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We think of a boundary whenever we think of an entity demarcated from its surroundings. There is a boundary (a line) separating Maryland and Pennsylvania. There is a boundary (a circle) isolating the interior of a disc from its exterior. There is a boundary (a surface) enclosing the bulk of this apple. Sometimes the exact location of a boundary is unclear or otherwise controversial (as when you try to trace out the margins of Mount Everest, or even the boundary of your own body). Sometimes the boundary lies skew to any physical discontinuity or qualitative differentiation (as with the border of Wyoming, or the boundary between the upper and lower halves of a homogeneous sphere). But whether sharp or blurry, natural or artificial, for every object there appears to be a boundary that marks it off from the rest of the world. Events, too, have boundaries — at least temporal boundaries. Our lives are bounded by our births and by our deaths; the soccer game began at 3pm sharp and ended with the referee’s final whistle at 4:45pm. It is sometimes suggested that even abstract entities, such as concepts or sets, have boundaries of their own, and Wittgenstein could emphatically proclaim that the boundaries of our language are the boundaries of our world (1921: 5.6). Whether all this boundary talk is coherent, however, and whether it reflects the structure of the world or simply the organizing activity of our mind, are matters of deep philosophical controversy.

Euclid defined a boundary as “that which is an extremity of anything” (Elements Bk I, Df 13), and Aristotle made this more precise by defining the extremity of a thing x as “the first point beyond which it is not possible to find any part [of x], and the first point within which every part [of x] is.” (Metaphysics V, 17, 1022a4–5) This definition is intuitive enough and may be regarded as the natural starting point for any investigation into the concept of a boundary. Indeed, although Aristotle’s definition was only meant to apply to material substances, it intuitively applies to events as well (insofar as they have mereological structure) and by extension also to abstract entities such as concepts and sets (compare the topologically standard notion of the boundary of a set x as the set of those points all of whose neighborhoods intersect both x and the complement of x). Nonetheless, precisely this intuitive characterization gives rise to several puzzles that justify philosophical concern, especially with respect to the boundaries of spatio-temporal particulars such as objects and events.

The first sort of puzzle relates to the intuition that a boundary separates two entities (or two parts of the same entity), which are then said to be continuous with each other. Imagine ourselves traveling from Maryland to Pennsylvania. What happens as we cross the Mason-Dixon line? Do we pass through a last point p in Maryland and a first point q in Pennsylvania? Clearly not, given the density of the continuum; for then we should have to admit an infinite number of further points between p and q that would be in neither State. But, equally clearly, we can hardly acknowledge the existence of just one of p and q, as is dictated by the standard mathematical treatment of the continuum, for either choice would amount to a peculiar privileging of one State over the other. And we cannot identify p with q, either, for we are speaking of two adjacent States, so their territories cannot have any parts in common. So, where is the Mason-Dixon line, and how does it relate to the two adjacent entities it separates?

The puzzle can be generalized. Consider the dilemma raised by Leonardo in his Notebooks: What is it that divides the atmosphere from the water? Is it air or is it water? (1938: 75–76). Or consider Suárez’s worry in Disputation 40 (Sect. V, §58), repeatedly echoed by Peirce (1892: 546; 1893: 7.127): What color is the line of demarcation between a black spot and its white background? Perhaps figure/ground considerations could be invoked to provide an answer in this second case, based on the principle that the boundary is always owned by the figure — the background is topologically open (Jackendoff 1987, App. B). But what is figure and what is ground when it comes to two adjacent halves of the same black spot? What is figure and what is ground when it comes to Maryland and Pennsylvania? What happens when we dive into the water? Besides, it would be natural to suppose that all entities of the same sort be treated alike, for instance, that all material bodies be construed as figure-like entities, each possessing its own boundary. But then, how could any two of them ever come into contact, short of penetrability? (In this last form, the question is widely discussed in recent literature; see e.g. Kline and Matheson 1987, Hazen 1990, Zimmerman 1996b, Kilborn 2007.)

Consider also Aristotle’s classical version of the puzzle in regard to temporal boundaries: When a moving object comes to rest, is it in motion or is it at rest? (Physics VI, 3, 234a ff.) Of course, one could maintain that there is no motion or rest at an instant, but only during an interval, as Aristotle himself held (232a32–34). Yet the question remains: Does the transitional moment belong to the motion interval or to the rest interval? (On this version of the puzzle, see Medlin 1963, Hamblin 1969, Kretzmann 1976, and Sorabji 1983, ch. 26.)

A second sort of puzzle relates to the fact that Aristotle’s mereological definition, and the common-sense intuition that it captures, only seem to apply to the realm of continuous entities. Modulo the above-mentioned difficulty, the thought that Maryland and Pennsylvania are bounded by the Mason-Dixon line is fair enough. But ordinary material objects — it might be observed — are not truly continuous (or dense) and speaking of an object’s boundary is like speaking of the “flat top” of a fakir’s bed of nails (Simons 1991: 91). On closer inspection, the spatial boundaries of physical objects are imaginary entities surrounding swarms of subatomic particles, and their exact shape and location involve the same degree of idealization of a drawing obtained by “connecting the dots”, the same degree of arbitrariness as any mathematical graph smoothed out of scattered and inexact data, the same degree of abstraction as the figures’ contours in a Seurat painting. Similarly, on closer inspection a body’s being in motion amounts to the fact that the vector sum of the motions of zillions of restless particles, averaged over time, is non-zero, hence it makes no sense to speak of the instant at which a body stops moving (Galton 1994: 4). All this may be seen as good news vis-à-vis the puzzles of Section 1.1, which would not even get off the ground (at least in the form given above; see S. R. Smith 2007 and Wilson 2008 for qualifications). But then the question arises: Is our boundary talk a mere façon de parler?

Even with reference to the Mason-Dixon line — and, more generally, those boundaries that demarcate adjacent parts of a continuous manifold, as when an individual cognitive agent conceptualizes a black spot as being made of two halves — one can raise the question of their ontological status. Such boundaries reflect to various degrees the organizing activity of our intellect, or of our social practices. And it might be argued that belief in their objectivity epitomizes a form of metaphysical realism that cries for justification. We may, in this connection, introduce a conceptual distinction between natural or bona fide boundaries, which are in some sense objective, i.e., grounded in some physical discontinuity or qualitative heterogeneity betwixt an entity and its surroundings, and artificial or fiat boundaries, which are not so grounded in the autonomous, mind-independent world (B. Smith 1995, building on Curzon 1907). Geo-political boundaries such as the Mason-Dixon line are obviously of the fiat sort, and on closer inspection it appears that even the surfaces of ordinary material objects such as apples and tables involve fiat articulations of some kind. So the question is, are there any bona fide boundaries? And, if not, is the fiat nature of our boundary talk a reason to justify an anti-realist attitude towards boundaries altogether? (Compare also how the issue arises in the realm of abstract entities: Are there concepts that carve the world “at the joints”, as per Plato’s recipe in the Phaedrus 265e?)

The question has deep ramifications. For once the fiat/bona fide opposition has been recognized, it is clear that it can be drawn in relation to whole objects and events also (Smith and Varzi 2000, Smith 2001). Insofar as (part of) the boundary of a whole is of the fiat sort, the whole itself may be viewed as a conceptual construction, hence the question of the ontological status of boundaries becomes of a piece with the more general issue of the conventional status of ordinary objects and events. Cfr. Goodman: “We make a star as we make a constellation, by putting its parts together and marking off its boundaries” (1980: 213) This is not to imply that we end up with imaginary or otherwise unreal wholes: as Frege wrote, the objectivity of the North Sea “is not affected by the fact that it is a matter of our arbitrary choice which part of all the water on the earth’s surface we mark off and elect to call the ‘North Sea’” (1884, §26). It does, however, follow that the entities in question would only enjoy an individuality as a result of our fiat, like the cookies carved out of the big dough: their objectivity is independent, but their individuality — their being what they are, perhaps even their having the identity and persistence conditions they have — would depend on the baker’s action (Sidelle 1989; Heller 1990; Jubien 1993; Varzi 2011).

A third puzzle relates to vagueness. Aristotle’s definition (as well as standard topology) suggests that there is always a sharp demarcation between the inside and the outside of a thing. Yet it may be observed that ordinary objects and events, as well as the extensions of many ordinary concepts, may have boundaries that are in some sense fuzzy or indeterminate. Clouds, deserts, mountains, let alone the figures of an impressionist painting, all seem to elude the idealized notion of a sharply bounded entity. Likewise, the temporal boundaries of many events, let alone their spatial boundaries, seem to be indeterminate. When exactly did the industrial revolution begin? When did it end? Where did it take place? And certainly the concepts corresponding to such predicates as ‘bald’ or ‘heap’ do not posses definite boundaries, either. As again Frege famously put it, to such concepts there seems to correspond “an area that ha[s] not a sharp boundary-line all around, but in places just vaguely fade[s] away into the background” (1903: §56).

How is such fuzziness to be construed? One option is insist on a purely epistemic account: the fuzziness would lie exclusively in our ignorance about the exact location of the relevant boundaries (Sorensen 1988, Williamson 1994). Alternatively, one may distinguish here between a de re account and a de dicto account. On the de re account, the fuzziness is truly ontological; the boundary of Mount Everest (say) would be vague insofar as there is no objective, determinate fact of the matter about which parcels of land lie on which side (Tye 1990; Copeland 1995; Akiba 2004; Hyde 2008). Likewise, on this account a predicate such as ‘bald’ would be vague because it stands for a vague set, a set with truly fuzzy boundaries. By contrast, the de dicto account corresponds to a purely linguistic (or conceptual) notion of vagueness. There is no vague boundary demarcating Mount Everest on this view; rather, there are many distinct parcels of land, each with a precise border, but our linguistic practices have not enforced a choice of any one of them as the official referent of the name ‘Everest’ (Mehlberg 1958; Lewis 1986; McGee 1997). Similarly, on this view the set of bald people does not have a fuzzy boundary; rather, our linguistic stipulations do not fully specify which set of people corresponds to the extension of ‘bald’ (Fine 1975; Keefe 2000). For boundaries of the fiat sort, a de dicto account suggests itself naturally: insofar as the process leading to the definition of a boundary may not be precise, the question of whether something lies inside or outside the boundary may be semantically indeterminate. But this account does not sit well with boundaries of the bona fide sort (if any); if any such boundary were vague, it would be so independently of our cognitive or social articulations, hence a de re account would seem to be necessary, which means that there would be genuine worldly indeterminacy.

A fourth source of concern relates to the intuition, implicit in Aristotle’s definition, that boundaries are lower-dimensional entities, i.e., have at least one dimension fewer than the entities they bound. The surface of a (continuous) sphere, for example, is two-dimensional (it has no “substance” or “divisible bulk”), the Mason-Dixon Line is one-dimensional (it has “length” but no “breadth”), and a boundary point such as the vertex of a cone is zero-dimensional (it extends in no direction). This intuition is germane to much of what we ordinarily say about boundaries. But it is problematic insofar as it contrasts with several independent intuitions that are of a piece with both common sense and philosophical theorizing. For instance, there is a standing tradition in epistemology (from Moore 1925 to Gibson 1979) according to which boundaries play a crucial role in perception: we see (opaque) physical objects indirectly by seeing their surfaces. Yet it is not clear how one could see entities that lack physical bulk. Likewise, we often speak of surfaces as of things that may be pitted, or damp, or that can be scratched, polished, sanded, and so on, and it is unclear whether such predicates can be applied at all to immaterial entities. In such cases, it would rather seem that surfaces (and boundaries more generally; see Jackendoff 1991) are to be construed as “thin layers” that are schematized as having fewer dimensions than the wholes to which they apply.

Arguably, this conceptual tension between boundaries understood as lower-dimensional entities and boundaries understood as thin layers reflects an irreducible ambiguity in ordinary speech (Stroll 1979, 1988). And, arguably, it is only the first conception that gives rise to the puzzles outlined in the foregoing sections; bulky boundaries can be treated as ordinary, extended parts of the bodies they bound. Yet a general theory of boundaries should have something to say about the second conception as well — and more generally about the interaction between the mathematical idealization associated with the former conception and the physical, cognitive, and philosophical significance of the latter (Galton 2007).

The issue also relates to the question of whether there can be extended mereological simples (raised by van Inwagen 1981 and Lewis 1991: 32, 76 and extensively discussed e.g. in Markosian 1998, Parsons 2000, Simons 2004, Braddon-Mitchell and Miller 2006, and McDaniel 2007, inter alia). For the characterization of boundaries as lower-dimensional extremities is compatible with the possibility that a boundary’s spatiotemporal extension transcend its mereological structure. A pointy boundary, for instance, could be pointy in that it has no proper parts, but that does not rule out that it be located at an extended region short of assuming that the mereological structure of a thing always matches perfectly that of its location (what Varzi 2007:1018 calls “mirroring” and Shaffer 2009:138, Uzquiano 2011 “harmony”).


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