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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: :x The axioms are: *Inclusion partially orders the domain. :G1. x \le x. ( reflexive) :G2. (x \le z \land z \le y) \rightarrow x \le y. ( transitive) WP4. :G3. (x \le y \land y \le x) \rightarrow x = y. ( antisymmetric) *Given any two regions, there exists a region that includes both of them. WP6. :G4. \exists z \le z \land y\le z *Proper Part densely orders the domain. WP5. :G5. x *Both atomic regions and a universal region do not exist. Hence the domain has neither an upper nor a lower bound. WP2. :G6. \exists y \exists z * Proper Parts Principle. If all the proper parts of ''x'' are proper parts of ''y'', then ''x'' is included in ''y''. WP3. :G7. \forall z \rightarrow x\le y. A
model A model is an informative representation of an object, person, or system. The term originally denoted the plans of a building in late 16th-century English, and derived via French and Italian ultimately from Latin , . Models can be divided in ...
of G1–G7 is an ''inclusion space''. Definition. Given some inclusion space S, an abstractive class is a class ''G'' of regions such that ''S\G'' is
totally ordered In mathematics, a total order or linear order is a partial order in which any two elements are comparable. That is, a total order is a binary relation \leq on some set X, which satisfies the following for all a, b and c in X: # a \leq a ( r ...
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 In mathematics, a Euclidean plane is a Euclidean space of Two-dimensional space, dimension two, denoted \textbf^2 or \mathbb^2. It is a geometric space in which two real numbers are required to determine the position (geometry), position of eac ...
, then the corresponding abstractive classes are points and lines. Inclusion-based point-free geometry (henceforth "point-free geometry") is essentially an axiomatization of Simons's system W. In turn, W formalizes a theory of Whitehead whose axioms are not made explicit. Point-free geometry is W with this defect repaired. Simons 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 In mathematics, especially order theory, a partial order on a set is an arrangement such that, for certain pairs of elements, one precedes the other. The word ''partial'' is used to indicate that not every pair of elements needs to be comparable; ...
. 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 In mathematics, a binary relation R on a set X is reflexive if it relates every element of X to itself. An example of a reflexive relation is the relation " is equal to" on the set of real numbers, since every real number is equal to itself. A ...
and so corresponds to G1. G3 establishes that inclusion, unlike Proper Part, is antisymmetric. Point-free geometry is closely related to a
dense linear order In mathematics, a partial order or total order < on a X is said to be dense if, for all x
D, whose axioms are G1-3, G5, and the totality axiom x \le y \lor y \le x. Hence inclusion-based point-free geometry would be a proper extension of D (namely D ∪ ), were it not that the D relation "≤" is a
total order In mathematics, a total order or linear order is a partial order in which any two elements are comparable. That is, a total order is a binary relation \leq on some set X, which satisfies the following for all a, b and c in X: # a \leq a ( re ...
.


Connection theory (mereotopology)

A different approach was proposed in Whitehead (1929), one inspired by De Laguna (1922). Whitehead took as primitive the
topological Topology (from the Greek words , and ) is the branch of mathematics concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, wit ...
notion of "contact" between two regions, resulting in a primitive "connection relation" between events. Connection theory C is a
first-order theory In mathematical logic, a theory (also called a formal theory) is a set of sentences in a formal language. In most scenarios a deductive system is first understood from context, giving rise to a formal system that combines the language with deduct ...
that distills the first 12 of Whitehead's 31 assumptions into 6 axioms, C1-C6. C is a proper fragment of the theories proposed by Clarke, who noted their mereological character. Theories that, like C, feature both inclusion and topological primitives, are called mereotopologies. C has one primitive
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 * ...
, binary "connection," denoted by the
prefix A prefix is an affix which is placed before the stem of a word. Particularly in the study of languages, a prefix is also called a preformative, because it alters the form of the word to which it is affixed. Prefixes, like other affixes, can b ...
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. *''C'' is reflexive. C.1. :C1. \ Cxx. *''C'' is
symmetric Symmetry () in everyday life refers to a sense of harmonious and beautiful proportion and balance. In mathematics, the term has a more precise definition and is usually used to refer to an object that is invariant under some transformations ...
. C.2. :C2. Cxy\rightarrow Cyx. *''C'' is extensional. C.11. :C3. \forall z zx \leftrightarrow Czy\rightarrow x = y. * All regions have proper parts, so that C is an atomless theory. P.9. :C4. \exists y *Given any two regions, there is a region connected to both of them. :C5. \exists z zx\land Czy *All regions have at least two unconnected parts. C.14. :C6. \exists y \exists z y\le x)\land (z\le x)\land\neg Cyz 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 chapter 2 of part 4 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 A model is an informative representation of an object, person, or system. The term originally denoted the plans of a building in late 16th-century English, and derived via French and Italian ultimately from Latin , . Models can be divided in ...
of Clarke's theory is a
Boolean algebra In mathematics and mathematical logic, Boolean algebra is a branch of algebra. It differs from elementary algebra in two ways. First, the values of the variable (mathematics), variables are the truth values ''true'' and ''false'', usually denot ...
, and models of such algebras cannot distinguish connection from overlap. It is doubtful whether either fact is faithful to Whitehead's intent.


See also

*
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 ...
* Mereotopology *
Pointless topology In mathematics, pointless topology, also called point-free topology (or pointfree topology) or topology without points and locale theory, is an approach to topology that avoids mentioning point (mathematics), points, and in which the Lattice (order ...


Notes


References


Bibliography

*Biacino L., and Gerla G., 1991,
Connection Structures
" ''Notre Dame Journal of Formal Logic'' 32: 242-47. * Casati, R., and Varzi, A. C., 1999. ''Parts and places: the structures of spatial representation''. MIT Press. * Clarke, Bowman, 1981,
A calculus of individuals based on 'connection'
" ''Notre Dame Journal of Formal Logic 22'': 204-18. * ------, 1985,
Individuals and Points
" ''Notre Dame Journal of Formal Logic 26'': 61-75. *De Laguna, T., 1922, "Point, line and surface as sets of solids," ''The Journal of Philosophy 19'': 449-61. * Gerla, G., 1995,
Pointless Geometries
in Buekenhout, F., Kantor, W. eds., ''Handbook of incidence geometry: buildings and foundations''. North-Holland: 1015-31. *--------, and Miranda A., 2008,
Inclusion and Connection in Whitehead's Point-free Geometry
" in Michel Weber and Will Desmond, (eds.),
Handbook of Whiteheadian Process Thought
', Frankfurt / Lancaster, ontos verlag, Process Thought X1 & X2. *Gruszczynski R., and Pietruszczak A., 2008,
Full development of Tarski's geometry of solids
" ''Bulletin of Symbolic Logic'' 14:481-540. The paper contains presentation of point-free system of geometry originating from Whitehead's ideas and based on Lesniewski's mereology. It also briefly discusses the relation between point-free and point-based systems of geometry. Basic properties of mereological structures are given as well. *Grzegorczyk, A., 1960, "Axiomatizability of geometry without points," ''Synthese 12'': 228-235. *Kneebone, G., 1963. ''Mathematical Logic and the Foundation of Mathematics''. Dover reprint, 2001. * Lucas, J. R., 2000. ''Conceptual Roots of Mathematics''. Routledge. Chpt. 10, on "prototopology," discusses Whitehead's systems and is strongly influenced by the unpublished writings of David Bostock. * Roeper, P., 1997, "Region-Based Topology," ''Journal of Philosophical Logic 26'': 251-309. * Simons, P., 1987. ''Parts: A Study in Ontology''. Oxford Univ. Press. * Whitehead, A.N., 1916, "La Theorie Relationiste de l'Espace," ''Revue de Metaphysique et de Morale 23'': 423-454. Translated as Hurley, P.J., 1979, "The relational theory of space," ''Philosophy Research Archives 5'': 712-741. *--------, 1919. ''An Enquiry Concerning the Principles of Natural Knowledge''. Cambridge Univ. Press. 2nd ed., 1925. *--------, 1920.
The Concept of Nature
'. Cambridge Univ. Press. 2004 paperback, Prometheus Books. Being the 1919 Tarner Lectures delivered at Trinity College. *--------, 1979 (1929). ''
Process and Reality ''Process and Reality'' is a book by Alfred North Whitehead, in which the author propounds a philosophy of organism, also called process philosophy. The book, published in 1929, is a revision of the Gifford Lectures he gave in 1927–28. Wh ...
''. Free Press. {{Alfred North Whitehead Alfred North Whitehead History of mathematics Mathematical axioms Mereology Ontology Topology