Abstract
In this paper I take up the task, begun by Lewis and Lewis in their seminal paper on the topic, of offering a theory of holes according to which a hole is simply its hole-lining. I begin by motivating the theory, arguing that it holds interest even absent its original animating concerns of nominalism and materialism, and present desiderata any such theory must satisfy. With this in place, I offer a definition both of a lining and of hole sameness, arguing that these can overcome the most prominent standing objections (due to Casati and Varzi) to identifying holes with linings. Finally, I present a more powerful counterargument in the spirit of Casati and Varzi, and conclude with an emended version of the theory meeting all our earlier desiderata. I conclude by dispelling some possible worries about the nominalist and materialist credentials of my theory.
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Notes
Given that this account is a restricted version of supersubstantivalism, its reliance on counterpart theory is a special case of the more general appeal counterpart theory has for supersubstantivalists, see Schaffer (2009).
The principles I am thinking of often go under the heading of "Mereological Harmony," which is given expression in e.g., Leonard (2016), Saucedo (2011), Uzquiano (2011) and Varzi (2007). While Mereological Harmony has come under criticism on various counts [see again (Saucedo, 2011), also (Hawthorne & Uzquiano, 2011; McDaniel, 2007; Newman, 2002) and others], it also has adamant defenders [see e.g., Schaffer (2009)], and much of this discussion concerns fringe cases like colocated bosons, universals, and tropes, which the relevant examples in this paper will try to avoid. The main non-fringe assumption I will make, conflicting with e.g., Cartwright (1975) and Uzquiano (2006), is that exact locations are free to be either closed or open. This will come up later explicitly.
This concept is, of course, a contested one in the literature on persistence: there is a debate between endurantists and exdurantists on the one hand, who think that ordinary material objects are exactly located at three-dimensional regions of zero temporal extent, and perdurantists on the other, who think that such objects are located instead at four-dimensional spacetime regions usually of more than zero temporal extent. (This way of framing the question is most natural in a pre-relativistic setting. For attempts to further this debate in a relativistic framework, see Balashov (2010) and Gilmore (2008). See Balashov (2010), Donnelly (2011), Eddon (2010), Gilmore (2008), Hawthorne et al. (2008) and Sider (2001).
I have tried, so far as is possible, to avoid framing my arguments in such a way as to presuppose any of these views of persistence. In particular, the main examples to be considered involve atemporal three-dimensional worlds, in which varying views about persistence should converge, and I have tried to leave my definitions general enough to suit any competitor in this debate. The main difficulty posed by my definitions will be for the endurantist, because in accordance with Mereological Harmony I take it that each material object has a unique location, and this is sometimes denied by endurantists who think that an object will have multiple exact locations at different times. This problem, though, is easily resolved by indexing all claims in my definitions about locations to a time. Despite the intrinsic interest of such debates, therefore, I think they will here be irrelevant.
One might also plausibly stipulate that a discontinuity must have measure \(>0\), and that a holed object’s region and its convex hull both be measurable. Given this stipulation and that we are working with the Lebesgue measure, or indeed any \(\sigma \)-finite measure, it follows that any object will have at most countably many discontinuities, since given such a measure no set (and in particular the convex hull of the object’s region minus the object’s region) with positive measure can be decomposed into uncountably many disjoint sets each with positive measure, and the discontinuities of an object will always be so disjoint from one another.
According to the common mathematical definition, a region will be connected just in case it is not the union/fusion of two disjoint nonempty open sets, and we can think of a connected spatial object as an object whose exact location is a connected region. Intuitively, a region or body is connected if one can draw a curve between any two points in it without leaving it. This intuitive gloss does not fit the definition perfectly, but in the topological spaces considered in this paper, the differences will be irrelevant. A maximally connected subregion of R is just a connected subregion of R with no connected proper superregion that is also a subregion of R.
The reason for the mention of infimal distances is that for some or all points p in some linings, such as one surrounding a discontinuity that is an open ball, there may be no set of points in the discontinuity least distant from p.
Note that, for the first time, we are appealing intuitively to principles in Mereological Harmony (discussed earlier in a footnote) whereby an object’s location places constraints on the mereology of the object, rather than vice versa. More specifically, if r(x) contains a point p, x must mereologically contain a part at least partially located at p. In the current setting, where we are concerned with objects composed of point particles, this definition could be rephrased thus (assuming that the location of a point-particle is a point), without relying on this component of Mereological Harmony:
Definition 1.3. A part x of o goes right up to a discontinuity d in o iff for any real \(\epsilon \) such that \(\epsilon \) is the infimum of the distances between some point in r(x) and all points in d, all point-particles p in o such that some \(\delta \le \epsilon \) is the infimum of the distances between r(p) and all points in d are also parts of x.
Argle: ”A hole is a hole not just by virtue of its own shape but also by virtue of the way it contrasts with the matter inside it and around it.”
Even those who disagree that shape is intrinsic do not usually contest such independence of a material object’s shape from the existence or location of other material objects, like the mereological complement of P in the ball, mereologically and spatially disjoint from them (Skow, 2007).
In this I differ from Argle himself, who declares, "The same is true of other shape-predicates; I wouldn’t say that any part of the cheese is a dodecahedron, though I admit that there are parts—parts that do not contrast with their surroundings—that are shaped like dodecahedra" (Lewis & Lewis, 1970). This pair of opinions—that filled linings are not holes and non-contrasting dodecahedrally shaped parts are not dodecahedra—is entirely natural; one’s views on the one question should mirror one’s views on the other.
Drawing on our earlier discussion, that is: holes in it rather than parts of it.
We will assume, again mainly for purposes of familiarity, the standard axioms of classical mereology.
Thanks to Erica Shumener for help on this point.
Strangely, Casati and Varzi actually object that Argle should count three holes, but this is apparently to beg the question against his claim to count holes otherwise than by strict identity.
It is important that this ball of removed particles be closed, since were it open there would remain a point-thick skin around it that we could identify as its only lining.
It might be objected that the argument thus far of this section rests on a mistake. Holed objects should be connected, as the fusion of Matryoshka-like spheres is not, or at least hole-linings should be. We thus have no reason to give up our earlier definition of a lining and accept this new one.
Suppose we accept a definition like 3.2 but build in the requirement that any lining must be connected. Then consider a sort of hybrid between the ball with just the unit radius hole in the centre and the onion object. One hemisphere is indiscernible from a hemisphere of the former, one indiscernible from a hemisphere of the latter. Thus, one half will be filled with point particles save for an empty unit half-ball at the centre, while the other half will consist in an infinite series of concentric domes of diminishing breadth joined to the first half as their base and surrounding an empty unit half-ball at the centre. This connected object will pose, with minor adjustments, all the problems for the connectedness-requiring definition as the Matryoshka object does for the official definition, and the same response I will offer later for my own stated problems will resolve these versions of the problems, too.
As noted in previous footnote, this definition also solves the problem mentioned above, that the old definition allowed linings that did not go "all the way around" the discontinuity. By 1.2, a lining must contain all the points at a distance to the discontinuity less than any other point in it, and thus cannot leave out any such points.
I owe this point to Harvey Lederman.
What do I mean by "essentially" here? As a first pass: if x is essentially F, then were x not F, a contradiction \(\bot \) would be true.
Where going-right-up-to-a-discontinuity-in is regarded as a two-place relation between a part and a whole, not a three-place relation between a part, whole, and actual discontinuity; the reference to "a discontinuity" is merely superficial and not ontologically committing.
For more on fictionalism, see Kalderon (2005).
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The author has no relevant financial or non-financial interests to disclose. She would like to thank Harvey Lederman for advice and comments on several drafts of this paper.
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L. Mollica: I would like to thank Harvey Lederman for extensive feedback on several drafts of this paper.
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Mollica, L. Argle victorious: a theory of holes as hole-linings. Synthese 200, 457 (2022). https://doi.org/10.1007/s11229-022-03952-z
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DOI: https://doi.org/10.1007/s11229-022-03952-z