Whitehead's point-free geometry
Encyclopedia
In mathematics
, point-free geometry is a geometry
whose primitive ontological
notion is region rather than point
. Two axiomatic system
s are set out below, one grounded in mereology
, the other in mereotopology
and known as connection theory. A point can mark a space or objects.
(1919, 1920), not as a theory of geometry
or of spacetime
, but of "events" and of an "extension relation" between events. Whitehead's purposes were as much philosophical as scientific and mathematical.
Whitehead did not set out his theories in a manner that would satisfy present-day canons of formality. The two formal first order theories described in this entry were devised by others in order to clarify and refine Whitehead's theories. The domain
for both theories consists of "regions." All unquantified variables in this entry should be taken as tacitly universally quantified; hence all axioms should be taken as universal closures. No axiom requires more than three quantified variables
; hence a translation of first order theories into relation algebra
is possible. Each set of axioms has but four existential quantifiers.
The fundamental primitive binary relation
is Inclusion, denoted by infix
"≤". (Inclusion corresponds to the binary Parthood relation that is a standard feature of all mereological theories.) The intuitive meaning of x≤y is "x is part of y." Assuming that identity
, denoted by infix "=", is part of the background logic, the binary relation
Proper Part, denoted by infix "<", is defined as:
The axioms are:
A model
of G1–G7 is an inclusion space.
Definition (Gerla and Miranda 2008: Def. 4.1). Given some inclusion space, an abstractive class is a class G of regions such that G is totally ordered
by Inclusion. Moreover, there does not exist a region included in all of the regions included in G.
Intuitively, an abstractive class defines a geometrical entity whose dimensionality is less than that of the inclusion space. For example, if the inclusion space is the Euclidean plane, then the corresponding abstractive classes are point
s and line
s.
Inclusion-based point-free geometry (henceforth "point-free geometry") is essentially an axiomatization of Simons's (1987: 83) system W. In turn, W formalizes a theory in Whitehead (1919) whose axioms are not made explicit. Point-free geometry is W with this defect repaired. Simons (1987) did not repair this defect, instead proposing in a footnote that the reader do so as an exercise. The primitive relation of W is Proper Part, a strict partial order. The theory of Whitehead (1919) has a single primitive binary relation K defined as xKy ↔ y<x. Hence K is the converse
of Proper Part. Simons's WP1 asserts that Proper Part is irreflexive and so corresponds to G1. G3 establishes that inclusion, unlike Proper Part, is anti-symmetric.
Point-free geometry is closely related to a dense linear order D, whose axioms are G1-3, G5, and the totality axiom
Hence inclusion-based point-free geometry would be a proper extension of D (namely D∪{G4, G6, G7}), were it not that the D relation "≤" is a total order
.
, A. N. Whitehead proposed a different approach, one inspired by De Laguna (1922). Whitehead took as primitive the topological notion of "contact" between two regions, resulting in a primitive "connection relation" between events. Connection theory C is a first order theory that distills the first 12 of the 31 assumptions in chpt. 2 of Process and Reality
into 6 axioms, C1-C6. C is a proper fragment of the theories proposed in Clarke (1981), who noted their mereological
character. Theories that, like C, feature both inclusion and topological
primitives, are called mereotopologies
.
C has one primitive relation
, binary "connection," denoted by the prefix
ed predicate letter C. That x is included in y can now be defined as x≤y ↔ ∀z[Czx→Czy]. Unlike the case with inclusion spaces, connection theory enables defining "non-tangential" inclusion, a total order that enables the construction of abstractive classes. Gerla and Miranda (2008) argue that only thus can mereotopology unambiguously define a point
.
The axioms C1-C6 below are, but for numbering, those of Def. 3.1 in Gerla and Miranda (2008).
A model of C is a connection space.
Following the verbal description of each axiom is the identifier of the corresponding axiom in Casati and Varzi (1999). Their system SMT (strong mereotopology) consists of C1-C3, and is essentially due to Clarke (1981). Any mereotopology
can be made atomless
by invoking C4, without risking paradox or triviality. Hence C extends the atomless variant of SMT by means of the axioms C5 and C6, suggested by chpt. 2 of Process and Reality
. For an advanced and detailed discussion of systems related to C, see Roeper (1997).
Biacino and Gerla (1991) showed that every model
of Clarke's theory is a Boolean algebra, and models of such algebras cannot distinguish connection from overlap. It is doubtful whether either fact is faithful to Whitehead's intent.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, point-free geometry is a geometry
Geometry
Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....
whose primitive ontological
Ontology
Ontology is the philosophical study of the nature of being, existence or reality as such, as well as the basic categories of being and their relations...
notion is region rather than point
Point (geometry)
In geometry, topology and related branches of mathematics a spatial point is a primitive notion upon which other concepts may be defined. In geometry, points are zero-dimensional; i.e., they do not have volume, area, length, or any other higher-dimensional analogue. In branches of mathematics...
. Two axiomatic system
Axiomatic system
In mathematics, an axiomatic system is any set of axioms from which some or all axioms can be used in conjunction to logically derive theorems. A mathematical theory consists of an axiomatic system and all its derived theorems...
s are set out below, one grounded in mereology
Mereology
In philosophy and mathematical logic, mereology treats parts and the wholes they form...
, the other in mereotopology
Mereotopology
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....
and known as connection theory. A point can mark a space or objects.
Motivation
Point-free geometry was first formulated in WhiteheadAlfred North Whitehead
Alfred North Whitehead, OM FRS was an English mathematician who became a philosopher. He wrote on algebra, logic, foundations of mathematics, philosophy of science, physics, metaphysics, and education...
(1919, 1920), not as a theory of geometry
Geometry
Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....
or of spacetime
Spacetime
In physics, spacetime is any mathematical model that combines space and time into a single continuum. Spacetime is usually interpreted with space as being three-dimensional and time playing the role of a fourth dimension that is of a different sort from the spatial dimensions...
, but of "events" and of an "extension relation" between events. Whitehead's purposes were as much philosophical as scientific and mathematical.
Whitehead did not set out his theories in a manner that would satisfy present-day canons of formality. The two formal first order theories described in this entry were devised by others in order to clarify and refine Whitehead's theories. The domain
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...
for both theories consists of "regions." All unquantified variables in this entry should be taken as tacitly universally quantified; hence all axioms should be taken as universal closures. No axiom requires more than three quantified variables
Quantification
Quantification has several distinct senses. In mathematics and empirical science, it is the act of counting and measuring that maps human sense observations and experiences into members of some set of numbers. Quantification in this sense is fundamental to the scientific method.In logic,...
; hence a translation of first order theories into relation algebra
Relation algebra
In mathematics and abstract algebra, a relation algebra is a residuated Boolean algebra expanded with an involution called converse, a unary operation...
is possible. Each set of axioms has but four existential quantifiers.
Inclusion-based point-free geometry
The axioms G1-G7 are, but for numbering, those of Def. 2.1 in Gerla and Miranda (2008). The identifiers of the form WPn, included in the verbal description of each axiom, refer to the corresponding axiom in Simons (1987: 83).The fundamental 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...
is Inclusion, denoted by infix
Infix
An infix is an affix inserted inside a word stem . It contrasts with adfix, a rare term for an affix attached to the end of a stem, such as a prefix or suffix.-Indonesian:...
"≤". (Inclusion corresponds to the binary Parthood relation that is a standard feature of all mereological theories.) The intuitive meaning of x≤y is "x is part of y." Assuming that identity
Identity function
In mathematics, an identity function, also called identity map or identity transformation, is a function that always returns the same value that was used as its argument...
, denoted by infix "=", is part of the background logic, the 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...
Proper Part, denoted by infix "<", is defined as:
The axioms are:
- Inclusion partially orders the domainDomain of discourseIn 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...
.
- G1.
(reflexiveReflexive relationIn 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:...
) - G2.
(transitiveTransitive relationIn 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....
) WP4. - G3.
(anti-symmetric)
- Given any two regions, there exists a region that includes both of them. WP6.
- G4.
- Proper Part densely ordersDense orderIn mathematics, a partial order ≤ on a set X is said to be dense if, for all x and y in X for which x In mathematics, a partial order ≤ on a set X is said to be dense if, for all x and y in X for which x...
the domainDomain of discourseIn 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...
. WP5.
- G5.
- Both atomic regionsAtomic (order theory)In the mathematical field of order theory, given two elements a and b of a partially ordered set, one says that b covers a, and writes a a, if a ...
and a universal regionMereologyIn philosophy and mathematical logic, mereology treats parts and the wholes they form...
do not exist. Hence the domainDomain of discourseIn 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...
has neither an upper nor a lower bound. WP2.
- G6.
- Proper Parts Principle. If all the proper parts of x are proper parts of y, then x is included in y. WP3.
- G7.
A model
Model theory
In mathematics, model theory is the study of mathematical structures using tools from mathematical logic....
of G1–G7 is an inclusion space.
Definition (Gerla and Miranda 2008: Def. 4.1). Given some inclusion space, an abstractive class is a class G of regions such that G is totally ordered
Total order
In set theory, a total order, linear order, simple order, or ordering is a binary relation on some set X. The relation is transitive, antisymmetric, and total...
by Inclusion. Moreover, there does not exist a region included in all of the regions included in G.
Intuitively, an abstractive class defines a geometrical entity whose dimensionality is less than that of the inclusion space. For example, if the inclusion space is the Euclidean plane, then the corresponding abstractive classes are point
Point (geometry)
In geometry, topology and related branches of mathematics a spatial point is a primitive notion upon which other concepts may be defined. In geometry, points are zero-dimensional; i.e., they do not have volume, area, length, or any other higher-dimensional analogue. In branches of mathematics...
s and line
Line (geometry)
The notion of line or straight line was introduced by the ancient mathematicians to represent straight objects with negligible width and depth. Lines are an idealization of such objects...
s.
Inclusion-based point-free geometry (henceforth "point-free geometry") is essentially an axiomatization of Simons's (1987: 83) system W. In turn, W formalizes a theory in Whitehead (1919) whose axioms are not made explicit. Point-free geometry is W with this defect repaired. Simons (1987) did not repair this defect, instead proposing in a footnote that the reader do so as an exercise. The primitive relation of W is Proper Part, a strict partial order. The theory of Whitehead (1919) has a single primitive binary relation K defined as xKy ↔ y<x. Hence K is the converse
Inverse relation
In mathematics, the inverse relation of a binary relation is the relation that occurs when you switch the order of the elements in the relation. For example, the inverse of the relation 'child of' is the relation 'parent of'...
of Proper Part. Simons's WP1 asserts that Proper Part is irreflexive and so corresponds to G1. G3 establishes that inclusion, unlike Proper Part, is anti-symmetric.
Point-free geometry is closely related to a dense linear order D, whose axioms are G1-3, G5, and the totality axiom
Hence inclusion-based point-free geometry would be a proper extension of D (namely D∪{G4, G6, G7}), were it not that the D relation "≤" is a total orderTotal order
In set theory, a total order, linear order, simple order, or ordering is a binary relation on some set X. The relation is transitive, antisymmetric, and total...
.
Connection theory
In his 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...
, A. N. Whitehead proposed a different approach, one inspired by De Laguna (1922). Whitehead took as primitive the topological notion of "contact" between two regions, resulting in a primitive "connection relation" between events. Connection theory C is a first order theory that distills the first 12 of the 31 assumptions in chpt. 2 of 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...
into 6 axioms, C1-C6. C is a proper fragment of the theories proposed in Clarke (1981), who noted their mereological
Mereology
In philosophy and mathematical logic, mereology treats parts and the wholes they form...
character. Theories that, like C, feature both inclusion and 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...
primitives, are called mereotopologies
Mereotopology
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....
.
C has one primitive 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...
, binary "connection," denoted by the prefix
Prefix
A prefix is an affix which is placed before the root of a word. Particularly in the study of languages,a prefix is also called a preformative, because it alters the form of the words to which it is affixed.Examples of prefixes:...
ed predicate letter C. That x is included in y can now be defined as x≤y ↔ ∀z[Czx→Czy]. Unlike the case with inclusion spaces, connection theory enables defining "non-tangential" inclusion, a total order that enables the construction of abstractive classes. Gerla and Miranda (2008) argue that only thus can mereotopology unambiguously define a point
Point (geometry)
In geometry, topology and related branches of mathematics a spatial point is a primitive notion upon which other concepts may be defined. In geometry, points are zero-dimensional; i.e., they do not have volume, area, length, or any other higher-dimensional analogue. In branches of mathematics...
.
The axioms C1-C6 below are, but for numbering, those of Def. 3.1 in Gerla and Miranda (2008).
- C is reflexiveReflexiveReflexive may refer to:In fiction:*MetafictionIn grammar:*Reflexive pronoun, a pronoun with a reflexive relationship with its self-identical antecedent*Reflexive verb, where a semantic agent and patient are the same...
. C.1.
- C1.
- C is symmetric. C.2.
- C2.
- C is extensionalExtensionalityIn logic, extensionality, or extensional equality refers to principles that judge objects to be equal if they have the same external properties...
. C.11.
- C3.
- All regions have proper parts, so that C is an atomlessAtomic (order theory)In the mathematical field of order theory, given two elements a and b of a partially ordered set, one says that b covers a, and writes a a, if a ...
theory. P.9.
- C4.
- Given any two regions, there is a region connected to both of them.
- C5.
- All regions have at least two unconnected parts. C.14.
- C6.
A model of C is a connection space.
Following the verbal description of each axiom is the identifier of the corresponding axiom in Casati and Varzi (1999). Their system SMT (strong mereotopology) consists of C1-C3, and is essentially due to Clarke (1981). Any mereotopology
Mereotopology
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....
can be made atomless
Atomic (order theory)
In the mathematical field of order theory, given two elements a and b of a partially ordered set, one says that b covers a, and writes a a, if a ...
by invoking C4, without risking paradox or triviality. Hence C extends the atomless variant of SMT by means of the axioms C5 and C6, suggested by chpt. 2 of 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 an advanced and detailed discussion of systems related to C, see Roeper (1997).
Biacino and Gerla (1991) showed that every model
Model theory
In mathematics, model theory is the study of mathematical structures using tools from mathematical logic....
of Clarke's theory is a Boolean algebra, and models of such algebras cannot distinguish connection from overlap. It is doubtful whether either fact is faithful to Whitehead's intent.