Refining Four-Dimensionalism

Abstract

Current formulations of Four-Dimensionalism may be objected to on grounds that they are too inflexible: the formulations do not seem to allow for enough variety in the views they are paired with. For instance, Kit Fine has noted that formulations of Four-Dimensionalism in terms of instantaneous parts may be too demanding for Four-Dimensionalists who believe nothing is instantaneous. And Trenton Merricks has argued that one can think something persists four-dimensionionally without taking it to have proper temporal parts (i.e., temporal parts distinct from the whole object), and claims that our formulation of Four-Dimensionalism should be revised to allow for this. I will add my own worries to those of Fine and Merricks. I will note that current formulations of Four-Dimensionalism are not sufficiently neutral with respect to the structure of time, with respect to how liberally objects decompose into parts, and with respect to whether objects and the regions they fill match in mereological structure. I will show that we can formulate Four-Dimensionalism in a sufficiently neutral way, while still producing a view that can do the work we typically require of Four-Dimensionalism.

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Notes

  1. 1.

    Fine (2006, p. 700).

  2. 2.

    Many four-dimensionalists will be happy to include additional primitive notions in their formulations of Four-Dimensionalism, such as in virtue of or grounded in. (For instance, many take Four-Dimensionalism to be a view about what entities persist in virtue of.) I will not argue against such views, but will show that we needn’t endorse them in order to give strikingly neutral formulations of Four-Dimensionalism that do the work we tend to demand, and which do not require the inclusion of relations like in virtue of. For precedent for separating talk of in virtue of from formulations of Four-Dimensionalism, see Sider (2001, Chap. 3). He notes that his formulation of Four-Dimensionalism lacks talk of in virtue of relations or property inheritance, and takes these to be additional claims about persistence that a four-dimensionalist can make.

  3. 3.

    We may instead take proper parthood to be primitive, and define parthood disjunctively, as proper parthood or identity.

  4. 4.

    We may instead take x to be a proper part of y iff it’s not the case that y is a proper part of x. For more on this, see Cotnoir (2010). Either definition will suit our purposes here.

  5. 5.

    I have taken this short list of definitions from Simons (1987) and Casati and Varzi (1999). Those texts draw from the formal theories presented by Stanislaw Leśniewski beginning in 1916 and Henry Leonard and Nelson Goodman’s 1940. For more on this, see the texts just listed as well as Varzi (2007) and the introduction of Kleinschmidt (2014).

  6. 6.

    Note: if you think objects are extended in time as well as in space, then their shapes and the regions they exactly occupy will be temporally extended as well.

  7. 7.

    For more discussion about why this fails as a definition of located at, see (Kleinschmidt, unpublished).

  8. 8.

    This is very nearly equivalent to Parsons’ weak location (2007, p. 203), and is similar to Casati and Varzi’s generic location (1999, pp. 120–121), but my definition allows for entities to be located at multiple regions.

  9. 9.

    This is exactly like Parsons’ entire location relation (2007, p. 203). It is extentionally equivalent to Casati and Varzi’s whole location (2007, pp. 120–121), though their definition relies on a preclusion of multilocation.

  10. 10.

    These definitions are loosely based on definitions from Casati and Varzi (1999), Parsons (2007), and Hudson (2006). For a more detailed overview of logics of location than the one given here, see Gilmore (2014) and Kleinschmidt (2014). For discussion of difficulties facing logics of location like these, see also (Kleinschmidt, forthcoming).

  11. 11.

    For simplicity and neutrality, I have refrained from relativization of times to reference frames. But the definitions can be thus amended for anyone wishing for such a change.

  12. 12.

    Note that since this does not require that it overlaps with a proper subinterval of T, which would have to be distinct from T. (The subinterval/proper subinterval distinction is analogous to the part/proper part distinction).

  13. 13.

    See, for instance, Markosian (1994).

  14. 14.

    See, for instance, Fine (2006), and Hawthorne (2006). For an argument that we cannot cash out Three-Dimensionalism merely mereologically, see (Kleinschmidt, unpublished).

  15. 15.

    See van Inwagen (1990).

  16. 16.

    See, for instance, Simons (1987) and Hawthorne (2006). For more references, see Sider (2001, p. 63). For an excellent discussion of how to define wholly present, as well as a survey of attempted definitions, see Crisp (2005). And if we wish to combine Three-Dimensionalism with gunky time, we can amend our formulation to say “persisting objects are wholly located in every time at which they are present.”

  17. 17.

    I tend to prefer the locative account because it does not require an extra primitive relation beyond what the Three-Dimensionalist’s opponent already (barring Supersubstantivalism) has reasons to posit. Though Kit Fine’s view has the advantage that it gives us a unified account of three-dimensional persistence for material objects and (if there are any) immaterial objects present in time (which, arguably, wouldn’t have sizes or locations, but may nonetheless exist in time; see Fine (2006) for other advantages). But we won’t need to take a stand on this topic here. We need something that sets Three-Dimensionalism apart from the other views, but regardless of our take on what this should be, there is work to be done in separating Four-Dimensionalism from The Spanning View.

  18. 18.

    This notion of exists \(_{\mathrm{Fine}}\) in time is not to be confused with the notion of existence-at-t defined by Heller (1984) and Sider (2001), according to which (roughly) an entity exists-at-t iff the entity exists and has a part present at t. For Fine, though exists \(_{\mathrm{Fine}}\) is not reducible to any notion of location, it is required that if an entity exists\(_{\mathrm{Fine}}\) at a time, all of its parts are present at that time.

  19. 19.

    Peter Simons defends a similar distinction between how ordinary objects (continuants) and events (occurrents) relate to time and space. However, he takes the different relations involved to be locative. For more on this, see Simons (2014).

  20. 20.

    Cody Gilmore (unpublished) has shown that the Spanning View is relevant to how we draw distinctions between alternative views of persistence. Parsons (2000) and (2007) defends this account of persistence, using ‘entending’ as synonymous for ‘spanning’, and Parsons (2000) has argued for its plausibility as a candidate view of persistence.

  21. 21.

    The term ‘spanner’ was first used by McDaniel (2003), to name a sort of entity that Cody Gilmore first presented to the literature and described in 2004 (see Gilmore, unpublished). For further discussion, see Hudson (2006, pp. 99–103).

  22. 22.

    With the exception of the addition of the notion of existence\(_{{\mathrm{Fine}}}\), this is nearly exactly the definition presented and discussed by Hudson (2006, p. 101). This is also nearly equivalent to the notion of entension presented in Parsons (2000).

  23. 23.

    For instance, if I am made of tiny, locatively non-overlapping, extended-simple parts, such as curved 1-dimensional strings. I will count as spanning each region that any such part is located at, in virtue of having no parts (proper or otherwise) located at any proper subregions of those regions. Another example: suppose I have a special post-it made of two extended simples of the same size. I fold it in half, but it’s made of special matter that allows me to make the halves colocate, so that the simple parts are now each located at the same region. This special post-it will now span the region at which it is located, though it is composite and colocated with its proper parts.

  24. 24.

    Again, spanners needn’t be simple; this simply makes our case simpler.

  25. 25.

    See Hoppolytus’s Refutation 9.10.4 \(=\) 22B59 and 9.10.5 \(=\) 22B61 (Cohen et al. 2005, pp. 29–30), John Tzetzes’ Notes on the Iliad, p. 126, Hermann \(=\) 22B126 (Cohen et al. 2005, p. 31), and Arius Didymus, Fr. 39.2 (Dox. gr. 471.4) \(=\) 22B12 (Cohen et al. 2005, p. 30).

  26. 26.

    Lewis (1986, pp. 203–204).

  27. 27.

    For more on distributional properties, see Parsons (2004).

  28. 28.

    Things are more complicated if you think four-dimensional objects or their parts can be multiply located. For a discussion of this, see Kleinschmidt (2011).

  29. 29.

    We might also say that it is incomplete because it does not say anything about what it is in virtue of that persistence occurs. Please feel free to add this to my formulations if you would like.

  30. 30.

    These can be contrasted with temporary proper parts, which I will discuss shortly.

  31. 31.

    I have substituted talk of presence at and containment in times for Sider’s talk of existence-at-times, just to avoid any confusion that this might require a denial of Eternalism, the view that past, present, and future objects all exist. Thomson (1983, p. 207) offers a slightly different account of temporal parts, on which, roughly, x is a temporal part of y at T iff x is located at the intersection of T and y’s location. Four-Dimensionalism when understood as involving temporal parts as defined by Thomson will face the same problems that I will raise for Four-Dimensionalism involving temporal parts as defined by Sider. For interesting further problems facing these accounts of temporal parts, see Effingham (2011).

  32. 32.

    To motivate this condition, recall the three-dimensionally persisting statue that is made of a distinct, instantaneous chunk of clay at each time. Arguably, this statue has temporal parts at each time. But with the added requirement that four-dimensionally persisting objects fuse all of their proper temporal parts, we avoid classifying this statue as a four-dimensionally persisting object.

  33. 33.

    Again, I’ve replaced talk of Siderian existence-at-times with talk of presence.

  34. 34.

    See Sider (2001).

  35. 35.

    Sider (2001, p. 60).

  36. 36.

    Hawthorne (2006, p. 87) formulates an even more demanding thesis for gunk theorists: “Gunky Plenitude: For any object x and any temporal interval of non-zero measure during which x exists, there is an object y such that y exists just in the interval and coincides with x at every time in the interval.” This requires then gunk theorist to posit even more objects than Four-Dimensionalism\(_{3}\) would require of them (since it requires objects at discontinuous intervals as well), but it does not require that these objects are parts of the persisting object.

  37. 37.

    van Inwagen (1981).

  38. 38.

    Mark Heller notes that one can reject DAUP while accepting Four-Dimensionalism, on p. 327 of his (1984). Many accept DAUP because they accept that objects have point-sized parts at every point at which they are present, and they accept Unrestricted Composition. But the Unrestricted Composition theorist has some work to do: see Dan Korman’s “Debunking Perceptual Beliefs About Ordinary Objects” for an interesting challenge to some widely endorsed motivation for Unrestricted Compositon.

  39. 39.

    Fine (2006, p. 700) also takes a stand against grouping claims of liberal decomposition with Four-Dimensionalism. He comes to a different conclusion than I do, though: this is one of his motivators for positing two different ways of being present in time, namely, extending through it, and existing in it.

  40. 40.

    That is, for each of the the non-overlapping, continuous fourths of the hour through which O persists, O has a temporal part located at that fourth of the hour. Similarly for sixteenths, sixty-fourths, and so on.

  41. 41.

    I want to emphasise that I do not just mean for this to describe merely the largest interval the entity persists through. This condition applies to any interval whatsoever that the thing fills; so, for instance, it applies to the first minute of your life, as well as the full interval your life fills. And it applies to the first second of that first minute. And so on.

  42. 42.

    The situation Merricks has us consider is slightly different: he has us imagine that everything is made up of four-dimensional cells which have temporal parts, but that the objects with the cells as proper parts do not have any additional, intermediate parts. So the composites of multiple cells do not have proper temporal parts, though parts of them have proper temporal parts. I’ve gone a different route simply because, on the view I’ve presented, no objects have proper temporal parts. So not only needn’t an object possess proper temporal parts for it to persist four-dimensionally, nothing in its world needs to have proper temporal parts.

  43. 43.

    For a presentation of this problem, see Olson (2007, pp. 78–98). The basic idea is this: It seems that the four-dimensionalist must accept the rather odd claim that, in addition to you thinking your thoughts, something distinct from you, say, your minute-long current temporal part, also thinks some of your thoughts. On the view I’ve presented, all of the proper parts of thinking creatures are too simple to think themselves.

  44. 44.

    This portion of the account is in some ways similar to Hudson’s account of pretension (Hudson 2006, p. 99): ‘\(x \hbox {pertends}\)\(\,= \hbox { df } x\) is a material object, and x is entirely located at (i.e., located at, and contained in) a non-point-sized region, r, and for each proper subregion of r, \(r^{*}\), x has a proper part entirely located at \(r^{*}\). But, importantly: pertension requires that objects obey DAUP, and my four-dimensional persistence does not.

  45. 45.

    Sider (2001) talks about interpreting ‘y is part of x at t’ to mean y has a temporal part at t that is part of x. I do not intend to pick out such a relation with my “at a time, x is part of y”. I intend the ordinary requirements for parthood to apply. So, for instance, there are no parts of y that are not also parts of x.

  46. 46.

    Further, if you believe that unextended objects can inhabit gunky regions, the conditions in Four-Dimensionalism\(_{7}\) will not be sufficient to guarantee that there is an adequate array of temporal parts. To briefly explain: consider a continuous, topologically closed, perfectly straight line segment 1\(^{\prime }\) long, made of uncountably many points. For each point, it has some exact distance in inches from the left end-point of the line-segment (for instance, the point exactly in the middle of the segment will be .5 in from the leftmost point). Consider all and only those points that have an exact distance that is a rational number of inches. Call these the Xs. The Xs do not make up the whole line-segment (for instance, there are some points with a distance from the leftmost point that is not a rational number of inches). Now suppose, keeping their size and arrangement fixed, that the Xs were embedded in a gunky, one-dimensional, inch-long region. Every subregion of that region would contain at least one of the Xs. If this sort of case is possible, you might think it is possible (or at least, conceivable) for a persisting, hour-long entity in gunky time to have a similar “dusting” of instantaneous temporal parts. Suppose that the hour-long entity is exactly located at a continuous, hour-long interval, and is not the fusion of the instantaneous temporal parts. This entity will count as having temporal parts contained within each interval it persists through, but you may think it isn’t persisting merely in virtue of those temporal parts, and that this should not count as a four-dimensionally persisting object. In this case, you will want a complication of condition (ii) similar to the one offered following the presentation of formulation Four-Dimensionalism\(_{8}\).

  47. 47.

    Some problems: It has difficulties with mixtures of Four-Dimensionalism and multiple-location, where an object bears the located at relation to more than one region (though issues of spatiotemporal multilocation raise enough problems that I leave discussion of this for another paper). And it may misclassify some three-dimensional objects that do not change their parts across time (though this could be avoided with some additional requirements).

  48. 48.

    It should be noted that this object would not count as persisting four-dimensionally relative to every reference frame. However, we can imagine a theorist who posits objects such as these and believes (perhaps for moving spotlight purposes) that there is a metaphysically privileged reference frame. All else being equal, it would be better for our formulation of Four-Dimensionalism to allow for such views, and to not presuppose that a privileged reference frame does not exist.

  49. 49.

    I.e., any subregion of a fusion of regions that x is located at. Note that this, as well as the more intuitive, simpler statement of condition (ii), will need adjustment if we wish to allow for spatiotemporal multilocation of (what we might otherwise think are) four-dimensionally persisting entities.

  50. 50.

    Interestingly, if we think objects can be smaller than any region they are contained within, we cannot use the definition of ‘containment’ given at the beginning of this paper (for these objects will not count as being located at any region). In fact, we may need multiple locative primitives, and even then we face problems. For more on this, see (Kleinschmidt, forthcoming). I will here rely on an intuitive notion of containment. For those who believe objects cannot be smaller than any region they are contained within, you are welcome to continue using the definitions given in Sect. 1.

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Correspondence to Shieva Kleinschmidt.

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Thanks to Yuri Balashov, Renee Bolinger, Matthew Davidson, Maegan Fairchild, Kit Fine, Michael Hall, John Hawthorne, Jake Ross, Mark Schroeder, Ted Sider, and the audience at the 2015 Central APA for helpful discussion on this topic.

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Kleinschmidt, S. Refining Four-Dimensionalism. Synthese 194, 4623–4640 (2017). https://doi.org/10.1007/s11229-016-1164-3

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Keywords

  • Persistence
  • Four-Dimensionalism
  • Parthood
  • Time