The Metaphysical Equivalence

**of Three and Four Dimensionalism***** **

Abstract

I argue that two competing accounts of persistence, three and four dimensionalism, are in fact metaphysically equivalent. I begin by clearly defining three and four dimensionalism, and then I show that the two theories are intertranslatable and equally simple. Through consideration of a number of different cases where intuitions about persistence are contradictory, I then go on to show that both theories describe these cases in the same manner. Further consideration of some empirical issues arising from the theory of special relativity lead me to conclude that the two theories are equally explanatory, and thus finally that they are metaphysically equivalent.

1 Introduction

When it comes to accounts of how objects persist through time, there are two main games in town: three dimensionalism and four dimensionalism. Four dimensionalists hold that objects are extended in time as well as space, and thus have four dimensions, while three dimensionalists hold that objects are extended in only three spatial dimensions. The most usual version of four dimensionalism is perdurantism, according to which objects persist by perduring–by being mereological fusions of temporal parts. Three dimensionalism or endurantism is the view that objects persist by enduring–by being wholly present at each time at which they exist. Both endurantists and perdurantists vigorously defend their views, often claiming that only *their* view is capable of solving various problems associated with change over time, or of respecting key folk intuitions. Some even go so far as to say that the opposing view is incoherent, indefinable, or just plain incomprehensible. In this paper I want to argue that appearances to the contrary, three and four dimensionalism are metaphysically equivalent theories, and thus neither one is to be preferred over the other.

2 What is Metaphysical Equivalence?

I say that three and four dimensionalism are metaphysically equivalent, but what does that mean? Briefly, the idea is that two theories are metaphysically equivalent just in case the theories are two ways of describing the same underlying metaphysics. Specifically, two theories x and y are metaphysically equivalent just in case:

(i) the set of worlds in which x is false is identical to the set of worlds in which y is false and

(ii) the set of worlds in which x is true is identical to the set of worlds in which y is true and

(iii) in every world w in which x is true, the part of w in virtue of which x is true, is identical to the part of w in virtue of which y is true and

(iv) in every world w1 in which x is false, the part of w1 in virtue of which x is false, is identical to the part of w1 in virtue of which y is false.

Elsewhere I have argued that there are a number of diagnostic criteria of metaphysical equivalence, and that two or more theories that meet each of these criteria should be considered metaphysically equivalent. Then two theories will be metaphysically equivalent if they are empirically equivalent theories for which there is a nonarbitrary mapping of the sentences of one theory onto the sentences of the other theory that preserves the truth values of those sentences (a translation manual) and where both theories share the same theoretical virtues to the same degree. Here I intend that theories are empirically equivalent if not only is there no observational prediction in the actual world that is a prediction of one theory and not the other, but further if there is no possible observational prediction that would be a prediction of one theory and not the other. Furthermore, let us say that we have a *nonarbitrary mapping* of the sentences of one theory onto the sentences of the other just in case if one theory recognises some difference in kind between certain phenomena, then so does the other theory. Finally, two theories will be metaphysically equivalent only if they share the same theoretical virtues to the same degree, that is, if they are equally simple, equally explanatorily powerful and so forth. Since I hold that some theoretical virtues supervene on a combination of other virtues, it will not be necessary in this context to show that two theories share the same degree of *all* theoretical virtues. Instead, I will focus on just two: simplicity and explanatory power, for these are two of the most central of the virtues, the combination of which plausibly provides the supervenience base for other virtues such as beauty.

3 Defining Endurance and Perdurance

Endurantism and perdurantism are but two metaphysical theses among a web of interrelated theories, and proponents of either of these views will almost certainly also be committed to some other of these interrelated views. The questions facing endurantists and perdurantists include whether to embrace presentism or eternalism, restricted or unrestricted composition, a tensed or a tenseless theory of the truth of propositions and so forth. Though each of these additional metaphysical theses–call them ancillary metaphysical commitments–is in some sense orthogonal to the issue of persistence, it is natural that some combinations of these views are more common than others.

I will not argue that every combination of these views is equivalent. An endurantist who is a presentist and believes in restricted composition no doubt has a view substantially different to a perdurantist who is an eternalist and believes in unrestricted composition. What I will argue is that for any bundle of views paired with perdurantism, the very same bundle paired with endurantism is metaphysically equivalent. Of course, if it turned out that endurantism is coherent only given presentism, and perdurantism is coherent only given eternalism, then it would turn out that endurantism and perdurantism are not metaphysically equivalent unless presentism and eternalism are too. But I do not think this latter is the case: I will argue that given plausible definitions of endurance and perdurance, each of these theories are compatible with a number of other metaphysical theses, and so long as we restrict ourselves to considering versions of endurantism and perdurantism that are committed to the same ancillary metaphysical theses, we see that they are indeed metaphysically equivalent.

So our first task is to define endurantism and perdurantism in such a way that they are compatible with as many ancillary metaphysical commitments as possible. Let us say that endurantism is the thesis that every actual object persists by enduring. Then we need to define what it is for an object to endure. In the following definition I introduce the term ‘partmb’ which I will define shortly. Our definition is then as follows:

E: An object *endures* iff it is wholly present at each moment at which it exists, where an object is ‘wholly present’ at a time just in case all of its parts are present at that time and where ‘P is part of O’ is true at any time t iff at t, P is partmb of O tly.

This definition captures the core endurantist intuition that in some sense all of an enduring object’s parts are present whenever it exists. Not surprisingly, this definition tells us nothing about which composite objects exist. It does not, for instance, rule out the possibility that there exist temporally scattered objects that endure, nor does it rule out the possibility that there exist odd gerrymandered objects such as trout-turkeys, which if they persist, do so by enduring. So the definition is compatible with either restricted or unrestricted composition.

Definition E is compatible with eternalism and a tenseless reading of propositions. The definition borrows from an adverbialist account of properties according to which the having of properties is temporally relativised such than an object that is red at t1 is red in a t1ly manner. Suppose that at t1 O is red, at t2 O is red, and at t3 O is blue. If we combine a tenseless theory of the truth of propositions with the endurantist view that persisting objects are strictly identical across time, then it follows that at t1, O has the property of being red t1ly, and also that O has the property of being red t2ly and blue t3ly. Clearly then, having the property of blue in a tnly manner is not sufficient for something to count as being blue in the everyday sense in which we attribute blueness. For O is blue t3ly at t1.

So the adverbialist needs to distinguish between what we will call "metaphysically basic" properties which are the temporally relativised properties of being, for instance, blue t3ly at t1, and the ordinary "English" sense of ‘property’ which involves being blue t3ly at t3. Call the metaphysically basic sense of having a property, having a propertymb or being Pmb. Then for the endurantist, having a property P at t in the ordinary sense corresponds to having propertymb P at t in a tly manner. That is, being blue at t1 in the ordinary sense is having the propertymb at t1, of being blue t1ly, thus the ordinary attribution of ‘O is blue’ made at t1 is true iff O is blue t1ly.

Now if E is to be compatible with a tenseless theory of the truth of propositions, we must understand ‘O is blue’ uttered at t1, not as a proposition in the narrow sense, but rather, as picking out different propositions at different times in the same manner as do indexicals. So if O is red at t1, then at t1 ‘O is red’ picks out the proposition ‘O is red t1ly’. Hence for any property P, and time t, ‘O is P’ uttered at t picks out the proposition ‘O is P tly’. Thus the property of being red names a relation between a time and having the propertymb of being red in a particular temporal way–red tnly. Thus the property of being red in this ordinary sense, picks out a different property at t1 than at t2. So there is no contradiction in saying that the very same object O is red at t1 and is not red at t3.

So too in definition E we perform much the same manoeuvre only in terms of parthood. We note that there is some metaphysically basic sense of having a *part* in a temporally modified manner. Call this having a partmb. For suppose that O is at t1 composed of A and B, and at t2 composed of A and C. At t1 O has part A and B t1ly, and has part C t2ly. So there is some technical sense–having a partmb–in which O has part C at t1. For the endurantist though, just as having a property is not the same as having a propertymb, so too having a part is not the same as having a partmb. The ordinary sense of having a part is captured by having some partmb P at t in a tly manner.

Then just as we understood that the ordinary property ‘red’ picks out a different proposition at different times, so too we must understand ‘P is part of O’ as picking out a different proposition at different times. At t, ‘P is part of O’ picks out the proposition ‘P is partmb of O tly’ and at t1 it picks out the proposition ‘P is partmb of O t1ly.’ So the property of having some part at a time is a different property to having that part at some other time. The property of having P at t is the property of having P tly at t. Thus there is no contradiction in holding that O has part P at t1 and lacks part P at t2, for the property of having part P at t1 is the property of having P t1ly at t1, and the property of having P at t2 is the property of having P t2ly at t2, and these are distinct properties. Hence definition E is compatible with eternalism and a tenseless theory of the truth of propositions.

E is also compatible with presentism, and matters are much more straightforward in that case. For if presentism is true, then the only time that exists is the present. So at whatever time t an object O exists, it is trivially true that all of its parts exist at t. For it is trivially true that all of O’s partsmb are had tly at t. Or to put it another way, O’s parts just are O’s partsmb. Hence E is compatible with a tensed reading of propositions, such that at one time the proposition ‘P is part of O’ may be true, while at another time it may be false.

So far then, we have a definition of endurance that is compatible with a number of ancillary metaphysical commitments. We can turn now to define perdurantism. In general, for perdurantists parthood is atemporal. If O is a perduring object and P is a (temporal or spatial) part of O, then ‘P is part of O’ is timelessly true: it is true when uttered at any temporal location. Almost all perdurantists are also eternalists. Berit Brogard, however, argues that not only is perdurantism compatible with presentism, but that such a combination has much to recommend it. His idea is that persisting objects are composed of temporal parts, but that at any time t, the only stage that exists is the t-stage. I cannot here defend this view, so I will simply assume that (given the coherence of presentism) perdurantism and presentism are consistent. Now a usual definition of perdurance would be as follows:

P: An object O perdures iff it is the mereological fusion of temporal parts, where x is an instantaneous temporal part of y at t iff x is part of y and x exists at, but only at t and x overlaps every part of y that exists at t.

Depending on how we construe the notion of a mereological fusion, definition P may or may not be consistent with presentism. Typically of course, we would tend to think that if there exists a fusion of objects at different times, then each of those times must be equally ontologically real: thus eternalism must be true. It is conceivable though, for the presentist to read ‘fusion’ such that there can exist a fusion of objects that did exist, do exist and will exist. Then a perduring object is an object composed of temporal stages some of which did exist, some of which do exist, and some of which will exist. Those who hold that ‘fusion’ cannot be read this way can amend the definition such that an object O perdures iff it is composed of a series of temporal parts.

It is not clear to me that either presentist endurantism or presentist perdurantism are in fact coherent: in what sense is an object temporally extended if every part of it that exists, exists in the present; in what sense is an object that exists now, strictly identical with a non-existent object in the past? I do think, however, that if sense can be made of presentism at all, it turns out that endurantist and perdurantist versions of presentism are equivalent. However, since arguably eternalism is a more oft accepted metaphysics, I will in general discuss eternalist versions of three and four dimensionalism. A major part of the project of demonstrating that these two theses are equivalent, though, rests on showing that the definitions of each are inter-translatable. Since the definitions can be read within a presentist metaphysics, such a demonstration would go a long way toward showing that presentist versions of three and four dimensionalism are metaphysically equivalent.

3.1 Further Ancillary Commitments

Another set of ancillary metaphysical commitments rests on the issue of which arrangements of basic particulars one takes to compose some composite object. One answer to this question is that every arrangement of basic particulars at times composes some persisting object. Call this the view that composition is unrestricted. Another answer to the question is that only some arrangements of basic particulars at times composes a persisting object. Call this the view that composition is restricted. The usual way, though as we shall see not the only way, to understand the claim that composition is unrestricted is in terms of mereological universalism, the thesis that there exists a mereological fusion of the members of every non-empty set S of concrete particulars. Then let us say that one is a non-universalist just if one holds that there exists a mereological fusion of the members of only some non-empty sets of concrete particulars.

Most commonly, perdurantists are universalists. Call this version of perdurantism "unrestricted perdurantism". But there are non-universalist perdurantists such as Storrs McCall who hold that odd gerrymandered objects like trout-turkeys do not exist. "Everyday" objects exist of course, and they persist by perduring. McCall’s view then, is a typical non-universalist view, since he holds that all of the persisting objects that exist are mereological fusions of instantaneous objects–temporal parts. Call a non-universalist view such as this "restricted perdurantism".

Endurantism on the other hand, is often associated with the rejection of universalism. So endurantists too will be non-universalists, though not of the same stripe as McCall. Endurantists can clearly hold that there exist some fusions of particulars: they can and do believe that there exist at least some fusions. The endurantist must deny, however, that everyday persisting objects–enduring objects–are mereological fusions of instantaneous objects. That endurantists are non-universalists, however, does not mean that they must take composition to be restricted.

Recently, Ted Sider has argued that mereological universalism entails perdurantism. The idea is that for any arbitrary set of objects S1, S2... at times t1, t2,,... there exists some fusion of the members of S1 at t1, and of S2 at t2, where x is a fusion of S at t iff every member of S is part of x at t, and each part of x at t overlaps at t some member of S. Such a fusion at a time will be an instantaneous object that has as spatial parts, each of the particulars that are fused at t. Call such an object a synchronic fusion. Then given that there exist synchronic fusions, it follows that some sets will have as members the particulars that are these synchronic fusions. A fusion of synchronic fusions will be a persisting object that has each of the synchronic fusions as parts. Since synchronic fusions are instantaneous objects, a persisting object O that is composed of such objects at times, and which has each of those objects as parts, just is a four dimensional object composed of temporal parts. Since universalism guarantees that there will exist fusions of synchronic fusions, universalism guarantees that perdurantism is true. So goes the argument.

Sider’s argument requires the introduction of the notion of a fusion at-a-time: it requires that we can fuse the members of S at t. The synchronic fusion of the members of S at t is an object that exists only at t, and which at t has as parts the members of S at t. But notice that the endurantist can also make use of this notion. The endurantist holds that an enduring object is wholly present at t: all of its parts exist at t. A synchronic fusion then, is simply the fusion at t, of those parts. With this notion of a synchronic fusion in mind, the endurantist can hold that for any two arbitrary times and particulars that exist at those times, there will exist some enduring object O whose temporal extent falls exactly between those times, and which at each time at which it exists, is composed of those particulars. Specifically, a non-universalist version of unrestricted composition can make use of the notion of a synchronic fusion, and of the relation of constitution. The constitution relation is one that the endurantist already employs. We will say that any two objects O and O* are related by constitution at time t just in case O and O* are wholly present at t, and O and O* are materially coincident at t. The endurantist who holds that composition is unrestricted can then hold that for any two or more arbitrary synchronic fusions F1, F2, F3,...Fn that exist at t1,t2,t3...tn there exists some enduring object O that is constituted by F1 at t1, F2 at t2, F3 at t3, .....Fn at tn.

The idea then, is that both synchronic fusions and persisting objects are wholly present whenever they exist: no synchronic fusion is a part of a persisting object. Rather, synchronic fusions constitute enduring objects at times. Call this non-universalist version of unrestricted composition "unrestricted endurantism". So too we can see that non-universalist versions of perdurantism such as McCall’s restricted perdurantism, also have an endurantist analog. Where McCall holds that everyday objects such as my dog D are the fusion of instantaneous temporal parts D1, D2, D3...Dn, an endurantist will hold that the enduring object D is constituted by D1, D2, D3..Dn at each time at which it exists. Call the view that only everyday objects exists, but are constituted by shorter-lived objects at times, "restricted endurantism".

Of course, even restricted endurantism is not perhaps not a usual version of endurantism, since most endurantists do not hold that my dog is constituted by objects at times. In fact I think this view has much to recommend it over typical endurantism, as I will briefly note throughout as we proceed. Call this "typical" version of endurantism "unitary endurantism". Then even unitary endurantism has a four dimensionalist analog. Josh Parsons, for instance, argues that the four dimensionalist need not be committed to the existence of temporal parts. He argues that just as it is possible for an object to be spatially extended and yet be mereologically simple, so too it is possible for an object to be temporally extended and be mereologically simple: that is, to lack temporal parts. Call the view according to which objects are temporally extended–are not wholly present at any time–but lack temporal parts "unitary four dimensionalism".

Now naturally restricted endurantism and unrestricted perdurantism are not metaphysically equivalent: by definition they posit the existence of different objects. But if we want to know whether three and four dimensionalism are equivalent, we had better compare versions of each that adopt the same view about composition. In what follows I will in general focus on an eternalist version of unrestricted endurantism and perdurantism. Much of the discussion of endurantism and perdurantism within the context of these ancillary metaphysical commitments, however, will hold true even if we consider restricted or unitary endurantism and perdurantism, as I will note as we proceed. For the general procedure of showing the two theories are intertranslatable is essentially the same procedure in any of these cases, though of course the various advantages and disadvantages of the views will be different given different ancillary commitments. I hope that even though I concentrate primarily on one version of each, it will be clear how the argument for metaphysical equivalence would run for each of the other versions.

4 Intertranslatability

The first criterion of metaphysical equivalence is empirical equivalence and it should be clear by now that no empirical data could reveal whether the filled region of space before me contains an O-stage, or a wholly present O. There is no possible observational prediction that would be a prediction of three, but not four dimensionalism or vice versa, thus we can conclude that the two theories are empirically equivalent.

So let us move on to consider whether three and four dimensionalism are intertranslatable. We will begin by returning to our two definitions below:

E: An object *endures* iff it is wholly present at each moment at which it exists, where an object is ‘wholly present’ at a time just in case all of its parts are present at that time and where ‘P is part of O’ is true at any time t iff at t, P is partmb of O tly.

P: An object O perdures iff it is the mereological fusion of temporal parts, where x is an instantaneous temporal part of y at t iff x is part of y and x exists at, but only at t and x overlaps every part of y that exists at t.

E and P appear to be jointly inconsistent. Given an eternalist reading of each, if an object O is wholly present at t then O is not composed of any parts that exist at times other than t. Hence O is not composed of temporal parts and thus does not perdure. Similarly, if O perdures then it is a mereological fusion of temporal parts, and thus at every time at which O exists it is composed of parts that are not present.

So how can these two theories be intertranslatable? They can be intertranslatable precisely because they have a different view about what it is to have a part. The endurantist holds that ‘P is part of O’ is true at t1 just if P is partmb of O t1ly. The perdurantist holds that ‘P is part of O’ is true at t1 just if there is *some* time t at which O exists, and P is part of O-at-t. So the perdurantist holds that ‘P is part of O’ is true at t1 just if, in endurantist terms, P is partmb of O tnly. Thus for the perdurantist, the everyday meaning of ‘part’ is captured by the metaphysically basic sense of having a partmb.

Although ‘part’ is partially defined by the axioms of mereology, these axioms do not tell us whether objects are composed of all and only parts that are present, or whether objects are the fusion of parts, some of which are non-present. It is precisely this matter that the endurantist and perdurantist disagree on. So ‘part’ is a theoretical term. There is no pre-established concept of parthood that both endurantist and perdurantist share. Rather, each of the theories implicitly defines the parthood relation via the role that the concept plays in each of the theories. For the endurantist, no object is composed of parts that exist in the past or in the future. For the perdurantist, all (persisting) objects are at every time at which they exist, composed of parts that do not exist at those times. So we should *not *conclude that the endurantist and perdurantist share the same understanding of parthood, but simply disagree about which parts particular objects have. The very notion of what it is to have a part is intimately bound up with the theory in which it figures. Thus each theory uses what appears to be the same terminology, but where in fact the terms mean different things in the mouth of the endurantist and perdurantist.

So we can see that when the endurantist claims at t1, that P is not part of O, she claims that P is not partmb of O t1ly. When the perdurantist affirms at t1, that P is part of O, she claims that P is partmb of O tnly (where tn is a time other than t1). These two statements though, are clearly not contradictory. Returning to our definitions, definition E states that an object endures just in case at every time at which it exists, all of its parts are present at that time. So O is wholly present at t1 just if every t1ly partmb is present at t1. And a four dimensional object O has a temporal part O-at-t1 just if every t1ly partmb of O is present at t1. So an object O has all of its parts at a time t1 in the endurantist sense, iff O has a temporal part present at t1 in the perdurantist sense. Hence we can interdefine ‘O is wholly present at t’ with ‘O has a temporal part present at t.’ Then it follows that ‘O is wholly present at every time at which it exists’ translates into ‘O has a temporal part present at every time at which it exists.’ Thus we translate ‘O is wholly present at every time at which it exists’ to ‘O is the mereological fusion of temporal parts.’ And this, of course, is definition P of perdurance. For to be the mereological sum of temporal parts in the perdurantist sense, just is to be an enduring object composed of all of its parts simpliciter at a time in the endurantist sense.

The translation between endurantism and perdurantism given a presentist reading of the definitions E and P is also straightforward and is much the same as on the eternalist reading: ‘O is wholly present at t’ will also translate as ‘O has a temporal part present at t.’ But for the perdurantist the parthood relation will be tensed, such that ‘P was part of O’ will be true just if there is some past time t* at which P was a part of the t*-temporal part of O. Then for the endurantist, ‘P is part of O’ will be true at t just if P exists at t, and P is part of wholly present O at t, and for the perdurantist, ‘P is part of O’ will be true at t just if P is part of O-at-t.

Given our general translation schema, we can also see how other endurantist and perdurantist notions are now interdefinable. Recalling that x and y are related by constitution at t just in case x and y materially coincide at t, and x and y are wholly present at t, we interdefine endurantist and perdurantist notions such that x and y are related by constitution at t iff x and y share a temporal part at t. Then x and y are contingently identical according to the perdurantist, just in case they are related by constitution at all times at which they both exist according to the endurantist.

4.1 Translating property talk

This brings us to the issue of property exemplification. Recall that given a tenseless theory of the truth of propositions, the endurantist adopts either an indexicalist or adverbialist analysis of properties such that either properties themselves are temporally relativised, (red-at-t) or the having of those properties is temporally relativised (red tly). Perdurantists analyse property exemplification in terms of there being some temporal part P of a perduring object that exemplifies a property simpliciter. Then ‘O is red’ is true at t just if there is some temporal part of O that is red, that is, if O-at-t is red. The difference between these two views then, is supposed to be that for the endurantist properties are in some sense relational, whereas for the perdurantist properties are had simpliciter.

How we should understand the relation between an endurantist and a perdurantist analysis of properties depends on which version of each we are considering. Suppose we consider unitary versions of endurantism and four dimensionalism. Since for the unitary four dimensionalist there do not exist any temporal parts, the analysis of properties will mirror exactly the usual endurantist analysis. If a unitary four dimensional object O is red at t, then there is no part of O–O-at-t–that is red simpliciter. So O must have the property of being red tly or red-at-t. And since O is a four dimensional object, that property is one that O has at every time at which it exists: that is, it is true of the whole temporally extended O that it is red tly. So the analysis of properties that the unitary four dimensionalist will adopt, will mirror exactly the analysis that the unitary endurantist will adopt.

Consider now an unrestricted version of endurantism and perdurantism. On the adverbial analysis, ‘O is red’ is true at t just if O is red tly at t. For the perdurantist, ‘O is red’ is true at t just if O-at-t is red, that is, just if O has a temporal part at t that is red. Given that we can translate talk about temporal parts into talk about being wholly present, we can inter-define these property terms such that ‘O is red tly at t’ iff ‘O-at-t is red’. What it is to be red tly at t, in the endurantist sense, just is what it is to have a temporal part that is red, in the perdurantist sense.

Now consider the indexicalist analysis according to which if O is red at t, the wholly present O has the property red-at-t, and it has this property at every time at which it exists. So O has no straightforward property of being red. For the perdurantist, the four dimensional object O is red at t in virtue of having some part that is red simpliciter at t. But O itself has no straightforward property of being red: the whole is red at t, just if it has the property of having-a-part-that-is-red-at-t. Once again since we know that having a temporal part at t, and being wholly present at t, are interdefinable, we can interdefine having the property of having-a-part-that-is-red-at-t and being red-at-t. No whole persisting object, be it four dimensional or three dimensional, ever has properties simpliciter in the usual sense of the word, that is, if we take ‘simpliciter’ to mean in an unqualified or absolute sense. A persisting object O that instantiates some property P at a time t instantiates either a temporally relativised property P-at-t, or instantiates the property of having a part that is P.

So what are the endurantist and perdurantist to say about instantiating properties simpliciter? Although neither perdurantist nor endurantist can hold that persisting objects instantiate properties simpliciter in the strict sense of the term, as we have noted, just as the perdurantist can truthfully say at t that ‘O is red’ just if O-at-t is red, so too the endurantist can say at t that ‘O is red’ just if O is red tly at t. What is it that undergirds these claims? In the case of the perdurantist, it is that there exists some object, O-at-t that is straightforwardly red. But what of the endurantist case? Call the object that overlaps O at t and exists only at t–the instantaneous temporal part O-at-t–‘Q’. Q is red simpliciter. Now consider an unrestricted version of endurantism. On this view, there also exists some object Q that exists at and only at t, and which at t overlaps every part of O at t. This Q is also red simpliciter. So for the endurantist, there exists some object at t that is red simpliciter, and that object is related to O in that at t, Q constitutes O (O and Q are related by the constitution relation at t). O is red at t in virtue of being red tly at t, and O is red tly at t in virtue of being constituted at t by Q, which is red simpliciter. This is just one advantage of the unrestricted composition version of endurantism over the unitary version.

We can see then, that the endurantist and perdurantist accounts of property exemplification are analogous. Both hold that only instantaneous objects ever exemplify properties simpliciter, and that whole objects exemplify properties by being related to those instantaneous objects in a particular manner: by having them as temporal parts, or by being constituted by them at times. Exactly the same will be true of restricted versions of endurantism and perdurantism. For a restricted version of each allows that only everyday objects exist, but that these objects are for the perdurantist, composed of temporal parts, and for the endurantist constituted by objects at times. So just as an unrestricted endurantist holds that O is red tly at t just if at t O is constituted by some object O* that is red simpliciter, so too will a restricted endurantist. Similarly, just as an unrestricted perdurantist holds that some object O is red at t just if O has a temporal part O-at-t that is red simpliciter, so too will the restricted perdurantist. So if unrestricted versions of endurantism and perdurantism are intertranslatable with respect to an analysis of property exemplification, then so too will restricted versions of each be intertranslatable.

So too it is easy to how property talk is intertranslatable if we consider presentist versions of endurantism and perdurantism. In that case since the truth of propositions is tensed, both endurantist and perdurantist can talk of objects having properties simpliciter whenever they exist. If t is the present, then both endurantist and perdurantist will say that ‘O is red’ is true at t, and may be false at earlier and later times. At any time t, ‘O is red’ will be true just if t is the present, and for the perdurantist O-at-t is red, or for the endurantist, wholly present O is red.

Finally, recall that I argued that any translation manual must be a nonarbitrary one: there must be a nonarbitrary mapping of the sentences of one theory onto the sentences of another. Is this the case here? Yes. Our two theories are isomorphic to a high degree, and there is no difference in kind recognised by one theory and not the other. This is hardly surprising since the core disagreement between the two resides only in how they understand what it is to be a part. So we can say that the mapping is a nonarbitrary one.

So let us now turn to consider whether endurantism and perdurantism share the same degree of theoretical virtues.

5 Theoretical Virtues: Simplicity

One major difficulty in considering the various theoretical virtues, is that there is no generally accepted account of most of them. This is particularly so with the notion of simplicity. Although there is a generally shared concept of simplicity, and often shared intuitions about what counts as simple, there is no formal definition or account that can be appealed to in determining which of two theories is the more simple. Rather, we tend to have a family of related notions, ranging from the parsimony of ontology, the descriptive length of the theory, the number of adjustable parameters in the theory so forth. Moreover, as De Vito notes, it is always possible to make one theory appear simpler by burying the complexity in the atomic predicates of the language. Of course, that there is no precisely formulated definition of simplicity is everyone’s problem, and all we can do is make use of a workable intuitive understanding of simplicity, in order to decide whether theories are equally simple or not.

It is plausible to hold that endurantism and perdurantism are equally simple. First, neither theory posits the existence of entities that the other does not. Second, the mapping between three and four dimensionalism is not disjunctive, that is, we do not map one sentence of one theory, onto a long disjunction of sentences of the other theory. Nor is there any reason to suppose that some additional complexity in one theory has been buried in the atomic predicates of the language: endurantism and perdurantism may construe the notion of parthood differently, but neither has a significantly more complicated notion. Further, although the endurantist introduces a term that the perdurantist does not: constitution, so too the perdurantist introduces a term that the endurantist does not: a temporal part. And as we have seen, we can inter-define these two terms.

So there is no reason to hold that either endurantism or perdurantism are more complex, for although both make use of terminology that the other does not, once we understand the core disagreement about what it is to be a part, we can see that each of these terminologies is interdefinable.

6 Theoretical Virtues: Explanatory Power

6.1 Case Studies

To show that three and four dimensionalism are metaphysically equivalent requires the additional task of showing that they are explanatorily equivalent, and this is a considerably more weighty task. Due to space constraints, is not possible to show that every version of endurantism and perdurantism are explanatorily equivalent, so I will concentrate only on an eternalist unrestricted version of each. Given that we have already shown how the other versions are intertranslatable, however, I hope the following discussion will give a feel for how the arguments for explanatory equivalence would proceed in the case of the other version. In particular, most of what is said in this section will apply to restricted versions of endurantism and perdurantism. Presentist and unitary versions of each would of course have different explanatory resources to unrestricted versions, and I leave as an exercise for the reader to show that presentist versions of endurantism and perdurantism are explanatorily equivalent and ditto for unitary versions of each.

We will begin by briefly discussing how three and four dimensionalism deal with a number of issues arising from the way objects change over time, and some folk intuitions we have about those objects. For in order to show that three and four dimensionalism are explanatorily equivalent, we need to show that they have essentially the same resources to deal with real cases. We will then move on to consider the theory of special relativity, and whether it provides a reason to prefer four over three dimensionalism in the explanatory stakes.

A cursory consideration of the interdefinition of constitution and temporal part sharing, tells us that at least in most instances, restricted and unrestricted versions of endurantism and perdurantism will deal with traditional puzzle cases in the same way. For example, consider the case of Tib and Tibbles, where Tibbles is a cat and Tib is that proper part of Tibbles that includes all of her but her tail. Now suppose that Tib’s tail is at some point amputated. Both restricted and unrestricted endurantists will describe the post-amputation case as one in which Tib comes to be related by constitution to Tibbles, while the restricted or unrestricted perdurantist will describe it as a case of Tib and Tibbles sharing temporal parts after the amputation. So both endurantist and perdurantist agree that Tib and Tibbles are distinct objects with different persistence conditions: but they are objects that at some times (namely after the amputation) wholly overlap, that is, are co-constituting or share a temporal part. Given that constitution and temporal part sharing are interdefinable, we can see that these two ways of describing the relation between Tib and Tibbles are the same, but merely couched in two different languages.

The same will be true for cases where two objects, such as the lump of clay called "Lumpl", and the statue Goliath, called "Goliath" are materially coincident at all times at which they exist. The endurantist holds that such objects are distinct, and are related by constitution at all times at which both exist. The perdurantist holds that Goliath and Lumpl are contingently identical. Both agree that Lumpl and Goliath have different modal properties, so both allow that had things gone differently in the actual world, then Goliath and Lumpl might not have coincided. Again, given that contingent identity and constitution-at-all-times are interdefinable, we can see that both theories treat these two cases in the same way.

There is one case, however, where we might think that matters are not so straightforward, and that is a case of fission. Consider the case of Will Riker, who undergoes the Star Trek transportation process wherein all of his matter is destroyed at one end, and new matter arranged in the appropriate way at the other end. Unfortunately two qualitatively identical persons each with Riker’s psychology are created at the destination end. Call the two resultant persons R and R*. Since by the transitivity of identity both R and R* cannot be Riker, either only one of them is Riker, or neither is Riker. Neither of these options seems very palatable, for we have two conflicting intuitions, the first that Riker survives transportation, and the second that there is no principled reason to identify one but not the other resulting persons with the original Riker. If we accept both of these intuitions, then it seems that we must conclude that what matters in survival is not the identity relation.

Perdurantism, however, provides a way of reconciling these competing intuitions whilst maintaining that what matters for survival is indeed identity. For the four dimensionalist holds that there are two four dimensional objects, one of which includes all of the temporal parts of R and of Riker, and one of which includes all of the temporal parts of R* and of Riker. So when we point to the temporal part that exists prior to fission and say that it is Riker, we are not pointing to a unique individual: for the name Riker refers ambiguously to the two spacetime worms, R and R*.

It seems that the endurantist is in trouble here, for she cannot say that there exist two four dimensional objects that share all of the same temporal parts prior to fission. If the endurantist cannot provide a way of understanding what happens in cases of fission, that is equally able to preserve our intuitions, then it will turn out that three and four dimensionalism are not equally explanatory, and thus not metaphysically equivalent. So what should the endurantist say? She could say that prior to fission there exist two materially coincident persons R and R*, to which the name "Riker" ambiguously refers. R and R* are related by constitution, and post-fission come to inhabit two different bodies. As with the perdurantist view, we can then say that both R and R* are identical post-fission with something that existed pre-fission: namely R and R*.

The problem for the endurantist is that since she holds that objects are wholly present whenever they exist, this means that prior to fission, two wholly present persons R and R* were in existence. But this seems implausible. Following Robinson, we could argue that we need to distinguish two different methods of counting: counting by identity and counting by constitution. Let us suppose that if the identity relation holds between R and R*, then R and R* are to be counted as one if we are counting by identity. So if we count by identity then there are two persons prior to fission, since by definition R and R* are distinct. But not all commonsense counting involves counting by identity, argues Robinson: sometimes we count by constitution. If someone points to a statue and asks "how many objects are there?" we expect the answer to be, "one", not three on the grounds that there is a statue, a lump and an aggregate of matter. So we will say that if the constitution relation holds between R and R*, then we should count them as one if we are counting by constitution. If we count by constitution prior to fission, we can say that there is only one person, since R and R* are related by constitution.

Now I think that the idea that we sometimes count by constitution rather than identity is very plausible. But there is an additional problem here. For suppose that Riker had undergone transportation and everything had gone smoothly. Would R and R* still have existed prior to, and indeed after, the transportation, or would "Riker" have uniquely referred to one individual?

The perdurantist has a perfectly good answer to this question. Since for him, Riker is a four dimensional object it makes sense that we cannot know, prior to fission, whether "Riker" refers uniquely or not, since we only know about part of the object Riker. Just as we cannot know whether all of an object is orange just by looking at one spatial part of it, so too we cannot know whether a name refers uniquely or not if we are not in a position to see the entire object to which the name refers.

But the endurantist does not accept that only a temporal part of Riker is present at any given time, and so the issue of whether or not "Riker" refers ambiguously or not, cannot be an issue whose resolution awaits being able to see the entire Riker. There must be some fact of the matter prior to fission, as to whether both R and R* exist: the act of fission cannot retrospectively make it the case that R and R* existed. But in that case, why does *everyone’s* name not ambiguously refer to two materially coincident persons P and P* that are related by constitution? Indeed, given that the transporter could have created seven qualitatively identical persons who were psychologically continuous with Riker, the endurantist would in that case have had to say that there were seven materially coincident persons prior to fission. So then on what basis can the endurantist deny that there are seven coincident persons "in" all of us right now?

To begin to solve this problem, we need to remember that we are talking about either a restricted or unrestricted version of perdurantism. For given unitary perdurantism, we cannot talk of R and R* sharing temporal parts prior to transportation, and it becomes more difficult for the perdurantist to make sense of this case. So we should be considering a restricted or unrestricted version of endurantism. (Notice that this is one more feature in favour of either of these versions of endurantism or perdurantism over a unitary versions). Now the problem just raised stems from the apparent implausibility in the claim that there could, or do, exist materially coincident persons in this manner. But consider that for the restricted or unrestricted endurantist, there exist multiple objects that materially coincide with Riker for periods of time. Some of them exist only for a moment, some of them exist for years, and one of them exists from the moment of Riker’s birth, up until the second before the transporter incident. This is true regardless of whether a fission event would ever occur or not. Are all of these objects persons? Some of them certainly aren’t. For some it is unclear: the object that is materially coincident with Riker at age nine, and which exists for only three minutes might not count as a person on some theories of personhood. But some of these objects are good candidates to be persons. All of the objects that exist for nearly as long as Riker does have all the characteristics associated with persons. Persons or not, these objects all exist. For the restricted perdurantist these objects are person-stages, and for the endurantist they are enduring objects that constitute a person at a time.

So would R and R* still have existed if fission had not occurred? Well there would still have existed a considerable number of coincident objects, among them an object that ceases to exist at age seventy at t, and one that ceases to exist at t1, and another at t2. Each of these objects is distinct. Call one of them R, and one of them R*. Of course, R and R* in this world have a considerably different life than in the world in which fission takes place: but in either world, these objects exist. So just as with the perdurantist analysis of fission, the name "Riker" is ambiguous between R and R*, (and also various other enduring objects): but this same ambiguity would have existed *even if* no fission had occurred. So the act of fission does not miraculously causes it to be the case that R and R* exist prior to fission, where they would not have existed had no fission taken place.

Both restricted and unrestricted versions of endurantism and perdurantism agree that prior to fission, the name "Riker" is ambiguous between two objects that share all the same matter and which post-fission do not share the same matter. And both agree that the pre-fission objects need not be *counted *as distinct prior to fission. So with respect to cases of fission, both theories have the same explanatory power. The only difference is that the perdurantist holds that prior to fission the two post-fission objects share a temporal part, while the endurantist holds that prior to fission the two post-fission objects are related by constitution. And again, as we have seen, these two notions are intertranslatable.

6.2 Special Relativity

Assuming that we can translate three dimensionalist talk into four dimensionalist talk and vice versa, are there any other reasons to suppose that these two theories provide different degrees of explanatory power? Recently it has been argued that considerations drawn from empirical findings about the nature of space and time, in particular the theory of special relativity, provide reasons to prefer four over three dimensionalism. Special relativity states that observers in different frames will disagree about the spatial distances between objects at times, and the durations between events at times. So space and time are not absolute as they are in Newtonian physics. Given special relativity, enduring objects will be wholly present in different locations in spacetime.

Early defenders of four dimensionalism such as Smart and Quine objected to three dimensionalism on the grounds that given the truth of special relativity, the same enduring object will have different temporal and spatial properties depending on the frame of reference of the observer. So for instance, relative to one frame of reference, some object O will measure 7cms, and relative to another frame of reference will measure 12 cms. This is not in and of itself a problem for the three dimensionalist. What appears to be problematic is that we have defined endurance in terms of an object being wholly present, where to be wholly present at a time means to have all of one’s parts present at that time. Now suppose that O has two proper parts A and B. O is wholly present at t only if O has A and B tly at t, that is, if A and B both exist at t and are part of O at t. The problem is that while relative to one frame of reference R, A and B exist simultaneously, relative to some other frame of reference R*, A and B do not exist simultaneously. From the perspective of R*, A comes into existence slightly before B. Thus depending on which frame of reference one occupies, O will come into existence at different times, for O’s very existence is defined in terms of the co-presence of its spatial parts, and these exist at different times relative to the frame of reference of the observer.

In response, the endurantist must concede that it is indeed a surprising fact that there is genuine relativity in the world. It is truly indeterminate at what time an object comes into existence. In this respect objects are "relativistic" in that there is no objective fact about when they begin and cease to exist, nor of which spatial parts they are composed at a time. The endurantist can maintain that relative to one frame of reference, an enduring object O is composed of simultaneously existing spacetime points, and relative to a different frame of reference, will be composed of a different set of simultaneously existing spacetime points. Suppose that relative to one frame of reference R, an object O is composed of one set of spacetime points S. Then suppose we move our frame of reference only a small amount, such that from the frame of reference of R*, O is composed of a quite different set of spacetime points. Given that the frames of reference are quite close together, the qualitative properties of O relative to R will be very similar to the properties of O relative to R*, and this is so even though O is composed of different spacetime points relative to R and R*.

While this might not traditionally have been the way in which endurantists conceptualised being wholly present, I do not see that the fact that objects turn out to be relativistic, undermines the central endurantist notion that objects are not temporally extended. Indeed, many four dimensionalists concede that three dimensionalism is consistent with special relativity. Being consistent with special relativity, however, and being equally explanatorily powerful do not, of course, amount to the same thing. While Balashov holds that both three and four dimensionalism can be understood in relativistic terms, he argues that in the context of truths about special relativity, four dimensionalism has explanatory resources that three dimensionalism lacks.

The idea is as follows. Given the truth of special relativity, it is not just the endurantist who must relativise the notion of being wholly present to a frame of reference: the same is true for the perdurantist notion of a temporal part. For an instantaneous temporal part just is a three dimensional object. In the relativistic framework then, endurantists and perdurantists agree that persisting objects occupy a particular volume in spacetime. That is, they agree of some object O, that O occupies volume x. Furthermore, both agree that there are three dimensional objects, and that those objects have different properties relative to difference frames of reference. The difference between the two is that the perdurantist holds that the spacetime volume that an object occupies is the mereological fusion of the various three dimensional parts, whereas the endurantist holds that the three dimensional objects are strictly identical across time, and wholly present at each frame of reference where their parts exist simultaneously.

Balashov then asks us to consider what it is that "stands behind" the various relativistic three dimensional objects. In the case of perdurantism, he points out, we can say that there is a relativistically invariant object, namely the four dimensional object that occupies the particular volume of spacetime, and it is this invariant four dimensional object that generates the various three dimensional shapes. According to Balashov, what needs to be explained is why all of the various relativistic three dimensional objects should fit together so nicely in a unified four dimensional volume. The perdurantist has explanatory resources with respect to this question that the endurantist does not, for the perdurantist can say that the three dimensional objects are carved out from a pre-existing entity, the four dimensional object. Think of the four dimensional volume as a big sausage. Then slice the sausage up in various ways to get different shaped sausage slices. What explains why those slices have the properties they do? The fact that they were sliced from the particular sausage they were, in the particular manner they were. So while three dimensional objects exemplify different properties relative to different frames of reference, there is some objective, invariant shape that stands behind, and thus explains each of these different shapes.

The endurantist has no recourse to this explanatory power, for she denies that there is any invariant four dimensional object: all there is, are the various relativistic three dimensional objects. As Balashov puts it, the endurantist has to start with the various three dimensional objects, and then discover that lo, they can be arranged into a unified four dimensional volume, and this arrangement must, for the endurantist, be nothing more than brute fact.

This way of thinking of things is a bit misleading. It brings to mind the image of finding variously shaped bits of meat around one’s house, and then discovering that they fit together into a sausage shape, despite the fact that they are not "sausage-slices". Let’s consider this idea of a "nice four dimensional volume" a little more closely. If one is a unrestricted perdurantist, then not all four dimensional objects fill nice four dimensional volumes. The object that is the mereological fusion of George Bush-at-t, and Helen Keller-at-t* does not fill a nice volume. Nor do any of the ‘gerrymandered’ objects that are fusions of non-causally continuous stages. Why do the fusions of some stages fill nice volumes? Well, because these stages are causally related such that each of the stages are spatially and temporally contiguous and so forth. Why do the various three dimensional shapes fit together to form a nice volume? Again, not all of them do: not if one accepts unrestricted composition. But those that do fit together in this manner do so because the various three dimensional shapes are not separate objects, like pieces of meat, that happen to fit together to form a sausage. Rather, the four dimensional volume just is the entire lifespan of the enduring object that fills that volume, and it neatly fills that volume because it is causally related to itself at every time at which it wholly exists. Various causal facts about an enduring object O at time t, make it the case that O will exist at t*. So there is no explanatory mystery here.

Another way of thinking about this is to consider what it is we take to be fundamental: the four dimensional volume, or the three dimensional frame relative "slices" of that volume. Balashov’s argument rests on the notion that fundamentally there exist four dimensional objects, and that the various three dimensional objects are frame relative slices of these objects. For if fundamentally there exist three dimensional objects, then why should they fit together in such a neat fashion?

Well, if all we had were relativistic three dimensional shapes, and no theory about how they "fit together", we would be surprised to discover that they fill the volumes that they do. The theory of special relativity, however, along with various other laws of nature, allows us to predict how objects that exist in the present, will exist in the future. That is, they allow us to predict what the four dimensional volume of an object will be. We do not take a bird’s eye view of the universe and see various four dimensional objects, which we can then use to explain the various frame relative shapes. Rather, we take as basic the three dimensional objects, and use the various "rules" in the form of the laws of nature, to predict what those objects will be like in the future. So it can hardly then come as a surprise when we discover that those objects fill nice four dimensional volumes: for that is precisely what we predicted given our theory. Thus the endurantist will turn on its head Balashov’s claim that it is four dimensional objects that are basic, and instead argue that three dimensional objects are basic, and we come to see that they fill certain four dimensional volumes in virtue of extrapolating certain principles and making certain theoretical predictions.

Conclusion

In conclusion then, we see that there is no good reason to hold that restricted or unrestricted versions of perdurantism have explanatory virtues that the same versions of endurantism lack. Further, we have seen that ancillary metaphysical commitments aside, three and four dimensionalism are intertranslatable in a nonarbitrary manner, and such that each theory is equally simple. Of course, it might still be true that sociologically speaking, most endurantists embrace one bundle of ancillary metaphysical commitments, perhaps presentism and restricted composition, while perdurantists embrace a different bundle, perhaps eternalism and unrestricted composition. These people then, do not have metaphysically equivalent views. But the point is that there is some analogue of such an endurantist bundle of views, that is metaphysically equivalent to a bundle that includes perdurantism. So long as the ancillary metaphysical commitments that endurantists *in fact* embrace, are consistent with perdurantism, and as long as the ancillary commitments that perdurantist in fact embrace are consistent with endurantism, as I hope I have showed, then this is mere sociological choice, interesting though it may be. For it shows that it is not a theory of persistence that sets these two parties apart, but rather is their choice of ancillary metaphysical commitments. Three and four dimensionalism are indeed metaphysically equivalent: they are simply two ways of describing the same underlying metaphysics, and hence they stand or fall together.

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