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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. History and motivation Mereotopology begins in philosophy with theories articulated by A. N. Whitehead in several books and articles he published between 1916 and 1929, drawing in part on the mereogeometry of De Laguna (1922). The first to have proposed the idea of a point-free definition of the concept of topological space in mathematics was Karl Menger in his book ''Dimensionstheorie'' (1928) -- see also his (1940). The early historical background of mereotopology is documented in Bélanger and Marquis (2013) and Whitehead's early work is discussed in Kneebone (1963: ch. 13.5) and Simons (1987: 2.9.1). The theory of Whitehead's 1929 ''Process and Reality'' augmented the part-whole relation with topological notions such as co ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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''. Point-free geometry was first formulated in Whitehead (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. 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 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 requi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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. Whitehead's ''Process and Reality'' Whitehead's background was an unusual one for a speculative philosopher. Educated as a mathematician, he became, through his coauthorship with his student and disciple Bertrand Russell and publication in 1913 of ''Principia Mathematica'', a major logician. Later he wrote extensively on physics and its philosophy, proposing a theory of gravity in Minkowski space as a logically possible alternative to Einstein's general theory of relativity. Whitehead's ''Process and Reality''Whitehead, A.N. (1929). ''Process and Reality. An Essay in Cosmology. Gifford Lectures Delivered in the University of Edinburgh During the Session 1927–1928'', Macmillan, New York, Cambridge University Press, Cambridge UK. is perhaps his ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 between a set and its elements, mereology emphasizes the meronomic relation between entities, which—from a set-theoretic perspective—is closer to the concept of inclusion between sets. Mereology has been explored in various ways as applications of predicate logic to formal ontology, in each of which mereology is an important part. Each of these fields provides its own axiomatic definition of mereology. A common element of such axiomatizations is the assumption, shared with inclusion, that the part-whole relation orders its universe, meaning that everything is a part of itself ( reflexivity), that a part of a part of a whole is itself a part of that whole ( transitivity), and that two distinct entities cannot each be a part of the othe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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''. Point-free geometry was first formulated in Whitehead (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. 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 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 require ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Topology
In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformations, such as Stretch factor, stretching, Twist (mathematics), twisting, crumpling, and bending; that is, without closing holes, opening holes, tearing, gluing, or passing through itself. A topological space is a set (mathematics), set endowed with a structure, called a ''Topology (structure), topology'', which allows defining continuous deformation of subspaces, and, more generally, all kinds of continuity (mathematics), continuity. Euclidean spaces, and, more generally, metric spaces are examples of a topological space, as any distance or metric defines a topology. The deformations that are considered in topology are homeomorphisms and homotopy, homotopies. A property that is invariant under such deformations is a topological property. Basic exampl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Barry Smith (ontologist)
Barry Smith (born 4 June 1952) is an academic working in the fields of ontology and biomedical informatics. Smith is the author of more than 700 scientific publications, including 15 authored or edited books, and he is one of the most widely cited contemporary philosophers. Education and career From 1970 to 1973 Smith studieMathematics and Philosophyat the University of Oxford. He obtained his PhD from the University of Manchester in 1976 for a dissertation on ontology and reference in Husserl and Frege. The dissertation was supervised by Wolfe Mays. Among the cohort of graduate students supervised by Mays in Manchester were Kevin Mulligan (Geneva/Lugano), and Peter Simons (Trinity College, Dublin). Both shared with Smith an interest in analytic metaphysics and in the contributions of certain turn-of-the-century Continental philosophers and logicians to central issues of analytic philosophy. In 1979 Mulligan, Simons and Smith together founded the Seminar for Austro-German P ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Achille Varzi (philosopher)
Achille C. Varzi (born May 8, 1958) is an Italian-born philosopher who is John Dewey Professor of philosophy at Columbia University. He graduated from the University of Trento and received his PhD in philosophy from the University of Toronto. Varzi is also Bruno Kessler Honorary Professor at the University of Trento and, since 2017, Visiting Professor at the University of Italian Switzerland. Work Varzi has made notable contributions to the fields of philosophical logic (mainly vagueness, supervaluationism, paraconsistency, formal semantics) and metaphysics (mainly mereology and mereotopology, causation, events, and issues relating to identity and persistence through time). His first book, ''Holes and Other Superficialities'' (1994, with Roberto Casati), was an exploration of the realist ontology of common sense and naive physics. His more recent work is inspired by a nominalist- conventionalist stance. Varzi is currently an editor of ''The Journal of Philosophy'' and an advis ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Interior (topology)
In mathematics, specifically in general topology, topology, the interior of a subset of a topological space is the Union (set theory), union of all subsets of that are Open set, open in . A point that is in the interior of is an interior point of . The interior of is the Absolute complement, complement of the closure (topology), closure of the complement of . In this sense interior and closure are Duality_(mathematics)#Duality_in_logic_and_set_theory, dual notions. The exterior of a set is the complement of the closure of ; it consists of the points that are in neither the set nor its boundary (topology), boundary. The interior, boundary, and exterior of a subset together partition of a set, partition the whole space into three blocks (or fewer when one or more of these is empty set, empty). Definitions Interior point If is a subset of a Euclidean space, then is an interior point of if there exists an open ball centered at which is completely contained in . (This is i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Region Connection Calculus
The region connection calculus (RCC) is intended to serve for qualitative spatial representation and reasoning. RCC abstractly describes regions (in Euclidean space, or in a topological space) by their possible relations to each other. RCC8 consists of 8 basic relations that are possible between two regions: * disconnected (DC) * externally connected (EC) * equal (EQ) * partially overlapping (PO) * tangential proper part (TPP) * tangential proper part inverse (TPPi) * non-tangential proper part (NTPP) * non-tangential proper part inverse (NTPPi) From these basic relations, combinations can be built. For example, proper part (PP) is the union of TPP and NTPP. Axioms RCC is governed by two axioms. * for any region x, x connects with itself * for any region x, y, if x connects with y, y will connects with x Remark on the axioms The two axioms describe two features of the connection relation, but not the characteristic feature of the connect relation.Dong 2008 For example, we can say ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebraic Structure
In mathematics, an algebraic structure consists of a nonempty set ''A'' (called the underlying set, carrier set or domain), a collection of operations on ''A'' (typically binary operations such as addition and multiplication), and a finite set of identities, known as axioms, that these operations must satisfy. An algebraic structure may be based on other algebraic structures with operations and axioms involving several structures. For instance, a vector space involves a second structure called a field, and an operation called ''scalar multiplication'' between elements of the field (called '' scalars''), and elements of the vector space (called '' vectors''). Abstract algebra is the name that is commonly given to the study of algebraic structures. The general theory of algebraic structures has been formalized in universal algebra. Category theory is another formalization that includes also other mathematical structures and functions between structures of the same type (homomor ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Model Theory
In mathematical logic, model theory is the study of the relationship between formal theories (a collection of sentences in a formal language expressing statements about a mathematical structure), and their models (those structures in which the statements of the theory hold). The aspects investigated include the number and size of models of a theory, the relationship of different models to each other, and their interaction with the formal language itself. In particular, model theorists also investigate the sets that can be defined in a model of a theory, and the relationship of such definable sets to each other. As a separate discipline, model theory goes back to Alfred Tarski, who first used the term "Theory of Models" in publication in 1954. Since the 1970s, the subject has been shaped decisively by Saharon Shelah's stability theory. Compared to other areas of mathematical logic such as proof theory, model theory is often less concerned with formal rigour and closer in spirit ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
