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 borders of a desert, the edges of a mountain, 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 the 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 (witness the popular method for representing the latter by means of simple closed curves encompassing their contents, as in Euler circles and Venn diagrams), and Wittgenstein could emphatically proclaim that the boundaries of our language are the boundaries of our world (1921: prop. 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, or of our collective practices and conventions, are matters of deep philosophical controversy.
Euclid defined a boundary as “that which is an extremity of anything” (Elements, I, def. 13). Aristotle defined the extremity of a thing as “the first thing beyond which it is not possible to find any part [of the given thing], and the first within which every part is” (Metaphysics, V, 1022a4–5). Together, these two definitions deliver the classic account of boundaries, an account that is both intuitive and comprehensive and offers the natural starting point for any further investigation into the boundary concept. Indeed, although Aristotle’s definition concerned primarily the extremities of spatial entities, it applies equally well in the temporal domain. Just as the Mason-Dixon line marks the boundary between Maryland and Pennsylvania insofar as no part of Maryland can be found on the northern side of the line, and no part of Pennsylvania on its southern side, so “the now is an extremity of the past (no part of the future being on this side of it), and again of the future (no part of the past being on that side of it)” (Physics, VI, 233b35–234a2). Similarly for concrete objects and events: just as the surface of an apple marks its spatial boundary insofar as the apple extends only up to it, so the referee’s whistle marks the temporal boundary of the game insofar as the game protracts only up to it. In the case of abstract entities, such as concepts and sets, the account is perhaps adequate only figuratively. Still, it is telling that one of the Greek words for ‘boundary’, ὅρος, is also a word for ‘definition’: as John of Damascus nicely put it, “definition is the term for the setting of land boundaries taken in a metaphorical sense” (The Fount of Knowledge, I, 8). Likewise, it is telling that in point-set topology the standard definition of a set’s boundary (from Hausdorff 1914, §7.2) reflects essentially the same intuition: the boundary, or frontier, of a set A is the set of those points all of whose neighborhoods intersect both A and the complement of A (where a neighborhood of a point p is, intuitively, a set of points that entirely “surround” p). It is not an exaggeration, therefore, to say that the Euclidean-Aristotelian characterization captures a general intuition about boundaries that applies across the board. Nonetheless, precisely this intuitive characterization gives rise to several puzzles that justify philosophical concern.
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 in contact with each other. Following Smith (1997a: 534), 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. As Aristotle put it, no two points can lie “in succession” to each other (Physics, VI, 231b6–9), so we should have to countenance an infinite number of further points between p and q, hence between the two States, contrary to their being in contact. 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 (Dedekind 1872); either choice would amount to a peculiar privileging of one State over the other, an unacceptably arbitrary asymmetry. And it would seem that we cannot identify p with q, either, for we are speaking of two adjacent States; their territories cannot have any parts in common, not even pointy parts. 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 da Vinci in his Notebooks (1938: 75–76): What is it that divides the atmosphere from the water? Is it air or is it water? Or consider Suárez’s worry in the Disputations (40, 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 a seabird dives into the water? Indeed, it would be natural to suppose that all entities of the same sort behave alike — for instance, that all material bodies be figure-like entities, each possessing its own boundary. But then, how could any two of them ever be or come in contact, short of penetrability? (In this last form, the question goes back to Sextus Empiricus, Against the Physicists, I, 258–266, and is widely discussed in recent literature; see e.g. Kline and Matheson 1987, Godfrey 1990, Hazen 1990, Zimmerman 1996b, Lange 2002, §1.3, Hudson 2005, §3.1, Kilborn 2007, Sherry 2015.)
Consider also Plato’s classical version of the puzzle in regard to temporal boundaries (Parmenides, 156c–e): When an object starts moving, or a moving object comes to rest, is it in motion or is it at rest? As Aristotle later put it, the question arises precisely because “the now that is the extremity of both times must be one and the same”, for again, “if each extremity were different, the one could not be in succession to the other” (Physics, VI, 234a5–6). Of course, one could maintain that there is no motion at an instant, but only during an interval, as Aristotle himself held (231b18–232a18, 234a24–b9). 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, Strang 1974, Kretzmann 1976, Sorabji and Kretzmann 1976, Sorabji 1983, ch. 26, Mortensen 1985.)
Besides, the problem is not specific to the transition between motion and rest and admits of several variants that would seem to resist Aristotle’s solution. Gellius, for instance, tells us that the Middle Platonists were much concerned with the parallel question of whether a dying person dies when they are already in the grasp of death or while they still live (Noctes Atticae, VI, xiii, 5–6). This was thought to be a genuine insolubilis, short of conceding the absurdity that no one ever dies (Sextus Empiricus, Against the Physicists, I, 269), and the same could be said of the many variants discussed by later Platonists and by medieval and modern philosophers (see Strobach 1998 and Goubier and Roques 2018). Deep down, the puzzle is no less than a primary illustration of the paradoxical nature of instantaneous change (about which see the entry on change and inconsistency).
A second sort of puzzle relates to the fact that Aristotle’s mereological (parthood-based) definition, and the common-sense intuition that it captures, seem to apply only 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 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 generalized puzzles of Section 1.1, which would not even get off the ground (at least in the form given above; see 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 are puzzling; but are they real? After all, they stem entirely from our social practices and from the organizing activity of our intellect. It might be argued, therefore, that belief in their objectivity epitomizes a form of metaphysical realism that cries for justification. Do such boundaries really exist?
We may, in this connection, introduce a conceptual distinction between “natural” or bona fide boundaries, which would be objective insofar as they are 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 (Smith 1995, building on Curzon 1907). The coastline of Britain, or the boundary separating a black spot from its white background, might be examples of the former, but geopolitical boundaries such as the Mason-Dixon line, or the boundary between the top and the bottom halves of the black spot, are clearly of the latter sort. Moreover, just as fiat boundaries may be involved in the partitioning of larger wholes into proper parts, so they are often at work when we circumclude a number of smaller entities into larger wholes: think of Benelux or Polynesia, but also of the easiness with which we represent the world as consisting of forests, swarms of bees, constellations, or when we group our actions into baseball games, electoral campaigns, wars, etc. (Smith 1999b). Now, insofar as they are not truly continuous, even the surfaces of ordinary material objects — hence the boundaries of the individual trees, the individual bees, the individual stars — may be said to involve fiat demarcations of just the same sort. (Cf. Goodman: “We make a star as we make a constellation, by putting its parts together and marking off its boundaries”; 1980: 213.) On closer inspection, even the boundaries of our individual actions may be seen in this light, indeed even a person’s birth and death may to some extent reflect conventional decisions and stipulations, witness the controversies on abortion and euthanasia (see the entries on life and the definition of death). On closer inspection, even the extolled coastline of Britain is to some extent fixed by us, witness the proverbial elusiveness of its objective length and location (Mandelbrot 1967). So the question arises: are there any natural, mind-independent, genuine bona fide boundaries? The natural/artificial distinction is intuitively clear; but how robust is it? Are there any concepts that truly carve the world “at the joints”, as per Plato’s butchering guidelines (Phaedrus, 265e)? And, if not, is the fiat nature of our boundary talk a reason to justify an anti-realist attitude towards boundaries altogether?
The question has deep ramifications. For once the fiat/bona fide opposition has been recognized, it is clear that it can be drawn not merely in relation to boundaries, but also in relation to those entities that may be said to have boundaries (Smith and Varzi 2000; Smith 2001, 2019; Tahko 2012; Davies 2019). If (part of) their boundaries are artificial — if they reflect an articulation of reality that is effected through human cognition and social practices — then those entities themselves may be viewed as conceptual constructions, a product of our worldmaking, hence the question of the ontological status of fiat boundaries becomes of a piece with the more general issue of the conventional status of ordinary objects and events. 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). Or as James wrote, echoing Michelangelo: “The mind works on the data it receives very much as a sculptor works on his block of stone; in a sense the statue stood there from eternity” (1890: I/188). It does, however, follow that the entities in question would only enjoy an individuality as a result of our selective strokes: their objectivity is independent of us, but their individuality — their being the sorts of thing they are, perhaps even their having the identity and persistence conditions they have — would depend on our choices and our identification and reidentification criteria (Sidelle 1989; Heller 1990; Zerubavel 1991; Jubien 1993; Varzi 2011, 2016; Azzouni 2017; Piras 2020).