In
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, point-free geometry is a
geometry
Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...
whose primitive
ontological
Ontology is the philosophical study of being. It is traditionally understood as the subdiscipline of metaphysics focused on the most general features of reality. As one of the most fundamental concepts, being encompasses all of reality and every ...
notion is ''
region
In geography, regions, otherwise referred to as areas, zones, lands or territories, are portions of the Earth's surface that are broadly divided by physical characteristics (physical geography), human impact characteristics (human geography), and ...
'' rather than
point. Two
axiomatic system
In mathematics and logic, an axiomatic system is a set of formal statements (i.e. axioms) used to logically derive other statements such as lemmas or theorems. A proof within an axiom system is a sequence of deductive steps that establishes ...
s are set out below, one grounded in
mereology
Mereology (; from Greek μέρος 'part' (root: μερε-, ''mere-'') and the suffix ''-logy'', 'study, discussion, science') is the philosophical study of part-whole relationships, also called ''parthood relationships''. As a branch of metaphys ...
, the other in
mereotopology and known as ''connection theory''.
Point-free geometry was first formulated by
Alfred North Whitehead
Alfred North Whitehead (15 February 1861 – 30 December 1947) was an English mathematician and philosopher. He created the philosophical school known as process philosophy, which has been applied in a wide variety of disciplines, inclu ...
, not as a theory of
geometry
Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...
or of
spacetime
In physics, spacetime, also called the space-time continuum, is a mathematical model that fuses the three dimensions of space and the one dimension of time into a single four-dimensional continuum. Spacetime diagrams are useful in visualiz ...
, but of "events" and of an "extension
relation
Relation or relations may refer to:
General uses
* International relations, the study of interconnection of politics, economics, and law on a global level
* Interpersonal relationship, association or acquaintance between two or more people
* ...
" between events. Whitehead's purposes were as much
philosophical
Philosophy ('love of wisdom' in Ancient Greek) is a systematic study of general and fundamental questions concerning topics like existence, reason, knowledge, Value (ethics and social sciences), value, mind, and language. It is a rational an ...
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 or universe of discourse (borrowing from the mathematical concept of ''universe'') is the set of entities over which certain variables of interest in some formal treatment may range.
It is also ...
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", "for all", "for every", or "given an arbitrary element". It expresses that a predicate can be satisfied by ev ...
; 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 re ...
is possible. Each set of axioms has but four
existential quantifier
Existentialism is a family of philosophy, philosophical views and inquiry that explore the human individual's struggle to lead an Authenticity (philosophy), authentic life despite the apparent Absurdity#The Absurd, absurdity or incomprehensibili ...
s.
Inclusion-based point-free geometry (mereology)
The fundamental primitive
binary relation
In mathematics, a binary relation associates some elements of one Set (mathematics), set called the ''domain'' with some elements of another set called the ''codomain''. Precisely, a binary relation over sets X and Y is a set of ordered pairs ...
is ''inclusion'', denoted by the
infix operator "≤", 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 the infix operator "=", is part of the background logic, the binary relation ''Proper Part'', denoted by the infix operator "<", is defined as:
: