Mereotopology
Encyclopedia
In formal ontology
, a branch of metaphysics
, and in ontological computer science
, mereotopology is a first-order theory, embodying mereological
and topological concepts, of the relations among wholes, parts, parts of parts, and the boundaries
between parts.
augmented the part-whole relation with topological notions such as contiguity
and connection
. Despite Whitehead's acumen as a mathematician, his theories were insufficiently formal, even flawed. By showing how Whitehead's theories could be fully formalized and repaired, Clarke (1981, 1985) founded contemporary mereotopology. The theories of Clarke and Whitehead are discussed in Simons (1987: 2.10.2), and Lucas (2000: chpt. 10). The entry Whitehead's point-free geometry
includes two contemporary treatments of Whitehead's theories, due to Giangiacomo Gerla, each different from the theory set out in the next section.
Although mereotopology is a mathematical theory, we owe its subsequent development to logic
ians and theoretical computer scientists
. Lucas (2000: chpt. 10) and Casati and Varzi (1999: chpts. 4,5) are introductions to mereotopology that can be read by anyone having done a course in first-order logic
. More advanced treatments of mereotopology include Cohn and Varzi (2003) and, for the mathematically sophisticated, Roeper (1997). For a mathematical treatment of point-free geometry, see Gerla (1995). Lattice
-theoretic (algebraic
) treatments of mereotopology as contact algebras have been applied to separate the topological
from the mereological
structure, see Stell (2000), Düntsch and Winter (2004).
Barry Smith
(1996), Anthony Cohn and his coauthors, and Varzi alone and with others, have all shown that mereotopology can be useful in formal ontology
and computer science
, by formalizing relations such as contact
, connection
, boundaries
, interior
s, holes, and so on. Mereotopology has been most useful as a tool for qualitative spatial-temporal reasoning
, with constraint calculi such as the Region Connection Calculus
(RCC).
. Casati and Varzi do not say if the models
of GEMTC include any conventional topological space
s.
We begin with some domain of discourse
, whose elements are called individual
s (a synonym
for mereology
is "the calculus of individuals"). Casati and Varzi prefer limiting the ontology to physical objects, but others freely employ mereotopology to reason about geometric figures and events, and to solve problems posed by research in machine intelligence.
An upper case Latin letter denotes both a relation
and the predicate letter referring to that relation in first-order logic
. Lower case letters from the end of the alphabet denote variables ranging over the domain; letters from the start of the alphabet are names of arbitrary individuals. If a formula begins with an atomic formula
followed by the biconditional, the subformula to the right of the biconditional is a definition of the atomic formula, whose variables are unbound. Otherwise, variables not explicitly quantified are tacitly universally quantified. The axiom Cn below corresponds to axiom C.n in Casati and Varzi (1999: chpt. 4).
We begin with a topological primitive, a binary relation
called connection; the atomic formula Cxy denotes that "x is connected to y." Connection is governed, at minimum, by the axioms:
C1.
(reflexive
)
C2.
(symmetric
)
Now posit the binary relation E, defined as:
Exy is read as "y encloses x" and is also topological in nature. A consequence of C1-2 is that E is reflexive
and transitive
, and hence a preorder
. If E is also assumed extensional
, so that:
then E can be proved antisymmetric
and thus becomes a partial order. Enclosure, notated xKy, is the single primitive relation of the theories in Whitehead (1919, 1925)
, the starting point of mereotopology.
Let parthood be the defining primitive binary relation
of the underlying mereology
, and let the atomic formula
Pxy denote that "x is part of y". We assume that P is a partial order. Call the resulting minimalist mereological theory M.
If x is part of y, we postulate that y encloses x:
C3.
C3 nicely connects mereological
parthood to topological enclosure.
Let O, the binary relation of mereological overlap, be defined as:
Let Oxy denote that "x and y overlap." With O in hand, a consequence of C3 is:
Note that the converse does not necessarily hold. While things that overlap are necessarily connected, connected things do not necessarily overlap. If this were not the case, topology
would merely be a model of mereology
(in which "overlap" is always either primitive or defined).
Ground mereotopology (MT) is the theory consisting of primitive C and P, defined E and O, the axioms C1-3, and axioms assuring that P is a partial order. Replacing the M in MT with the standard extensional mereology GEM
results in the theory GEMT.
Let IPxy denote that "x is an internal part of y." IP is defined as:
Let σx φ(x) denote the mereological sum (fusion) of all individuals in the domain satisfying φ(x). σ is a variable binding prefix
operator. The axioms of GEM assure that this sum exists if φ(x) is a first-order formula
. With σ and the relation IP in hand, we can define the interior
of x,
as the mereological sum of all interior parts z of x, or:
Two easy consequences of this definition are:
where W is the universal individual, and
C5.
(Inclusion)
The operator i has two more axiomatic properties:
C6.
(Idempotence
)
C7.
where a×b is the mereological product of a and b, not defined when Oab is false. i distributes over product.
It can now be seen that i is isomorphic to the interior operator of topology
. Hence the dual
of i, the topological closure operator
c, can be defined in terms of i, and Kuratowski's axioms for c are theorems. Likewise, given an axiomatization of c that is analogous to C5-7, i may be defined in terms of c, and C5-7 become theorems. Adding C5-7 to GEMT results in Casati and Varzi's preferred mereotopological theory, GEMTC.
x is self-connected if it satisfies the following predicate:
Note that the primitive and defined predicates of MT alone suffice for this definition. The predicate SC enables formalizing the necessary condition given in Whitehead's Process and Reality
for the mereological sum of two individuals to exist: they must be connected. Formally:
C8.
Given some mereotopology X, adding C8 to X results in what Casati and Varzi call the Whiteheadian extension of X, denoted WX. Hence the theory whose axioms are C1-8 is WGEMTC.
The converse of C8 is a GEMTC theorem. Hence given the axioms of GEMTC, C is a defined predicate if O and SC are taken as primitive predicates.
If the underlying mereology is atom
less and weaker than GEM, the axiom that assures the absence of atoms (P9 in Casati and Varzi 1999) may be replaced by C9, which postulates that no individual has a topological boundary
:
C9.
When the domain consists of geometric figures, the boundaries can be points, curves, and surfaces. What boundaries could mean, given other ontologies, is not an easy matter and is discussed in Casati and Varzi (1999: chpt. 5).
Formal ontology
In philosophy, the term formal ontology is used to refer to an ontology defined by axioms in a formal language with the goal to provide an unbiased view on reality, which can help the modeler of domain- or application-specific ontologies to avoid possibly erroneous ontological assumptions...
, a branch of metaphysics
Metaphysics
Metaphysics is a branch of philosophy concerned with explaining the fundamental nature of being and the world, although the term is not easily defined. Traditionally, metaphysics attempts to answer two basic questions in the broadest possible terms:...
, and in ontological computer science
Ontology (computer science)
In computer science and information science, an ontology formally represents knowledge as a set of concepts within a domain, and the relationships between those concepts. It can be used to reason about the entities within that domain and may be used to describe the domain.In theory, an ontology is...
, mereotopology is a first-order theory, embodying mereological
Mereology
In philosophy and mathematical logic, mereology treats parts and the wholes they form...
and topological concepts, of the relations among wholes, parts, parts of parts, and the boundaries
Boundary (topology)
In topology and mathematics in general, the boundary of a subset S of a topological space X is the set of points which can be approached both from S and from the outside of S. More precisely, it is the set of points in the closure of S, not belonging to the interior of S. An element of the boundary...
between parts.
History and motivation
Mereotopology begins with theories A. N. Whitehead articulated in several books and articles he published between 1916 and 1929. Whitehead's early work is discussed in Kneebone (1963: chpt. 13.5) and Simons (1987: 2.9.1). The theory of Whitehead's 1929 Process and RealityProcess and Reality
In philosophy, especially metaphysics, the book Process and Reality by Alfred North Whitehead sets out its author's philosophy of organism, also called process philosophy...
augmented the part-whole relation with topological notions such as contiguity
Contact (mathematics)
In mathematics, contact of order k of functions is an equivalence relation, corresponding to having the same value at a point P and also the same derivatives there, up to order k. The equivalence classes are generally called jets...
and connection
Connected space
In topology and related branches of mathematics, a connected space is a topological space that cannot be represented as the union of two or more disjoint nonempty open subsets. Connectedness is one of the principal topological properties that is used to distinguish topological spaces...
. Despite Whitehead's acumen as a mathematician, his theories were insufficiently formal, even flawed. By showing how Whitehead's theories could be fully formalized and repaired, Clarke (1981, 1985) founded contemporary mereotopology. The theories of Clarke and Whitehead are discussed in Simons (1987: 2.10.2), and Lucas (2000: chpt. 10). The entry Whitehead's point-free geometry
Whitehead's point-free geometry
In mathematics, point-free geometry is a geometry whose primitive ontological notion is region rather than point. Two axiomatic systems are set out below, one grounded in mereology, the other in mereotopology and known as connection theory...
includes two contemporary treatments of Whitehead's theories, due to Giangiacomo Gerla, each different from the theory set out in the next section.
Although mereotopology is a mathematical theory, we owe its subsequent development to logic
Logic
In philosophy, Logic is the formal systematic study of the principles of valid inference and correct reasoning. Logic is used in most intellectual activities, but is studied primarily in the disciplines of philosophy, mathematics, semantics, and computer science...
ians and theoretical computer scientists
Computer science
Computer science or computing science is the study of the theoretical foundations of information and computation and of practical techniques for their implementation and application in computer systems...
. Lucas (2000: chpt. 10) and Casati and Varzi (1999: chpts. 4,5) are introductions to mereotopology that can be read by anyone having done a course in first-order logic
First-order logic
First-order logic is a formal logical system used in mathematics, philosophy, linguistics, and computer science. It goes by many names, including: first-order predicate calculus, the lower predicate calculus, quantification theory, and predicate logic...
. More advanced treatments of mereotopology include Cohn and Varzi (2003) and, for the mathematically sophisticated, Roeper (1997). For a mathematical treatment of point-free geometry, see Gerla (1995). Lattice
Lattice (order)
In mathematics, a lattice is a partially ordered set in which any two elements have a unique supremum and an infimum . Lattices can also be characterized as algebraic structures satisfying certain axiomatic identities...
-theoretic (algebraic
Algebraic structure
In abstract algebra, an algebraic structure consists of one or more sets, called underlying sets or carriers or sorts, closed under one or more operations, satisfying some axioms. Abstract algebra is primarily the study of algebraic structures and their properties...
) treatments of mereotopology as contact algebras have been applied to separate the topological
Topology
Topology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing...
from the mereological
Mereology
In philosophy and mathematical logic, mereology treats parts and the wholes they form...
structure, see Stell (2000), Düntsch and Winter (2004).
Barry Smith
Barry Smith (ontologist)
Barry Smith is a Julian Park Distinguished Professor of Philosophy in the University at Buffalo and Research Scientist in the New York State . From 2002 to 2006 he was Director of the Institute for Formal Ontology and Medical Information Science in Leipzig and Saarbrücken, GermaUny...
(1996), Anthony Cohn and his coauthors, and Varzi alone and with others, have all shown that mereotopology can be useful in formal ontology
Formal ontology
In philosophy, the term formal ontology is used to refer to an ontology defined by axioms in a formal language with the goal to provide an unbiased view on reality, which can help the modeler of domain- or application-specific ontologies to avoid possibly erroneous ontological assumptions...
and computer science
Ontology (computer science)
In computer science and information science, an ontology formally represents knowledge as a set of concepts within a domain, and the relationships between those concepts. It can be used to reason about the entities within that domain and may be used to describe the domain.In theory, an ontology is...
, by formalizing relations such as contact
Contact (mathematics)
In mathematics, contact of order k of functions is an equivalence relation, corresponding to having the same value at a point P and also the same derivatives there, up to order k. The equivalence classes are generally called jets...
, connection
Connected space
In topology and related branches of mathematics, a connected space is a topological space that cannot be represented as the union of two or more disjoint nonempty open subsets. Connectedness is one of the principal topological properties that is used to distinguish topological spaces...
, boundaries
Boundary (topology)
In topology and mathematics in general, the boundary of a subset S of a topological space X is the set of points which can be approached both from S and from the outside of S. More precisely, it is the set of points in the closure of S, not belonging to the interior of S. An element of the boundary...
, interior
Interior (topology)
In mathematics, specifically in topology, the interior of a set S of points of a topological space consists of all points of S that do not belong to the boundary of S. A point that is in the interior of S is an interior point of S....
s, holes, and so on. Mereotopology has been most useful as a tool for qualitative spatial-temporal reasoning
Spatial-temporal reasoning
Spatial–temporal reasoning is used in both the fields of psychology and computer science.-Spatial–temporal reasoning in psychology:Spatial-temporal reasoning is the ability to visualize spatial patterns and mentally manipulate them over a time-ordered sequence of spatial transformations.This...
, with constraint calculi such as the Region Connection Calculus
Region Connection Calculus
The region connection calculus serves for qualitative spatial representation and reasoning. RCC abstractly describes regions by their possible relations to each other...
(RCC).
Preferred approach of Casati & Varzi
Casati and Varzi (1999: chpt.4) set out a variety of mereotopological theories in a consistent notation. This section sets out several nested theories that culminate in their preferred theory GEMTC, and follows their exposition closely. The mereological part of GEMTC is the conventional theory GEMMereology
In philosophy and mathematical logic, mereology treats parts and the wholes they form...
. Casati and Varzi do not say if the models
Model theory
In mathematics, model theory is the study of mathematical structures using tools from mathematical logic....
of GEMTC include any conventional topological space
Topological space
Topological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connectedness, and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion...
s.
We begin with some domain of discourse
Domain of discourse
In the formal sciences, the domain of discourse, also called the universe of discourse , is the set of entities over which certain variables of interest in some formal treatment may range...
, whose elements are called individual
Individual
An individual is a person or any specific object or thing in a collection. Individuality is the state or quality of being an individual; a person separate from other persons and possessing his or her own needs, goals, and desires. Being self expressive...
s (a synonym
Synonym
Synonyms are different words with almost identical or similar meanings. Words that are synonyms are said to be synonymous, and the state of being a synonym is called synonymy. The word comes from Ancient Greek syn and onoma . The words car and automobile are synonyms...
for mereology
Mereology
In philosophy and mathematical logic, mereology treats parts and the wholes they form...
is "the calculus of individuals"). Casati and Varzi prefer limiting the ontology to physical objects, but others freely employ mereotopology to reason about geometric figures and events, and to solve problems posed by research in machine intelligence.
An upper case Latin letter denotes both a relation
Relation (mathematics)
In set theory and logic, a relation is a property that assigns truth values to k-tuples of individuals. Typically, the property describes a possible connection between the components of a k-tuple...
and the predicate letter referring to that relation in first-order logic
First-order logic
First-order logic is a formal logical system used in mathematics, philosophy, linguistics, and computer science. It goes by many names, including: first-order predicate calculus, the lower predicate calculus, quantification theory, and predicate logic...
. Lower case letters from the end of the alphabet denote variables ranging over the domain; letters from the start of the alphabet are names of arbitrary individuals. If a formula begins with an atomic formula
Atomic formula
In mathematical logic, an atomic formula is a formula with no deeper propositional structure, that is, a formula that contains no logical connectives or equivalently a formula that has no strict subformulas. Atoms are thus the simplest well-formed formulas of the logic...
followed by the biconditional, the subformula to the right of the biconditional is a definition of the atomic formula, whose variables are unbound. Otherwise, variables not explicitly quantified are tacitly universally quantified. The axiom Cn below corresponds to axiom C.n in Casati and Varzi (1999: chpt. 4).
We begin with a topological primitive, a binary relation
Binary relation
In mathematics, a binary relation on a set A is a collection of ordered pairs of elements of A. In other words, it is a subset of the Cartesian product A2 = . More generally, a binary relation between two sets A and B is a subset of...
called connection; the atomic formula Cxy denotes that "x is connected to y." Connection is governed, at minimum, by the axioms:
C1.
(reflexiveReflexive relation
In mathematics, a reflexive relation is a binary relation on a set for which every element is related to itself, i.e., a relation ~ on S where x~x holds true for every x in S. For example, ~ could be "is equal to".-Related terms:...
)
C2.
(symmetricSymmetric relation
In mathematics, a binary relation R over a set X is symmetric if it holds for all a and b in X that if a is related to b then b is related to a.In mathematical notation, this is:...
)
Now posit the binary relation E, defined as:
Exy is read as "y encloses x" and is also topological in nature. A consequence of C1-2 is that E is reflexive
Reflexive relation
In mathematics, a reflexive relation is a binary relation on a set for which every element is related to itself, i.e., a relation ~ on S where x~x holds true for every x in S. For example, ~ could be "is equal to".-Related terms:...
and transitive
Transitive relation
In mathematics, a binary relation R over a set X is transitive if whenever an element a is related to an element b, and b is in turn related to an element c, then a is also related to c....
, and hence a preorder
Preorder
In mathematics, especially in order theory, preorders are binary relations that are reflexive and transitive.For example, all partial orders and equivalence relations are preorders...
. If E is also assumed extensional
Axiom of extensionality
In axiomatic set theory and the branches of logic, mathematics, and computer science that use it, the axiom of extensionality, or axiom of extension, is one of the axioms of Zermelo-Fraenkel set theory.- Formal statement :...
, so that:
then E can be proved antisymmetric
Antisymmetric relation
In mathematics, a binary relation R on a set X is antisymmetric if, for all a and b in Xor, equivalently,In mathematical notation, this is:\forall a, b \in X,\ R \and R \; \Rightarrow \; a = bor, equivalently,...
and thus becomes a partial order. Enclosure, notated xKy, is the single primitive relation of the theories in Whitehead (1919, 1925)
Whitehead's point-free geometry
In mathematics, point-free geometry is a geometry whose primitive ontological notion is region rather than point. Two axiomatic systems are set out below, one grounded in mereology, the other in mereotopology and known as connection theory...
, the starting point of mereotopology.
Let parthood be the defining primitive binary relation
Binary relation
In mathematics, a binary relation on a set A is a collection of ordered pairs of elements of A. In other words, it is a subset of the Cartesian product A2 = . More generally, a binary relation between two sets A and B is a subset of...
of the underlying mereology
Mereology
In philosophy and mathematical logic, mereology treats parts and the wholes they form...
, and let the atomic formula
Atomic formula
In mathematical logic, an atomic formula is a formula with no deeper propositional structure, that is, a formula that contains no logical connectives or equivalently a formula that has no strict subformulas. Atoms are thus the simplest well-formed formulas of the logic...
Pxy denote that "x is part of y". We assume that P is a partial order. Call the resulting minimalist mereological theory M.
If x is part of y, we postulate that y encloses x:
C3.
C3 nicely connects mereological
Mereology
In philosophy and mathematical logic, mereology treats parts and the wholes they form...
parthood to topological enclosure.
Let O, the binary relation of mereological overlap, be defined as:
Let Oxy denote that "x and y overlap." With O in hand, a consequence of C3 is:
Note that the converse does not necessarily hold. While things that overlap are necessarily connected, connected things do not necessarily overlap. If this were not the case, topology
Topology
Topology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing...
would merely be a model of mereology
Mereology
In philosophy and mathematical logic, mereology treats parts and the wholes they form...
(in which "overlap" is always either primitive or defined).
Ground mereotopology (MT) is the theory consisting of primitive C and P, defined E and O, the axioms C1-3, and axioms assuring that P is a partial order. Replacing the M in MT with the standard extensional mereology GEM
Mereology
In philosophy and mathematical logic, mereology treats parts and the wholes they form...
results in the theory GEMT.
Let IPxy denote that "x is an internal part of y." IP is defined as:
Let σx φ(x) denote the mereological sum (fusion) of all individuals in the domain satisfying φ(x). σ is a variable binding prefix
Substring
A subsequence, substring, prefix or suffix of a string is a subset of the symbols in a string, where the order of the elements is preserved...
operator. The axioms of GEM assure that this sum exists if φ(x) is a first-order formula
First-order logic
First-order logic is a formal logical system used in mathematics, philosophy, linguistics, and computer science. It goes by many names, including: first-order predicate calculus, the lower predicate calculus, quantification theory, and predicate logic...
. With σ and the relation IP in hand, we can define the interior
Interior (topology)
In mathematics, specifically in topology, the interior of a set S of points of a topological space consists of all points of S that do not belong to the boundary of S. A point that is in the interior of S is an interior point of S....
of x,
as the mereological sum of all interior parts z of x, or:Two easy consequences of this definition are:
where W is the universal individual, and
C5.
(Inclusion)The operator i has two more axiomatic properties:
C6.
(IdempotenceIdempotence
Idempotence is the property of certain operations in mathematics and computer science, that they can be applied multiple times without changing the result beyond the initial application...
)
C7.
where a×b is the mereological product of a and b, not defined when Oab is false. i distributes over product.
It can now be seen that i is isomorphic to the interior operator of topology
Topology
Topology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing...
. Hence the dual
Duality (mathematics)
In mathematics, a duality, generally speaking, translates concepts, theorems or mathematical structures into other concepts, theorems or structures, in a one-to-one fashion, often by means of an involution operation: if the dual of A is B, then the dual of B is A. As involutions sometimes have...
of i, the topological closure operator
Closure operator
In mathematics, a closure operator on a set S is a function cl: P → P from the power set of S to itself which satisfies the following conditions for all sets X,Y ⊆ S....
c, can be defined in terms of i, and Kuratowski's axioms for c are theorems. Likewise, given an axiomatization of c that is analogous to C5-7, i may be defined in terms of c, and C5-7 become theorems. Adding C5-7 to GEMT results in Casati and Varzi's preferred mereotopological theory, GEMTC.
x is self-connected if it satisfies the following predicate:
Note that the primitive and defined predicates of MT alone suffice for this definition. The predicate SC enables formalizing the necessary condition given in Whitehead's Process and Reality
Process and Reality
In philosophy, especially metaphysics, the book Process and Reality by Alfred North Whitehead sets out its author's philosophy of organism, also called process philosophy...
for the mereological sum of two individuals to exist: they must be connected. Formally:
C8.
Given some mereotopology X, adding C8 to X results in what Casati and Varzi call the Whiteheadian extension of X, denoted WX. Hence the theory whose axioms are C1-8 is WGEMTC.
The converse of C8 is a GEMTC theorem. Hence given the axioms of GEMTC, C is a defined predicate if O and SC are taken as primitive predicates.
If the underlying mereology is atom
Atomic formula
In mathematical logic, an atomic formula is a formula with no deeper propositional structure, that is, a formula that contains no logical connectives or equivalently a formula that has no strict subformulas. Atoms are thus the simplest well-formed formulas of the logic...
less and weaker than GEM, the axiom that assures the absence of atoms (P9 in Casati and Varzi 1999) may be replaced by C9, which postulates that no individual has a topological boundary
Boundary (topology)
In topology and mathematics in general, the boundary of a subset S of a topological space X is the set of points which can be approached both from S and from the outside of S. More precisely, it is the set of points in the closure of S, not belonging to the interior of S. An element of the boundary...
:
C9.
When the domain consists of geometric figures, the boundaries can be points, curves, and surfaces. What boundaries could mean, given other ontologies, is not an easy matter and is discussed in Casati and Varzi (1999: chpt. 5).
See also
- MereologyMereologyIn philosophy and mathematical logic, mereology treats parts and the wholes they form...
- Pointless topologyPointless topologyIn mathematics, pointless topology is an approach to topology that avoids mentioning points. The name 'pointless topology' is due to John von Neumann...
- Point-set topology
- TopologyTopologyTopology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing...
- Topological spaceTopological spaceTopological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connectedness, and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion...
(with links to T0 through T6) - Whitehead's point-free geometryWhitehead's point-free geometryIn mathematics, point-free geometry is a geometry whose primitive ontological notion is region rather than point. Two axiomatic systems are set out below, one grounded in mereology, the other in mereotopology and known as connection theory...
External links
- Stanford Encyclopedia of PhilosophyStanford Encyclopedia of PhilosophyThe Stanford Encyclopedia of Philosophy is a freely-accessible online encyclopedia of philosophy maintained by Stanford University. Each entry is written and maintained by an expert in the field, including professors from over 65 academic institutions worldwide...
: Boundary -- by Achille Varzi. With many references.