## 1. Issues



Euclid defined a boundary as "that which is an extremity of
anything" (*Elements*, I, def. 13). Aristotle defined the
extremity of a thing as "the first thing beyond which it is not
possible to find any part [of the given thing], and the first within
which every part is" (*Metaphysics*, V, 1022a4-5).
Together, these two definitions deliver the classic account of
boundaries, an account that is both intuitive and comprehensive and
offers the natural starting point for any further investigation into
the boundary concept. Indeed, although Aristotle's definition
concerned primarily the extremities of spatial entities, it applies
equally well in the temporal domain. Just as the Mason-Dixon line
marks the boundary between Maryland and Pennsylvania insofar as no
part of Maryland can be found on the northern side of the line, and no
part of Pennsylvania on its southern side, so "the now is an
extremity of the past (no part of the future being on this side of
it), and again of the future (no part of the past being on that side
of it)" (*Physics*, VI, 233b35-234a2). Similarly
for concrete objects and events: just as the surface of an apple marks
its spatial boundary insofar as the apple extends only up to it, so
the referee's whistle marks the temporal boundary of the game
insofar as the game protracts only up to it. In the case of abstract
entities, such as concepts and sets, the account is perhaps adequate
only figuratively. Still, it is telling that one of the Greek words
for 'boundary', oros, is also a
word for 'definition': as John of Damascus nicely put it,
"definition is the term for the setting of land boundaries taken
in a metaphorical sense" (*The Fount of Knowledge*, I,
8). Likewise, it is telling that in point-set topology the standard
definition of a set's boundary (from Hausdorff 1914, SS7.2)
reflects essentially the same intuition: the boundary, or frontier, of
a set *A* is the set of those points all of whose neighborhoods
intersect both *A* and the complement of *A* (where a
neighborhood of a point *p* is, intuitively, a set of points
that entirely "surround" *p*). It is not an
exaggeration, therefore, to say that the Euclidean-Aristotelian
characterization captures a general intuition about boundaries that
applies across the board. Nonetheless, precisely this intuitive
characterization gives rise to several puzzles that justify
philosophical concern.


### 1.1 Owned vs. Unowned Boundaries



The first sort of puzzle relates to the intuition that a boundary
*separates* two entities, or two parts of the same entity,
which are then said to be *in contact* with each other.
Following Smith (1997a: 534), imagine ourselves traveling from
Maryland to Pennsylvania. What happens as we cross the Mason-Dixon
line? Do we pass through a last point *p* in Maryland and a
first point *q* in Pennsylvania? Clearly not, given the density
of the continuum. As Aristotle put it, no two points can lie "in
succession" to each other (*Physics*, VI, 231b6-9),
so we should have to countenance an infinite number of further points
between *p* and *q*, hence between the two States,
contrary to their being in contact. But, equally clearly, we can
hardly acknowledge the existence of just one of *p* and
*q*, as is dictated by the standard mathematical treatment of
the continuum (Dedekind 1872); either choice would amount to a
peculiar privileging of one State over the other, an unacceptably
arbitrary asymmetry. And it would seem that we cannot identify
*p* with *q*, either, for we are speaking of two
*adjacent* States; their territories cannot have any parts in
common, not even pointy parts. So, *where* is the Mason-Dixon
line, and how does it relate to the two adjacent entities it
separates?



The puzzle can be generalized. Consider the dilemma raised by Leonardo
da Vinci in his *Notebooks* (1938: 75-76): What is it
that divides the atmosphere from the water? Is it air or is it water?
Or consider Suarez's worry in the *Disputations*
(40, V, SS58), repeatedly echoed by Peirce (1892: 546; 1893:
7.127): What color is the line of demarcation between a black spot and
its white background? Perhaps figure/ground considerations could be
invoked to provide an answer in this second case, based on the
principle that the boundary is always owned by the figure -- the
background is topologically "open" (Jackendoff 1987, App.
B). But what is figure and what is ground when it comes to two
adjacent halves of the same black spot? What is figure and what is
ground when it comes to Maryland and Pennsylvania? What happens when a
seabird dives into the water? Indeed, it would be natural to suppose
that all entities of the same sort behave alike -- for instance,
that all material bodies be figure-like entities, each possessing its
own boundary. But then, how could any two of them ever be or
*come* in contact, short of penetrability? (In this last form,
the question goes back to Sextus Empiricus, *Against the
Physicists*, I, 258-266, and is widely discussed in recent
literature; see e.g. Kline and Matheson 1987, Godfrey 1990, Hazen
1990, Zimmerman 1996b, Lange 2002, SS1.3, Hudson 2005, SS3.1,
Kilborn 2007, Sherry 2015.)



Consider also Plato's classical version of the puzzle in regard
to temporal boundaries (*Parmenides*, 156c-e): When an
object starts moving, or a moving object comes to rest, is it in
motion or is it at rest? As Aristotle later put it, the question
arises precisely because "the now that is the extremity of both
times must be one and the same", for again, "if each
extremity were different, the one could not be in succession to the
other" (*Physics*, VI, 234a5-6). Of course, one
could maintain that there is no motion *at* an instant, but
only *during* an interval, as Aristotle himself held
(231b18-232a18, 234a24-b9). Yet the question remains: Does
the transitional moment belong to the motion interval or to the rest
interval? (On this version of the puzzle, see Medlin 1963, Hamblin
1969, Strang 1974, Kretzmann 1976, Sorabji and Kretzmann 1976, Sorabji
1983, ch. 26, Mortensen 1985.)



Besides, the problem is not specific to the transition between motion
and rest and admits of several variants that would seem to resist
Aristotle's solution. Gellius, for instance, tells us that the
Middle Platonists were much concerned with the parallel question of
whether a dying person dies when they are already in the grasp of
death or while they still live (*Noctes Atticae*, VI, xiii,
5-6). This was thought to be a genuine *insolubilis*,
short of conceding the absurdity that no one ever dies (Sextus
Empiricus, *Against the Physicists*, I, 269), and the same
could be said of the many variants discussed by later Platonists and
by medieval and modern philosophers (see Strobach 1998 and Goubier and
Roques 2018). Deep down, the puzzle is no less than a primary
illustration of the paradoxical nature of instantaneous change (about
which see the entry on
 change and inconsistency).


### 1.2 Natural vs. Artificial Boundaries



A second sort of puzzle relates to the fact that Aristotle's
mereological (parthood-based) definition, and the common-sense
intuition that it captures, seem to apply only to the realm of
continuous entities. Modulo the above-mentioned difficulty, the
thought that Maryland and Pennsylvania are bounded by the Mason-Dixon
line is fair enough. But ordinary material objects -- it might be
observed -- are not truly continuous and speaking of an
object's boundary is like speaking of the "flat top"
of a fakir's bed of nails (Simons 1991: 91). On closer
inspection, the spatial boundaries of physical objects are imaginary
entities surrounding swarms of subatomic particles, and their exact
shape and location involve the same degree of idealization of a
drawing obtained by "connecting the dots", the same degree
of arbitrariness as any mathematical graph smoothed out of scattered
and inexact data, the same degree of abstraction as the figures'
contours in a Seurat painting. Similarly, on closer inspection a
body's being in motion amounts to the fact that the vector sum
of the motions of zillions of restless particles, averaged over time,
is non-zero, hence it makes no sense to speak of *the instant*
at which a body stops moving (Galton 1994: 4). All this may be seen as
good news *vis-a-vis* the generalized puzzles of Section
1.1, which would not even get off the ground (at least in the form
given above; see Smith 2007 and Wilson 2008 for qualifications). But
then the question arises: Is our boundary talk a mere *facon
de parler*? Even with reference to the Mason-Dixon line --
and, more generally, those boundaries that demarcate adjacent parts of
a continuous manifold, as when an individual cognitive agent
conceptualizes a black spot as being made of two halves -- one
can raise the question of their ontological status. Such boundaries
*are* puzzling; but are they real? After all, they stem
entirely from our social practices and from the organizing activity of
our intellect. It might be argued, therefore, that belief in their
objectivity epitomizes a form of metaphysical realism that cries for
justification. Do such boundaries really exist?



We may, in this connection, introduce a conceptual distinction between
"natural" or *bona fide* boundaries, which would be
objective insofar as they are grounded in some physical discontinuity
or qualitative heterogeneity betwixt an entity and its surroundings,
and "artificial" or *fiat* boundaries, which are
not so grounded in the autonomous, mind-independent world (Smith 1995,
building on Curzon 1907). The coastline of Britain, or the boundary
separating a black spot from its white background, might be examples
of the former, but geopolitical boundaries such as the Mason-Dixon
line, or the boundary between the top and the bottom halves of the
black spot, are clearly of the latter sort. Moreover, just as
*fiat* boundaries may be involved in the partitioning of larger
wholes into proper parts, so they are often at work when we
circumclude a number of smaller entities into larger wholes: think of
Benelux or Polynesia, but also of the easiness with which we represent
the world as consisting of forests, swarms of bees, constellations, or
when we group our actions into baseball games, electoral campaigns,
wars, etc. (Smith 1999b). Now, insofar as they are not truly
continuous, even the surfaces of ordinary material objects --
hence the boundaries of the individual trees, the individual bees, the
individual stars -- may be said to involve *fiat*
demarcations of just the same sort. (Cf. Goodman: "We make a
star as we make a constellation, by putting its parts together and
marking off its boundaries"; 1980: 213.) On closer inspection,
even the boundaries of our individual actions may be seen in this
light, indeed even a person's birth and death may to some extent
reflect conventional decisions and stipulations, witness the
controversies on abortion and euthanasia (see the entries on
 life
 and
 the definition of death).
 On closer inspection, even the extolled coastline of Britain is to
some extent fixed by us, witness the proverbial elusiveness of its
objective length and location (Mandelbrot 1967). So the question
arises: are there *any* natural, mind-independent, genuine
*bona fide* boundaries? The natural/artificial distinction is
intuitively clear; but how robust is it? Are there any concepts that
truly carve the world "at the joints", as per
Plato's butchering guidelines (*Phaedrus*, 265e)? And, if
not, is the *fiat* nature of our boundary talk a reason to
justify an anti-realist attitude towards boundaries altogether?



The question has deep ramifications. For once the *fiat/bona
fide* opposition has been recognized, it is clear that it can be
drawn not merely in relation to boundaries, but also in relation to
those entities that may be said to *have* boundaries (Smith and
Varzi 2000; Smith 2001, 2019; Tahko 2012; Davies 2019). If (part of)
their boundaries are artificial -- if they reflect the
articulation of reality that is effected through human cognition and
social practices -- then those entities themselves may be viewed
as conceptual constructions, a product of our worldmaking, hence the
question of the ontological status of *fiat* boundaries becomes
of a piece with the more general issue of the conventional status of
ordinary objects and events. This is not to imply that we end up with
imaginary or otherwise unreal wholes: as Frege wrote, the objectivity
of the North Sea "is not affected by the fact that it is a
matter of our arbitrary choice which part of all the water on the
earth's surface we mark off and elect to call the 'North
Sea'" (1884, SS26). Or as James wrote, echoing
Michelangelo: "The mind works on the data it receives very much
as a sculptor works on his block of stone; in a sense the statue stood
there from eternity" (1890: I/188). It does, however, follow
that the entities in question would only enjoy an individuality as a
result of our selective strokes: their objectivity is independent of
us, but their individuality -- their being the sorts of thing
they are, perhaps even their having the identity and persistence
conditions they have -- would depend on our choices and our
identification and reidentification criteria (Sidelle 1989; Heller
1990; Zerubavel 1991; Jubien 1993; Varzi 2011, 2016; Azzouni 2017;
Piras 2020).



These ramifications go well beyond the concerns of metaphysics; they
extend across the board, especially insofar as we have a tendency to
call "natural" exactly those boundaries that best suit our
parochial interests. In the geopolitical domain, for instance, the
rhetoric with which certain frontiers have been *claimed* to be
drawn by the hand of Nature, if not by the hand of God, has been
responsible 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 on
this topic, stemming mostly from the classic works of Reclus
(1905-1908), Febvre (1922), and Ancel (1938) (see Broek 1941
and, for recent overviews, Rankin and Schofield 2004 and Fall 2010).
But similar considerations apply to all sorts of atrocities that we
committed and continue to commit in all sorts of domains. That
rhetoric is at work every time we invoke Plato's carving
guidelines fraudulently and surreptitiously, every time we stigmatize
as a "bad butcher" whoever holds views or follows
practices that don't match our own as though they were
"against nature". This does not mean that the guidelines
themselves are corrupt; the realist is still free to take them
seriously, if cautiously. But is there any viable way to implement
them irrespective of our ideological, cultural, and individual biases?
We can hardly be wrong in attributing *fiat* status to a
boundary that is manifestly drawn by us; but who can vet the
objectivity of a natural joint? Recent work in feminist metaphysics
and epistemology and in the philosophy and sociology of race has
brought these issues to the fore (see the entries on
 feminist metaphysics,
 feminist epistemology and philosophy of science,
 race,
 and the
 critical 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 (as
well as standard topology) suggests that there is always a sharp
demarcation 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 have
boundaries that are in some sense fuzzy or indeterminate. Clouds,
deserts, mountains, let alone the figures of an impressionist
painting, all seem to elude the idealized notion of a sharply bounded
entity. Likewise, the temporal boundaries of many events, let alone
their spatial boundaries, seem to be indeterminate. When exactly did
the industrial revolution begin? When did it end? Where did it take
place? And certainly the concepts corresponding to many ordinary
predicates, such as 'bald' or 'heap', do not
possess definite boundaries, either. As again Frege famously put it,
to such concepts there seems to correspond "an area that ha[s]
not a sharp boundary-line all around, but in places just vaguely
fade[s] away into the background" (1903: SS56). How is such
fuzziness to be construed?



One radical option would be to dismiss the question altogether.
Wittgenstein, for instance, urged that "an indefinite boundary
is not really a boundary at all" (1953: 45e). Or one might as
well insist that the issue is purely epistemic: the fuzziness would
lie exclusively in our ignorance about the exact location of the
relevant 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 broadly
between *de re* accounts and *de dicto* accounts. On a
*de re* account, the fuzziness is truly ontological. Thus,
*pace* Evans (1978), the boundary of, say, Mount Everest would
be fuzzy insofar as there is no objective, determinate fact of the
matter about which parcels of land lie on which side (Tye 1990;
Copeland 1995; Morreau 2002; Hyde 2008). Similarly, a predicate such
as 'bald' would be vague because it stands for a vague
set, a set with a truly fuzzy boundary (Zadeh 1965; Goguen 1969). It
is an open question, then, whether such boundary fuzziness would
entail a corresponding indeterminacy in the identity conditions of the
relevant bounded entities (see e.g. Noonan 2008 vs. Paganini 2017),
just as it may be an open question whether it would entail an
objective indeterminacy in the existence conditions of certain (other)
entities, say, the mereological difference between Everest and the sum
of its definite parts (see e.g. Hawley 2002 and Donnelly 2009). By
contrast, a *de dicto* account would correspond to a purely
linguistic or conceptual notion of vagueness. There is no fuzzy
boundary demarcating Mount Everest on this view; rather, there are
many distinct parcels of land, each with a precise border and precise
existence and identity conditions, but our linguistic practices have
not enforced a choice of any one of them as the official referent of
the term 'Mount Everest' (Mehlberg 1958; Lewis 1986; McGee
1997). Similarly, on this view there is no such thing as a vaguely
demarcated set of bald people; rather, our linguistic stipulations do
not precisely specify which set of people, among the many that exist,
corresponds to the extension of 'bald' (Fine 1975; Keefe
2000).



Whether the fuzziness of a given boundary should be construed *de
re* or *de dicto* may be a case-by-case matter. No one
thinks that all indeterminacy is a matter of worldly unsettledness,
and the contrary claim, to the effect that boundary indeterminacy is
always the product of semantic or representational imprecision, is
highly contentious despite its distinguished pedigree (Russell 1923).
Moreover, there is room for a middle third-way, according to which
there is genuine ontological indeterminacy, but it doesn't lie
in the boundaries themselves; it would lie, rather, in the
determinable/determinate structure of boundary properties (Wilson
20123), or in the fact that it is ontically indeterminate which
precise boundaries things have (Akiba 2004; Barnes 2010; Abasnezhad
and Hosseini 2014). Generally speaking, however, it would seem that
the *de re/de dicto* alternative aligns rather closely with the
*bona fide/fiat* opposition. With fuzzy boundaries of the
*fiat* sort, a *de dicto* account suggests itself
naturally: insofar as the process leading to the definition of a
boundary may not be precise, the question of whether something lies
inside or outside the boundary may be semantically indeterminate. On
the other hand, this account does not sit well with boundaries of the
*bona fide* sort. If any such boundary were fuzzy, it would be
so independently of our cognitive or social articulations, hence a
*de re* account would seem inevitable; the fuzziness of the
boundary would lie in the way the world is (or isn't). Thus, it
would appear that a boundary may suffer from *de dicto*
fuzziness if and only if it is of the *fiat* sort, and from
*de re* fuzziness if and only if it is of the *bona
fide* sort.



Either way, making room for fuzzy boundaries will open the door to the
many difficulties that arise whenever vagueness phenomena are taken at
face value. In particular, both *de re* and *de dicto*
accounts will have to deal with the challenge of providing a way out
of the sorites paradox: how can we ever cross a fuzzy boundary, if no
single step can make a difference? And both accounts will have to
provide a way of accommodating phenomena of higher order vagueness:
not only may it be indeterminate whether something lies on one side or
the other of a fuzzy boundary; it may also be indeterminate whether it
is indeterminate whether it lies on one side or the other -- and
so on. (On these issues, see the entries on the
 Sorites paradox
 and
 vagueness.)
 Some authors think such challenges are insurmountable, and that
*de re* and *de dicto* conceptions are equally
misguided. There would be no words or entities with fuzzy boundaries;
whenever we think there are, we are, rather, representing the world
through concepts that are strictly speaking "boundaryless"
(Sainsbury 1990).


### 1.4 Bodiless vs. Bulky Boundaries



A fourth source of concern relates to the intuition, implicit in
Aristotle's and Euclid's definitions, that boundaries are
lower-dimensional entities, i.e., have at least one dimension fewer
than the entities they bound. The surface of a (continuous) sphere,
for example, is two-dimensional (it has no "substance" or
"divisible bulk"), the Mason-Dixon Line is one-dimensional
(it has "length" but no "breadth"), and a
boundary point such as the vertex of a cone is zero-dimensional (it
extends in no direction). This intuition has become common currency in
contemporary philosophy through Johnson (1922: 168; 1924:
163-164) and is germane to much of what we ordinarily say about
boundaries. Yet it is problematic insofar as it contrasts with several
independent intuitions that are of a piece with both common sense and
philosophical theorizing. For instance, there is a standing tradition
in epistemology (from Moore 1925 to Jackson 1977 and Gibson 1979)
according to which boundaries play a crucial role in perception: we
see (opaque) physical objects indirectly by seeing their surfaces. Yet
it is not clear how one could see a whole just by seeing part of it
(Chisholm 1957; Clarke 1965), even less how one could literally
*see* parts that lack physical bulk altogether (Stroll 1986a,
1986b). Likewise, we often speak of surfaces as of things that may be
pitted, or damp, or that can be scratched, polished, sanded, and so
on, and it is unclear whether such predicates can be applied at all to
immaterial entities. In such cases, it would rather seem that surfaces
(and boundaries more generally; see Jackendoff 1991) are to be
construed as "thin layers" that are *schematized*
as having fewer dimensions than the wholes to which they apply. The
same sort of schematization is commonplace in the geopolitical domain,
where the boundary lines that we find depicted in ordinary maps and
atlases may in fact correspond to border "belts", or
"zones", of various kinds and widths (Prescott 1965). The
very fact that the geographic notion of a natural boundary is often
illustrated with reference to rivers, ravines, mountain ranges, etc.
is indicative of this tendency. As Ratzel famously put it: "The
border area is the reality, the border line the abstraction
thereof" (1897: 448).



Arguably, this conceptual tension between boundaries understood as
lower-dimensional entities and boundaries understood as thin layers,
or zones, reflects an irreducible ambiguity in ordinary speech (Stroll
1977, 1979). And, arguably, it is only the first conception that gives
rise to the puzzles outlined in the foregoing sections; bulky
boundaries and border zones can be treated as ordinary, extended
proper parts of the bodies and regions they bound. Yet a general
theory of boundaries should have something to say about the second
conception as well -- and more generally about the interaction
between the mathematical idealization associated with the former
conception and the physical, cognitive, and philosophical significance
of the latter (Stroll 1988; Simons 1991; Galton 2007).



The issue also relates to the question of whether there can be
extended mereological simples. The ancient atomists had no doubts,
with Democritus famously holding that the ultimate constituent of
reality, while indivisible, come in an infinite variety
"differing in size and in shape" (Aristotle,
*Metaphysics*, III, 203b2). Today it is more common to assume
that the mereological structure of a thing always matches perfectly
that of its spatio-temporal receptacle (what Varzi 2007 calls
"mirroring" and Uzquiano 2011 "harmony"),
which would imply the opposite answer. Nevertheless, in recent years
several authors have rejected the assumption, seriously reviving the
hypothesis of mereological simples that occupy extended regions of
space and/or time (see e.g. Markosian 1998, Parsons 2000, Simons 2004,
Braddon-Mitchell and Miller 2006, and McDaniel 2007 along with the
entry on
 location and mereology).
 Insofar as the lower-dimensional nature of boundaries may be
construed in terms of their being *indivisible* along one or
more dimensions, as opposed to *unextended* along those
dimensions (a conception encouraged also by some medieval
philosophers, e.g. Ibn S&imacr;n&amacr;, the
*Il&amacr;hiyy&amacr;t* of *Kit&amacr;b
al-Sif&amacr;*, II, ii, 4-6), such a view would suggest
that also a boundary's spatio-temporal extension may transcend
its 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, though
it could extend over a (small) one-dimensional, a two-dimensional, or
even a three-dimensional region. Again, such extended boundaries would
be refreshingly unproblematic. As with their divisible bulky cousins,
one could easily explain how we can see them, for instance, or polish
them, scratch them, etc. However, they would also raise questions of
their own. Above all: How would such boundaries connect to the things
they bound? Would there be a further, unextended boundary separating
the two (and matching the unextended boundary between the
corresponding receptacles)? In the case of boundaries with divisible
bulk, the answer to the latter question is obviously in the
affirmative. It is an open question, however, whether the same answer
should apply to extended indivisible boundaries.


## 2. Theories



So boundaries are, on the one hand, central to the common-sense
picture of the world and yet, on the other, deeply problematic. We may
accordingly distinguish two main sorts of theories, depending on
whether one is willing to take the problems at face value (realist
theories) or to bypass them altogether, treating boundaries as
fictional abstractions of some sort (non-realist theories).


### 2.1 Realist Theories



Most realist theories about boundaries, construed as lower-dimensional
entities, share the view that such entities are ontological parasites:
points, lines, and surfaces cannot be separated and cannot exist in
isolation from the entities they bound (*pace* Suarez,
who thought that God would be capable of such marvels; see
*Disputation* 40, V, SS41). This view does justice to the
intuition that boundaries, if real, are somewhat "less
real" than the entities they bound, and as such it goes back to
Aristotle (*Metaphysics*, XI, 1060b12-16) and has been
defended 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 his
pupils, especially Husserl in his third *Logical Investigation*
(1901). Of course, the notion of ontological dependence is itself open
to several interpretations (see Koslicki 2014 and the entry on
 ontological dependence).
 In the case of boundaries, it is generally agreed that the relevant
relation is one of rigid existential dependence: if *x* is a
boundary of *y*, then necessarily *x* exists only if
*y* exists (Chisholm 1984, 1994). Realist theories may differ
significantly, however, with regard to how such rigidly dependent,
lower-dimensional entities relate to the extended entities they
bound.



To illustrate, consider again the first puzzle of Section 1 (owned vs.
unowned boundaries). Let *A* and *B* be any two extended
entities separated by a common boundary, such as Maryland and
Pennsylvania. Then we may distinguish four main views (Varzi 1997;
Strobach 1998, pt. II).



**(1)** The boundary may belong *neither* to *A*
*nor* to *B*. This was, ultimately, Leonardo's
view (1938: 76) and is perhaps the least popular among recent
philosophers (though see e.g. Hestevold 1986 and Schmolze 1996). It
implies that contact may obtain between *A* and *B* even
if both *A* and *B* are topologically open, so long as
nothing lies between them except for their common, outer boundary
-- i.e., so long as the closure of *A* (*A* plus
its boundary) overlaps the closure of *B* (*B* plus its
boundary). So, on this view, there is no last point *p* of
Maryland just as there is no first point *q* of Pennsylvania:
the States of the Union do not, strictly speaking, use up the whole
territory. In the temporal realm, the view is somewhat more popular
(Sorensen 1986), for it may be seen as providing a quietist way out of
the transition puzzles mentioned in section 1.1. Cf. Epicurus:
"Death is nothing to us, since so long as we exist, it is not
with us; but when death comes, then we do not exist" (*Letter
to Menoeceus*, 1926: 85). On the other hand, it is precisely on
such grounds that some early modern Islamic philosophers, e.g.
Mull&amacr; Sadr&amacr;, argued that change of substantial
nature can *never* be instantaneous and must therefore be
gradual: insofar as it is impossible for there to be
"formless" matter, there can never be an instant when
something has lost the previous form and not yet gained the new one
(*Asf&amacr;r*, IV, 274).



**(2)** The boundary belongs *either* to *A*
*or* to *B*, though it may be indeterminate to which of
*A* and *B* it belongs. This view builds on
Bolzano's classic analysis of the continuum (1851: SS66),
which in turn is mirrored by the standard account of point-set
topology (see e.g. Kelley 1955). It implies that contact may obtain
between *A* and *B* only when either *A* or
*B* 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 matter
unsettled. This indeterminacy, in turn, may be construed as semantic
or as epistemic, if not metaphysical, depending on whether the
relevant boundary is of the *fiat* sort, as with the
Mason-Dixon line, or of the *bona fide* sort. (Similarly in the
temporal realm; see e.g. Kamp 1980, Le Poidevin 2000, Pickup 2022.)
All of this can be explained without resorting to the machinery of
abstract set theory; a mereotopological account, where the relevant
topological distinctions are elucidated in terms of the part-whole
relation, provides a viable alternative (Smith 1996; Casati and Varzi
1999, Ch. 5; Varzi 2007, SS2.4.1) and can be suitably adjusted to
fit even the most liberal views of spatio-temporal receptacles (Hudson
2001). Nonetheless the view has its costs. From a naive perspective,
the distinction between open and (semi-)closed entities may seem
unwarranted, if not "monstrous" (as Brentano 1976: 146 put
it). From a metaphysical perspective, it introduces asymmetries that
may be found disturbing, e.g. when it comes to explaining *why*
contact is only possible between entities of opposite sorts (Zimmerman
1996b), or why each sort will have different part-whole grounding
properties (Smid 2015; Calosi 2018).



**(3)** The boundary may belong *both* to *A*
*and* to *B*, but the relevant overlap would be *sui
generis* precisely insofar as it would involve lower-dimensional
parts. Modulo the "extended simples" view mentioned in
Section 1.4, boundaries do not *take up* space-time and so, on
this view, it is not implausible to say that (for example) the
Mason-Dixon line belongs to both Maryland and Pennsylvania. This was
probably Aristotle's own position, according to which
"things are called continuous when the touching limits of each
become one", i.e., "one in number though two in
definition" (*Physics*, V, 227a11, VIII, 262a20). In some
cases, however, this theory may require a blunt rejection of the
principle of non-contradiction (Priest 2006, chs. 11ff; Cotnoir and
Weber 2015; Weber 2021, SS1.3). With reference to the
Suarez-Peirce puzzle, for instance, if the line of demarcation
between a black spot and its white background belongs to both, then it
must be both white and black. A way out would be to deny that
boundaries, *qua* lower-dimensional, can enjoy the same sort of
properties that characterize extended bodies, color properties being
among them (Galton 2003: 167-168). It is unclear, however,
whether this conservative strategy can be generalized. For instance, a
contradiction would seem to resurface with reference to Plato's
puzzle and its variants in the temporal realm (Priest 1982, 1985;
Tanaka 1998): at the instant when a (homogeneous) object undergoes the
transition from being stationary to moving it would have to be both
stationary and moving; at the instant when someone dies, the self-same
person would have to be both alive and dead; and so on. Thus, on the
face of it, this is really a theory that would take us straight to
Hegel: "The limit is the mediation in virtue of which something
and other each *both is and is not*" (1833: 99).



**(4)** There really are *two* boundaries, one belonging to
*A* and one belonging to *B*, and these two boundaries
are co-located -- that is, they coincide spatially without
overlapping mereologically. This view may be traced back to
Suarez, *Disputation* 40 (though see Schmaltz 2019 and
2020, 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 (1984, 1993, 1996), Smith (1997a,
1999a), and Baumann *et al.* (2014, 2016). It may also have
been another aspect of Aristotle's conception, insofar as he
held that things can be in contact, or contiguous, without being
continuous in the sense mentioned in (3): in that case, Aristotle
writes, "their extremities are together", i.e., "are
in one primary place" (*Physics*, V, 226b22-23; see
Bartha 2001, Pfeiffer 2018, ch. 7, Shatalov 2020, Sattler 2020, ch. 7,
and Katz 2022 for extensive discussion). Apart from Aristotle's
peculiar distinction, however, the main motivation for this view is to
do away with the "monstrous" ontological differentiation
between closed and open entities while preserving the important
distinction between contact and overlap. The spatial coincidence of
boundaries would of course violate Locke's impenetrability
principle, 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 be *sui generis*
insofar as the entities in question do not take up any space, so one
need not find it as outrageous as Bayle, for instance, made it sound
(see *Dictionary*, art. *Zeno*, remark G). Indeed, it
could be argued that a coincidence-based account has several
derivative advantages when it comes to modeling a number of phenomena
that are topologically problematic, such as the possibility of
collision 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 that
lines can be colored, it would follow -- as Brentano (1976: 41)
explicitly acknowledged -- that the presence of a black spot on a
white background would deliver two coincident boundaries, one black
and one white. Would that result in something grey? (For discussion,
see Massin 2018 and Nunez Erices 2019; see also Chisholm 1980
and Strobach 1998, ch. II.2, for parallel worries in the temporal
realm.)



These four views are mutually exclusive, but they need not be
exhaustive and can be further articulated or integrated to address the
issues raised by the other puzzles of Section 1. For example, with
reference to the second puzzle (Section 1.2), Smith and Varzi (2000)
consider a double-barred theory that is of type (2) with respect to
*bona fide* boundaries and of type (4) with respect to
*fiat* boundaries (so there is no coincidence of real
boundaries but merely of *fiat* articulations). Similarly, the
indeterminacy hypothesis advocated by type-(2) theories can be
regarded as being of a piece with the sort of indeterminacy that is
involved in the phenomenon of boundary fuzziness (Section 1.3). For
*fiat* boundaries, for instance, a *de dicto* account
may be applied in both cases: statements about such boundaries are
true if and only if they are super-true, i.e., true under every
admissible way of precisifying the relevant *fiat*
articulations (Varzi 2001 and references therein). There are other
options, too. For instance, Hudson (2001; 2005, ch. 3) offers an
account of boundaries that permits instances of each of (1), (2) and
(3) (and without forfeiting the principle of non-contradiction). It is
also possible to reject the basic mereotopological setting that leads
to the taxonomy above and, with it, the need to choose among
(1)-(4). See, for instance, Breysse and De Glas (2007) for a
formal proposal in this spirit and Galton (1996, 2004), Donnelly and
Smith (2003), and Donnelly (2003, 2004) for theories where the
treatment of boundaries of different dimensions (e.g. points as
opposed to lines or surfaces) results in "layered"
mereotopological systems. Finally, see Kraus (2016) for a theory that
does away altogether with the assumption that boundaries must be
dependent *parts*.


### 2.2 Non-realist Theories



Non-realist theories come in a large variety, especially insofar as
the banishment of boundaries can be part of broader projects concerned
with the social and cognitive construction of everyday reality.
Perhaps the most radical view is that boundaries should not be reified
for 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 as *a surface*; there are
only solid things thus and thus *surfaced*" (1932: 106).
By extension, there would really be no such things as boundaries, only
entities thus and thus bounded. This is straightforward ontological
eliminativism, as when one says that there are no holes but only
objects thus and thus holed (Lewis and Lewis 1970: 206), or no colors
but only things thus and thus colored (Quine 1948: 29-30). The
main challenge, then, is to go beyond the informal suggestion and
articulate a systematic way of dealing in a similar fashion with every
ordinary sentence that seems to call for genuine reification. Can
every boundary-referring noun-phrase be de-nominalized this way? Can
every quantification over boundaries be reinterpreted as a
quantification over bounded objects? Etc.



For the most part, however, non-realist accounts have concentrated on
the more limited task of reinterpreting ordinary boundary talk in the
context of geometric and topological theorizing. Generally speaking,
such accounts are not necessarily expression of an eliminativist
ontology. They are, rather, boundary-free. More precisely, they are
free from any genuine commitment to boundaries understood as the
lower-dimensional entities endorsed by the realist theories outlined
in Section 2.1. The common idea, here, is that deep down all such talk
involves some sort of "abstraction" -- an idea that
can be found already in the medieval and modern debates on
anti-indivisibilism (Zupko 1993; Zimmerman 1996a; Holden 2004; Shapiro
and Hellman 2021) and that made its way into contemporary formal
theories via Lobacevskij's *New Principles*
(1835-1938). It is this idea that has recently become the focus
of renewed attention in connection with the hypothesis that the
universe 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 abstraction
would be involved here? And how can one account for our ordinary and
mathematical talk about boundaries if these are to be conceived of as
mere abstractions?



With special reference to the boundaries of spatio-temporal
particulars, we may distinguish two main approaches. (Our taxonomy,
here, will be rather sketchy; for a fuller picture, see Vakarelov
2007, Hahmann and Gruninger 2012, and the monographs by Gorzka
2003 and Gruszczy&nacute;ski 2016.)



**(A)** Substantivalists about space-time may see the abstraction
as stemming from the relationship between a particular and its
spatio-temporal receptacle, relying on the topology of space-time to
account for our boundary talk when it comes to specific cases. It has
been held, for instance, that material bodies are the material content
of open regions of space, more specifically regular open regions,
boundary contact between bodies being explained in terms of overlap
between the closures of their receptacles. (An open region is regular
if it is equal to the interior of its own closure. This rules out, for
instance, regions lacking what would commonly be described as a single
interior point, or regions with ingrowing boundary
"cracks".) This view can be traced back to Descartes
(*Principles* II.15) and has been explicitly articulated by
Cartwright (1975). It does, to be sure, yield a hybrid account, an
account that does away only with the boundaries of material bodies
(and, by extension, events); their receptacles are typically subject
to a standard topology in which boundaries are treated as per theory
(2) above. Yet this account is enough to bypass the puzzles of Section
1 (especially 1.1) insofar as there is no pressing problem in assuming
a standard topology for space-time and, with it, the distinction
between open and closed regions. The main problem for the theory is,
rather, to justify the claim that only *some* regions (open
regular regions, for instance) are receptacles. (See Hudson 2002 for a
challenge to this view, and Uzquiano 2006 for a defense.) On the other
hand, there are more radical, non-hybrid theories that do away with
boundaries 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, and
Lando and Scott (2019) for a robust approach combining both topology
and measure theory. In the temporal realm, the same approach has
inspired 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/or
time are themselves abstractions emerging from concrete relations
among material bodies and/or events, then, as Whitehead put it, it
seems illegitimate to "assume *sub silentio*" that
boundary elements are among the latter (1919: 5). In particular, it
seems natural to treat them as mere "conceptual limits"
emerging from the progressive contemplation of ever thinner layers of
the bounded entities with which we are directly acquainted (1916:
730). The best formulation of this idea is Whitehead's own
theory of "extensive abstraction", originally developed in
purely mereological terms and later refined in mereotopological terms
from suggestions by De Laguna (1921, 1922) and Nicod (1924: I.4) (see
Whitehead 1929, Part IV) and variously endorsed by Broad (1923: ch.
1), Russell (1927: ch. 28), Stebbing (1930: ch. 23), and eventually
Menger (1940) and others. On this account, boundary elements are no
longer viewed as lower-dimensional entities; they are, rather,
*higher-order* constructs, viz. equivalence classes of
convergent series of nested bodies "packed one within the other
like the nest of boxes of a Chinese toy" (Whitehead 1920: 61),
but with no smallest box. For example, the series of all concentric
spheres included in a given sphere converges to the *point* at
the center, the series of all concentric right cylinders of equal
length included in a given cylinder converge to the axial
*line*, and so on. Call a convergent series of this sort an
*abstractive class* if and only if it has no bottom, i.e., if
and only if no object is part of every member of the class. And call
two co-convergent abstractive classes *equivalent* if and only
if every member of the first class has a member of the second as part,
and vice versa. For instance, an abstractive class of spheres is
equivalent to the class of all the cubes inscribed in the spheres,
which converge to the same point at the center, etc. Then each
boundary element can be viewed as an equivalence class of suitably
defined converging abstractive classes, and if things are done
properly, one can eventually reconstruct ordinary talk about
lower-dimensional boundaries as talk about such higher-order entities.
(For details, see Clarke 1985 and Varzi 2021.) This approach has
analogues also in the temporal realm, where instants can be construed
more directly as convergent classes of nested time intervals, which in
turn can be construed as sets of overlapping events. (The *locus
classicus*, directly inspired by Whitehead's thinking, is
Russell 1914, but see also Walker 1947, Kamp 1979, van Benthem 1983,
and Vakarelov 2015.) Moreover, the approach itself is amenable to
several variants, depending on exactly which primitives one assumes.
Whitehead himself eventually settled on a framework based on a
primitive binary relation of topological connection; a popular
alternative formulation, following Tarski (1929), uses instead
parthood along with as a special unary primitive '*x* is
a sphere', defining e.g. a geometric point as the class of all
spheres that are concentric -- in a suitably defined sense
-- with a given one. (For details and developments of this
alternative approach, see Bennett 2001, Gruszczy&nacute;ski and
Pietruszczak 2008, Betti and Loeb 2012, and Clay 2017; for extensions
and further variants, Borgo and Masolo 2010, Borgo 2013, and Gerla and
Gruszczy&nacute;ski 2017.) Another influential variant, though
developed 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 to
Whitehead's (see Biacino and Gerla 1996) and has recently been
worked out in full detail by Gruszczy&nacute;ski and Pietruszczak
(2018, 2019). (Point-free topologies were initially called
*pointless* topologies, following Johnstone 1983, but the
terminology has aptly been updated.)



From a technical viewpoint, these two approaches tend to overlap in
many ways. In particular, Whitehead's technique is often used
also in the context of substantivalists theories, as with the
"Region Connection Calculus" of Randell *et al.*
(1992), which has proved highly influential in the computational
treatment of boundary-free spatial reasoning. Moreover, both sorts of
theories face a number of common questions when it comes to specifying
the ontological assumptions on which they rest. Formally they are all
boundary-free, which means that, in a logical setting, their variables
are meant to range exclusively over extended regions or bodies. Yet
not all such things are equal; is there room for interesting
distinctions? For instance, should these theories have room for the
standard topological distinction between open and closed entities?
Should they preserve the distinction between regular entities and
irregular ones? Some authors find such distinctions unintelligible
from a boundary-free perspective. Randell *et al.* (1992), for
instance, rule them out at the outset, and Bostock (2010) rules them
out by argument. However others disagree, beginning with Clarke
(1981), who actually provides detailed instructions on how to retrieve
all standard topological distinctions within the framework of
Whitehead's theory. (For other theories, see Cohn and Varzi
2003.) Of course, retrieving them formally doesn't mean much;
the distinctions may still collapse. But do they -- or should
they -- collapse?



Other questions may be more or less pressing depending on one's
general philosophical concerns. For example, a common objection to
type-(B) theories is that the abstractness of boundaries would seem to
run afoul of the abstractness of set-theoretic constructions. One can
see and paint the surface of a table, and in principle one could even
see and paint an infinite series of ever thinner layers of
table-parts; but one cannot see or paint the *set* of these
parts, even less an equivalence class of such sets (Simons 1991: 96).
Even de Laguna, one of the very first sponsors of Whitehead's
method, remarked that the identification of points and other
boundaries with abstractive sets is open to serious misinterpretation:
"Although we perceive solids, we perceive no abstractive sets of
solids [...] In accepting the abstractive set, we are as
veritably going beyond experience as in accepting the solid of
zero-length" (1922: 460). This is not a formal objection.
Indeed, there are philosophers who, such as Maddy (1980), believe that
sets *can* be perceived, at least sets of physical objects such
as tables. Nonetheless, the perceptibility of an infinite equivalence
class remains problematic. It is especially problematic if, as in
Whitehead's case, the motivation for going boundary-free is
grounded on a broadly empiricist epistemology.



Finally, there is room for speculations concerning different
non-realist strategies one might consider besides type-(A) and
type-(B) theories. At the moment, the main option that received some
attention in the literature appears to be an
"operationalist" account of the sort advocated by Adams
(1984, 1996), where the abstractive process by which boundary elements
are constructed out of concrete observables is explained in terms of
"operational tests". Such an account might meet De
Laguna's worries. Arguably, however, it is best regarded as a
parallel story. It may offer an explanation of our empirical knowledge
concerning boundaries, but it remains ultimately neutral with regard
to *their* ontological status.


## 3. Other Topics



The issues and theories outlined above define the main coordinates
within which the nature of boundaries has been a subject of
philosophical concern since antiquity. To this, we may add that in
recent times the range of concerns has broadened significantly, with
ramifications that extend well beyond the scope of formal metaphysical
theorizing.


### 3.1 Border Studies



We already mentioned, in this regard, the growing interest in the use
and abuse of boundary concepts in the geopolitical domain,
particularly in reference to the traditional opposition between
so-called natural borders and artificial borders (Section 1.2).
Geopolitical borders are the spatial records of very complex patterns
of interaction -- through political agreement, unilateral
stipulation, or warfare -- between neighboring communities. They
are, as Foucher (2007: 28) put it, "traces of time inscribed in
space", 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). Even
in times of peace there are entire industries (of real estate law,
cadastral registration, land surveying) as well as a host of
administrative offices (of passport control, customs and immigration
check, 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 of
their 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 abstract
*facon de parler*. But it is equally hard to locate the
*source* of its enormous importance in our political and social
lives, disentangling myth from science, culture from nature, power
from government. The emergence of "border studies" as a
broadly interdisciplinary field of research bears witness to the
increasing awareness of these difficulties and of their practical and
theoretical importance (see Wastl-Walter 2011 and Wilson and Donnan
2012 for comprehensive handbooks). And while the interdisciplinary
nature of the field draws primarily from geography, anthropology,
sociology, environmental psychology, economics, political science, and
law, deep down the issues belong to the very core of political
philosophy and deliver further perspectives from which to assess the
philosophical significance of borders, frontiers, and boundaries more
generally. (See e.g. the recent publication of Cooper and Tinning
2019.)


### 3.2 Boundaries in the Life Sciences



Similar considerations apply to the growing interest in
boundary-related issues in the life sciences and the philosophy
thereof. By themselves, the remarks of section 1.2 on the
natural/artificial opposition are already enough to suggest a
boundary-based reading of many contemporary debates on, say, the
foundations of biology. Biological taxonomies are the result of
"circumscribing" and classifying individual organisms into
taxa of various ranks (species, genera, families, etc.), so any
controversy between realists vs. antirealists, essentialists vs.
conventionalists, or even monists vs. pluralists may to some extent be
seen as stemming from different ways of assessing the ontological
status of the relevant taxonomic boundaries (cf. Hull 1974,
Dupre 1993, Kitcher 2001, Ereshefsky 2001). The vicissitudes of
the chimera, in ancient mythology (Ebbesen 1986) as well as in
contemporary genetics (Bonnicksen 2009, Hyun and De Los Angeles 2019),
are perhaps the best illustration of science's constant
struggling with the status of such boundaries, with all that it
entails, on ethical and scientific grounds, in the special case of the
boundaries that define the human species (see the entry on
 human/non-human chimeras).



Yet this is only one way in which boundaries are at work, and open to
philosophical theorizing, in the field of biology. Another, equally
important way emerges in the study of the organism-environment
relations that are so central to the understanding of evolution.
Consider, for instance, the partitioning of the ecological environment
effected by what biologists call "niche construction"
(Odling-Smee et al. 2003; Sultan 2015). There are niches that are
fully enclosed within a physical retainer, such as an egg, a closed
oyster shell, or a larval chamber; niches that, like a kangaroo pouch
or a bear's cave, are bound partly by a physical boundary and
partly by an immaterial boundary marking (more or less vaguely) the
opening through which the organism is free to leave; niches that are
bounded by a physical retainer to a very low extent, as with the niche
of 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 bordered
entirely by boundaries of the *fiat* sort, as with a fish
orbiting underwater or a falcon in the sky circling above the area
where its prey is to be found.



It is remarkable how closely such a variety of niche structures
mirrors the variety of boundary structures that we find in the
geopolitical world (Smith and Varzi 1999, 2002). The *prima
facie* strength and protective function of any particular niche
-- what Gibson (1979) calls its "affordance
character" -- is to a large degree determined by the sort
of boundaries that delimit it. And just as the history of geopolitical
boundaries 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 the
history of evolution and biological survival is, in many ways, a
history of the growth in complexity of "the constant interplay
of the organism and the environment" (Lewontin 1978: 215). The
mechanisms of what has come to be called "autopoiesis"
(Maturana and Varela 1972) can similarly be described in terms of a
system's ability to maintain and renew itself by regulating its
composition and conserving its boundaries. And once we see things from
this perspective, it is a short step to see the same mechanisms at
work also in the structuring of our relations with the social
environment, from personal space (Hull 1966) to human territoriality
at large (Sack 1986; Taylor 1988). In all these cases, the question is
not only what boundaries are; it is how they function and how they
serve the things they bound (Ludu 2016; Werner 2021, 2022).


### 3.3 Boundary Perception



Lastly, it needs stressing that boundaries occupy a prominent position
also in the mapping of reality that emerges from our individual
cognitive acts. Above all, the contents of perception tend to be
structured in terms of figure-ground organization, which is to say the
organization of the visual field into objects that stand out from
their surroundings (Rubin 1915), and here again we find a tension
between *bona fide* and *fiat* articulations whose
philosophical significance extends well beyond the remarks of Section
1.2.



On the one hand, the objects we see and track most easily are those
that stand out by virtue of the natural boundaries they seem to enjoy.
The use of edge-detection techniques for object recognition in
computer vision (see e.g. Davies 2012) is motivated precisely by this
consideration, for in normal circumstances such boundaries correspond
to salient discontinuities in the depth and brightness of the
perceived scene. The very fact that humans, from the time they are
infants, tend to reify such non-entities as holes and shadows just as
easily as they do with regard to material objects (Giralt and Bloom
2000; Nelson and Palmer 2001) lends evidence to the importance of
"natural" discontinuities in the segmentation of the
visual scene: the possession of a genuine boundary is a sign of
objecthood.



On the other hand, it is also a fact that sometimes we parse the
visual scene in terms of boundaries that involve the creative
contribution of our perceptual apparatus, which, as we know from
Schumann's classic work on illusory contours, tends to
articulate reality in terms of continuous borders even when such
borders "are objectively not there" (1900: 26).
Kanizsa's triangle (1955) is perhaps the example that best
epitomizes this phenomenon. But the same could be said of many other
ways in which *fiat* boundaries emerge from the figure-ground
organization of the visual field through the basic factors studied by
Gestalt psychologists, such as proximity, continuity, closure, color
and texture similarity, good form, etc. (Wertheimer 1923; Koffka
1935). 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 the
metaphysical question of whether all boundaries -- and, with
them, the identity and persistence conditions of those things that
have boundaries -- are on closer look the product of our own
worldmaking, a sign of that mismatch between appearance and reality
that may lead one to embrace an across-the-board anti-realist stance
towards the ontology of ordinary objects and events. That is one
important line of philosophical inquiry. But one may also want to know
more about the structure of appearance as such. One may be interested
in the mechanisms of perceptual organization insofar as they may
reveal something about *us*. This is a distinguished subject of
inquiry in philosophy over and above its significance for psychology
(see Wagemans 2015 for a representative collection of recent work). It
is, for sure, an important subject of further inquiry for anyone
interested in the surreptitious, pervasive, stubborn, enigmatic
intrusion of boundaries in our picture of the world.


## 4. Excerpts from the Literature



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



Zhuangzi, *Essentials for Nurturing Life*, 2 (1994:
26-27)
 The cook put down his cleaver and responded. [...] In accord
with the natural grain, I slice at the great crevices, lead the blade
through the great cavities. Following its inherent structure, I never
encounter the slightest obstacle even where the veins and arteries
come together or where the ligaments and tendons join, much less from
obvious big bones. A good cook changes his cleaver once a year because
he chops. An ordinary cook changes his cleaver once a month because he
hacks. Now I've been using my cleaver for nineteen years and
have cut up thousands of oxen with it, but the blade is still as fresh
as 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 each
kind according to its species along its natural joints, and to try not
to 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 rest
and later in motion or being previously in motion and later at rest
without changing. [...] Yet there is no time in which something
can, simultaneously, be neither in motion nor at rest. [...] Yet
surely it also doesn't change without changing. [...] Is
there, then, this queer thing in which it might be, just when it
changes? -- What queer thing? -- The instant. The instant
seems to signify something such that changing occurs from it to each
of two states. For a thing doesn't change from rest while rest
continues, or from motion while motion continues. Rather, this queer
creature, 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 anything
be at rest. [...] Inasmuch as it is the same now that belongs to
both the times [past and future], and it is possible for a thing to be
in 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 be
in motion or at rest in any part of it in which it is of such a nature
as to be in motion or at rest: it [would] follow that the same thing
can at the same time be at rest and in motion; for both the times have
the same extremity, viz. the now. [...] It follows then that the
motion of that which is in motion and the rest of that which is at
rest must occupy time.


Aristotle, *Metaphysics*, V, 1022a4-5 (1984:
1613).
 We call a limit the last point of each thing, i.e., the first
point beyond which it is not possible to find any part, and the first
point within which every part is. [Note: This standard translation
differs from the one given in Section 1 above, which did not use the
word 'point'. The Greek text only says 'the
last' and 'the first', without accompanying sortals,
so a generic word such as 'thing' seemed more appropriate
in 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 the
primary surfaces) to be principles, these at least are not separable
substances, but sections and divisions -- the former of surfaces,
the latter of bodies (while points are sections and divisions of
lines); and further they are limits of these same things; and all
these 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 that
which has length and breadth only. The extremities of a surface are
lines. [...] A boundary is that which is an extremity of
anything.


Ovid, *Fasti*, II, 657-660 (2000: 47)
 The simple neighborhood meets and celebrates the feast, and
chants your praises, holy Terminus: You confine peoples and cities and
great kingdoms; all land would be disputed without you. You seek
none's favour, you are bribed by no gold; you guard entrusted
lands with pledge of law. 


Plutarch, *Numa*, XVI, 1-2 (1914: 363)
 Terminus signifies *boundary*, and to this god they make
public and private sacrifices where their fields are set off by
boundaries; of living victims nowadays, but anciently the sacrifice
was a bloodless one, since Numa reasoned that the god of boundaries
was a guardian of peace and a witness of just dealing, and should
therefore be clear of slaughter. [...] He knew that a boundary,
if observed, fetters lawless power; and if not observed, convicts of
injustice. 


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 time
he actually expired, or when he was on the point of expiring? When a
person rising could be said to rise; when he actually stood, or when
he was but just sitting? He who learned any art, at what time he
became 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 and
ridiculous; yet it will appear more absurd, if you assert both or
allow 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 is
alive; but again, he does not die having died, since then he will be
dying 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 whole
touches a whole, it will not be touching but a unification of them
both, and the two bodies will be one body. [...] Then again, it
is not possible for a part to touch a part. For the part is conceived
as a part in virtue of its state in relation to the whole, but in
terms of its own limits it is a whole. [...] Then again, nor does
a whole touch a part. For if the whole is going to touch the part, the
whole, being shrunk down together with the part, will be a part, and
the 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 taken
in a metaphorical sense. For, just as the boundary separates that
which belongs to one from that which belongs to another, so does the
definition set off the nature of one thing from that of any other.



Boethius, *Second Commentary on Porphyry's Isagoge*
I, xi, 14-21 (1994: 23-24)
 A line is something in a body. And what it is it owes to the
body. That is, it keeps its being through the body. This is explained
as follows. If it is separated from the body, it does not subsist. Who
by 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 God
and the soul, can endure in their incorporeality outside sensibles.
But others, such as a line without subject body, are entirely unable
to be outside the sensibles they are in. 


Ibn S&imacr;n&amacr; (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 to
begin by postulating in it a dimension in whatever manner you desire.
[...] It is due to the body's having this description that
one refers to body as being long, wide, and deep, just as it is said
that body is that which is divisible in [terms] of all dimensions.



Ibn S&imacr;n&amacr; (Avicenna), *The Book of the Healing
- Physics*, III, ii, 8-10 (2009: 269-271)
 *Continuous* is an equivocal term that is said in three
senses [...] One [...] is said of the magnitude that it is
continuous with another when its limit and that of the other are one.
[...] The second [...] is said of that which, when one side
of the continuous thing is moved in a direction away from the other,
the other follows it. [...] Something is said to be continuous in
itself when it is such that you can posit parts for it between which
there is the continuity that is in the first sense. 


Ibn Rusd (Averroes), *On Aristotle's
Metaphysics*, I, 7 (2010: 36)
 [In the language of] the masses, 'one' generally
signifies such things only inasmuch as they are isolated from other
things and set apart by their essence [...] Some of these things
are isolated by the places which encompass them (this is the commonest
meaning of being-isolated), others are isolated by their limits only
(this [applies to] the contiguous), again others are those which are
isolated only by imagination (this is how number is attached to what
is continuous). 


John Duns Scotus, *Ordinatio*, II, d. 2/2, q. 5, ad 2
(1973: 323-324; Eng. trans. by Peter L. P. Simpson)
 If a point is only a privation, line too will be only a
privation, as well as surface and solid; for a termed thing is defined
by what terminates it and something positive does not essentially
include a privation. [...] And from this further is inferred
something unacceptable, that if a surface is only the privation of
depth, how will a point be the privation of a privation? [...] In
addition, 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 with
a part that is such that each of *its* parts touches the flat
body. Therefore, it does not touch it primarily with some part that is
prior to all the other touching parts. Rather, any given touching part
is still such that a half of it does not touch immediately, and a half
of 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 is
neither air nor water but is without substance, because a body
interposed between two bodies prevents their contact, and this does
not happen in water with air. [...] Therefore a surface is the
common boundary of two bodies which are not continuous, and does not
form part of either one or the other, for if the surface formed part
of it, it would have divisible bulk, whereas, however, it is not
divisible and nothingness divides these bodies the one from the other.



Mull&amacr; Sadr&amacr;, *al-Asf&amacr;r
al-Arba'a*, IV, 274 (Eng. trans. from Rezaei and Bozorgi
2001: 34)
 If there is no common boundary between water and air which is the
warmest instance of water and the coldest of air, it will require that
in an instance, that is when water transforms into air, *hyle*
stays formless, which is impossible. 


Michelangelo Buonarroti, *Sonnet* 151, 1-4 (1996:
195)
 The greatest artist does not have any concept which a single
piece of marble does not itself contain within its excess, though only
a hand that obeys the intellect can discover it. 


Francisco Suarez, *Metaphysical Disputations* 40, V,
SS41 (1861: 562-563; Eng. trans. by Robert Pasnau)
 [I]t could plausibly be said that a line can be conserved by God
without a limiting point, and a surface without a line, and a body
without a surface. 


Francisco Suarez, *Metaphysical Disputations* 40, V,
SS58 (1861: 567; Eng. trans. by Robert Pasnau)
 Suppose that, in the continuous surface of a wall, one half is
maximally white and the other half is maximally black, and those
qualities each have their own indivisible boundaries
[*termini*] by which they are contained and limited. It will
then be necessary that, from that part in virtue of which they are
contiguous, they each have their own boundaries inhering at the same
time in the same line connecting the quantitative surface. [...]
From this it further happens that in the same indivisible boundary of
matter there would be, at the same, time two specific substantial
forms with respect to something that is indivisible in extension.



Rene Descartes*, Principles of Philosophy*, II.15
(1985: 229)
 'Surface' here does not mean any part of the
surrounding body but merely the boundary between the surrounding and
surrounded bodies, which is no more than a mode. Or rather what is
meant is simply the common surface, which is not a part of one body
rather than the other but is always reckoned to be the same, provided
it 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 nature
hath intermixed Mathematical points with the parts divisible *in
infinitum*, to the end that they may serve to connect them, and
compose the extremities of bodies. They thought by that means to
answer 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 surfaces
conformable to the definition, their existence is certainly possible:
but if we have no such idea, 'tis impossible we can ever
conceive the termination of any figure; without which conception there
can be no geometrical demonstration. 


Margaret Cavendish, *Observations upon Experimental
Philosophy*, I, xxxi (2001: 129)
 [T]he opinion of atoms, is fitter for a poetical fancy, than for
serious philosophy [...] there cannot be atoms in nature, or else
nature would be like a beggar's coat full of lice: Neither would
she be able to rule those wandering and straggling atoms, because
[...] each is a single body by itself, having no dependence upon
each other. Wherefore, if there should be a composition of atoms, it
would not be a body made of parts, but of so many whole and entire
single bodies, meeting together as a swarm of bees. 


Leonhard Euler, *Lettres a une princesse
d'Allemagne*, CII (1795: 453)
 As a general notion contains an infinite number of individual
objects, we may consider it as a space in which they are all
contained. Thus, for the notion of *man* we form a space
[...] in which we conceive all men to be comprehended. For the
notion of *mortal* we form another, [...] in which we
conceive every thing mortal to be comprehended. And when I affirm,
*all men are mortal*, it is the same thing with affirming, that
the first figure is contained in the second. 


John Venn, 'On the Diagrammatic and Mechanical
Representation of Propositions and Reasonings' (1880: 1)
 What we here represent is, of course, the extent or scope of each
term of the proposition. We draw two circles, and make them include or
exclude or intersect one another, according as the classes denoted by
the terms happen to stand in relation to one another in this respect.



Jean Jacques Rousseau, *Discourse on the Origin of
Inequality* (1992: 44)
 The first person who, having enclosed a plot of land, took it
into his head to say *this is mine* and found people simple
enough to believe him, was the true founder of civil society. What
crimes, wars, murders, what miseries and horrors would the human race
have been spared, had someone pulled up the stakes or filled the ditch
and cried out to his fellow men: 'Do not listed to this
impostor'. 


Alphonse de Lamartine, 'La Marseillaise de la paix'
(1841: 796)
 Do we see any trace of boundaries in the sky? Does its dome have
a wall, a marker, a center? Nations! A pompous word to say barbarity!



Immanuel Kant, *Prolegomena to Any Future Metaphysics*,
SS57 (1783: 94)
 In all bounds there is something positive (e.g., a surface is the
boundary of corporeal space, and is therefore itself a space, a line
is a space, which is the boundary of the surface, a point the boundary
of the line, but yet always a place in space), whereas limits contain
mere 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 with
respect to another something; but it has this limit *in it* and
is something through the mediation of that limit, which is just as
much its non-being. The limit is the mediation in virtue of which
something and other each *both is and is not*. 


Lobacevskij, *New Principles of Geometry*
(1835-1938: 21-22)
 Surfaces, lines, points, as geometry defines them, exist only in
our imagination. [...] We will attach ourselves only to ideas
which in our mind unite themselves directly to the conception of
bodies to which our imagination is habituated. 


Bernard Bolzano *Paradoxes of the Infinite*, SS66
(1851: 167-168)
 I define the *limit* of a body as the aggregate of all the
*extreme* [*ausserst*] ether-atoms which still
*belong to it*. [...] A closer consideration further shows
that many bodies are at certain places altogether devoid of limiting
atoms; none of their atoms can be described as the *extreme*
ones among those which still belong to it and would accompany it if it
started to move. [Two bodies are in contact] when the extreme atoms of
the one, [...] together with certain atoms of the other, form a
continuous extension. 


Franz Brentano, 'Nativistic, Empiricist, and Anoetistic
Theories of our Presentation of Space' (1976: 146)
 One of the two lines into which the line would be split upon
division would [...] have an end point, but the other no
beginning point. This inference has been quite correctly drawn by
Bolzano, who was led thereby to his monstrous doctrine that there
would exist bodies with and without surfaces, the one class containing
just so many as the other, because contact would be possible only
between a body with a surface and another without. He ought, rather,
to have had his attention drawn by such consequences to the fact that
the whole conception of the line and of other continua as sets of
points runs counter to the concept of contact and thereby abolishes
precisely what makes up the essence of the continuum. 


Franz Brentano, 'On What Is Continuous' (1976: 12 and
41)
 Because a boundary, even when itself continuous, can never exist
except as belonging to something continuous of more dimensions
[...], it is, considered for itself, nothing other than a
universal, to which -- as to other universals -- more than
one thing can correspond. And the geometer's proposition that
only *one* straight line is conceivable between two points, is
strictly speaking false. [...] If a red surface and a blue
surface are in contact with each other, then a red and a blue line
coincide. 


Charles S. Peirce, 'The Logic of Quantity' (1893:
7.127)
 A drop of ink has fallen upon the paper and I have walled it
round. Now every point of the area within the walls is either black or
white; and no point is both black and white. That is plain. The black
is, however, all in one spot or blot; it is within bounds. There is a
line of demarcation between the black and the white. Now I ask about
the points of this line, are they black or white? Why one more than
the other? Are they (A) both black and white or (B) neither black nor
white? Why A more than B, or B more than A? [...] The logical
conclusion [...] 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 line
between two nations, separating the imaginary rights of one from the
imaginary rights of the other. 


Gottlob Frege, *The Foundations of Arithmetic* SS26
(1884: 35)
 One calls the equator an imaginary line, but it would be wrong to
call it a line that has merely been thought up. It was not created by
thought as the result of a psychological process, but is only
apprehended or grasped by thought. If its being apprehended were a
matter of its coming into being, then we could not say anything
positive about the equator for any time prior to this supposed coming
into being. 


Gottlob Frege, *The Fundamental Laws of Arithmetic,* Vol.
II, SS56 (1903: 159)
 A definition of a concept (of a possible predicate) must
[...] unambiguously determine, as regards any object, whether or
not it falls under the concept. [...] We may express this
metaphorically as follows: the concept must have a sharp boundary. To
a concept without sharp boundary there would correspond an area that
had not a sharp boundary-line all around, but in places just vaguely
faded away into the background. [...] The law of the excluded
middle is really just another form of the requirement that the concept
should have a sharp boundary. 


George Nathaniel Curzon, *Frontiers* (1907: 13 and 54)
 I have accepted the broad distinction between Natural and
Artificial Frontiers, both as generally recognized, and as
scientifically the most exact. [...] There is also a class of
so-called Natural Frontiers which I have been obliged to omit, as
possessing no valid claim to the title, namely those which are claimed
by nations as natural on grounds of ambition, or expediency, or more
often sentiment. The attempt to realize Frontiers of this type has
been responsible for many of the wars, and some of the most tragic
vicissitudes 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 the
color-line. 


Lucien Febvre, *La Terre et l'evolution
humaine* (1922: 297)
 For the whole problem is, or appears to us to be, a question of
boundaries. Within us, so deeply implanted that we no longer notice
its hold on us, there is a certain idea of the 'natural
limits' of the great States which causes us to think of their
boundaries as things in themselves, having an actual value, a kind of
mechanical virtue, and a compulsory and at the same time creative
power. [...] A whole philosophy of history [is] comprised in that
word 'natural'. 


William James, *Pragmatism* (1907: 247)
 Hence, even in the field of sensation, our minds exert a certain
arbitrary choice. By our inclusions and omissions we trace the
field's extent; by our emphasis we mark its foreground and its
background; by our order we read it in this direction or in that. We
receive in short the block of marble, but we carve the statue
ourselves. 


Ernst Cassirer, *Philosophie der symbolischen Formen*
(1923: 280)
 The beginning of thought and speech is not this: we do not simply
seize on and name certain distinctions that are somewhere present in
feeling or intuition; on the contrary, on our own initiative we draw
certain dividing lines, effect certain separations and connections, by
virtue of which distinct individual configurations emerge from the
uniform flux of consciousness. 


Ludwig Wittgenstein, *Tractatus Logico-Philosophicus*,
5.6
 The limits [*Grenzen*] of my language mean the limits of
my world. 


Ludwig Wittgenstein, *Philosophical Investigations*, I,
SS99 (1953: 45e)
 An indefinite boundary is not really a boundary at all. Here one
thinks perhaps: if I say 'I have locked the man up fast in the
room -- there is only one door left open' -- then I
simply haven't locked him in at all; his being locked in is a
sham. 


Ludwig Wittgenstein, *Remarks on the Philosophy of
Psychology*, II, SS626 (1980: 108e)
 It is unnatural to draw a conceptual boundary line where there is
not some special justification for it, where similarities would
constantly draw us across the arbitrarily drawn line. 


Alfred N. Whitehead, *An Enquiry Concerning the Principles of
Natural Knowledge* (1919: 5)
 An investigation into the foundations of geometry has to explain
space as a complex of relations between things. It has to describe
what a point is, and has to show how the geometric relations between
points issue from the ultimate relations between the ultimate things
which are the immediate objects of knowledge. Thus the starting point
of a discussion on the foundations of geometry is a discussion of the
character of the immediate data of perception. It is not now open to
mathematicians to assume *sub silentio* that points are among
these data. 


William Ernest Johnson, *Logic, Part III: The Logical
Foundations of Science* (1924: 163-164)
 We are thus led to distinguish between the parts of a whole, and
the boundaries between these parts. The parts of a line are lines, the
parts of an area are areas, the parts of a region are regions; but the
boundary between contiguous parts of a line is a point, and the
boundary between contiguous parts of an area is a line, and the
boundary between contiguous parts of a region is an area or surface.
The parts of a whole, therefore, are homogeneous with one another and
with the whole; but [...] the boundaries between two contiguous
parts are always of one lower order of dimensions than the parts.



Susan Stebbing, *A Modern Introduction to Logic* (1930: 452
and 451)
 [The] success of physics in dealing with the sensible world is
inexplicable *unless* such expressions as 'point',
'line', 'moment', 'instantaneous
space', 'momentary configurations' *can* be
expressed in terms of what is sensible. [...] It does not matter
that a point should prove to be a complicated structure provided that
its logical relations are *simple* and have the necessary
formal properties. [...] All that matters in deduction are the
purely formal, or logical, properties of the terms that enter into the
deduction; the *nature* 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, which
means by a more or less hostile act against its surroundings: without
the Pope there would have been no Luther, and without the pagans no
Pope. 


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 really
no such entity as *a surface*; there are only solid things thus
and thus *surfaced*. 'Surface,' it is true, is a
substantive in grammar; but it is not the name of a particular
existent, but of an attribute. 


Percy Stanley Fritz, *Colorado, the Centennial State*
(1941: 3)
 The State of Colorado is primarily a mental concept. It is an
arbitrary political designation made for convenience sake. There are
no natural boundaries whatsoever. If, when you leave the State of
Colorado in your automobile, you should miss seeing the sign by the
roadside which says 'Colorado-Kansas', you would
never know that you had passed from one State to the other. [...]
If a windstorm happened to carry away the state boundary sign, it
would require a surveyor with precision instruments to tell where the
boundary was. The State of Colorado is simply a convenient legal
fiction. 


Jan O. M. Broek, 'The Problem of Natural Frontiers'
(1941: 8)
 One of the most powerful arguments to make a frontier seem just
is to stamp it as a natural frontier.


Stephen Jones, *Boundary-Making* (1945: 3)
 The goodness or badness of a boundary depends as much upon the
general situation as upon the details of delimitation and demarcation.
A boundary, like the human skin, may have diseases of its own or may
reflect the illnesses of the body. 


Martin Heidegger, 'Building Dwelling Thinking' (1952:
154)
 A boundary is not that at which something stops but, as the
Greeks recognized, the boundary is that from which something
*begins 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? 


J. J. Gibson, *The Ecological Approach to Visual
Perception* (1979: 23)
 The surface is where most of the action is. The surface is where
light is reflected or absorbed, not the interior of the substance. The
surface is what touches the animal, not the interior. The surface is
where chemical reactions mostly take place. The surface is where
vaporization or diffusion of substances into the medium occurs. And
the surface is where vibrations of the substance are transmitted into
the medium. 


George Spencer-Brown, *Laws of Form* (1969: 1)
 Distinction is perfect continence. That is to say, a distinction
is drawn by arranging a boundary with separate sides so that a point
on 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 articulated
into discrete objects, thus proves to be a correct one, so long as we
make 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 putting
together certain stars rather than others, so we make stars by drawing
certain boundaries rather than others. Nothing dictates whether the
skies shall be marked off into constellations or other objects. We
have to make what we find, be it the Great Dipper, Sirius, food, fuel,
or a stereo system. 


Alan Sidelle, 'Rigidity, Ontology, and Semantic
Structure' (1992: 172)
 The world is capable of being cut up in so many ways, and
whenever we consider such a cut (some principle of individuation), we
are considering the world cut that way, i.e., so articulated. An
articulation will specify both actual conditions which must be met for
something to be (an) F, and identity conditions for tracing Fs through
space, time, and possible worlds. If there are portions of the world
which meet the actual conditions, then there are Fs. 


Roderick Chisholm, 'Boundaries as Dependent
Particulars' (1984: 88)
 If the continuous object is cut in half, then does the one
boundary [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? And
does this mean that when two things become continuous, then two things
that had been diverse *become identical with each other*, two
things thus becoming one thing? 


David K. Lewis, *The Plurality of Worlds* (1986: 212)
 The reason why it's vague where the outback begins is not
that there's this thing, the outback, with imprecise borders;
rather there are many things, with different borders, and nobody has
been fool enough to try to enforce a choice of one of them as the
official 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 boundaries
are fuzzy. Some molecules are inside Everest and some molecules
outside. But some have an indefinite status: there is no objective,
determinate fact of the matter about whether they are inside or
outside. 


Mark Sainsbury, 'Concepts without Boundaries' (1990:
257)
 A vague concept is boundaryless in that no boundary marks the
things which fall under it from the things which do not, and no
boundary marks the things which definitely fall under it from the
things which do not definitely do so; and so on. Manifestations are
the unwillingness of knowing subjects to draw any such boundaries, the
cognitive impossibility of identifying such boundaries, and the
needlessness 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 between
the [particles] and thus would be both am 'imaginary'
rather than real entity (like saying a fakir bed of nails has a flat
top) and to some extent arbitrary (like the curves scientists draw
through 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 out
of it mental chunks we then treat as if they were discrete, totally
detached from their surroundings. The way we mark off islands of
property is but one example of the general process by which we create
meaningful social entities. 


Catherine Elgin, 'Unnatural Science' (1995: 297)
 Our parochial concerns pretty clearly influence our choice of
categories for macroscopic objects and kinds. [...] It would be
rather remarkable if, for example, a taxonomy that draws the
distinction between horses and zebras where we do aligned at all well
with 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 portions
of autonomous reality. Only the respective external boundaries are
created by us; it is these which are the products of our mental and
linguistic activity, and of associated conventional laws, norms and
habits. 


Barry Smith, 'On Substances, Accidents and Universals'
(1997b: 122)
 Delineation is, be it noted, an immensely powerful faculty of
cognition; the scope of delineatory intentionality, the effortlessness
with which we can comprehend highly complex wholes -- which may
be scattered throughout the length and breadth of the universe, in
both space and time -- with a simple delineatory act ('the
legacy of the Renaissance', 'the Austro-Hungarian Empire
and its successor states', 'English poetry') is
wondrous to behold, and bears comparison with the magic of
single-rayed intentionality, whereby, on the basis of a list of
entries which might be drawn up entirely at random, we can be
directed, 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 across
which motion is to be prevented, and you have one of the key themes of
history. With a closed line (i.e., a curve enclosing a figure), and
the prevention of motion from outside the line to its inside, you
derive the idea of property. With the same line, and the prevention of
motion from inside to outside, you derive the idea of prison. With an
open line (i.e., a curve that does not enclose a figure), and the
prevention of motion in either direction, you derive the idea of
border. Properties, prisons, borders: it is through the prevention of
motion that space enters history. 


Gloria E. Anzaldua, *Borderlands/La Frontera* (1987:
3)
 Borders are set up to define the places that are safe and unsafe,
to distinguish *us* from *them*. A border is a dividing
line, a narrow strip along a steep edge. A borderland is a vague and
undetermined place created by the emotional residue of an unnatural
boundary. It is in a constant state of transition. The prohibited and
forbidden are its inhabitants. *Los atravesados* live here: the
squint-eyed, the perverse, the queer, the troublesome, the mongrel,
the mulato, the half-breed, the half dead; in short, those who cross
over, pass over, or go through the confines of the
'normal'. 


Sally Haslanger, 'Feminism in Metaphysics: Negotiating the
Natural' (2000: 116; 2012: 148-149)
 [T]here is an unmistakable pattern of projecting onto women and
people of color, as their 'nature' or as
'natural', features that are instead (if manifested at
all) a product of social forces. This projective error has led
feminists to be extremely suspicious of natural kinds and objective
types: if one function of references to 'nature' or
'natures' is to mark the boundaries of what is socially
possible, thereby 'justifying' pernicious institutions, we
must be wary of the suggestion that any category is
'natural'. [...] Is there any meaningful (and
politically viable) distinction between the natural and the social,
and if so, where does the line fall? Is there any way to theorize
about what's natural that does not depend on the projection of
our political biases? If so, how? 


Sarah Green, 'A Sense of Border' (2012: 580)
 While it is clear that borders do not independently exist as
self-evident entities in the landscape [...] it is also the case
that once constructed (which includes all the various associated
bordering practices, both formal and informal), borders can take on
thing-like qualities, both in practice and in people's
imaginations. 


Jody Azzouni, *Ontology without Borders* (2017: 143)
 [T]here are no worldly ontological borders between purported
objects; there are no properties, no relations -- no aspects of
*anything* in the world -- that constitute or amount to
worldly individuation conditions for objects [...] we project
object boundaries onto the world.