In
mathematics, point-free geometry is a
geometry
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is ...
whose primitive
ontological
In metaphysics, ontology is the philosophical study of being, as well as related concepts such as existence, becoming, and reality.
Ontology addresses questions like how entities are grouped into categories and which of these entities ex ...
notion is ''
region
In geography, regions, otherwise referred to as zones, lands or territories, are areas that are broadly divided by physical characteristics (physical geography), human impact characteristics (human geography), and the interaction of humanity and t ...
'' rather than
point. Two
axiomatic system
In mathematics and logic, an axiomatic system is any set of axioms from which some or all axioms can be used in conjunction to logically derive theorems. A theory is a consistent, relatively-self-contained body of knowledge which usually contai ...
s are set out below, one grounded in
mereology
In logic, philosophy and related fields, mereology ( (root: , ''mere-'', 'part') and the suffix ''-logy'', 'study, discussion, science') is the study of parts and the wholes they form. Whereas set theory is founded on the membership relation bet ...
, the other in
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 bet ...
and known as ''connection theory''.
Point-free geometry was first formulated in
Whitehead (1919, 1920), not as a theory of
geometry
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is ...
or of
spacetime
In physics, spacetime is a mathematical model that combines the three dimensions of space and one dimension of time into a single four-dimensional manifold. Spacetime diagrams can be used to visualize relativistic effects, such as why differen ...
, but of "events" and of an "extension
relation" between events. Whitehead's purposes were as much
philosophical
Philosophy (from , ) is the systematized study of general and fundamental questions, such as those about existence, reason, knowledge, values, mind, and language. Such questions are often posed as problems to be studied or resolved. Some s ...
as scientific and mathematical.
Formalizations
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 of discourse
In the formal sciences, the domain of discourse, also called the universe of discourse, universal set, or simply universe, is the set of entities over which certain variables of interest in some formal treatment may range.
Overview
The domai ...
for both theories consists of "regions." All
unquantified variables in this entry should be taken as tacitly
universally quantified
In mathematical logic, a universal quantification is a type of quantifier, a logical constant which is interpreted as "given any" or "for all". It expresses that a predicate can be satisfied by every member of a domain of discourse. In other ...
; 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
In mathematics and abstract algebra, a relation algebra is a residuated Boolean algebra expanded with an involution called converse, a unary operation. The motivating example of a relation algebra is the algebra 2''X''² of all binary relations ...
is possible. Each set of axioms has but four
existential quantifier
In predicate logic, an existential quantification is a type of quantifier, a logical constant which is interpreted as "there exists", "there is at least one", or "for some". It is usually denoted by the logical operator symbol ∃, which, whe ...
s.
Inclusion-based point-free geometry (mereology)
The fundamental primitive
binary relation
In mathematics, a binary relation associates elements of one set, called the ''domain'', with elements of another set, called the ''codomain''. A binary relation over sets and is a new set of ordered pairs consisting of elements in and i ...
is ''inclusion'', denoted by
infix
An infix is an affix inserted inside a word stem (an existing word or the core of a family of words). It contrasts with ''adfix,'' a rare term for an affix attached to the outside of a stem, such as a prefix or suffix.
When marking text for int ...
"≤", which corresponds to the binary ''Parthood'' relation that is a standard feature in
mereological theories. The intuitive meaning of ''x'' ≤ ''y'' is "''x'' is part of ''y''." Assuming that equality, denoted by infix "=", is part of the background logic, the binary relation ''Proper Part'', denoted by infix "<", is defined as:
: