on the right-hand side of (*) has the same extension as the path inclusion relation and any temporal part of an object o is also a temporal segment of o, we still need one further assumption to guarantee that the relation defined in (*) is co-extensional with the temporal segment relation:
\( \begin{gathered} {\text{for any objects }}x{\text{ and }}y,{\text{ if }}x{\text{ is located at region }}r_{x} {\text{ and }}y{\text{ is located at region }}r_{y} {\text{ where }}r_{x} \, \cap \, r_{y} \ne \emptyset , \hfill \\ {\text{then some object }}z{\text{ is located in region }}r_{x} \cap r_{y} . \hfill \\ \end{gathered} \)
I think there are good reasons for adopting this extra assumption (so as to avoid ‘overlapping’ objects which share no parts) that are independent of our concerns with temporal parts. In any case, both 3D-ers and 4D-ers are liable to run into trouble with a (*)-style definition of temporal parts if this extra assumption is eschewed. In both cases, dropping the extra assumption allows for cases in which b is a perfectly good temporal segment of object o, but b does not count as a temporal part of o according to (*) because some other part of o spills over into b’s spatiotemporal location (and so also into the times at which b is present), but does not happen to share any parts with b.
This being the case, there are two other intermediate positions which I find at least as plausible as the one under consideration: (i) a 3D-er might hold that all objects have indefinitely many proper temporal segments, but deny that there is any atemporal parthood relation (since the only likely candidates fail to preserve important ordinary assumptions about parthood); or (ii) a 4D-er might hold that all objects have indefinitely many proper temporal segments, but deny that there is any atemporal parthood relation (since the only likely candidates fail to preserve important ordinary assumptions about parthood).
Fine (2006, p. 704) presents a more sophisticated defense of the three-dimensionalist account of motion, as well as interesting linguistic evidence that there is an intuitive distinction between the way in which objects are thought to be present in space and time and the way in which events are thought to be present in space and time.
See also the discussion of the problem of the rotating homogenous disk in Sider (2001, 6.5). Here, Sider concedes: “There is no denying that endurance theories do a better job of capturing our intuitions about motion in homogeneous substances.” (2001, p. 235).
Admittedly, as e.g. Sider (2000) has pointed out, Lewis’ solution does not allow that persisting objects are bent or shape simpliciter. To that extent, Lewis’ treatment of temporary properties may seem less intuitive than options available to stage theorists, like Sider, or to presentists. But, unlike the perdurantist positions on (TP) and (PART), neither presentism nor stage theory is consistent with (3D) (or, for that matter, with (4D)). Thus, I think that a Lewis-style account might be the best the 3D-er can do if he wishes to preserve an intuition that properties like bentness and straightness are had simpliciter by any objects at all.
Compare this to Kit Fine’s distinction between the three-dimensionalist account of persistence as existence at different times versus the four-dimensionalist account of persistence as extension in time (Fine 2006). I think that Fine is getting at roughly the same kind of distinction as I intend (3D) and (4D) to capture. But Fine does not link his existence/extension contrast to different ways of being located in spacetime and denies that existence and extension can be defined in terms of more basic location relations (2006, p. 701, note 4).
I say “roughly analogous” because the brick fills the interior of my office by occupying multiple spatially extended regions. For the analogy to be exact, we must require the brick (or some object which could conceivably be so small) to occupy multiple spatially unextended regions. We could do this, but the example would lose its visual appeal. As it stands, we are still preserving the central contrast between spanning a spatial/temporal expanse by occupying multiple smaller units as opposed to spanning a spatial/temporal expanse by occupying a single unit that covers the entire expanse.
More carefully: the issue of whether objects stand in atemporal parthood relations seems extraneous to the debate over persistence if eternalism is true (as has been assumed throughout this paper). The case may be different if presentism is taken into consideration since the presentist may have access to a non-relative notion of parthood which is not available to either 3D-ers or 4D-ers.
See Fine (2006, p. 700) for a similar point. Like Olson and myself, Fine thinks that the difference between endurantist and perdurantist accounts of persistence does not hinge on issue (TP) alone.
Note, however, that Olson himself does not identify three-dimensionalism with (3D) or four-dimensionalism with (4D). Olson proposes five theses which he takes to jointly characterize four-dimensionalism (2006, pp. 749–750). I do not have room to present each of these here—among other things, they require that each object has a temporal part (in the sense of (*)) for each time at which it is present and that objects have only atemporal properties. However, Olson’s four-dimensionalist theses do not include any counterpart of (4D). It seems to me that, taken together, they are compatible with either (3D) or (4D). For this reason, it seems to me that, taken together, they do not characterize a distinct mode of persistence.
Note that it is open to both the proponent and the opponent of these sorts non-relativized parts (parts simpliciter) to introduce a different sense in which objects have parts at a spatial region at a time. In fact, given the possibility of multi-location at a time, a temporally and spatially relative parthood relation would seem to be most natural and useful.
I have been arguing for the strong claim that on their own (3D) and (4D) amount to distinct accounts of persistence (though not the only possible accounts of persistence). I think this is right and, if it is, then it follows from the compatibility of both (3D) and (4D) with the standard perdurantist position on (TP) that this latter position does not specify what persistence amounts to (since it is compatible with distinct accounts of persistence). However, even if my claim that (3D) and (4D) amount to independent accounts of persistence is too strong, I still think we have good reason to deny that the temporal parts position tells us on its own what it is for an object to persist. For, even if we were to insist that an object with a full array of temporal parts does not persist in the same way as any object which lacks temporal parts, it still seems to make an important difference in our story of how this object is present through time whether it occupies a single temporally extended region or multiple temporally unextended regions. (Just as the large singly located brick does not fill up my office in the same way as does the small multiply located brick, even if they both have parts at all subregions of the larger spatial region.) If we take the issue concerning parts to be involved at all in accounts of persistence, then we would seem to have a distinct account of persistence for every combination of a position on location with a position on temporal parts. And thus we should still at least deny that the debate over persistence hinges solely on the issue of temporal parts.
It is useful to compare my point here to that of Parsons (2007). Like me, Parsons holds that a position along the lines of (3D)—which, as Parsons points out, is endorsed by some endurantists—is compatible with the thesis that objects have proper temporal parts. Parsons takes this compatibility as one reason for denying that (3D) alone captures the endurantist position. (Parson also doubts whether the notion of exact location assumed in (3D) makes sense—see footnote 11 above.) For, according to Parsons, “what unifies endurantists…is their opposition to temporal parts” (2007, p. 219).
I think that Parsons’ claim about endurantists’ opposition to temporal parts is misleading—whatever they may have said about temporal parts, endurantists like Thomson, Simons, Fine, and Lowe have all held that objects have parts which, on certain reasonable definitions of temporal parts, count as their temporal parts. But even if this were not so, my point is that, unlike the thesis that objects lack temporal parts, (3D) specifies an account of how objects persist. And the dispute between endurantists and perdurantists was supposed to have something to do with how objects persist.
Granted, the presentist might prefer to put this somewhat differently, since for the presentist—but not for the eternalist—to be present at the current time is to exist. So a presentist might rather say that each object exactly occupies a three-dimensional region whenever it exists.
But it is possible that the presentist might make sense of temporal parts in some other way. See the suggestions in Sider ( 2001, 3.4).
I am grateful for the very helpful comments of Eric Olson and an anonymous reviewer on an earlier draft of this paper.
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Donnelly, M. Endurantist and perdurantist accounts of persistence. Philos Stud 154, 27–51 (2011). https://doi.org/10.1007/s11098-010-9526-z
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DOI: https://doi.org/10.1007/s11098-010-9526-z