We think of a boundary whenever we think of an entity demarcated fromits surroundings. There is a boundary (a line) separating Maryland andPennsylvania. There is a boundary (a circle) isolating the interior ofa disc from its exterior. There is a boundary (a surface) enclosingthe bulk of this apple. Sometimes the exact location of a boundary isunclear or otherwise controversial (as when you try to trace out theborders of a desert, the edges of a mountain, or even the boundary ofyour own body). Sometimes the boundary lies skew to any physicaldiscontinuity or qualitative differentiation (as with the border ofWyoming, or the boundary between the upper and the lower halves of ahomogeneous sphere). But whether sharp or blurry, natural orartificial, for every object there appears to be a boundary that marksit off from the rest of the world. Events, too, have boundaries— at least temporal boundaries. Our lives are bounded by ourbirths and by our deaths; the soccer game began at 3pm sharp and endedwith the referee’s final whistle at 4:45pm. It is sometimessuggested that even abstract entities, such as concepts or sets, haveboundaries of their own (witness the popular method for representingthe latter by means of simple closed curves encompassing theircontents, as in Euler circles and Venn diagrams), and Wittgensteincould emphatically proclaim that the boundaries of our language arethe boundaries of our world (1921: prop. 5.6). Whether all thisboundary talk is coherent, however, and whether it reflects thestructure of the world or simply the organizing activity of our mind,or of our collective practices and conventions, are matters of deepphilosophical controversy.

• 1. Issues
  • 1.1 Owned vs. Unowned Boundaries
  • 1.2 Natural vs. Artificial Boundaries
  • 1.3 Sharp vs. Fuzzy Boundaries
  • 1.4 Bodiless vs. Bulky Boundaries
• 2. Theories
  • 2.1 Realist Theories
  • 2.2 Non-realist Theories
• 3. Other Topics
  • 3.1 Border Studies
  • 3.2 Boundaries in the Life Sciences
  • 3.3 Boundary Perception
• 4. Excerpts from the Literature
• Bibliography
• Academic Tools
• Other Internet Resources
• Related Entries

1. Issues

Euclid defined a boundary as “that which is an extremity ofanything” (Elements, I, def. 13). Aristotle defined theextremity of a thing as “the first thing beyond which it is notpossible to find any part [of the given thing], and the first withinwhich every part is” (Metaphysics, V, 1022a4–5).Together, these two definitions deliver the classic account ofboundaries, an account that is both intuitive and comprehensive andoffers the natural starting point for any further investigation intothe boundary concept. Indeed, although Aristotle’s definitionconcerned primarily the extremities of spatial entities, it appliesequally well in the temporal domain. Just as the Mason-Dixon linemarks the boundary between Maryland and Pennsylvania insofar as nopart of Maryland can be found on the northern side of the line, and nopart of Pennsylvania on its southern side, so “the now is anextremity of the past (no part of the future being on this side ofit), and again of the future (no part of the past being on that sideof it)” (Physics, VI, 233b35–234a2). Similarlyfor concrete objects and events: just as the surface of an apple marksits spatial boundary insofar as the apple extends only up to it, sothe referee’s whistle marks the temporal boundary of the gameinsofar as the game protracts only up to it. In the case of abstractentities, such as concepts and sets, the account is perhaps adequateonly figuratively. Still, it is telling that one of the Greek wordsfor ‘boundary’, ὅρος, is also aword for ‘definition’: as John of Damascus nicely put it,“definition is the term for the setting of land boundaries takenin a metaphorical sense” (The Fount of Knowledge, I,8). Likewise, it is telling that in point-set topology the standarddefinition of a set’s boundary (from Hausdorff 1914, §7.2)reflects essentially the same intuition: the boundary, or frontier, ofa setA is the set of those points all of whose neighborhoodsintersect bothA and the complement ofA (where aneighborhood of a pointp is, intuitively, a set of pointsthat entirely “surround”p). It is not anexaggeration, therefore, to say that the Euclidean-Aristoteliancharacterization captures a general intuition about boundaries thatapplies across the board. Nonetheless, precisely this intuitivecharacterization gives rise to several puzzles that justifyphilosophical concern.

1.1 Owned vs. Unowned Boundaries

The first sort of puzzle relates to the intuition that a boundaryseparates two entities, or two parts of the same entity,which are then said to bein contact with each other.Following Smith (1997a: 534), imagine ourselves traveling fromMaryland to Pennsylvania. What happens as we cross the Mason-Dixonline? Do we pass through a last pointp in Maryland and afirst pointq in Pennsylvania? Clearly not, given the densityof the continuum. As Aristotle put it, no two points can lie “insuccession” to each other (Physics, VI, 231b6–9),so we should have to countenance an infinite number of further pointsbetweenp andq, hence between the two States,contrary to their being in contact. But, equally clearly, we canhardly acknowledge the existence of just one ofp andq, as is dictated by the standard mathematical treatment ofthe continuum (Dedekind 1872); either choice would amount to apeculiar privileging of one State over the other, an unacceptablyarbitrary asymmetry. And it would seem that we cannot identifyp withq, either, for we are speaking of twoadjacent States; their territories cannot have any parts incommon, not even pointy parts. So,where is the Mason-Dixonline, and how does it relate to the two adjacent entities itseparates?

The puzzle can be generalized. Consider the dilemma raised by Leonardoda Vinci in hisNotebooks (1938: 75–76): What is itthat divides the atmosphere from the water? Is it air or is it water?Or consider Suárez’s worry in theDisputations(40, V, §58), repeatedly echoed by Peirce (1892: 546; 1893:7.127): What color is the line of demarcation between a black spot andits white background? Perhaps figure/ground considerations could beinvoked to provide an answer in this second case, based on theprinciple that the boundary is always owned by the figure — thebackground is topologically “open” (Jackendoff 1987, App.B). But what is figure and what is ground when it comes to twoadjacent halves of the same black spot? What is figure and what isground when it comes to Maryland and Pennsylvania? What happens when aseabird dives into the water? Indeed, it would be natural to supposethat all entities of the same sort behave alike — for instance,that all material bodies be figure-like entities, each possessing itsown boundary. But then, how could any two of them ever be orcome in contact, short of penetrability? (In this last form,the question goes back to Sextus Empiricus,Against thePhysicists, I, 258–266, and is widely discussed in recentliterature; see e.g. Kline and Matheson 1987, Godfrey 1990, Hazen1990, Zimmerman 1996b, Lange 2002, §1.3, Hudson 2005, §3.1,Kilborn 2007, Sherry 2015.)

Consider also Plato’s classical version of the puzzle in regardto temporal boundaries (Parmenides, 156c–e): When anobject starts moving, or a moving object comes to rest, is it inmotion or is it at rest? As Aristotle later put it, the questionarises precisely because “the now that is the extremity of bothtimes must be one and the same”, for again, “if eachextremity were different, the one could not be in succession to theother” (Physics, VI, 234a5–6). Of course, onecould maintain that there is no motionat an instant, butonlyduring an interval, as Aristotle himself held(231b18–232a18, 234a24–b9). Yet the question remains: Doesthe transitional moment belong to the motion interval or to the restinterval? (On this version of the puzzle, see Medlin 1963, Hamblin1969, Strang 1974, Kretzmann 1976, Sorabji and Kretzmann 1976, Sorabji1983, ch. 26, Mortensen 1985.)

Besides, the problem is not specific to the transition between motionand rest and admits of several variants that would seem to resistAristotle’s solution. Gellius, for instance, tells us that theMiddle Platonists were much concerned with the parallel question ofwhether a dying person dies when they are already in the grasp ofdeath or while they still live (Noctes Atticae, VI, xiii,5–6). This was thought to be a genuineinsolubilis,short of conceding the absurdity that no one ever dies (SextusEmpiricus,Against the Physicists, I, 269), and the samecould be said of the many variants discussed by later Platonists andby medieval and modern philosophers (see Strobach 1998 and Goubier andRoques 2018). Deep down, the puzzle is no less than a primaryillustration of the paradoxical nature of instantaneous change (aboutwhich see the entry onchange and inconsistency).

1.2 Natural vs. Artificial Boundaries

A second sort of puzzle relates to the fact that Aristotle’smereological (parthood-based) definition, and the common-senseintuition that it captures, seem to apply only to the realm ofcontinuous entities. Modulo the above-mentioned difficulty, thethought that Maryland and Pennsylvania are bounded by the Mason-Dixonline is fair enough. But ordinary material objects — it might beobserved — are not truly continuous and speaking of anobject’s boundary is like speaking of the “flat top”of a fakir’s bed of nails (Simons 1991: 91). On closerinspection, the spatial boundaries of physical objects are imaginaryentities surrounding swarms of subatomic particles, and their exactshape and location involve the same degree of idealization of adrawing obtained by “connecting the dots”, the same degreeof arbitrariness as any mathematical graph smoothed out of scatteredand inexact data, the same degree of abstraction as the figures’contours in a Seurat painting. Similarly, on closer inspection abody’s being in motion amounts to the fact that the vector sumof the motions of zillions of restless particles, averaged over time,is non-zero, hence it makes no sense to speak ofthe instantat which a body stops moving (Galton 1994: 4). All this may be seen asgood newsvis-à-vis the generalized puzzles of Section1.1, which would not even get off the ground (at least in the formgiven above; see Smith 2007 and Wilson 2008 for qualifications). Butthen the question arises: Is our boundary talk a merefaçonde parler? Even with reference to the Mason-Dixon line —and, more generally, those boundaries that demarcate adjacent parts ofa continuous manifold, as when an individual cognitive agentconceptualizes a black spot as being made of two halves — onecan raise the question of their ontological status. Such boundariesare puzzling; but are they real? After all, they stementirely from our social practices and from the organizing activity ofour intellect. It might be argued, therefore, that belief in theirobjectivity epitomizes a form of metaphysical realism that cries forjustification. Do such boundaries really exist?

We may, in this connection, introduce a conceptual distinction between“natural” orbona fide boundaries, which would beobjective insofar as they are grounded in some physical discontinuityor qualitative heterogeneity betwixt an entity and its surroundings,and “artificial” orfiat boundaries, which arenot so grounded in the autonomous, mind-independent world (Smith 1995,building on Curzon 1907). The coastline of Britain, or the boundaryseparating a black spot from its white background, might be examplesof the former, but geopolitical boundaries such as the Mason-Dixonline, or the boundary between the top and the bottom halves of theblack spot, are clearly of the latter sort. Moreover, just asfiat boundaries may be involved in the partitioning of largerwholes into proper parts, so they are often at work when wecircumclude a number of smaller entities into larger wholes: think ofBenelux or Polynesia, but also of the easiness with which we representthe world as consisting of forests, swarms of bees, constellations, orwhen we group our actions into baseball games, electoral campaigns,wars, etc. (Smith 1999b). Now, insofar as they are not trulycontinuous, even the surfaces of ordinary material objects —hence the boundaries of the individual trees, the individual bees, theindividual stars — may be said to involvefiatdemarcations of just the same sort. (Cf. Goodman: “We make astar as we make a constellation, by putting its parts together andmarking off its boundaries”; 1980: 213.) On closer inspection,even the boundaries of our individual actions may be seen in thislight, indeed even a person’s birth and death may to some extentreflect conventional decisions and stipulations, witness thecontroversies on abortion and euthanasia (see the entries onlife andthe definition of death). On closer inspection, even the extolled coastline of Britain is tosome extent fixed by us, witness the proverbial elusiveness of itsobjective length and location (Mandelbrot 1967). So the questionarises: are thereany natural, mind-independent, genuinebona fide boundaries? The natural/artificial distinction isintuitively clear; but how robust is it? Are there any concepts thattruly carve the world “at the joints”, as perPlato’s butchering guidelines (Phaedrus, 265e)? And, ifnot, is thefiat nature of our boundary talk a reason tojustify an anti-realist attitude towards boundaries altogether?

The question has deep ramifications. For once thefiat/bonafide opposition has been recognized, it is clear that it can bedrawn not merely in relation to boundaries, but also in relation tothose entities that may be said tohave boundaries (Smith andVarzi 2000; Smith 2001, 2019; Tahko 2012; Davies 2019). If (part of)their boundaries are artificial — if they reflect anarticulation of reality that is effected through human cognition andsocial practices — then those entities themselves may be viewedas conceptual constructions, a product of our worldmaking, hence thequestion of the ontological status offiat boundaries becomesof a piece with the more general issue of the conventional status ofordinary objects and events. This is not to imply that we end up withimaginary or otherwise unreal wholes: as Frege wrote, the objectivityof the North Sea “is not affected by the fact that it is amatter of our arbitrary choice which part of all the water on theearth’s surface we mark off and elect to call the ‘NorthSea’” (1884, §26). Or as James wrote, echoingMichelangelo: “The mind works on the data it receives very muchas a sculptor works on his block of stone; in a sense the statue stoodthere from eternity” (1890: I/188). It does, however, followthat the entities in question would only enjoy an individuality as aresult of our selective strokes: their objectivity is independent ofus, but their individuality — their being the sorts of thingthey are, perhaps even their having the identity and persistenceconditions they have — would depend on our choices and ouridentification and reidentification criteria (Sidelle 1989; Heller1990; Zerubavel 1991; Jubien 1993; Varzi 2011, 2016; Azzouni 2017;Piras 2020).

These ramifications go well beyond the concerns of metaphysics; theyextend across the board, especially insofar as we have a tendency tocall “natural” exactly those boundaries that best suit ourparochial interests. In the geopolitical domain, for instance, therhetoric with which certain frontiers have beenclaimed to bedrawn by the hand of Nature, if not by the hand of God, has beenresponsible for some of the most tragic vicissitudes in our history,as even Lord Curzon — the epitome of British imperialism —recognized (1907: 54). There is by now a large body of literature onthis topic, stemming mostly from the classic works of Reclus(1905–1908), Febvre (1922), and Ancel (1938) (see Broek 1941and, for recent overviews, Rankin and Schofield 2004 and Fall 2010).But similar considerations apply to all sorts of atrocities that wecommitted and continue to commit in all sorts of domains. Thatrhetoric is at work every time we invoke Plato’s carvingguidelines fraudulently and surreptitiously, every time we stigmatizeas a “bad butcher” whoever holds views or followspractices that don’t match our own as though they were“against nature”. This does not mean that the guidelinesthemselves are corrupt; the realist is still free to take themseriously, if cautiously. But is there any viable way to implementthem irrespective of our ideological, cultural, and individual biases?We can hardly be wrong in attributingfiat status to aboundary that is manifestly drawn by us; but who can vet theobjectivity of a natural joint? Recent work in feminist metaphysicsand epistemology and in the philosophy and sociology of race hasbrought these issues to the fore (see the entries onfeminist metaphysics,feminist epistemology and philosophy of science,race, andcritical philosophy of race). No theory of boundaries can ignore them.

1.3 Sharp vs. Fuzzy Boundaries

A third worry relates to vagueness. Aristotle’s definition (aswell as standard topology) suggests that there is always a sharpdemarcation between the inside (interior) and the outside (exterior)of a thing. Yet it may be observed that ordinary objects and events,as well as the extensions of many ordinary concepts, may haveboundaries that are in some sense fuzzy or indeterminate. Clouds,deserts, mountains, let alone the figures of an impressionistpainting, all seem to elude the idealized notion of a sharply boundedentity. Likewise, the temporal boundaries of many events, let alonetheir spatial boundaries, seem to be indeterminate. When exactly didthe industrial revolution begin? When did it end? Where did it takeplace? And certainly the concepts corresponding to many ordinarypredicates, such as ‘bald’ or ‘heap’, do notpossess 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 vaguelyfade[s] away into the background” (1903: §56). How is suchfuzziness to be construed?

One radical option would be to dismiss the question altogether.Wittgenstein, for instance, urged that “an indefinite boundaryis not really a boundary at all” (1953: 45e). Or one might aswell insist that the issue is purely epistemic: the fuzziness wouldlie exclusively in our ignorance about the exact location of therelevant boundaries, which in themselves would be perfectly sharp(Sorensen 1988; Williamson 1994). When taken at face value, however,the question is highly controversial. One may distinguish broadlybetweende re accounts andde dicto accounts. On ade re account, the fuzziness is truly ontological. Thus,pace Evans (1978), the boundary of, say, Mount Everest wouldbe fuzzy insofar as there is no objective, determinate fact of thematter about which parcels of land lie on which side (Tye 1990;Copeland 1995; Morreau 2002; Hyde 2008). Similarly, a predicate suchas ‘bald’ would be vague because it stands for a vagueset, a set with a truly fuzzy boundary (Zadeh 1965; Goguen 1969). Itis an open question, then, whether such boundary fuzziness wouldentail a corresponding indeterminacy in the identity conditions of therelevant bounded entities (see e.g. Noonan 2008 vs. Paganini 2017),just as it may be an open question whether it would entail anobjective indeterminacy in the existence conditions of certain (other)entities, say, the mereological difference between Everest and the sumof its definite parts (see e.g. Hawley 2002 and Donnelly 2009). Bycontrast, ade dicto account would correspond to a purelylinguistic or conceptual notion of vagueness. There is no fuzzyboundary demarcating Mount Everest on this view; rather, there aremany distinct parcels of land, each with a precise border and preciseexistence and identity conditions, but our linguistic practices havenot enforced a choice of any one of them as the official referent ofthe term ‘Mount Everest’ (Mehlberg 1958; Lewis 1986; McGee1997). Similarly, on this view there is no such thing as a vaguelydemarcated set of bald people; rather, our linguistic stipulations donot precisely specify which set of people, among the many that exist,corresponds to the extension of ‘bald’ (Fine 1975; Keefe2000).

Whether the fuzziness of a given boundary should be construeddere orde dicto may be a case-by-case matter. No onethinks that all indeterminacy is a matter of worldly unsettledness,and the contrary claim, to the effect that boundary indeterminacy isalways the product of semantic or representational imprecision, ishighly contentious despite its distinguished pedigree (Russell 1923).Moreover, there is room for a middle third-way, according to whichthere is genuine ontological indeterminacy, but it doesn’t liein the boundaries themselves; it would lie, rather, in thedeterminable/determinate structure of boundary properties (Wilson20123), or in the fact that it is ontically indeterminate whichprecise boundaries things have (Akiba 2004; Barnes 2010; Abasnezhadand Hosseini 2014). Generally speaking, however, it would seem thatthede re/de dicto alternative aligns rather closely with thebona fide/fiat opposition. With fuzzy boundaries of thefiat sort, ade dicto account suggests itselfnaturally: insofar as the process leading to the definition of aboundary may not be precise, the question of whether something liesinside or outside the boundary may be semantically indeterminate. Onthe other hand, this account does not sit well with boundaries of thebona fide sort. If any such boundary were fuzzy, it would beso independently of our cognitive or social articulations, hence ade re account would seem inevitable; the fuzziness of theboundary would lie in the way the world is (or isn’t). Thus, itwould appear that a boundary may suffer fromde dictofuzziness if and only if it is of thefiat sort, and fromde re fuzziness if and only if it is of thebonafide sort.

Either way, making room for fuzzy boundaries will open the door to themany difficulties that arise whenever vagueness phenomena are taken atface value. In particular, bothde re andde dictoaccounts will have to deal with the challenge of providing a way outof the sorites paradox: how can we ever cross a fuzzy boundary, if nosingle step can make a difference? And both accounts will have toprovide a way of accommodating phenomena of higher order vagueness:not only may it be indeterminate whether something lies on one side orthe other of a fuzzy boundary; it may also be indeterminate whether itis indeterminate whether it lies on one side or the other — andso on. (On these issues, see the entries on theSorites paradox andvagueness.) Some authors think such challenges are insurmountable, and thatde re andde dicto conceptions are equallymisguided. There would be no words or entities with fuzzy boundaries;whenever we think there are, we are, rather, representing the worldthrough concepts that are strictly speaking “boundaryless”(Sainsbury 1990).

1.4 Bodiless vs. Bulky Boundaries

A fourth source of concern relates to the intuition, implicit inAristotle’s and Euclid’s definitions, that boundaries arelower-dimensional entities, i.e., have at least one dimension fewerthan 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 aboundary point such as the vertex of a cone is zero-dimensional (itextends in no direction). This intuition has become common currency incontemporary philosophy through Johnson (1922: 168; 1924:163–164) and is germane to much of what we ordinarily say aboutboundaries. Yet it is problematic insofar as it contrasts with severalindependent intuitions that are of a piece with both common sense andphilosophical theorizing. For instance, there is a standing traditionin epistemology (from Moore 1925 to Jackson 1977 and Gibson 1979)according to which boundaries play a crucial role in perception: wesee (opaque) physical objects indirectly by seeing their surfaces. Yetit is not clear how one could see a whole just by seeing part of it(Chisholm 1957; Clarke 1965), even less how one could literallysee parts that lack physical bulk altogether (Stroll 1986a,1986b). Likewise, we often speak of surfaces as of things that may bepitted, or damp, or that can be scratched, polished, sanded, and soon, and it is unclear whether such predicates can be applied at all toimmaterial entities. In such cases, it would rather seem that surfaces(and boundaries more generally; see Jackendoff 1991 or Hestevold 2020, ch. 6) are to beconstrued as “thin layers” that areschematizedas having fewer dimensions than the wholes to which they apply. Thesame sort of schematization is commonplace in the geopolitical domain,where the boundary lines that we find depicted in ordinary maps andatlases may in fact correspond to border “belts”, or“zones”, of various kinds and widths (Prescott 1965). Thevery fact that the geographic notion of a natural boundary is oftenillustrated with reference to rivers, ravines, mountain ranges, etc.is indicative of this tendency. As Ratzel famously put it: “Theborder area is the reality, the border line the abstractionthereof” (1897: 448).

Arguably, this conceptual tension between boundaries understood aslower-dimensional entities and boundaries understood as thin layers,or zones, reflects an irreducible ambiguity in ordinary speech (Stroll1977, 1979). And, arguably, it is only the first conception that givesrise to the puzzles outlined in the foregoing sections; bulkyboundaries and border zones can be treated as ordinary, extendedproper parts of the bodies and regions they bound. Yet a generaltheory of boundaries should have something to say about the secondconception as well — and more generally about the interactionbetween the mathematical idealization associated with the formerconception and the physical, cognitive, and philosophical significanceof the latter (Stroll 1988; Simons 1991; Galton 2007).

The issue also relates to the question of whether there can beextended mereological simples. The ancient atomists had no doubts,with Democritus famously holding that the ultimate constituent ofreality, while indivisible, come in an infinite variety“differing in size and in shape” (Aristotle,Metaphysics, III, 203b2). Today it is more common to assumethat the mereological structure of a thing always matches perfectlythat of its spatio-temporal receptacle (what Varzi 2007 calls“mirroring” and Uzquiano 2011 “harmony”),which would imply the opposite answer. Nevertheless, in recent yearsseveral authors have rejected the assumption, seriously reviving thehypothesis of mereological simples that occupy extended regions ofspace and/or time (see e.g. Markosian 1998, Parsons 2000, Simons 2004,Braddon-Mitchell and Miller 2006, and McDaniel 2007 along with theentry onlocation and mereology). Insofar as the lower-dimensional nature of boundaries may beconstrued in terms of their beingindivisible along one ormore dimensions, as opposed tounextended along thosedimensions (a conception encouraged also by some medievalphilosophers, e.g. Ibn Sīnā, theIlāhiyyāt ofKitābal-Šifā, II, ii, 4–6), such a view would suggestthat also a boundary’s spatio-temporal extension may transcendits mereological structure; the boundary would lack divisible bulk,but it might nonetheless possess divisible extent. A boundary point,for instance, would be pointy in that it has no proper parts, thoughit could extend over a (small) one-dimensional, a two-dimensional, oreven a three-dimensional region. Again, such extended boundaries wouldbe refreshingly unproblematic. As with their divisible bulky cousins,one could easily explain how we can see them, for instance, or polishthem, scratch them, etc. However, they would also raise questions oftheir own. Above all: How would such boundaries connect to the thingsthey bound? Would there be a further, unextended boundary separatingthe two (and matching the unextended boundary between thecorresponding receptacles)? In the case of boundaries with divisiblebulk, the answer to the latter question is obviously in theaffirmative. It is an open question, however, whether the same answershould apply to extended indivisible boundaries.

2. Theories

So boundaries are, on the one hand, central to the common-sensepicture of the world and yet, on the other, deeply problematic. We mayaccordingly distinguish two main sorts of theories, depending onwhether one is willing to take the problems at face value (realisttheories) or to bypass them altogether, treating boundaries asfictional abstractions of some sort (non-realist theories).

2.1 Realist Theories

Most realist theories about boundaries, construed as lower-dimensionalentities, share the view that such entities are ontological parasites:points, lines, and surfaces cannot be separated and cannot exist inisolation from the entities they bound (pace Suárez,who thought that God would be capable of such marvels; seeDisputation 40, V, §41). This view does justice to theintuition that boundaries, if real, are somewhat “lessreal” than the entities they bound, and as such it goes back toAristotle (Metaphysics, XI, 1060b12–16) and has beendefended e.g. by Boethius (Second Commentary I, xi,14–21), revived by Abelard (Glosses on Porphyry,8.1–4), and eventually backed up by Brentano (1976) and hispupils, especially Husserl in his thirdLogical Investigation(1901). Of course, the notion of ontological dependence is itself opento several interpretations (see Koslicki 2014 and the entry onontological dependence). In the case of boundaries, it is generally agreed that the relevantrelation is one of rigid existential dependence: ifx is aboundary ofy, then necessarilyx exists only ify exists (Chisholm 1983, 1994). Realist theories may differsignificantly, however, with regard to how such rigidly dependent,lower-dimensional entities relate to the extended entities theybound.

To illustrate, consider again the first puzzle of Section 1 (owned vs.unowned boundaries). LetA andB be any two extendedentities separated by a common boundary, such as Maryland andPennsylvania. Then we may distinguish four main views (Varzi 1997;Strobach 1998, pt. II).

(1) The boundary may belongneither toAnor toB. This was, ultimately, Leonardo’sview (1938: 76) and is perhaps the least popular among recentphilosophers (though see e.g. Hestevold 1986 and Schmolze 1996). Itimplies that contact may obtain betweenA andB evenif bothA andB are topologically open, so long asnothing lies between them except for their common, outer boundary— i.e., so long as the closure ofA (A plusits boundary) overlaps the closure ofB (B plus itsboundary). So, on this view, there is no last pointp ofMaryland just as there is no first pointq of Pennsylvania:the States of the Union do not, strictly speaking, use up the wholeterritory. In the temporal realm, the view is somewhat more popular(Sorensen 1986), for it may be seen as providing a quietist way out ofthe transition puzzles mentioned in section 1.1. Cf. Epicurus:“Death is nothing to us, since so long as we exist, it is notwith us; but when death comes, then we do not exist” (Letterto Menoeceus, 1926: 85). On the other hand, it is precisely onsuch grounds that some early modern Islamic philosophers, e.g.Mullā Ṣadrā, argued that change of substantialnature cannever be instantaneous and must therefore begradual: insofar as it is impossible for there to be“formless” matter, there can never be an instant whensomething has lost the previous form and not yet gained the new one(Asfār, IV, 274).

(2) The boundary belongseither toAor toB, though it may be indeterminate to which ofA andB it belongs. This view builds onBolzano’s classic analysis of the continuum (1851: §66),which in turn is mirrored by the standard account of point-settopology (see e.g. Kelley 1955). It implies that contact may obtainbetweenA andB only when eitherA orB is topologically closed (i.e., contains its own boundary)in the relevant contact area while the other is topologically open,though the appeal to indeterminacy allows one to leave the matterunsettled. This indeterminacy, in turn, may be construed as semanticor as epistemic, if not metaphysical, depending on whether therelevant boundary is of thefiat sort, as with theMason-Dixon line, or of thebona fide sort. (Similarly in thetemporal realm; see e.g. Kamp 1980, Le Poidevin 2000, Pickup 2022.)All of this can be explained without resorting to the machinery ofabstract set theory; a mereotopological account, where the relevanttopological distinctions are elucidated in terms of the part-wholerelation, provides a viable alternative (Smith 1996; Casati and Varzi1999, ch. 5; Varzi 2007, §2.4.1) and can be suitably adjusted tofit even the most liberal views of spatio-temporal receptacles (Hudson2001). Nonetheless the view has its costs. From a naive perspective,the distinction between open and (semi-)closed entities may seemunwarranted, if not “monstrous” (as Brentano 1976: 146 putit). From a metaphysical perspective, it introduces asymmetries thatmay be found disturbing, e.g. when it comes to explainingwhycontact is only possible between entities of opposite sorts (Zimmerman1996b), or why each sort will have different part-whole groundingproperties (Smid 2015; Calosi 2018; Sciacca 2025).

(3) The boundary may belongboth toAand toB, but the relevant overlap would besuigeneris precisely insofar as it would involve lower-dimensionalparts. Modulo the “extended simples” view mentioned inSection 1.4, boundaries do nottake up space-time and so, onthis view, it is not implausible to say that (for example) theMason-Dixon line belongs to both Maryland and Pennsylvania. This wasprobably Aristotle’s own position, according to which“things are called continuous when the touching limits of eachbecome one”, i.e., “one in number though two indefinition” (Physics, V, 227a11, VIII, 262a20). In somecases, however, this theory may require a blunt rejection of theprinciple of non-contradiction (Priest 2006, chs. 11ff; Cotnoir andWeber 2015; Weber 2021, §1.3). With reference to theSuárez-Peirce puzzle, for instance, if the line of demarcationbetween a black spot and its white background belongs to both, then itmust be both white and black. A way out would be to deny thatboundaries,qua lower-dimensional, can enjoy the same sort ofproperties that characterize extended bodies, color properties beingamong them (Galton 2003: 167–168). It is unclear, however,whether this conservative strategy can be generalized. For instance, acontradiction would seem to resurface with reference to Plato’spuzzle and its variants in the temporal realm (Priest 1982, 1985;Tanaka 1998): at the instant when a (homogeneous) object undergoes thetransition from being stationary to moving it would have to be bothstationary and moving; at the instant when someone dies, the self-sameperson would have to be both alive and dead; and so on. Thus, on theface of it, this is really a theory that would take us straight toHegel: “The limit is the mediation in virtue of which somethingand other eachboth is and is not” (1833: 99).

(4) There really aretwo boundaries, one belonging toA and one belonging toB, and these two boundariesare co-located — that is, they coincide spatially withoutoverlapping mereologically. This view may be traced back toSuárez,Disputation 40 (though see Schmaltz 2019 and2020, ch. 3, vs. Zimmerman 1996a), was distinctly endorsed by Brentano(1976), and in recent times has found many a follower, from Rhees(1938) to Sanford (1967) to Chisholm (1983, 1993, 1996), Smith (1997a,1999a), and Baumannet al. (2014, 2016). It may also havebeen another aspect of Aristotle’s conception, insofar as heheld that things can be in contact, or contiguous, without beingcontinuous in the sense mentioned in (3): in that case, Aristotlewrites, “their extremities are together”, i.e., “arein one primary place” (Physics, V, 226b22–23; seeBartha 2001, Pfeiffer 2018, ch. 7, Shatalov 2020, Sattler 2020, ch. 7,and Katz 2022 for extensive discussion). Apart from Aristotle’speculiar distinction, however, the main motivation for this view is todo away with the “monstrous” ontological differentiationbetween closed and open entities while preserving the importantdistinction between contact and overlap. The spatial coincidence ofboundaries would of course violate Locke’s impenetrabilityprinciple, to the effect that whatever exists anywhere at any time“excludes all of the same kind” (Essay,II-xxvii-1). Yet, again, the violation would besui generisinsofar as the entities in question do not take up any space, so oneneed not find it as outrageous as Bayle, for instance, made it sound(seeDictionary, art.Zeno, remark G). Indeed, itcould be argued that a coincidence-based account has severalderivative advantages when it comes to modeling a number of phenomenathat are topologically problematic, such as the possibility ofcollision or the fusion and fission of material bodies (Cotnoir 2019).Whether such considerations are enough to block other worries,however, remains to be determined. For instance, to the extent thatlines can be colored, it would follow — as Brentano (1976: 41)explicitly acknowledged — that the presence of a black spot on awhite background would deliver two coincident boundaries, one blackand one white. Would that result in something grey? (For discussion,see Massin 2018 and Nuñez Erices 2019; see also Chisholm 1980and Strobach 1998, ch. II.2, for parallel worries in the temporalrealm.)

These four views are mutually exclusive, but they need not beexhaustive and can be further articulated or integrated to address theissues raised by the other puzzles of Section 1. For example, withreference to the second puzzle (Section 1.2), Smith and Varzi (2000)consider a double-barreled theory that is of type (2) with respect tobona fide boundaries and of type (4) with respect tofiat boundaries (so there is no coincidence of realboundaries but merely offiat articulations). Similarly, theindeterminacy hypothesis advocated by type-(2) theories can beregarded as being of a piece with the sort of indeterminacy that isinvolved in the phenomenon of boundary fuzziness (Section 1.3). Forfiat boundaries, for instance, ade dicto accountmay be applied in both cases: statements about such boundaries aretrue if and only if they are super-true, i.e., true under everyadmissible way of precisifying the relevantfiatarticulations (Varzi 2001 and references therein). There are otheroptions, too. For instance, Hudson (2001; 2005, ch. 3) offers anaccount of boundaries that permits instances of each of (1), (2) and(3) (and without forfeiting the principle of non-contradiction). It isalso possible to reject the basic mereotopological setting that leadsto the taxonomy above and, with it, the need to choose among(1)–(4). See, for instance, Breysse and De Glas (2007) for aformal proposal in this spirit and Galton (1996, 2004), Donnelly andSmith (2003), and Donnelly (2003, 2004) for theories where thetreatment of boundaries of different dimensions (e.g. points asopposed to lines or surfaces) results in “layered”mereotopological systems. Finally, see Kraus (2016) for a theory thatdoes away altogether with the assumption that boundaries must bedependentparts.

2.2 Non-realist Theories

Non-realist theories come in a large variety, especially insofar asthe banishment of boundaries can be part of broader projects concernedwith the social and cognitive construction of everyday reality.Perhaps the most radical view is that boundaries should not be reifiedfor the simple reason that their character is purely qualitative,natural or artificial as the case may be. As H. H. Price once put it,“there is really no such entity asa surface; there areonly solid things thus and thussurfaced” (1932: 106).By extension, there would really be no such things as boundaries, onlyentities thus and thus bounded. This is straightforward ontologicaleliminativism, as when one says that there are no holes but onlyobjects thus and thus holed (Lewis and Lewis 1970: 206), or no colorsbut only things thus and thus colored (Quine 1948: 29–30). Themain challenge, then, is to go beyond the informal suggestion andarticulate a systematic way of dealing in a similar fashion with everyordinary sentence that seems to call for genuine reification. Canevery boundary-referring noun-phrase be de-nominalized this way? Canevery quantification over boundaries be reinterpreted as aquantification over bounded objects? Etc.

For the most part, however, non-realist accounts have concentrated onthe more limited task of reinterpreting ordinary boundary talk in thecontext of geometric and topological theorizing. Generally speaking,such accounts are not necessarily expression of an eliminativistontology. They are, rather, boundary-free. More precisely, they arefree from any genuine commitment to boundaries understood as thelower-dimensional entities endorsed by the realist theories outlinedin Section 2.1. The common idea, here, is that deep down all such talkinvolves some sort of “abstraction” — an idea thatcan be found already in the medieval and modern debates onanti-indivisibilism (Zupko 1993; Zimmerman 1996a; Holden 2004; Shapiroand Hellman 2021) and that made its way into contemporary formaltheories via Lobačevskij’sNew Principles(1835–1938). It is this idea that has recently become the focusof renewed attention in connection with the hypothesis that theuniverse might consist of “atomless gunk”, as Lewis (1991:20) calls it, i.e., that either space and time or matter, or both,divide forever into smaller and smaller parts (Zimmerman 1996b;McDaniel 2006; Hudson 2007; Russell 2008). What sort of abstractionwould be involved here? And how can one account for our ordinary andmathematical talk about boundaries if these are to be conceived of asmere abstractions?

With special reference to the boundaries of spatio-temporalparticulars, we may distinguish two main approaches. (Our taxonomy,here, will be rather sketchy; for a fuller picture, see Vakarelov2007, Hahmann and Grüninger 2012, and the monographs by Gorzka2003 and Gruszczyński 2016.)

(A) Substantivalists about space-time may see the abstractionas stemming from the relationship between a particular and itsspatio-temporal receptacle, relying on the topology of space-time toaccount for our boundary talk when it comes to specific cases. It hasbeen held, for instance, that material bodies are the material contentof open regions of space, more specifically regular open regions,boundary contact between bodies being explained in terms of overlapbetween the closures of their receptacles. (An open region is regularif it is equal to the interior of its own closure. This rules out, forinstance, regions lacking what would commonly be described as a singleinterior point, or regions with ingrowing boundary“cracks”.) This view can be traced back to Descartes(Principles II.15) and has been explicitly articulated byCartwright (1975). It does, to be sure, yield a hybrid account, anaccount that does away only with the boundaries of material bodies(and, by extension, events); their receptacles are typically subjectto a standard topology in which boundaries are treated as per theory(2) above. Yet this account is enough to bypass the puzzles of Section1 (especially 1.1) insofar as there is no pressing problem in assuminga standard topology for space-time and, with it, the distinctionbetween open and closed regions. The main problem for the theory is,rather, to justify the claim that onlysome regions (openregular regions, for instance) are receptacles. (See Hudson 2002 for achallenge to this view, and Uzquiano 2006 for a defense.) On the otherhand, there are more radical, non-hybrid theories that do away withboundaries also with regard to the structure of space-time.Topological theories of this sort may be found e.g. in Johanson(1981), Randell, Cui, and Cohn 1992), Asher and Vieu (1995), Forrest(1996), Roeper (1997), Pratt-Hartmann (2007), and Hellman and Shapiro(2018), but see also Gerla (1990) for a geometric account, Skyrms(1993) and Arntzenius (2008) for measure-theoretic alternatives, andLando and Scott (2019) for a robust approach combining both topologyand measure theory. In the temporal realm, the same approach hasinspired much work in so-called interval semantics for tense logic,beginning with Hamblin (1971) and Humberstone (1979).

(B) If one is not a substantivalist about space and/or time,the abstractionist approach is even more tempting. For if space and/ortime are themselves abstractions emerging from concrete relationsamong material bodies and/or events, then, as Whitehead put it, itseems illegitimate to “assumesub silentio” thatboundary elements are among the latter (1919: 5). In particular, itseems natural to treat them as mere “conceptual limits”emerging from the progressive contemplation of ever thinner layers ofthe bounded entities with which we are directly acquainted (1916:730). The best formulation of this idea is Whitehead’s owntheory of “extensive abstraction”, originally developed inpurely mereological terms and later refined in mereotopological termsfrom suggestions by De Laguna (1921, 1922) and Nicod (1924: I.4) (seeWhitehead 1929, Part IV) and variously endorsed by Broad (1923: ch.1), Russell (1927: ch. 28), Stebbing (1930: ch. 23), and eventuallyMenger (1940) and others. On this account, boundary elements are nolonger viewed as lower-dimensional entities; they are, rather,higher-order constructs, viz. equivalence classes ofconvergent series of nested bodies “packed one within the otherlike the nest of boxes of a Chinese toy” (Whitehead 1920: 61),but with no smallest box. For example, the series of all concentricspheres included in a given sphere converges to thepoint atthe center, the series of all concentric right cylinders of equallength included in a given cylinder converge to the axialline, and so on. Call a convergent series of this sort anabstractive class if and only if it has no bottom, i.e., ifand only if no object is part of every member of the class. And calltwo co-convergent abstractive classesequivalent if and onlyif every member of the first class has a member of the second as part,and vice versa. For instance, an abstractive class of spheres isequivalent to the class of all the cubes inscribed in the spheres,which converge to the same point at the center, etc. Then eachboundary element can be viewed as an equivalence class of suitablydefined converging abstractive classes, and if things are doneproperly, one can eventually reconstruct ordinary talk aboutlower-dimensional boundaries as talk about such higher-order entities.(For details, see Clarke 1985 and Varzi 2021.) This approach hasanalogues also in the temporal realm, where instants can be construedmore directly as convergent classes of nested time intervals, which inturn can be construed as sets of overlapping events. (Thelocusclassicus, directly inspired by Whitehead’s thinking, isRussell 1914, but see also Walker 1947, Kamp 1979, van Benthem 1983,and Vakarelov 2015.) Moreover, the approach itself is amenable toseveral variants, depending on exactly which primitives one assumes.Whitehead himself eventually settled on a framework based on aprimitive binary relation of topological connection; a popularalternative formulation, following Tarski (1929), uses insteadparthood along with as a special unary primitive ‘x isa sphere’, defining e.g. a geometric point as the class of allspheres that are concentric — in a suitably defined sense— with a given one. (For details and developments of thisalternative approach, see Bennett 2001, Gruszczyński andPietruszczak 2008, Betti and Loeb 2012, and Clay 2017; for extensionsand further variants, Borgo and Masolo 2010, Borgo 2013, and Gerla andGruszczyński 2017.) Another influential variant, thoughdeveloped independently, is the point-free topology of Grzegorczyk(1960), which is based on parthood along with a primitive notion of‘spatial body’ and a binary relation of separation, i.e.,the opposite of connection. This theory is closely related toWhitehead’s (see Biacino and Gerla 1996) and has recently beenworked out in full detail by Gruszczyński and Pietruszczak(2018, 2019). (Point-free topologies were initially calledpointless topologies, following Johnstone 1983, but theterminology has aptly been updated.)

From a technical viewpoint, these two approaches tend to overlap inmany ways. In particular, Whitehead’s technique is often usedalso in the context of substantivalists theories, as with the“Region Connection Calculus” of Randellet al.(1992), which has proved highly influential in the computationaltreatment of boundary-free spatial reasoning. Moreover, both sorts oftheories face a number of common questions when it comes to specifyingthe ontological assumptions on which they rest. Formally they are allboundary-free, which means that, in a logical setting, their variablesare meant to range exclusively over extended regions or bodies. Yetnot all such things are equal; is there room for interestingdistinctions? For instance, should these theories have room for thestandard topological distinction between open and closed entities?Should they preserve the distinction between regular entities andirregular ones? Some authors find such distinctions unintelligiblefrom a boundary-free perspective. Randellet al. (1992), forinstance, rule them out at the outset, and Bostock (2010) rules themout by argument. However others disagree, beginning with Clarke(1981), who actually provides detailed instructions on how to retrieveall standard topological distinctions within the framework ofWhitehead’s theory. (For other theories, see Cohn and Varzi2003.) Of course, retrieving them formally doesn’t mean much;the distinctions may still collapse. But do they — or shouldthey — collapse?

Other questions may be more or less pressing depending on one’sgeneral philosophical concerns. For example, a common objection totype-(B) theories is that the abstractness of boundaries would seem torun afoul of the abstractness of set-theoretic constructions. One cansee and paint the surface of a table, and in principle one could evensee and paint an infinite series of ever thinner layers oftable-parts; but one cannot see or paint theset of theseparts, even less an equivalence class of such sets (Simons 1991: 96).Even de Laguna, one of the very first sponsors of Whitehead’smethod, remarked that the identification of points and otherboundaries with abstractive sets is open to serious misinterpretation:“Although we perceive solids, we perceive no abstractive sets ofsolids […] In accepting the abstractive set, we are asveritably going beyond experience as in accepting the solid ofzero-length” (1922: 460). This is not a formal objection.Indeed, there are philosophers who, such as Maddy (1980), believe thatsetscan be perceived, at least sets of physical objects suchas tables. Nonetheless, the perceptibility of an infinite equivalenceclass remains problematic. It is especially problematic if, as inWhitehead’s case, the motivation for going boundary-free isgrounded on a broadly empiricist epistemology.

Finally, there is room for speculations concerning differentnon-realist strategies one might consider besides type-(A) andtype-(B) theories. At the moment, the main option that received someattention in the literature appears to be an“operationalist” account of the sort advocated by Adams(1984, 1996), where the abstractive process by which boundary elementsare constructed out of concrete observables is explained in terms of“operational tests”. Such an account might meet DeLaguna’s worries. Arguably, however, it is best regarded as aparallel story. It may offer an explanation of our empirical knowledgeconcerning boundaries, but it remains ultimately neutral with regardtotheir ontological status.

3. Other Topics

The issues and theories outlined above define the main coordinateswithin which the nature of boundaries has been a subject ofphilosophical concern since antiquity. To this, we may add that inrecent times the range of concerns has broadened significantly, withramifications that extend well beyond the scope of formal metaphysicaltheorizing.

3.1 Border Studies

We already mentioned, in this regard, the growing interest in the useand abuse of boundary concepts in the geopolitical domain,particularly in reference to the traditional opposition betweenso-called natural borders and artificial borders (Section 1.2).Geopolitical borders frame the “nomos” of the earth, as Schmitt (1950) called it. They are the spatial records of very complex patternsof interaction — through political agreement, unilateralstipulation, or warfare — between neighboring communities. Theyare, in Foucher’s (2007: 28) words, “traces of time inscribed inspace”, always in flux between division and union, between the“scars” of history (Mozer 1973: 14) and the“seams” of geography (Cinalli and Jacobson 2020: 43). Evenin times of peace there are entire industries (of real estate law,cadastral registration, land surveying) as well as a host ofadministrative offices (of passport control, customs and immigrationcheck, etc.) that are devoted to the maintenance of such traces(Crawford 2022). The ancient Romans actually had a god, Terminus,presiding over all boundary markers, with a major“Terminalia” festival celebrated yearly on the last day oftheir calendar (Dionysius,Roman Antiquities, II, 74; Ovid,Fasti, II, 639–684; Plutarch,Roman Questions,XV,Numa, XVI).

It is hard to regard all of this as being merely about some abstractfaçon de parler. But it is equally hard to locate thesource of its enormous importance in our political and sociallives, disentangling myth from science, culture from nature, powerfrom government. The emergence of “border studies” as abroadly interdisciplinary field of research bears witness to theincreasing awareness of these difficulties and of their practical andtheoretical importance (see Wastl-Walter 2011, Wilson and Donnan2012, and Scott 2020 for comprehensive handbooks). And while the interdisciplinarynature of the field draws primarily from geography, anthropology,sociology, environmental psychology, economics, political science, andlaw, deep down the issues belong to the very core of politicalphilosophy and deliver further perspectives from which to assess thephilosophical significance of borders, frontiers, and boundaries moregenerally. (See e.g. the recent publication of Cooper and Tinning2019.)

3.2 Boundaries in the Life Sciences

Similar considerations apply to the growing interest inboundary-related issues in the life sciences and the philosophythereof. By themselves, the remarks of section 1.2 on thenatural/artificial opposition are already enough to suggest aboundary-based reading of many contemporary debates on, say, thefoundations of biology. Biological taxonomies are the result of“circumscribing” and classifying individual organisms intotaxa of various ranks (species, genera, families, etc.), so anycontroversy between realists vs. antirealists, essentialists vs.conventionalists, or even monists vs. pluralists may to some extent beseen as stemming from different ways of assessing the ontologicalstatus of the relevant taxonomic boundaries (cf. Hull 1974,Dupré 1993, Kitcher 2001, Ereshefsky 2001). The vicissitudes ofthe chimera, in ancient mythology (Ebbesen 1986) as well as incontemporary genetics (Bonnicksen 2009, Hyun and De Los Angeles 2019),are perhaps the best illustration of science’s constantstruggling with the status of such boundaries, with all that itentails, on ethical and scientific grounds, in the special case of theboundaries that define the human species (see the entry onhuman/non-human chimeras).

Yet this is only one way in which boundaries are at work, and open tophilosophical theorizing, in the field of biology. Another, equallyimportant way emerges in the study of the organism-environmentrelations that are so central to the understanding of evolution.Consider, for instance, the partitioning of the ecological environmenteffected by what biologists call “niche construction”(Odling-Smee et al. 2003; Sultan 2015). There are niches that arefully enclosed within a physical retainer, such as an egg, a closedoyster shell, or a larval chamber; niches that, like a kangaroo pouchor a bear’s cave, are bound partly by a physical boundary andpartly by an immaterial boundary marking (more or less vaguely) theopening through which the organism is free to leave; niches that arebounded by a physical retainer to a very low extent, as with the nicheof the oxpecker removing ticks from the back of an African rhinoceros(bounded by a part of the rhinoceros’s hide); and, finally,niches that lack a physical retainer altogether and are borderedentirely by boundaries of thefiat sort, as with a fishorbiting underwater or a falcon in the sky circling above the areawhere its prey is to be found.

It is remarkable how closely such a variety of niche structuresmirrors the variety of boundary structures that we find in thegeopolitical world (Smith and Varzi 1999, 2002). Theprimafacie strength and protective function of any particular niche— what Gibson (1979) calls its “affordancecharacter” — is to a large degree determined by the sortof boundaries that delimit it. And just as the history of geopoliticalboundaries is, to a great extent, a history of the“razor’s edge” on which hang suspended all issues“of life or death to nations” (Curzon 1907: 7), so thehistory of evolution and biological survival is, in many ways, ahistory of the growth in complexity of “the constant interplayof the organism and the environment” (Lewontin 1978: 215). Themechanisms of what has come to be called “autopoiesis”(Maturana and Varela 1972) can similarly be described in terms of asystem’s ability to maintain and renew itself by regulating itscomposition and conserving its boundaries. And once we see things fromthis perspective, it is a short step to see the same mechanisms atwork also in the structuring of our relations with the socialenvironment, from personal space (Hall 1966) to human territorialityat large (Sack 1986; Taylor 1988). In all these cases, the question isnot only what boundaries are; it is how they function and how theyserve the things they bound (Ludu 2016; Werner 2021, 2022).

3.3 Boundary Perception

Lastly, it needs stressing that boundaries occupy a prominent positionalso in the mapping of reality that emerges from our individualcognitive acts. Above all, the contents of perception tend to bestructured in terms of figure-ground organization, which is to say theorganization of the visual field into objects that stand out fromtheir surroundings (Rubin 1915), and here again we find a tensionbetweenbona fide andfiat articulations whosephilosophical significance extends well beyond the remarks of Section1.2.

On the one hand, the objects we see and track most easily are thosethat stand out by virtue of the natural boundaries they seem to enjoy.The use of edge-detection techniques for object recognition incomputer vision (see e.g. Davies 2012) is motivated precisely by thisconsideration, for in normal circumstances such boundaries correspondto salient discontinuities in the depth and brightness of theperceived scene. The very fact that humans, from the time they areinfants, tend to reify such non-entities as holes and shadows just aseasily as they do with regard to material objects (Giralt and Bloom2000; Nelson and Palmer 2001) lends evidence to the importance of“natural” discontinuities in the segmentation of thevisual scene: the possession of a genuine boundary is a sign ofobjecthood.

On the other hand, it is also a fact that sometimes we parse thevisual scene in terms of boundaries that involve the creativecontribution of our perceptual apparatus, which, as we know fromSchumann’s classic work on illusory contours, tends toarticulate reality in terms of continuous borders even when suchborders “are objectively not there” (1900: 26).Kanizsa’s triangle (1955) is perhaps the example that bestepitomizes this phenomenon. But the same could be said of many otherways in whichfiat boundaries emerge from the figure-groundorganization of the visual field through the basic factors studied byGestalt psychologists, such as proximity, continuity, closure, colorand texture similarity, good form, etc. (Wertheimer 1923; Koffka1935). There is, we said, no bona fide boundary in a Seurat painting,except perhaps around each individual color spot; yet we see each“figure” as though it possessed a regular contour.

In Section 1.2 we noted that such considerations are relevant to themetaphysical question of whether all boundaries — and, withthem, the identity and persistence conditions of those things thathave boundaries — are on closer look the product of our ownworldmaking, a sign of that mismatch between appearance and realitythat may lead one to embrace an across-the-board anti-realist stancetowards the ontology of ordinary objects and events. That is oneimportant line of philosophical inquiry. But one may also want to knowmore about the structure of appearance as such. One may be interestedin the mechanisms of perceptual organization insofar as they mayreveal something aboutus. This is a distinguished subject ofinquiry in philosophy over and above its significance for psychology(see Wagemans 2015 for a representative collection of recent work). Itis, for sure, an important subject of further inquiry for anyoneinterested in the surreptitious, pervasive, stubborn, enigmaticintrusion of boundaries in our picture of the world.

4. Excerpts from the Literature

The entry concludes with a sample of significant passages from thephilosophical literature, listed in approximate chronological order.Some come from texts cited explicitly in the foregoing sections; theothers provide further evidence of the many dimensionsalong which the boundary concept has been analyzed and subjected tophilosophical inquiry throughout history.

Zhuangzi,Essentials for Nurturing Life, 2 (1994:26–27)

The cook put down his cleaver and responded. […] In accordwith the natural grain, I slice at the great crevices, lead the bladethrough the great cavities. Following its inherent structure, I neverencounter the slightest obstacle even where the veins and arteriescome together or where the ligaments and tendons join, much less fromobvious big bones. A good cook changes his cleaver once a year becausehe chops. An ordinary cook changes his cleaver once a month because hehacks. Now I’ve been using my cleaver for nineteen years andhave cut up thousands of oxen with it, but the blade is still as freshas though it had just come from the grindstone.

Plato,Phaedrus, 265e (1997: 542)

[The goal of the method of division] is to be able to cut up eachkind according to its species along its natural joints, and to try notto splinter any part, as a bad butcher might do.

Plato,Parmenides, 156c–d (1997: 388)

[A thing] won’t be able to undergo being previously at restand later in motion or being previously in motion and later at restwithout changing. […] Yet there is no time in which somethingcan, simultaneously, be neither in motion nor at rest. […] Yetsurely it also doesn’t change without changing. […] Isthere, then, this queer thing in which it might be, just when itchanges? — What queer thing? — The instant. The instantseems to signify something such that changing occurs from it to eachof two states. For a thing doesn’t change from rest while restcontinues, or from motion while motion continues. Rather, this queercreature, the instant, lurks between motion and rest.

Aristotle,Physics, VI, 234a24–b9 (1984:395–396)

[N]othing can be in motion in a now. […] Nor can anythingbe at rest. […] Inasmuch as it is the same now that belongs toboth the times [past and future], and it is possible for a thing to bein motion throughout one time and to be at rest throughout the other,and that which is in motion or at rest for the whole of a time will bein motion or at rest in any part of it in which it is of such a natureas to be in motion or at rest: it [would] follow that the same thingcan at the same time be at rest and in motion; for both the times havethe same extremity, viz. the now. […] It follows then that themotion of that which is in motion and the rest of that which is atrest must occupy time.

Aristotle,Metaphysics, V, 1022a4–5 (1984:1613).

We call a limit the last point of each thing, i.e., the firstpoint beyond which it is not possible to find any part, and the firstpoint within which every part is. [Note: This standard translationdiffers from the one given in Section 1 above, which did not use theword ‘point’. The Greek text only says ‘thelast’ and ‘the first’, without accompanying sortals,so a generic word such as ‘thing’ seemed more appropriatein the present context than ‘point’.]

Aristotle,Metaphysics, XI, 1060b12–17 (1984:1675–1676)

If we are to suppose lines or what comes after these (I mean theprimary surfaces) to be principles, these at least are not separablesubstances, but sections and divisions — the former of surfaces,the latter of bodies (while points are sections and divisions oflines); and further they are limits of these same things; and allthese are in other things and none is separable.

Euclid,Elements, I, defs. 1–3, 5–6, 13(1926: 153)

A point is that which has no part. A line is breadthless length.The extremities of a line are points. […] A surface is thatwhich has length and breadth only. The extremities of a surface arelines. […] A boundary is that which is an extremity ofanything.

Ovid,Fasti, II, 657–660 (2000: 47)

The simple neighborhood meets and celebrates the feast, andchants your praises, holy Terminus: You confine peoples and cities andgreat kingdoms; all land would be disputed without you. You seeknone’s favour, you are bribed by no gold; you guard entrustedlands with pledge of law.

Plutarch,Numa, XVI, 1–2 (1914: 363)

Terminus signifiesboundary, and to this god they makepublic and private sacrifices where their fields are set off byboundaries; of living victims nowadays, but anciently the sacrificewas a bloodless one, since Numa reasoned that the god of boundarieswas a guardian of peace and a witness of just dealing, and shouldtherefore be clear of slaughter. […] He knew that a boundary,if observed, fetters lawless power; and if not observed, convicts ofinjustice.

Aulus Gellius,Noctes Atticae, VI.xiii.5–6 (1795:2/32)

It was asked, When a dying man could be said to die; at the timehe actually expired, or when he was on the point of expiring? When aperson rising could be said to rise; when he actually stood, or whenhe was but just sitting? He who learned any art, at what time hebecame an artist; when he was really one, or when he was just not one?If you assert any one of these, you assert what is absurd andridiculous; yet it will appear more absurd, if you assert both orallow neither.

Sextus Empiricus,Against the Physicists, I, 269 (2012:52–53)

Socrates dies either while being or while not being. […]Well, when he is and is alive, he does not die; after all, he isalive; but again, he does not die having died, since then he will bedying twice, which is absurd. So Socrates does not die.

Sextus Empiricus,Against the Physicists, I,260–264 (2012: 51–52)

[A] whole does not reasonably touch a whole; for if a wholetouches a whole, it will not be touching but a unification of themboth, and the two bodies will be one body. […] Then again, itis not possible for a part to touch a part. For the part is conceivedas a part in virtue of its state in relation to the whole, but interms of its own limits it is a whole. […] Then again, nor doesa whole touch a part. For if the whole is going to touch the part, thewhole, being shrunk down together with the part, will be a part, andthe part, being stretched out alongside the whole, will be a whole.[…] And the same argument applies to the converse. […]Hence if neither the whole touches the whole, nor the part the part,nor the whole the part, nor the reverse, nothing touches anything.

John of Damascus,The Fount of Knowledge, I, 8 (1958:27)

Definition is the term for the setting of land boundaries takenin a metaphorical sense. For, just as the boundary separates thatwhich belongs to one from that which belongs to another, so does thedefinition set off the nature of one thing from that of any other.

Boethius,Second Commentary on Porphyry’s IsagogeI, xi, 14–21 (1994: 23–24)

A line is something in a body. And what it is it owes to thebody. That is, it keeps its being through the body. This is explainedas follows. If it is separated from the body, it does not subsist. Whoby any sense faculty ever grasped a line separated from a body?

Peter Abelard,Glosses on Porphyry, 8.1–4 (1994:26)

There are two species of incorporeals. Some of them, such as Godand the soul, can endure in their incorporeality outside sensibles.But others, such as a line without subject body, are entirely unableto be outside the sensibles they are in.

Ibn Sīnā (Avicenna),The Book of the Healing– Metaphysics, II, ii, 4–6 (2005: 50)

[T]he body is the substance for which it is possible for you tobegin by postulating in it a dimension in whatever manner you desire.[…] It is due to the body’s having this description thatone refers to body as being long, wide, and deep, just as it is saidthat body is that which is divisible in [terms] of all dimensions.

Ibn Sīnā (Avicenna),The Book of the Healing– Physics, III, ii, 8–10 (2009: 269–271)

Continuous is an equivocal term that is said in threesenses […] One […] is said of the magnitude that it iscontinuous with another when its limit and that of the other are one.[…] The second […] is said of that which, when one sideof the continuous thing is moved in a direction away from the other,the other follows it. […] Something is said to be continuous initself when it is such that you can posit parts for it between whichthere is the continuity that is in the first sense.

Ibn Rušd (Averroes),On Aristotle’sMetaphysics, I, 7 (2010: 36)

[In the language of] the masses, ‘one’ generallysignifies such things only inasmuch as they are isolated from otherthings and set apart by their essence […] Some of these thingsare isolated by the places which encompass them (this is the commonestmeaning of being-isolated), others are isolated by their limits only(this [applies to] the contiguous), again others are those which areisolated only by imagination (this is how number is attached to whatis continuous).

John Duns Scotus,Ordinatio, II, d. 2/2, q. 5, ad 2(1973: 323–324; English translation by Peter L. P. Simpson)

If a point is only a privation, line too will be only aprivation, as well as surface and solid; for a termed thing is definedby what terminates it and something positive does not essentiallyinclude a privation. […] And from this further is inferredsomething unacceptable, that if a surface is only the privation ofdepth, how will a point be the privation of a privation? […] Inaddition, there are on a surface many corporeal or sensible qualities,as it seems. Therefore a surface is not merely a privation.

William of Ockham,Quodlibetal Questions, I, q. 9, ad 2(1991: 53)

[T]he spherical body does not touch the flat body primarily witha part that is such that each ofits parts touches the flatbody. Therefore, it does not touch it primarily with some part that isprior to all the other touching parts. Rather, any given touching partis still such that a half of it does not touch immediately, and a halfof that half does not touch immediately, and so on ad infinitum.

Leonardo da Vinci,Notebooks (1938: 76)

What is it […] that divides the atmosphere from the water?It is necessary that there should be a common boundary which isneither air nor water but is without substance, because a bodyinterposed between two bodies prevents their contact, and this doesnot happen in water with air. […] Therefore a surface is thecommon boundary of two bodies which are not continuous, and does notform part of either one or the other, for if the surface formed partof it, it would have divisible bulk, whereas, however, it is notdivisible and nothingness divides these bodies the one from the other.

Mullā Ṣadrā,al-Asfāral-Arba‘a, IV, 274 (English translation from Rezaei and Bozorgi2001: 34)

If there is no common boundary between water and air which is thewarmest instance of water and the coldest of air, it will require thatin an instance, that is when water transforms into air,hylestays formless, which is impossible.

Michelangelo Buonarroti,Sonnet 151, 1–4 (1996:195)

The greatest artist does not have any concept which a singlepiece of marble does not itself contain within its excess, though onlya hand that obeys the intellect can discover it.

Francisco Suárez,Metaphysical Disputations 40, V,§41 (1861: 562–563; English translation by Robert Pasnau)

[I]t could plausibly be said that a line can be conserved by Godwithout a limiting point, and a surface without a line, and a bodywithout a surface.

Francisco Suárez,Metaphysical Disputations 40, V,§58 (1861: 567; English translation by Robert Pasnau)

Suppose that, in the continuous surface of a wall, one half ismaximally white and the other half is maximally black, and thosequalities each have their own indivisible boundaries[termini] by which they are contained and limited. It willthen be necessary that, from that part in virtue of which they arecontiguous, they each have their own boundaries inhering at the sametime in the same line connecting the quantitative surface. […]From this it further happens that in the same indivisible boundary ofmatter there would be, at the same time, two specific substantialforms with respect to something that is indivisible in extension.

René Descartes, Principles of Philosophy, II.15(1985: 229)

‘Surface’ here does not mean any part of thesurrounding body but merely the boundary between the surrounding andsurrounded bodies, which is no more than a mode. Or rather what ismeant is simply the common surface, which is not a part of one bodyrather than the other but is always reckoned to be the same, providedit keeps the same size and shape.

Pierre Bayle,Historical and Critical Dictionary, art.‘Zeno’, remark G.IV (1734–1738: 5/613)

[S]ome scholastic Philosophers […] suppose, that naturehath intermixed Mathematical points with the parts divisibleininfinitum, to the end that they may serve to connect them, andcompose the extremities of bodies. They thought by that means toanswer also the objection of the penetrative contact of two surfaces:but this evasion is so absurd, that it doth not deserve to be refuted.

David Hume,A Treatise of Human Nature, I, ii, 4 (1978:44)

[I]f we have the idea of indivisible points, lines and surfacesconformable to the definition, their existence is certainly possible:but if we have no such idea, ’tis impossible we can everconceive the termination of any figure; without which conception therecan be no geometrical demonstration.

Margaret Cavendish,Observations upon ExperimentalPhilosophy, I, xxxi (2001: 129)

[T]he opinion of atoms, is fitter for a poetical fancy, than forserious philosophy […] there cannot be atoms in nature, or elsenature would be like a beggar’s coat full of lice: Neither wouldshe be able to rule those wandering and straggling atoms, because[…] each is a single body by itself, having no dependence uponeach other. Wherefore, if there should be a composition of atoms, itwould not be a body made of parts, but of so many whole and entiresingle bodies, meeting together as a swarm of bees.

Leonhard Euler,Lettres à une princessed’Allemagne, CII (1795: 453)

As a general notion contains an infinite number of individualobjects, we may consider it as a space in which they are allcontained. Thus, for the notion ofman we form a space[…] in which we conceive all men to be comprehended. For thenotion ofmortal we form another, […] in which weconceive every thing mortal to be comprehended. And when I affirm,all men are mortal, it is the same thing with affirming, thatthe first figure is contained in the second.

John Venn, ‘On the Diagrammatic and MechanicalRepresentation of Propositions and Reasonings’ (1880: 1)

What we here represent is, of course, the extent or scope of eachterm of the proposition. We draw two circles, and make them include orexclude or intersect one another, according as the classes denoted bythe terms happen to stand in relation to one another in this respect.

Jean Jacques Rousseau,Discourse on the Origin ofInequality (1992: 44)

The first person who, having enclosed a plot of land, took itinto his head to saythis is mine and found people simpleenough to believe him, was the true founder of civil society. Whatcrimes, wars, murders, what miseries and horrors would the human racehave been spared, had someone pulled up the stakes or filled the ditchand cried out to his fellow men: ‘Do not listen to thisimpostor’.

Alphonse de Lamartine, ‘La Marseillaise de la paix’(1841: 796)

Do we see any trace of boundaries in the sky? Does its dome havea wall, a marker, a center? Nations! A pompous word to say barbarity!

Immanuel Kant,Prolegomena to Any Future Metaphysics,§57 (1783: 94)

In all bounds there is something positive (e.g., a surface is theboundary of corporeal space, and is therefore itself a space, a lineis a space, which is the boundary of the surface, a point the boundaryof the line, but yet always a place in space), whereas limits containmere negations.

Georg Wilhelm F. Hegel,The Science of Logic, I, i,1.2.B.b (1833: 98–99)

[I]n limit, something marks the boundary of its other. […]Something, as an immediate existence, is therefore the limit withrespect to another something; but it has this limitin it andis something through the mediation of that limit, which is just asmuch its non-being. The limit is the mediation in virtue of whichsomething and other eachboth is and is not.

Nikolaj I. Lobačevskij,New Principles of Geometry(1835–1938: 21–22)

Surfaces, lines, points, as geometry defines them, exist only inour imagination. […] We will attach ourselves only to ideaswhich in our mind unite themselves directly to the conception ofbodies to which our imagination is habituated.

Bernard BolzanoParadoxes of the Infinite, §66(1851: 167–168)

I define thelimit of a body as the aggregate of all theextreme [äusserst] ether-atoms which stillbelong to it. […] A closer consideration further showsthat many bodies are at certain places altogether devoid of limitingatoms; none of their atoms can be described as theextremeones among those which still belong to it and would accompany it if itstarted to move. [Two bodies are in contact] when the extreme atoms ofthe one, […] together with certain atoms of the other, form acontinuous extension.

Franz Brentano, ‘Nativistic, Empiricist, and AnoetisticTheories of our Presentation of Space’ (1976: 146)

One of the two lines into which the line would be split upondivision would […] have an end point, but the other nobeginning point. This inference has been quite correctly drawn byBolzano, who was led thereby to his monstrous doctrine that therewould exist bodies with and without surfaces, the one class containingjust so many as the other, because contact would be possible onlybetween a body with a surface and another without. He ought, rather,to have had his attention drawn by such consequences to the fact thatthe whole conception of the line and of other continua as sets ofpoints runs counter to the concept of contact and thereby abolishesprecisely what makes up the essence of the continuum.

Franz Brentano, ‘On What Is Continuous’ (1976: 12 and41)

Because a boundary, even when itself continuous, can never existexcept as belonging to something continuous of more dimensions[…], it is, considered for itself, nothing other than auniversal, to which — as to other universals — more thanone thing can correspond. And the geometer’s proposition thatonlyone straight line is conceivable between two points, isstrictly speaking false. […] If a red surface and a bluesurface are in contact with each other, then a red and a blue linecoincide.

Charles S. Peirce, ‘The Logic of Quantity’ (1893:7.127)

A drop of ink has fallen upon the paper and I have walled itround. Now every point of the area within the walls is either black orwhite; and no point is both black and white. That is plain. The blackis, however, all in one spot or blot; it is within bounds. There is aline of demarcation between the black and the white. Now I ask aboutthe points of this line, are they black or white? Why one more thanthe other? Are they (A) both black and white or (B) neither black norwhite? Why A more than B, or B more than A? […] The logicalconclusion […] is that the points of the boundary do not exist.

Ambrose Bierce, ‘The Devil’s Dictionary’ (1881:323)

Boundary, n. In political geography, an imaginary linebetween two nations, separating the imaginary rights of one from theimaginary rights of the other.

Gottlob Frege,The Foundations of Arithmetic §26(1884: 35)

One calls the equator an imaginary line, but it would be wrong tocall it a line that has merely been thought up. It was not created bythought as the result of a psychological process, but is onlyapprehended or grasped by thought. If its being apprehended were amatter of its coming into being, then we could not say anythingpositive about the equator for any time prior to this supposed cominginto being.

Gottlob Frege,The Fundamental Laws of Arithmetic, Vol.II, §56 (1903: 159)

A definition of a concept (of a possible predicate) must[…] unambiguously determine, as regards any object, whether ornot it falls under the concept. […] We may express thismetaphorically as follows: the concept must have a sharp boundary. Toa concept without sharp boundary there would correspond an area thathad not a sharp boundary-line all around, but in places just vaguelyfaded away into the background. […] The law of the excludedmiddle is really just another form of the requirement that the conceptshould have a sharp boundary.

George Nathaniel Curzon,Frontiers (1907: 13 and 54)

I have accepted the broad distinction between Natural andArtificial Frontiers, both as generally recognized, and asscientifically the most exact. […] There is also a class ofso-called Natural Frontiers which I have been obliged to omit, aspossessing no valid claim to the title, namely those which are claimedby nations as natural on grounds of ambition, or expediency, or moreoften sentiment. The attempt to realize Frontiers of this type hasbeen responsible for many of the wars, and some of the most tragicvicissitudes in history.

W. E. B. Du Bois,The Souls of Black Folk (1903: vi, 13,40); also in ‘The Color Line Belts the World’ (1906:30)

The problem of the Twentieth Century is the problem of thecolor-line.

Lucien Febvre,La Terre et l’évolutionhumaine (1922: 297)

For the whole problem is, or appears to us to be, a question ofboundaries. Within us, so deeply implanted that we no longer noticeits hold on us, there is a certain idea of the ‘naturallimits’ of the great States which causes us to think of theirboundaries as things in themselves, having an actual value, a kind ofmechanical virtue, and a compulsory and at the same time creativepower. […] A whole philosophy of history [is] comprised in thatword ‘natural’.

Georg Simmel,The View of Life (1918: 2)

Although the boundary as such is necessary, yet every single specific boundary can be stepped over, every fixity can be displaced, every enclosure can be burst, and every such act, of course, finds or creates a new boundary.

William James,Pragmatism (1907: 247)

Hence, even in the field of sensation, our minds exert a certainarbitrary choice. By our inclusions and omissions we trace thefield’s extent; by our emphasis we mark its foreground and itsbackground; by our order we read it in this direction or in that. Wereceive in short the block of marble, but we carve the statueourselves.

Ernst Cassirer,Philosophie der symbolischen Formen(1923: 280)

The beginning of thought and speech is not this: we do not simplyseize on and name certain distinctions that are somewhere present infeeling or intuition; on the contrary, on our own initiative we drawcertain dividing lines, effect certain separations and connections, byvirtue of which distinct individual configurations emerge from theuniform flux of consciousness.

Ludwig Wittgenstein,Tractatus Logico-Philosophicus,5.6

The limits [Grenzen] of my language mean the limits ofmy world.

Ludwig Wittgenstein,Philosophical Investigations, I,§99 (1953: 45e)

An indefinite boundary is not really a boundary at all. Here onethinks perhaps: if I say ‘I have locked the man up fast in theroom — there is only one door left open’ — then Isimply haven’t locked him in at all; his being locked in is asham.

Ludwig Wittgenstein,Remarks on the Philosophy ofPsychology, II, §626 (1980: 108e)

It is unnatural to draw a conceptual boundary line where there isnot some special justification for it, where similarities wouldconstantly draw us across the arbitrarily drawn line.

Alfred N. Whitehead,An Enquiry Concerning the Principles ofNatural Knowledge (1919: 5)

An investigation into the foundations of geometry has to explainspace as a complex of relations between things. It has to describewhat a point is, and has to show how the geometric relations betweenpoints issue from the ultimate relations between the ultimate thingswhich are the immediate objects of knowledge. Thus the starting pointof a discussion on the foundations of geometry is a discussion of thecharacter of the immediate data of perception. It is not now open tomathematicians to assumesub silentio that points are amongthese data.

William Ernest Johnson,Logic, Part III: The LogicalFoundations of Science (1924: 163–164)

We are thus led to distinguish between the parts of a whole, andthe boundaries between these parts. The parts of a line are lines, theparts of an area are areas, the parts of a region are regions; but theboundary between contiguous parts of a line is a point, and theboundary between contiguous parts of an area is a line, and theboundary between contiguous parts of a region is an area or surface.The parts of a whole, therefore, are homogeneous with one another andwith the whole; but […] the boundaries between two contiguousparts are always of one lower order of dimensions than the parts.

Susan Stebbing,A Modern Introduction to Logic (1930: 452and 451)

[The] success of physics in dealing with the sensible world isinexplicableunless such expressions as ‘point’,‘line’, ‘moment’, ‘instantaneousspace’, ‘momentary configurations’can beexpressed in terms of what is sensible. […] It does not matterthat a point should prove to be a complicated structure provided thatits logical relations aresimple and have the necessaryformal properties. […] All that matters in deduction are thepurely formal, or logical, properties of the terms that enter into thededuction; thenature of the term itself has no relevance.

Robert Musil,The Man Without Qualities (1930: 22)

Ultimately a thing exists only by virtue of its boundaries, whichmeans by a more or less hostile act against its surroundings: withoutthe Pope there would have been no Luther, and without the pagans noPope.

H. H. Price,Perception (1932: 106)

A surface must be the surface of something; and not merely that,it must be the surface of some material thing. Indeed, there is reallyno such entity asa surface; there are only solid things thusand thussurfaced. ‘Surface,’ it is true, is asubstantive in grammar; but it is not the name of a particularexistent, but of an attribute.

Percy Stanley Fritz,Colorado, the Centennial State(1941: 3)

The State of Colorado is primarily a mental concept. It is anarbitrary political designation made for convenience sake. There areno natural boundaries whatsoever. If, when you leave the State ofColorado in your automobile, you should miss seeing the sign by theroadside which says ‘Colorado–Kansas’, you wouldnever know that you had passed from one State to the other. […]If a windstorm happened to carry away the state boundary sign, itwould require a surveyor with precision instruments to tell where theboundary was. The State of Colorado is simply a convenient legalfiction.

Jan O. M. Broek, ‘The Problem of Natural Frontiers’(1941: 8)

One of the most powerful arguments to make a frontier seem justis to stamp it as a natural frontier.

Stephen Jones,Boundary-Making (1945: 3)

The goodness or badness of a boundary depends as much upon thegeneral situation as upon the details of delimitation and demarcation.A boundary, like the human skin, may have diseases of its own or mayreflect the illnesses of the body.

Martin Heidegger, ‘Building Dwelling Thinking’ (1952:154)

A boundary is not that at which something stops but, as theGreeks recognized, the boundary is that from which somethingbegins its presencing.

John L. Austin,Sense and Sensibilia (1962: 100)

It is […] wrong to imply that everything has a surface.Where and what exactly is the surface of a cat?

James J. Gibson,The Ecological Approach to VisualPerception (1979: 23)

The surface is where most of the action is. The surface is wherelight is reflected or absorbed, not the interior of the substance. Thesurface is what touches the animal, not the interior. The surface iswhere chemical reactions mostly take place. The surface is wherevaporization or diffusion of substances into the medium occurs. Andthe surface is where vibrations of the substance are transmitted intothe medium.

George Spencer-Brown,Laws of Form (1969: 1)

Distinction is perfect continence. That is to say, a distinctionis drawn by arranging a boundary with separate sides so that a pointon one side cannot reach the other side without crossing a boundary.For example, in a plane space a circle draws a distinction.

Michael Dummett,Frege. Philosophy of Language (1973:577)

The picture of reality as an amorphous lump, not yet articulatedinto discrete objects, thus proves to be a correct one, so long as wemake the right use of it.

Nelson Goodman, ‘Notes on the Well-Made World’ (1983:104)

Now as we thus make constellations by picking out and puttingtogether certain stars rather than others, so we make stars by drawingcertain boundaries rather than others. Nothing dictates whether theskies shall be marked off into constellations or other objects. Wehave to make what we find, be it the Great Dipper, Sirius, food, fuel,or a stereo system.

Alan Sidelle, ‘Rigidity, Ontology, and SemanticStructure’ (1992: 172)

The world is capable of being cut up in so many ways, andwhenever we consider such a cut (some principle of individuation), weare considering the world cut that way, i.e., so articulated. Anarticulation will specify both actual conditions which must be met forsomething to be (an) F, and identity conditions for tracing Fs throughspace, time, and possible worlds. If there are portions of the worldwhich meet the actual conditions, then there are Fs.

Roderick Chisholm, ‘Boundaries as DependentParticulars’ (1983: 88)

If the continuous object is cut in half, then does the oneboundary [that demarcates two adjacent parts] become two boundaries,one thing thus becoming two things? […] But how can one thing— even if it is only a boundary — become two things? Anddoes this mean that when two things become continuous, then two thingsthat had been diversebecome identical with each other, twothings thus becoming one thing?

David K. Lewis,The Plurality of Worlds (1986: 212)

The reason why it’s vague where the outback begins is notthat there’s this thing, the outback, with imprecise borders;rather there are many things, with different borders, and nobody hasbeen fool enough to try to enforce a choice of one of them as theofficial referent of the word ‘outback’.

Michael Tye, ‘Vague Objects’ (1990: 535)

There is no line which sharply divides the matter composing[Mount] Everest from the matter outside it. Everest’s boundariesare fuzzy. Some molecules are inside Everest and some moleculesoutside. But some have an indefinite status: there is no objective,determinate fact of the matter about whether they are inside oroutside.

Mark Sainsbury, ‘Concepts without Boundaries’ (1990:257)

A vague concept is boundaryless in that no boundary marks thethings which fall under it from the things which do not, and noboundary marks the things which definitely fall under it from thethings which do not definitely do so; and so on. Manifestations arethe unwillingness of knowing subjects to draw any such boundaries, thecognitive impossibility of identifying such boundaries, and theneedlessness and even disutility of such boundaries.

Peter Simons, ‘Faces, Boundaries, and Thin Layers’(1991: 91)

If a macroscopic body consists of spatially separate particles[…] a connected boundary would need to bridge the gaps betweenthe [particles] and thus would be both am ‘imaginary’rather than real entity (like saying a fakir bed of nails has a flattop) and to some extent arbitrary (like the curves scientists drawthrough the scattered and inexact data to give a smooth graph).

Eviatar Zerubavel,The Fine Line (1991: 5)

We transform the natural world into a social one by carving outof it mental chunks we then treat as if they were discrete, totallydetached from their surroundings. The way we mark off islands ofproperty is but one example of the general process by which we createmeaningful social entities.

Catherine Elgin, ‘Unnatural Science’ (1995: 297)

Our parochial concerns pretty clearly influence our choice ofcategories for macroscopic objects and kinds. […] It would berather remarkable if, for example, a taxonomy that draws thedistinction between horses and zebras where we do aligned at all wellwith natural categories suitable for describing the cosmos as a whole,but indifferent to human faculties and ends.

Barry Smith, ‘On Drawing Lines on a Map’ (1995:479)

The interiors of fiat objects are […] autonomous portionsof autonomous reality. Only the respective external boundaries arecreated by us; it is these which are the products of our mental andlinguistic activity, and of associated conventional laws, norms andhabits.

Barry Smith, ‘On Substances, Accidents and Universals’(1997b: 122)

Delineation is, be it noted, an immensely powerful faculty ofcognition; the scope of delineatory intentionality, the effortlessnesswith which we can comprehend highly complex wholes — which maybe scattered throughout the length and breadth of the universe, inboth space and time — with a simple delineatory act (‘thelegacy of the Renaissance’, ‘the Austro-Hungarian Empireand its successor states’, ‘English poetry’) iswondrous to behold, and bears comparison with the magic ofsingle-rayed intentionality, whereby, on the basis of a list ofentries which might be drawn up entirely at random, we can bedirected, in succession, to mountains in Siberia, teapots in Halifax,and black holes in the galaxy of Mog.

Raviel Netz, 2009,Barbed Wire (2009: xi)

Define, on the two-dimensional surface of the earth, lines acrosswhich motion is to be prevented, and you have one of the key themes ofhistory. With a closed line (i.e., a curve enclosing a figure), andthe prevention of motion from outside the line to its inside, youderive the idea of property. With the same line, and the prevention ofmotion from inside to outside, you derive the idea of prison. With anopen line (i.e., a curve that does not enclose a figure), and theprevention of motion in either direction, you derive the idea ofborder. Properties, prisons, borders: it is through the prevention ofmotion that space enters history.

Gloria E. Anzaldúa,Borderlands/La Frontera (1987:3)

Borders are set up to define the places that are safe and unsafe,to distinguishus fromthem. A border is a dividingline, a narrow strip along a steep edge. A borderland is a vague andundetermined place created by the emotional residue of an unnaturalboundary. It is in a constant state of transition. The prohibited andforbidden are its inhabitants.Los atravesados live here: thesquint-eyed, the perverse, the queer, the troublesome, the mongrel,the mulato, the half-breed, the half dead; in short, those who crossover, pass over, or go through the confines of the‘normal’.

Judith Butler,Bodies that Matter (1993: 11)

[T]o “refer” naively or directly to such an extra-discursive object will always require the prior delimitation of the extra-discursive. And insofar as the extra-discursive is delimited, it is formed by the very discourse from which it seeks to free itself. This delimitation […] marks a boundary that includes and excludes, that decides, as it were, what will and will not be the stuff of the object to which we then refer.

Sally Haslanger, ‘Feminism in Metaphysics: Negotiating theNatural’ (2000: 116; 2012: 148–149)

[T]here is an unmistakable pattern of projecting onto women andpeople of color, as their ‘nature’ or as‘natural’, features that are instead (if manifested atall) a product of social forces. This projective error has ledfeminists to be extremely suspicious of natural kinds and objectivetypes: if one function of references to ‘nature’ or‘natures’ is to mark the boundaries of what is sociallypossible, thereby ‘justifying’ pernicious institutions, wemust be wary of the suggestion that any category is‘natural’. […] Is there any meaningful (andpolitically viable) distinction between the natural and the social,and if so, where does the line fall? Is there any way to theorizeabout what’s natural that does not depend on the projection ofour political biases? If so, how?

Sarah Green, ‘A Sense of Border’ (2012: 580)

While it is clear that borders do not independently exist asself-evident entities in the landscape […] it is also the casethat once constructed (which includes all the various associatedbordering practices, both formal and informal), borders can take onthing-like qualities, both in practice and in people’simaginations.

Jody Azzouni,Ontology without Borders (2017: 143)

[T]here are no worldly ontological borders between purportedobjects; there are no properties, no relations — no aspects ofanything in the world — that constitute or amount toworldly individuation conditions for objects […] we projectobject boundaries onto the world.

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Acknowledgements

Many thanks to the editors and to Roberto Casati, Dan Fass, Gottfried Heinemann,and Benjamin Wilck for good advice and helpful comments and suggestions.