## Abstract

The paper presents a thorough exploration of the problem of persistence in a relativistic context. Using formal methods such as mereology, formal theories of location and the so called intrinsic formulation of special relativity we provide a new, more rigorous and more comprehensive taxonomy of persisting entities. This new taxonomy differs significantly from the ones that are present in the recent literature.

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## Notes

- 1.
We are being deliberately vague here. By things we simply mean whatever entities someone is committed to in their ontology.

- 2.
We will use endurantism and perdurantism interchangeably for three and four-dimensionalism respectively. Those familiar with the debate will probably see that we will not talk about what is sometimes referred as exdurantism or stage-view. This is because we do believe, although we cannot argue for it here, that exdurantism has the same ontological commitments as four-dimensionalism, i.e. exdurantism commits to a four-dimensionalist ontology.

- 3.
- 4.
From here on STR. We will not really enter into details of the mathematical structure of STR. We will stick to the presentation of it offered in Malament (2009).

- 5.
A note on notation, which is in line with Malament (2009). Points in the affine space

*A*will be denoted by lower case letters, \( p,q, \ldots ,t \). A vector is indicated with \( \overrightarrow {u} \). Given two distinct points in*A*,*q*,*p*we write \( q = p + \overrightarrow {u} = p + \overrightarrow {pq} \). This might seem a little bit abstract but there is very easy and helpful way to understand it. Here it is. The point*q*is the point you get to if you start from point*p*and “walk along” the vector \( \overrightarrow {pq} \). We take the signature of Minkoski spacetime to be (1, 3), thus adopting the sign convention (−, −, −, +). Technically this is done by defining an inner product operation between vectors, i.e. a map:\( \left\langle , \right\rangle :V \times V \to {\mathbb{R}} \) from vectors to real numbers. This map allows us (i) to define a notion of distance between two arbitrary spacetime points*p*,*q*given by \( \sqrt {\left\langle {\overrightarrow {pq} ,\overrightarrow {pq} } \right\rangle } = \,||\overrightarrow {pq} || \), and (ii) induces the infamous classification of vectors in Minkowski spacetime. In particular for any non-zero vector \( \overrightarrow {u} \), \( \overrightarrow {u} \) is timelike iff \( \left\langle {\overrightarrow {u} ,\,\overrightarrow {u} } \right\rangle > 0 \), spacelike iff \( \left\langle {\overrightarrow {u} ,\,\overrightarrow {u} } \right\rangle < 0 \) and null iff \( \left\langle {\overrightarrow {u} ,\,\overrightarrow {u} } \right\rangle = 0. \) Two points are said to be timelike, spacelike or null related just in case the vector conjoining them is timelike, spacelike or null respectively. Here is a visualization of this classification. At every point*p*there is a double-cone with vertex at*p*. Points that are timelike related to*p*are those*on the inside*of the double-cone, points that are spacelike separated from*p*are*outside*the double-cone and points that are null related to*p*are the ones*on*the double-cone. It is one of the most significant fact about STR that whereas we can specify an invariant temporal order for timelike separated points it is impossible to do so for spacelike separated points. See also the discussion in footnote 23. - 6.
*Proper Parthood*: \( x \prec \prec y =_{df} x \prec y \wedge x \ne y \);*Overlap*\( O(x,y) =_{df} (\exists z)(z \prec x \wedge z \prec y) \). - 7.
- 8.
Though the latter is strictly speaking redundant.

- 9.
*Weak Supplementation*: \( x \prec \prec y \to (\exists z)(z \prec y \wedge \sim O(x,z)) \) - 10.
We will sometimes write \( Sum(z,\varphi (x)) = z = Sum(x_{1} , \ldots ,x_{n} ) \) for

*z*= mereological sum of \( (x_{1} , \ldots ,x_{n} ) \). In this case we have that \( \varphi (x) = (x = x_{1} \vee \ldots \vee x_{n} ) \). - 11.
A number of philosophers understand location in exactly these terms. See for example Casati and Varzi (1999), Hudson (2001), Sattig (2006), Gilmore (2007), Balashov (2008), Balashov (2010), Hawthorne (2008) and Donnelly (2010).Thus the Sistine Chapel is not the exact location of either the Atlantic Ocean or of a copy of the Aristotelian Physics. Actually a copy of Aristotelian Physics

*could*in principle be shaped like he interior of the Sistine Chapel. We take it that most (if not all!) of the copies of Aristotelian Physics are not shaped that way. Surely the copy we have in our office is not and as such (if that copy does not alter significantly its shape properties) it could not be exactly located there. Thanks to an anonymous referee here. - 12.
We are making the simplifying assumption that the first argument of every location relation is a material object, whereas the second is a spacetime region. The first argument of the location relation need not be a material object. Some philosophers think for example it could be an event, such as the performance of Samuel Beckett’s

*Krapp*’*s Last Tape*. The second argument of the location relation could be a material object rather than a spacetime region, as in “My copy of*Krapp*’*s Last Tape*is in that box”. This raises serious questions about what really counts as “that box”. We cannot really enter into these delicate questions here. But note that Leibnizian relationists about space(time) would probably want to cash out spatiotemporal relations such as location in terms of relations holding between material objects rather than material objects and spacetime regions (Naturally they will have their fair share of problems with STR, as our assumption of*Special Relativistic Realism*should suggest). Finally note that we could have a relation of location holding between an event and a material object as in “The first performance of*Krapp*’*s Last Tape*was held at the Royal Court Theatre in London”. Thanks to an anonymous referee of this journal. - 13.
This formulation of

*Expansivity*is adapted to the present framework. In a theory of location that does not have*Exactness*among its axioms a weaker formulation should be adopted such as*Inexact Expansivity*: \( x \prec \prec y \to ((\exists R)(ExL(x,R) \to (\exists R_{1} )(ExL(y,R_{1} ) \wedge R \prec \prec R_{1} )) \). Also, in a mereological theory that does not have the*Weak Supplementation*principle we should want to allow for the case in which a composite object and its unique proper part are exactly located at the same region. Then*Expansivity*should be replaced with*Weak Expansivity*: \( x \prec y \wedge ExL(x,R) \to (\exists R_{1} )(ExL(y,R_{1} ) \wedge R \prec R_{1} ) \) where we note that the proper parthood relation of the*Expansivity*axiom has been replaced with that of parthood. - 14.
Despite their intuitiveness, they can be considered contentious. Several interpretations of quantum mechanics would deem

*Exactness*wrong. Relativity however does support a classical notion of spacetime trajectory so that it seems, at first sight, safe. Also in an atomistic spacetime inhabited by gunky objects, both*Exactness*and*Expansivity*will be violated. For more on this issue (see Saucedo 2011). - 15.
To see that this set of axioms is coherent think of \( (x_{1} , \ldots ,x_{n} ),(R_{1} , \ldots ,R_{n} ) \) as open and closed intervals of the real line and interpret ‘\( \prec \)’ as set-theoretic inclusion and ‘\( ExL \)’ as set-theoretic membership.

- 16.
*Weak Location*: \( WL(x,R) =_{df} (\exists R_{1} )(ExL(x,R_{1} ) \wedge O(R,R_{1} )) \). Another interesting locative notion is that of*Overfilling*which can be defined via: \( OvF(x,R) =_{df} (\exists R_{1} )(ExL(x,R_{1} ) \wedge R \prec R_{1} ) \). - 17.
Background dependence is a crucial notion in contemporary physics. More technically in a background dependent theory spatial and temporal distances are represented by independent variables

*x*and*t*that coordinatize the background. In background independent theories spatial and temporal distances have to be extracted somehow from the dynamical variables describing the evolution of various fields. See for example Colosi et al. (2005). - 18.
Moreover the standard set-theoretic account of spacetime, according to which spacetime is composed of mereological atoms, i.e. spacetime points, such that any collection of them forms a unique spacetime region, entails that the mereological theory that regiment the notion of parthood among spacetime regions is a far stronger theory, namely General Extensional Mereology. In this case it can be shown that the notion of product is equivalent to the set-theoretic notion of intersection and the notion of sum to the set-theoretic notion of union. We will thus be justified in using such notions for the sake of clarity when referring solely to spacetime regions.

- 19.
A note on terminology. Gilmore calls our \( 3D_{L} \) objects “saints”, whereas he calls our \( 3D_{M} \) objects “non-segmented objects”. On the other hand he calls our \( 4D_{L} \) objects “worms”, whereas he calls our \( 4D_{M} \) objects “segmented objects”.

- 20.
This construction is specific for relativistic spacetimes. In fact the notion of temporal precedence can be grounded in the mathematical structure of the inner product for such spacetimes. If spacetime is Newtonian you will have to take the notion of absolute precedence as primitive and then go on to define the notion of an achronal region via that notion.

- 21.
It is possible to divide timelike vectors into future and past directed as we have already mentioned in footnote 5. What direction we choose as the future is conventional. On the other hand it is not conventional whether two timelike vectors are co-oriented. Two distinct timelike vectors \( \overrightarrow {u} ,\overrightarrow {v} \) are co-oriented, i.e. both future or past directed, iff \( \left\langle {\overrightarrow {u} ,\overrightarrow {v} } \right\rangle > 0 \).

- 22.
There is no need to be formal here.

- 23.
This raises an interesting issue, i.e. whether the notion of achronality is able to capture and vindicate the intuitive notion behind “being temporally un-extended”. Consider a flat acronal region

*R*, where*Flat*is defined as in footnote 36, and consider two arbitrary points \( p,q \in R \). Then there is a “temporal perspective” associated with a particular observer (or reference frame) according to which events happening at*p*,*q*are simultaneous. The problem is that there is a second “temporal perspective” according to which the event at*p*happens*before*the event at*q*and even a third one according to which*p*happens*after**q*. This is because, as we mentioned in footnote 5, there is no invariant temporal order for spacelike separated points. As far as we can see there are two strategies that might be adopted here: (i) insist that this analysis shows that in relativistic spacetimes there is no meaningful notion of “being temporally unextended” whatsoever, (ii) insist that this analysis shows that when passing to a relativistic spacetime we should take the notion of “being temporally unextended” to mean that “there is at a least one temporal perspective according to which all events happening at an achronal region count as simultaneous”. Now, clearly (i) is bad news for three-dimensionalists. For, if there is no sensible notion of “being temporally unextended” their entire metaphysics, whose main (informal) tenet is that material objects are temporally unextended, is at risk of crumbling down. A charitable attitude is then to go with (ii). This is still not enough. For achronal regions need not be flat. In such a case there is not even a single temporal perspective according to which all events happening at every point in that region count as simultaneous. And this sets us back to the previous dilemma: either we endorse (i) or we weaken (ii) further and get to: (iii) when passing to a relativistic spacetime we should interpret “being temporally unextended” as simply “being achronal”. This is the best fighting chance three-dimensionalism has in a relativistic setting and it is a sensible one. So, in the rest of the paper, we will be charitable and stick to that. For another related worry see also footnote 25. Many thanks to an anonymous referee of this journal for having pushed this important point. - 24.
We have already said that we can resort to set theoretic notions when dealing only with spacetime regions.

- 25.
Maybe this is too strong. Consider a faster-than light particle. Its path could be achronal, yet we could be interested in saying it is a persisting object. The existence of such particles would however seriously undermine our

*Special Relativistic Realism*assumption so that we will not pursue this line of argument here. Furthermore note that a light ray counts as a persisting object in our account. - 26.
But see Gilmore (2006).

- 27.
The last clause is needed to rule out the possibility of a four-dimensional object exactly multilocated at its path and at one of its proper non achronal subregions. This might seem an abstruse possibility, yet it counts as a metaphysical possibility.

- 28.
See Sider (2001) and reference therein.

- 29.
Balashov (2008: 64) uses a new three-place primitive to be read as “

*x*is part of*y*at achronal region*R*”. Two things are worth noting. First of all the intrinsic formulation of STR renders the introduction of this new primitive avoidable. Second the use of this new primitive seems to raise important and difficult questions. Gilmore (2009) introduces the following “Restriction question”: What is the set of necessary and sufficient conditions a spacetime region*R*has to meet for a material object to be part of another at that region? Gilmore explores many different plausible candidates and argues convincingly that they all face major drawbacks. He then concludes that foes of*Absolutism*, roughly the thesis that parthood is a two-place relation, should use a four-place relation of parthood \( P(x,y,R_{x} ,R_{y} ) \) to be read as “*x*at \( R_{x} \) is part of*y*at \( R_{y} \)”, where \( R_{x} ,R_{y} \) are the exact locations of*x*and*y*respectively. Balashov’s proposal seems to fall prey to Gilmore’s arguments. - 30.
Every union (mereological sum) of maximal subregions is again a maximal subregion, i.e.: \( Max(R_{1} ) \wedge \ldots \wedge Max(R_{n} ) \to Max\left( {\bigcup\limits_{1}^{n} {R_{i} } } \right) \). It is worth spending a few words on this definition of

*Maximality*. It makes crucial references to Cauchy surfaces. These surfaces are inextensible hyper-surfaces, that can be regarded, with due precautions, as the closest analogue to time instants. As such they are achronal regions, and we have seen in footnote 23 that this might raise some worries. It also raises the important question about whether all relativistic spacetime can be divided, or “foliated” by these kind of surfaces. It turns out that Minkowski spacetime always admits a foliation via Cauchy surfaces. But many general relativistic spacetimes do not, the most prominent example being probably the so called Gödel’s spacetime. Since we are restricting our attention to Minkowski spacetime we will stick to this definition of*Maximality*here, being well-aware that it resists bold generalizations. There might seem to be another worry here, even modulo the concerns about achronality that we dealt with in the aforementioned footnote 23. It is that this definition of*Maximality*, using the closest relativistic analogues of time-instants, raises some problems for the possibility of what we label later on*Spatiotemporally Extended Simples*, i.e. material objects exactly located at non-achronal regions that do not “divide” into temporal parts exactly located at achronal subregions of their exact location represented by the intersection of that very exact location with Cauchy surfaces. We believe that this worry could be set aside. For those who believe in the possibility, or even in the existence of the so called spatiotemporally extended simples, also believe that the mereological structure of material objects needs not mirror, and in fact does not mirror, the mereological structure of spacetime regions they are exactly located at (See our discussion of the violation of*Division*in footnote 42.*Division*would in fact ensure that mirroring). Whether a fully blown “mereological harmony”, as Schaffer (2009) and Gilmore (2013) calls it, is at least desirable is another interesting question, but one that we cannot address here. Thanks to an anonymous referee here. - 31.
This definition seems to leave out something we mentioned in our informal rendering, namely that the exact locations of the temporal parts are proper parts of the persisting object’s path. However this follows directly from

*Expansivity*and definition (3.10). Formally: \( TP(x,y) \to (ExL(x,R) \to R \prec \prec Path(y)) \). It could be objected that this argument takes for granted that the exact location of*y*is \( Path(y) \), and this is the case only if the persisting objects are locational four-dimensional objects. This objection is fair. We will however argue that mereological four-dimensional objects cannot be locationally three-dimensional, so that this concern can be set aside. - 32.
This will entail that locational three (four)-dimensionalism shouldn’t be phrased as universal claims.

- 33.
Merricks (2005) argues that there cannot be any such ontology. His argument depends however upon a particular metaphysics of time, namely presentism, that is at best moot in a relativistic context.

- 34.
Note that

*z*could not qualify as a locational four-dimensional object either, for it could be exactly multilocated at non achronal regions. We will consider briefly these issues later on. - 35.
Let us see just one example. Balashov (2010) restricts the possible locations of temporal parts to flat non achronal subregions of the objects path. In the simplest case of a material object extended in just one spacelike dimension we could define: (

*Flat*) \( Flat(R) =_{df} (\forall p,q,r \in R)(r = p + \overrightarrow {pr} = p + a\overrightarrow {pq} ) \) With \( a \in {\mathbb{R}} \). We could then define the notion of flat temporal by part adding the clause that every exact location of a temporal part is a flat region. - 36.
Though we will briefly discuss a complication of this definition in Sect. 3.4.

- 37.
He calls (i) “segmented saints” and (ii) “non-segmented worms”.

- 38.
- 39.
This definition works iff (i) there are spatiotemporal atoms and (ii) these atoms are unextended. Both clauses seem satisfied within STR.

- 40.
Naturally they could also be locational three-dimensional objects.

- 41.
We will give a formal rendering in the following section.

- 42.
Note that extended simples, independently of being persisting entities, all violate the following locative axiom that can be labeled

*Division*: \( OvF(x,R) \to (\exists y)(y \prec x \wedge ExL(y,R)) \). Informally it says that something has parts that are exactly located at every region it overfills. In various forms it is discussed in Casati and Varzi (1999: 122), discussed and not endorsed in Parsons (2006: 10) and criticized in Van Inwagen (1981, reprinted in Rea 1997: 191). - 43.
Gilmore is explicit in claiming that from a logical and metaphysical point of view the distinction between segmented and non-segmented objects (mereological four and three-dimensional objects in our terminology) cross-cuts (his words) the distinction between worms and saints (locational four and three-dimensional objects in our terminology), thus yielding four types of persisting entities. He is clear that this does not depend on further metaphysical assumptions regarding for example coinciding objects. It is however an interesting question whether it is possible to maintain that (i) the clay is a mereological four-dimensional object, whereas (ii) it constitutes different statues which are locational three-dimensional objects. This is a point that is worth exploring further in its own right, but it would not undermine our main argument. Our argument is that “one and the same object cannot be both mereologically four-dimensional and locationally three-dimensional”. This scenario crucially depends however on the constitution view according to which “constitution is not identity”. Hence the statues are not numerically identical with the portion of clay that constitutes them and hence there is nothing (neither the clay, nor the statues) that is both mereologically four-dimensional and locationally three-dimensional. Thanks to an anonymous referee for having pushed this point.

- 44.
If, on the other hand, these ontologies are not tenable, problems of composition will be less interesting. It is possible to show however that three-dimensional ontologies are not compatible with any mereological theory that features the so called

*Unrestricted Composition*axiom, such as Classical or General Mereology. This is because the mereological sum of two distinct three-dimensional objects that are exactly located at distinct regions*R*_{1},*R*_{2}such that \( R_{1} \cup R_{2} \) is not achronal, on pain of contradicting*Expansivity*, would not be exactly located at an achronal region. As such it will not qualify as a three-dimensional object. - 45.
This is because entities that are uniquely exactly located do not violate the following locative axiom which we might label

*Additivity*: \( ExL(x,R_{1} ) \wedge ExL(y,R_{2} ) \to ((\exists z)(z = Sum(x,y)) \to ExL(z,R_{1} \cup R_{2} )). \) - 46.
Hudson (2001) contemplates this possibility.

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## Acknowledgments

We are deeply grateful to Cody Gilmore, Yuri Balashov, Mark Hinchliff, Fred Muller, Dennis Dieks, Andrea Bottani, Giuliano Torrengo and Stefano Bordoni for their invaluable help in commenting and discussing the various drafts of the paper. We also want to thank an anonymous referee of this journal whose careful and insightful comments improved the paper greatly.

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Calosi, C., Fano, V. A New Taxonomy of Persisting (Relativistic) Objects.
*Topoi* **34, **283–294 (2015). https://doi.org/10.1007/s11245-013-9212-9

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### Keywords

- Persistence
- Mereology
- Location
- Minkowski Spacetime