Abstract
According to three-dimensionalism, objects persist in time by being wholly present at each time they exist; on the contrary, four-dimensionalism asserts that objects persist by having different temporal parts at different times or that they are instantaneous temporal parts of four-dimensional aggregates. Le Poidevin has argued that four-dimensionalism better accommodates two common assumptions concerning persistence and continuity; namely, that time itself is continuous and that objects persist in time in a continuous way. To this purpose, he has offered two independent arguments, each of which moves from a different understanding of continuity. In this paper, both of Le Poidevin’s arguments will be discussed and rejected.
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Notes
Properly speaking, the concept of an atomless gunk is a mereological one, not an ordinal one: in fact, it denotes an object all of whose parts have proper parts in their turn (Lewis 1991: 20). Stuchlik, however, employs it in relation to time as a synonym of the ordinal concept of a densely ordered set: in this sense, time is gunky just in case every finite interval of time can be subdivided into denumerably many proper subintervals. Stuchlik is accordingly claiming that the stage theory presupposes that time is either discrete or continuous but not dense.
Still, this paper is not intended to provide a limited defence of three-dimensionalism. To see this, it is sufficient to notice that my rebuttal of Le Poidevin’s arguments can be subscribed to by those who deny that there is any metaphysical difference between three-dimensionalism and four-dimensionalism (see for example McCall and Lowe 2003, 2006). Rather, this paper is meant to be part of a more general investigation concerning the metaphysical implications of the idea of continuity.
Le Poidevin’s own treatment of this problem can be found in 2003 (111–115).
Le Poidevin does not clearly distinguish between perdurantism and the stage theory, so his argument might seem to get off the ground only if one does not take the latter into consideration. The stage theory, in fact, might seem to offer a rather easy solution to the problem of the moment of becoming: for any object there is unambiguously a first moment of existence and a last moment of existence, and both coincide with the one moment during which that object exists. On these grounds, one may conclude that the stage theory does not need to resort to any form of vagueness in order to make room for the hypothesis that time is continuous in the mathematical sense, and that as such it must be preferred to both three-dimensionalism and perdurantism, thus completely sidestepping Le Poidevin’s argumentation. (I am grateful to an anonymous referee for having pointed this out.) However, this conclusion would overlook a crucial aspect of the stage theory, namely that, as a theory of persistence, it aims to offer a consistent story about what we ordinarily think to be persisting objects. To this purpose, it needs to be able to tell unambiguously, for any instantaneous object x, which other instantaneous objects are its temporal counterparts. To the stage theorist, in consequence, the problem of the moment of becoming reappears in the fashion of the problem whether there is a last moment of time at which there is a temporal counterpart of x, or rather a first time at which there is no such counterpart.
Considerations of this kind may also explain why Le Poidevin’s paper has received comparatively scarce attention from both three-dimensionalists and four-dimensionalists.
This latter clause is meant to accommodate otherwise questionable cases, such as the one of a clock that is temporally taken apart and then reconstructed using its own pieces.
This assumption has been interestingly called into question by Petkov (2005: 162–163), who suggests that quantum particles possibly persist by periodically going in and out of existence in a discontinuous fashion.
Clearly, ‘spatiotemporal continuity’ is to be understood here in the non-technical sense. Notice that, in this sense, objects can persist continuously in time even if time is discrete (whether or not space is mathematically continuous). However, in that case, they could only be spatiotemporally continuous if at rest or, if space is discrete, if moving at the constant speed of one minimal unit of time per one minimal unit of space; else, they will have to move between non-adjacent places at subsequent times (Mazzola 2014).
In reality, condition (ii) is not mentioned in Le Poidevin’s formulation of (2). However, that omission is evidently due to a typo.
Incidentally, the physical dimensions envisaged in (iv) are required to be continuous in the technical or mathematical sense. This surreptitiously makes the non-technical sense of continuity parasitic upon its technical counterpart. Notice, however, that every reference to the mathematical understanding of continuity can be dropped from (iv) by simply replacing the phrase ‘different positions along a continuous dimension’ with the phrase ‘different positions along the same dimension’. This will suffice to make (iv) compatible with the possibility that physical dimensions such as distance, mass, temperature, etc. be discontinuous or quantized.
For the pedantic, for any s, x is not a temporal part of s or y is not a temporal part of s or s is not four-dimensional.
I do not myself think that it is. However, what matters here is just that, in any case, (1) is not worse as an analysis of diachronic identity than (2), (3), and (4) are as principles for the unification of temporal parts.
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Acknowledgments
I would like to thank an anonymous referee for her constructive comments. This research has been funded through a UQ Postdoctoral Research Fellowship granted by the University of Queensland.
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Mazzola, C. On Continuity and Endurance. Acta Anal 30, 133–147 (2015). https://doi.org/10.1007/s12136-014-0236-6
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DOI: https://doi.org/10.1007/s12136-014-0236-6