Three-Dimensionalism

An object o persists over time if and only if it is present at multiple distinct times (where o is present at a time t iff its complete spatiotemporal path—i.e., its career trajectory—passes through t.) For example, my dog Laney Lou persists over time—she has been present at each moment of the past five years, is present at the current moment, and will hopefully remain present at many future moments. Three-Dimensionalism is a position on how objects persist over time. Although, as has been pointed out elsewhere,¹ it is difficult to pin down exactly what positive account of persisting objects three-dimensionalists endorse, it is agreed on nearly all sides that Three-Dimensionalism is opposed to Four-Dimensionalism—the claim, roughly, that objects persist by being extended in time in the same way they are extended in space.² Four-dimensionalists would say that Laney Lou is spread out over her five-plus years of time in the same way she is spread out over four cubic feet of space. They might add that, just as she has parts at different subregions of her spatial extent (e.g., she has ears and a nose on one end and a tail and hind quarters on the other end), so also she has parts in different subregions of her temporal extent. For example, four-dimensionalists might claim that Laney Lou has a temporal part that extends just over her puppyhood, another temporal part that extends just over her adulthood, as well as even shorter-lived temporal parts that extend only over particular moments of her life. Philosophers have commonly supposed that three-dimensionalists must deny some, or all, of these standard four-dimensionalist claims about objects like Laney Lou.

How exactly to make sense of Three-Dimensionalism as an alternative to a four-dimensionalist account of persistence is the topic of the current article. In what follows, I focus on the distinction between Three- and Four-Dimensionalism, highlighting serious shortcomings of familiar characterizations of this distinction. I propose that Three-Dimensionalism is best understood as a position on the grounds for facts about objects’ presence over time.

The most common characterization of Three-Dimensionalism invokes a distinction between (i) persisting by being “wholly present” at multiple times versus (ii) persisting by having distinct temporal parts at distinct times. Entities that persist in the former fashion are said to “perdure,” while entities that persist in the latter fashion are said to “endure.” On the standard characterization, Three-Dimensionalism (or, “Endurantism”) is the thesis that all concrete objects (as opposed, perhaps, to events like weddings) endure, while Four-Dimensionalism (or, “Perdurantism”) is the thesis that all concrete objects perdure. Among others, Lewis makes this distinction when he introduces opposing accounts of persistence in On the Plurality of Worlds.³ Lewis goes on to explain that the four-dimensionalist account of persistence treats extension in time as analogous with extension in space—the perduring object extends through time by having different temporal parts at different times, just as it extends through space by having different (spatial) parts at different places. For example, a road extends from Brooklyn to Queens by having a segment in Brooklyn and a segment in Queens. Analogously, on the four-dimensionalist account, a person persists from 1961 to 2015 by having distinct temporal parts—person-sized temporal segments—in each year of this interval.

The three-dimensionalist account of persistence does not treat presence over an interval of time as analogous to presence over an interval of space. The road extending from Brooklyn into Queens is not spatially “wholly present” in either borough (since, e.g., the Queens segment of the road lies outside of Brooklyn). But what exactly does whole presence at a time consist in? To fill out this alternative to the four-dimensionalist account, Lewis here merely gestures at an analogy with the way in which universals (if there are such things) are supposed be wholly present in each of their instances (1986, p. 202). But this analogy gives us little to go on. Even if they do exist, universals would seem to differ from objects like dogs or roads in respects relevant to accounts of persistence—for example, universals presumably neither gain nor lose parts and so might be present in their entirety wherever they are present in a way that mereologically variable objects like dogs are not.

The crucial notion of whole presence at a time invoked in the standard characterization of Three-Dimensionalism has proven notoriously difficult to pin down. See, for example, Crisp & Smith (2005, pp. 320–332) for a good discussion of several commonly proposed definitions along with convincing reasons for thinking that each is unsatisfactory.⁴ Below, I discuss problems with two of the most influential types of accounts of whole presence at a time—mereological accounts and locative accounts.⁵

Some three-dimensionalists have responded to difficulties in making sense of whole presence at a time by simply dropping the positive “whole presence” requirement from their accounts of persistence. For example, in “The Definition of Endurance,” McCall and Lowe propose the following definition of endurance:

An object endures iff (i) it lacks temporal parts, and (ii) it exists at more than one time.⁶

Since condition (ii) of McCall and Lowe’s definition just stipulates that the enduring object persists, McCall and Lowe construe the three-dimensionalist account of persistence (endurance) as the mere negation of the standard four-dimensionalist account of persistence. On this thin reading of their position, three-dimensionalists are supposed to claim only that objects persist somehow or other in a way that does not require temporal parts.

More attention has been paid to purported advantages and disadvantages of Three-Dimensionalism, as compared to Four-Dimensionalism, than to precise formulations of the opposing positions themselves. The primary alleged advantage of Three-Dimensionalism is that it is supposed to match our ordinary intuitions of the world much better than does Four-Dimensionalism. For example, Thomson highlights the unintuitiveness of the four-dimensionalist claim that, wherever there is any persisting object, indefinitely many distinct spatially co-extensive objects are constantly popping in and out of existence.⁷ Fine presents linguist evidence for an intuitive distinction between the way things (dogs or roads) are supposed to exist at times but extend over space, while events (weddings or ball games) are supposed to extend over both time and space.⁸

Much more numerous are the alleged disadvantages of Three-Dimensionalism. Lewis complains that three-dimensionalists are forced to construe apparently monadic temporary properties as relations to times.⁹ Heller claims that three-dimensionalists are at a serious disadvantage in accounting for apparent cases of spatial coincidence.¹⁰ Given that, for example, a statue is distinct from, but temporarily coincident with, the lump of clay constituting it, the three-dimensionalist is forced to admit that distinct objects may be wholly present at the same place at the same time. The four-dimensionalist, on the other hand, may construe temporary coincidence as an unproblematic case of overlap—the common segments of the statue and the lump are shared temporal parts of both objects. In addition, Heller claims an advantage for Four-Dimensionalism in accounting for apparent indeterminacy in objects’ temporal boundaries.¹¹ The idea is that given their plethora of spatiotemporally overlapping objects, four-dimensionalists may explain apparent durational vagueness as indeterminacy of reference. (We haven’t decided whether “the Empire State Building” refers to an object whose career began with the laying of a foundation or a spatiotemporally overlapping object whose career began later, with the completion of 102 enclosed stories.) Sider also relies on the assumption that the three-dimensionalist cannot admit so many objects as does the four-dimensionalist in claiming that, unlike the latter, the former must accept either sharp cut-offs in whether composition occurs or indeterminacy in what objects exist.¹² Other philosophers criticize Three-Dimensionalism for clinging to intuitive, but scientifically illegitimate, disanalogies between space and time.

A comprehensive discussion of any one of the issues mentioned in the preceding paragraphs is beyond the scope of this article. However, it is helpful to bear in mind assumed advantages and disadvantages of Three-Dimensionalism as we evaluate potential characterizations of it. If a potential characterization of Three-Dimensionalism is too weak to bear out an alleged advantage or disadvantage, it may be that the proposed characterization overlooks some important, though perhaps implicit, aspect of the distinction between Three- and Four-Dimensionalism. Alternatively, it may be that the assumed advantage or disadvantage is based in fuzzy ideas that do not hold up in a more rigorous treatment of Three-Dimensionalism. Three-Dimensionalism’s alleged disadvantage in handling cases of coincidence has been an especially contentious issue in recent literature.¹³ I recur to issues surrounding coincidence throughout this article in relation to various proposed characterizations of Three-Dimensionalism.

In this section, I focus on accounts in which the distinction between Three- and Four-Dimensionalism boils down to a difference in claims about the kinds of parts persisting objects have or lack. There are three general types of mereological characterizations of Three-Dimensionalism: (a) those that retain the positive whole presence at a time component of the standard characterization and offer a mereological analysis of this notion; (b) those that rely solely on the negative claim that persisting objects lack (a plenum of) temporal parts; (c) those that construe Three-Dimensionalism as a distinctive position on the nature of the parthood relation holding among persisting objects. I discuss each type of mereological account separately in the three following subsections.

a.

Mereological Accounts of Whole Presence at a Time

This approach sticks to the standard account, construing Three-Dimensionalism as the thesis that objects persist by being wholly present whenever they are present. The crucial notion of whole presence at a time is here understood mereologically—the whole object is present when all of its parts are present.¹⁴ As has been pointed out elsewhere,¹⁵ such an account of whole presence at a time cannot be construed as a restriction merely on the parts the persisting object has at a fixed time. In particular, the following temporalized analysis of whole presence at a time is much too weak.

(D_TMer-WP) object o is wholly present at time t iff o is present at t and, for any object o*, if o* is part of o at t, then o* is present at t.

On any reasonable construal of what it is for one object to be part of another at a time, it should turn out that o* is part of o at t only if both o* and o are present at t. A cell that was destroyed a year ago cannot now be part of my dog, even though it was part of her a year ago. In particular, the standard four-dimensionalist account of parthood at a time—according to which o* is part of o at time t iff o*’s temporal part at t is part of o’s temporal part at t—entails that o* is part of o at t only if both o* and o are present at t. Thus, (D_TMer-WP) fails to introduce a sense of “whole presence” on which three- and four-dimensionalists might disagree over whether persisting objects are wholly present whenever they are present.

If parthood is not restricted to a single time as in (D_TMer-WP), the most likely alternative mereological account of whole presence at a time construes it as requiring that any object that is ever part of the persisting object is present when the persisting object is wholly present.¹⁶

(D_AMer-WP) object o is wholly present at time t iff for any object o* and any time t*, if o* is part of o at t*, then o* is present at t.

But most three-dimensionalists deny that persisting objects are generally wholly present in the strong sense of (D_AMer-WP). We ordinarily assume that organisms, buildings, and so on, are constantly gaining and losing parts. At times after a former part has been destroyed, or before a future part has been created, the persisting object cannot be wholly present in the sense of (D_AMer-WP). Laney Lou is present now, even though many of her initial cells are no longer present and many of her future cells are not yet present.

What to do? Most three-dimensionalists respond to the difficulty of producing a satisfactory mereological account of whole presence at a time by either abandoning this notion entirely or by construing it in locational, rather than mereological, terms. Both approaches are discussed below. Note, however, that a three-dimensionalist could retain a mereological account of whole presence by endorsing an extreme ontological theory—one that conflicts with ordinary assumptions in denying that persisting objects survive the loss or gain of parts. Chisholm is a notable example of such a three-dimensionalist. According to Chisholm, objects that persist in the “strict and philosophical sense” have exactly the same parts throughout their careers.¹⁷ Such objects must be wholly present at each moment of their careers in the strong sense of (D_AMer-WP). Chisholm’s mereological essentialism entails that apparently mereologically volatile objects survive only for short intervals. Such an unintuitive ontological theory seems to most philosophers too high a price to pay for a mereological account of whole presence at a time.

b.

Negative mereological characterizations of Three-Dimensionalism

A common response to the difficultly of making sense of the notion of whole presence at a time is to drop it entirely and characterize Three-Dimensionalism only through negative claims about the existence of temporal parts. To consider such proposals in more detail, we need to introduce more rigor into our discussion of temporal parts. As will be seen in the next subsection, some philosophers claim that a satisfactory distinction between Three- and Four-Dimensionalism hinges on an appropriate notion of what it is for one thing to be a temporal part of another.

Probably the most influential definition of temporal parthood in current discussions is that of Sider (2001, 3.2). Sider’s temporal parthood relation holds between a temporally unextended object and a persisting object at the instant the instantaneous temporal part is present and part of the longer-lived object. For our purposes, it is more useful to generalize Sider’s approach by introducing a temporal parthood relation that is not restricted to instantaneous temporal parts.

(D_TP-Time) o* is a temporal part of o over time-interval I = _def (i) o* is present at each instant in I and only at instants in I; (ii) o* is part of o at each instant in I; (iii) at each instant t in I, o* overlaps at t each object that is part of o at t.

Time intervals are here understood as ranges of time instants, including discontinuous ranges and “instantaneous” ranges consisting of just one instant. Objects x and y overlap at instant t iff some object z is a part of both x and y at t. Thus, condition (iii) of definition (D_TP-Time) requires that if o* is a temporal part of o over I, then o* is coextensive with o throughout I in the sense that at each instant in I, o* extends to all parts of o’s parts.

A time-independent temporal parthood relation may be introduced in terms of (D_TP-Time)’s time-relative relation as follows.

(D_TP-Atemp) o* is a temporal part of o = _def for some time-interval I, o* is a temporal part of o over I.

It follows from the definitions (D_TP-Time) and (D_TP-Atemp) that every object is, trivially, a temporal part of itself. A proper temporal part of o is any temporal part of o other than o itself.

A strong negative characterization of Three-Dimensionalism construes it as the claim that persisting objects have no proper temporal parts. This is the characterization of Three-Dimensionalism proposed in McCall and Lowe (2009). As Sider points out (2001, 64–66), one important problem with the strong negative characterization of Three-Dimensionalism is that it seems to conflict with the claim, endorsed by many three-dimensionalists including Lowe, that distinct objects may temporarily coincide.¹⁸

Purported cases of temporary coincidence are those in which distinct objects temporarily share the same matter. If the three-dimensionalist admits any cases of temporary coincidence at all, it seems that he must admit cases in which a short-lived object coincides throughout its career with a longer-lived object. For example, a clay statue and a lump of clay may be constituted by the same clay particles from their joint creation until a small bit of clay is broken away from the statue, destroying the lump but not the statue. In this case, it seems that the lump should count as a proper temporal part of the statue. Note that while they coincide, the lump must overlap each of the statue’s parts since both objects, and all of their parts, consist of the same particles at these times. If we further assume, as seems plausible, that the lump is part of the statue while the two objects coincide, then it follows immediately from definitions (D_TP-Time) and (D_TP-Atemp) that the lump is indeed a proper temporal part of the statue.

Weaker negative characterizations of Three-Dimensionalism allow three-dimensionalists to admit that some objects have some proper temporal parts. Three-Dimensionalism may be characterized either as (i) the denial of the claim that each persisting object actually has a temporal part over each interval of its career; or (weaker still) (ii) the denial of the claim that, necessarily, each persisting object has a temporal part over each interval of its career. Sider reluctantly endorses the latter characterization of Three-Dimensionalism (2001, 68). As Eric Olson points out, such a thin characterization of Three-Dimensionalism turns the debate between three- and four-dimensionalists into a question of how many coinciding objects there might be.¹⁹ According to Sider’s extremely weak negative characterization of Three-Dimensionalism, the disagreement between a coincidence-permissive three-dimensionalist and a four-dimensionalist could reduce to the question of whether a single persisting object might lack a temporal part at a single instant of its career. It is not obvious how such minor differences in ontological commitments could amount to distinct accounts of persistence.

Significantly, some three-dimensionalists—including Thomson and Fine—have endorsed ontological assumptions that entail that cases of temporary coincidence are very common.²⁰ Many more philosophers, though perhaps not themselves proponents of coincidence-permissive Three-Dimensionalism, admit that such a position is coherent.²¹

Leaving aside concerns over whether a prohibition on temporal parts is consistent with coincidence-friendly versions of Three-Dimensionalism, there are important further reasons to be skeptical of the proposal that the debate between three- and four-dimensionalists boils down to differences in their positions on the existence of temporal parts. First, and most obviously, given a purely negative characterization of it, Three-Dimensionalism would not tell us how objects persist. At best it would tell us how (some) objects do not persist—not by having a distinct temporal part at each interval of their careers.

Second, it is not even clear why the claim that o has a distinct temporal part at each interval of its career should count as an explanation of how o remains present throughout its career. In other words, it is not clear why the positive position that the proposed characterization of Three-Dimensionalism is supposed to negate should itself count as an explanation of how objects persist, and not merely a position on what objects there are. Consider a spatial analogy. Philosophers may disagree over what kinds of parts objects have at a time. For example, philosophers might disagree over whether objects are composed at a time of spatially extended simples, extensionless simples, or gunk. But these do not seem to be distinct positions on how objects are extended in space—on what it is to, for example, fill up a spherical region now. A basketball would seem to extend over a spherical region in the same way whether or not it has indivisible (or extensionless) parts. But if this is right, then by analogous reasoning the ball should extend in the same way over a stretch of time regardless of whether or not it has a full array of shorter-lived parts.

For all of the reasons considered in the preceding paragraphs the project of characterizing Three-Dimensionalism as a mere prohibition on temporal parts seems problematic.²² In the next subsection, I consider one final proposal for a mereological characterization of Three-Dimensionalism. This proposal construes Three-Dimensionalism as a distinctive claim about the kind of parthood relation holding among persisting objects, and not as a position on when objects’ parts are present or what shorter-lived objects are distributed through intervals of persisting objects’ careers. However, as we shall see, some philosophers take the proposed restriction on parthood to entail that persisting objects have no temporal parts.

c.

Three-Dimensionalism and Time-Relative Parthood

In How Things Persist, Hawley endorses the following characterization of Three-Dimensionalism (in her terminology, “endurance theory”):

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…. [E]ndurance theory is the claim that ordinary material objects—animals, fruit, furniture, and the like—endure.²³

Elsewhere in the same passage, she rejects the standard “temporal parts” characterizations of Four-Dimensionalism (28). Instead, she takes the central four-dimensionalist thesis to be that objects are temporally extended, where an object counts as temporally extended “just if the appropriate basic notion of parthood for that object is an atemporal notion” (28). By contrast, “an object persists without being temporally extended if there is no atemporal notion of parthood appropriate for that object” (28).

As I understand the proposed dual characterizations of Three- and Four-Dimensionalism, the distinction between the two positions is here supposed to reduce to the question of whether or not a parthood relation that is atemporal (or, time-independent)—in the sense that it neither links pairs of objects to specific times at which one is part of the other nor varies in its application from one time to the next—holds between persisting objects and their parts. Note that Hawley does not take the distinction to hinge on the question of whether a time-relative parthood relation holds between persisting objects and their parts. Since four-dimensionalists can introduce a time-relative parthood relation in terms of atemporal parthood relations among appropriate temporal parts of persisting objects, they are allowed both a basic atemporal parthood relation and a secondary time-relative parthood relation (26–27). It is only the three-dimensionalists who are supposed to be mereologically restricted in having no access to a parthood relation that holds eternally and independently of particular times.

I know of no philosopher besides Hawley who has characterized Three-Dimensionalism solely through its position on the time-relativization of parthood claims. However, several other philosophers, including Sider, Olson, Merricks and McKinnon, have claimed that, except for mereological essentialists like Chisholm, three-dimensionalists must deny that there is any time-independent sense in which one persisting object is part of another.²⁴

Relevant to considerations raised in the previous subsection, both Hawley and Olson take the three-dimensionalist’s rejection of atemporal parthood to entail that she must also deny the existence of (appropriately defined) temporal parts. In objecting to Sider’s definition of temporal parthood in terms of a time-relative parthood relation, Olson claims that we “need to understand parthood timelessly in order to understand the concept of temporal parthood” (2006, 748). Hawley claims that three-dimensionalists “cannot make sense of the notion of a temporal part, because they cannot accept atemporal talk about parthood” (2001, 26). Thus, Hawley and Olson deny that temporal parthood can be defined as in (D_TP-Time)—which, as noted, is a generalization of Sider’s own definition—but must be instead introduced in terms of a parthood relation that is not relativized to times. For example, the following emendation of (D_TP-Time) may be acceptable.²⁵

(D_TP) o* is a temporal part of o over time-interval I = _def (i) o* is present at each instant in I and only at instants in I; (ii) o* is part of o; (iii) o* overlaps each object that is part of o and present during I.

Importantly, both Hawley and Olson allow that the three-dimensionalist might endorse an ontology that is nearly as full of short-lived objects as that of the four-dimensionalist.²⁶ Thus, both seem to take the three-dimensionalist’s rejection of temporal parts to depend solely on her commitment to time-relativized parthood, and not at all on her ontological commitments. In principle, the three-dimensionalist could hold that for nearly every interval I of its career, persisting object o coincides with an object o_I that is present only throughout I. Nonetheless, say Hawley and Olson, the three-dimensionalist cannot construe any of the shorter-lived objects as temporal parts of o because she cannot construe them as parts, in the relevant timeless sense, of o.

The problem with this formulation of the three-dimensionalist position is that it is hard to see what could possibly prohibit the three-dimensionalist, as opposed to the four-dimensionalist, from counting segments of persisting objects as atemporal parts of the longer-lived objects (and so also as temporal parts in Hawley’s and Olson’s sense). The purported restriction on parthood seems especially unmotivated if the three-dimensionalist may adopt an expansive ontology, including a lot of shorter-lived objects that look very much like the four-dimensionalist’s temporal parts. But even given a more parsimonious three-dimensionalist ontology, it is not obvious why, say, a muscle cell that lives out its entire career as a part of my dog’s body should not count in an atemporal sense as a part of my dog, and not just a part of her at specific times.

Like four-dimensionalists, three-dimensionalists can perfectly well make sense of a binary relation PAR holding between objects o* and o just in case o* is throughout its career materially included in o. Moreover, there is no reason why the three-dimensionalist, any more than the four-dimensionalist, should say that PAR holds between objects at specific times. After all, PAR is not a relation like loves that changes over time—there is no sense in which o* can be materially contained in o throughout its entire career at some times and not at other times. But PAR is just the four-dimensionalist’s atemporal parthood relation. If the four-dimensionalist can construe PAR as a kind of parthood relation (even as, in some sense, the parthood relation), why can’t the three-dimensionalist do so as well? True, PAR does not line up well with our ordinary use of the term “part of.” The tires on my bicycle are ordinarily described as “parts” of the bike even though they do not stand in the PAR relation to it (since they spent time in a shop before being becoming parts of the bike). But no one takes this sort of mismatch between the atemporal PAR relation and ordinary time-relative parthood as a reason why the four-dimensionalist cannot construe PAR as a kind of parthood. Why should it be any more problematic for three-dimensionalists?²⁷

Before leaving our discussion of mereological accounts of Three-Dimensionalism, note finally that if, as Thomson, Fine, Lowe and others suggest, the three-dimensionalist might adopt an expansive ontology, which is almost as full of short-lived objects as that of the standard four-dimensionalist, and if, as is argued in the current subsection, atemporal parthood is equally available to three- and four-dimensionalists, then one influential argument for a four-dimensionalist advantage in cases of coincidence fails.²⁸ For then the expansive three-dimensionalist might, like the expansive four-dimensionalist, construe temporary coincidence as mere overlap.²⁹

A different kind of approach distinguishes Three- and Four-Dimensionalism partly or wholly through claims about objects’ locations in spacetime.

(3D) Each persisting object occupies multiple temporally unextended regions of spacetime.

(4D) Each persisting object occupies a single temporally extended region of spacetime.

In standard locative accounts, Three-Dimensionalism is (partially) characterized by a commitment to (3D), while Four-Dimensionalism is (partially) characterized by a commitment to (4D).³⁰

Proponents of the locative approach typically cash out the standard whole presence at a time component of Three-Dimensionalism as the claim that the persisting object occupies a region within each time in the span of its career. In this context, times are just timeslices—maximal temporally unextended slices of spacetime—and the regions occupied by a persisting object are the intersections of its career path with the time(slice)s through which it persists. The persisting object is wholly present at a time, in the sense that it is wholly located within that time. Such a locative characterization of Three-Dimensionalism thus seems to overcome an important shortcoming of standard mereological characterizations of Three-Dimensionalism. Here, we seem to have a positive three-dimensionalist account of persistence—enduring objects remain present through multiple times by occupying a succession of regions within those times.

Unfortunately, there is serious objection to such locative characterizations of Three- and Four-Dimensionalism. It is not obvious that there is any concept of occupation (or, location) about which three- and four-dimensionalists might disagree. Here are two seemingly legitimate interpretations of the term “occupies” as used to relate persisting objects to spatiotemporal regions.

o occupies_Pathr iff r is the smallest region of spacetime that includes all of o over the course of o’s career.

o occupies_Timer at time t iff r is the smallest subregion of timeslice t that includes all of o within t.³¹

Note that the binary relation occupies_Path does not match the ordinary use of the term “occupies.” For example, a basketball stands in occupies_Path not to any spherical region, but only to a single four-dimensional region that tracks the ball throughout its career. Nonetheless, occupies_Path seems a natural relation to use when locating objects from a “God’s-eye” perspective, which takes the whole content of spacetime (including any object’s entire career) into account.

By contrast, occupies_Time does correspond more or less to the ordinary temporally situated use of “occupies.” The basketball stands in occupies_Time to specific spherical regions at specific times throughout its career. Notice further that, although occupies_Time relates objects to regions at specific times, variations on this relation need not invoke specific times.

o occupies_SomeTimer iff for some time t, o occupies_Timer at time t.

The binary relation occupies_SomeTime holds between the basketball and multiple spherical regions. More generally, any persisting object stands in occupies_SomeTime to multiple temporally unextended spacetime regions.

Three- and four-dimensionalists cannot disagree over the sorts of regions to which persisting objects stand in the relations occupies_Path, occupies_Time, or occupies_SomeTime. Both sides agree that: (i) occupies_Path holds between a persisting object and a unique temporally extended region; (ii) occupies_Time links objects and times to temporally unextended regions; and (iii) occupies_SomeTime links each persisting object to multiple temporally unextended regions. Thus, if occupies_Path captures the intended meaning of “occupies” in (3D) and (4D), then both three- and four-dimensionalists agree that (3D) is false and (4D) is true. If instead occupies_SomeTime captures the intended meaning of “occupies” in (3D) and (4D), then both three- and four-dimensionalists agree that (3D) is true and (4D) is false.³²

What is required for (3D) and (4D) to distinguish Three- from Four-Dimensionalism is a sense of “occupies” that, unlike the relations considered above, is neutral as to whether persisting objects occupy multiple temporally unextended regions or, instead, occupy unique temporally extended regions. Could there be some such further occupation relation? Perhaps, but so far no proponent of locative characterizations of Three- and Four-Dimensionalism has explained just what it is supposed to be. Typically the intended occupation relation is indicated only very roughly, usually through associations between the geometrical properties of objects and regions. For example, Gilmore explains that his exact occupation relation “is said to hold between a thing and a region just in case, loosely put, the thing exactly fits into the region, where this is meant to guarantee that the thing and the region have precisely the same shape, size, and position” (2006, 200).³³ Gilmore goes on to list four further assumptions about exact occupation, but these are all negative assumptions—denials of claims that exact occupation must be restricted in certain ways—and so do little to fill out the picture of what the relation is supposed to be. Importantly, Gilmore’s loose positive characterization in terms of geometric properties may be interpreted differently depending on whether shape, size, and position are supposed to be properties had by objects at times or, instead, properties characterizing objects’ entire careers. Given the former interpretation of geometric properties, exact occupation seems to amount to occupies_SomeTime, making (3D) true and (4D) false. Given the latter interpretation, exact occupation seems to amount to occupies_Path, making (4D) true and (3D) false.

Are proponents of locative characterizations required to give any transparent account of the occupation relation invoked in (3D) and (4D)? It has been suggested that for such accounts to work at all, this relation must be taken on board as a primitive—that is, a notion that is not defined (within a given theory) in terms of simpler properties or relations.³⁴ However, even relational primitives are typically endowed with some kind of meaning through examples of their intended applications. With neither a definition nor examples, it is hard to see what this special occupation relation is supposed to be. The crux of the problem for the proposed locative characterizations of Three- and Four-Dimensionalism is that the intended occupation relation can distinguish the opposing accounts only if there are no noncontroversial examples illustrating how it is supposed to link persisting objects to spatiotemporal regions. For, three- and four-dimensionalists are supposed to support incompatible claims about the regions occupied by persisting objects. Thus, it seems in principle impossible for the proponent of the kind of locative account under consideration to indicate the intended sense of “occupies” in (3D) and (4D) by pointing to neutral examples.

Without an account of the intended sense of “occupies” in claims (3D) and (4D), it is hard to see what the purported locative disagreement between three- and four-dimensionalists is supposed to be about or what might count as a reason for favoring one or the other side of the dispute. For this reason, it seems best to set aside locative characterizations of Three- and Four-Dimensionalism until they can be framed in terms of a clearer notion of occupation.

Finally, note that without a disagreement over locational claims along the lines of (3D) and (4D), it is not clear why three-dimensionalists would be at a special disadvantage in handling purported cases of coincidence, where coincidence is construed locationally in such a way that o and o* coincide iff o and o* occupy the same region. Just like coincidence-friendly three-dimensionalists, four-dimensionalists generally hold that distinct objects may stand in occupies_SomeTime to the same region and in occupies_Time to the same region at the same time. Though four-dimensionalists generally deny that distinct objects stand in occupies_Path to the same region, we have not yet seen a reason why coincidence-friendly three-dimensionalists cannot just as easily prohibit path-sharing. Such a reason will be suggested in the next section with one final strategy for distinguishing between Three- and Four-Dimensionalism.

A promising proposal is to distinguish between Three- and Four-Dimensionalism not primarily (if at all) on the basis of claims about what parts persisting objects have or what regions they occupy, but instead on the basis of claims about the grounds (or lack of grounds) for facts about how persisting objects are situated in the world, at a time and over time. For example, Hawthorne suggests that the central distinction between Three- and Four-Dimensionalism boils down to differing positions on what kinds of objects or facts are fundamental—in particular, whether facts about spatiotemporally unextended objects are more fundamental than facts about extended persisting objects.³⁵ On Hawthorne’s reading of the debate, the issue is not just whether persisting objects are composed of short-lived parts, but, more importantly, whether facts about unextended objects suffice on their own to ground all facts about persisting objects. A different suggestion from Hofweber and Velleman is that the special three-dimensionalist form of persistence (endurance) has to do, not with an object’s parts or locations, but with the grounds for the instantiation of identity properties. Roughly, Hofweber and Velleman construe endurance in such a way that object o endures only if at each time t of o’s career the contents of timeslice t alone fully determine that o is present at t.³⁶ Hofweber and Velleman take their account of endurance to be very restrictive, entailing that the only endurants are objects that may have discontinuous career paths. They suggest, in particular, that it follows from their account that persons and artifacts do not endure, since they must have continuous paths.

Like Hofweber and Velleman, Hawley proposes in her 2008 paper “Persistence and Determination” that Three-Dimensionalism posits spatiotemporal restrictions on potential grounds for facts about what objects are present at a time.³⁷ Hawley takes three-dimensionalists to be committed at least to the following claim.

(3D-Det) Facts about what objects are present at time t do not depend on what occurs after t.³⁸

Notice that (3D-Det) leaves open the possibility that there may be no grounds at all for facts about what objects are present at time t. It only restricts possible grounds for object o’s presence at t to the contents of spacetime up to, and including, t. (3D-Det) is thus less drastic than Hofweber and Velleman’s proposal that the contents of t alone fully determine what enduring objects are present at t.

Hawley takes four-dimensionalists to be committed instead to the following claim.

(4D-Det) Facts about what objects are present at time t depend in part on what occurs after t.

According to (4D-Det), future events can play a role in determining what objects are present now. According to (3D-Det), facts about what objects are present now are fixed independently of how things pan out in the future—a future event cannot require that different objects are present now than would be if that event did not occur.

Most four-dimensionalists do seem to be committed to (4D-Det). We can see this in at least two ways. First, many, if not most, four-dimensionalists are committed to what I will call a “simple expansive ontology.” A simple expansive ontology is one that claims that any matter-filled region of spacetime is the career path of exactly one object.³⁹ Proponents of a simple expansive ontology seem to be committed to (4D-Det). To see this, suppose that all matter is instantaneously annihilated (perhaps through an act of God) five seconds from now. In this Annihilation Scenario, the only objects currently present are objects whose careers end in five seconds or less. Now consider a less pessimistic scenario, in which everything is roughly the same as in the Annihilation Scenario up to four seconds from now, but, unlike in the Annihilation Scenario, matter is not annihilated in five seconds—instead, matter remains in the world for years to come.⁴⁰ Then, given a simple expansive ontology, roughly the same almost-departed objects—objects with less than five seconds left in their careers—are now present in both the Annihilation Scenario and the Less Pessimistic Scenario. But in the latter scenario, very many additional objects would now be present—objects whose careers continue minutes, days, or years into the future.

Even if a four-dimensionalist does not endorse a simple expansive ontology, she seems committed to (4D-Det) if she denies that distinct objects may share the same career path (i.e., stand in occupies_Path to the same region), but allows that distinct objects may share an initial path segment.⁴¹ We can see this by again contrasting an annihilation scenario, in which all matter is destroyed before a potentially branched initial path segment has a chance to split, with a less pessimistic scenario, in which matter remains long enough for the same initial path segment to develop into the beginning of the careers of two distinct objects. Less dramatically, suppose a statue and a lump of clay were created simultaneously five minutes ago when an artist joined the statue’s clay head to its clay body. An hour from now, displeased with her work, she smashes the clay, destroying the statue but not the lump. In this Smashing Scenario, two objects—a statue and a distinct lump—are now present, occupying now the same three-dimensional region. In an alternative scenario, the artist expresses her dissatisfaction by ripping the statue in half instead of smashing it. Both statues and lumps are the sorts of things that perish when they are ripped in half. Thus, in the Ripping Scenario, there is a single combined statue/lump career trajectory, where in the Smashing Scenario there are distinct statue and lump career trajectories. The four-dimensionalist who allows that statues and lumps may share an initial path segment, but denies that distinct objects may share the same complete path, must hold that different objects would now be present in the Ripping Scenario than those now present in the Smashing Scenario. Where in the Smashing Scenario had been a statue and a distinct lump, there would be in the Ripping Scenario a single object that qualifies as both a statue and a lump.

Though their commitments regarding claims along the lines of (3D-Det) are in some ways less clear, it is plausible that most three-dimensionalists are at least implicitly committed to (3D-Det). Hawley herself sees this as a straightforward implication of standard three-dimensionalist claims about persisting objects’ “whole presence” at a time. She says, for example, that three-dimensionalists “should baulk at the claim that temporally distant goings-on can determine whether an object is present at a time at which, if it were present at all, it would be wholly present” (2008, 9). Hawley’s point is that whatever exactly “whole presence” amounts to, it doesn’t seem that three-dimensionalists would say that an object is “wholly present” now if they allow the fact that it is present now to depend on how things turn out in the future. In addition, (3D-Det) clearly fits ordinary assumptions about how objects are present over time better than does (4D-Det). We do not ordinarily think that future events retroactively account for what is in the world now—that, for example, what objects are now on my desk depends on what will happen in the future. As noted in section 2, Three-Dimensionalism, unlike Four-Dimensionalism, is supposed to roughly preserve ordinary assumptions about how objects are present over time.

An important advantage of the proposed distinction between Three- and Four-Dimensionalism is that, unlike the mereological and locative characterizations considered in sections 3 and 4, (3D-Det) and (4D-Det) can make sense of the assumed disadvantage of Three-Dimensionalism in handling cases of apparent coincidence. We have already seen this in the statue and lump example. Unlike the proponent of (4D-Det), the proponent of (3D-Det) cannot claim that the number of objects created by joining a clay head to a clay body depends on whether the clay is later smashed or, instead, ripped in half. But then it seems that the proponent of (3D-Det) must either deny that a statue and a distinct lump of clay share an initial path segment in the Smashing Scenario or allow that a statue and a distinct lump share the same complete career path in the Ripping Scenario. The former option significantly restricts the three-dimensionalist’s options for admitting cases of temporary coincidence in way that preserves ordinary assumptions about the circumstances under which objects like statues or lumps are created. The latter option leaves the three-dimensionalist open to the especially difficult grounding problems surrounding cases of permanent coincidence. If the statue and the lump share the same path, then they must share all basic physical properties throughout their careers. What, then, could account for the distinction between the two objects?⁴²

Aside from the special problems raised by potential cases of permanent coincidence, the proponent of (3D-Det) is at a disadvantage even in accounting for cases of mere temporary coincidence. If what objects are now present does not depend on what occurs in the future, then even if initially coincident objects differ eventually in their basic physical profiles, such future differences cannot provide grounds for the distinction between the objects—that they are distinct does not depend on future differences. Thus, if three- and four-dimensionalists differ in their commitments to (3D-Det) and (4D-Det), then they are not equally free to claim that distinctions between temporarily coinciding objects may be grounded in distinctions in the objects’ future parts and properties, as is claimed in Wasserman (2002) and Mcgrath (2007).⁴³

(3D-Det) may also help explain other alleged disadvantages of Three-Dimensionalism. Most obviously, (3D-Det) seems to impose a significant disanalogy between space and time. What objects are present at a given spatial region depends on how matter is distributed around the region in all directions and not just in one privileged direction. There is no spatial analogy for the special irrelevance that (3D-Det) accords to the future. Also, (3D-Det) may help explain the presumption that three-dimensionalists are at a disadvantage in handling cases of apparent indeterminacy in diachronic composition or in objects’ temporal boundaries. Standard four-dimensionalist accounts assume either a simple expansive ontology, or at least an ontology expansive enough to guarantee that there are indefinitely many objects with slightly different temporal boundaries wherever there is any object. Three-dimensionalists have claimed that they may, if they wish, adopt the same kind of approach as do four-dimensionalists so long as they are willing to endorse similarly expansive ontologies.⁴⁴ However, as we saw in comparing the Annihilation Scenario to the Less Pessimistic Scenario, the claim that there are indefinitely many objects with slightly different temporal boundaries wherever there is any object seems to conflict with (3D-Det).

Hawley suggests that three-dimensionalists are committed not only to claims like (3D-Det), which limit potential grounds for facts about what objects are present at a region at a time, but also to restrictions on potential grounds for facts about objects’ sortal properties.

(3D-Sort) If an object o is present at time t, then o’s sortal properties do not depend on what occurs after t.

If the fact that a particular object o is present at a given time may be determinate without it also being determinate what sort of object o is, then (3D-Sort) is independent of (3D-Det). Even if (3D-Sort) is a consequence of (3D-Det), the former claim is worth considering separately because it suggests an appealing positive three-dimensionalist account of persistence.⁴⁵ Three-dimensionalists might say that objects remain present over intervals of time by realizing potentialities characteristic of their sorts. If an object l present at time t is a lump of clay and it is characteristic of lumps to persist so long as, and only so long as, all of their underlying material hangs together, then l’s continued presence after t is explained partly by the fact that l’s material continues to hang together and partly by the fact that l is a lump. If, say, l had been a statue rather than a lump and if l’s material were squashed immediately after t, then l would not be present after t even though its material continues to hang together. In this sense, l’s persistence is a realization of its potentiality as a lump to remain present under specific conditions.⁴⁶ Note that such an explanation of o’s continuing presence from one time to the next is not available if facts about o’s sortal properties are themselves grounded in facts about o’s career-wide properties.

Standard characterizations of Three-Dimensionalism rely on an underspecified notion of whole presence at a time. The most common proposals for cashing out this notion through restrictions on persisting objects’ parts or locations are plagued by the sorts of problems described in this article. It is not obvious why any account of persistence should involve distinctive claims about the parthood relation or the spatiotemporal regions occupied by persisting objects. Further, although standard three- or four-dimensionalist claims about temporal parts do entail distinctive ontological commitments, it is not obvious why such ontological commitments would amount to a particular account of how objects persist over time.

A more promising approach is to distinguish between Three- and Four-Dimensionalism on the basis of claims about the grounds for facts about persisting objects’ presence over time. I have tried to show how one such characterization of Three-Dimensionalism might help make sense of otherwise obscure assumptions regarding the disadvantages of Three-Dimensionalism in handling cases of apparent coincidence (and perhaps also assumed limitations of Three-Dimensionalism in handling cases of apparent vagueness in temporal boundaries or in diachronic composition). The grounding characterization also suggests a much-needed positive three-dimensionalist account of persistence—objects remain present over time by realizing their potentialities. Obviously, much more work is required to develop this suggestion into a viable account of persistence—most importantly, it needs to be aligned with a more general account of grounding relations among facts.⁴⁷ But given that standard mereological or locative characterizations of Three- and Four-Dimensionalism have led to a sometimes unproductive discussion, in which one side’s alleged advantage is immediately co-opted by the other side, it is well worth exploring a new way of framing the debate between competing accounts of persistence.⁴⁸