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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] |
Gunk (mereology)
In mereology, an area of philosophical logic, the term gunk applies to any whole whose parts all have further proper parts. That is, a gunky object is not made of indivisible ''atoms'' or '' simples''. Because parthood is transitive, any part of gunk is itself gunk. If point-sized objects are always simple, then a gunky object does not have any point-sized parts. By usual accounts of gunk, such as Alfred Tarski's in 1929, three-dimensional gunky objects also do not have other degenerate parts shaped like one-dimensional curves or two-dimensional surfaces. (See also ''Whitehead's point-free geometry''.) Gunk is an important test case for accounts of the composition of material objects: for instance, Ted Sider has challenged Peter van Inwagen's account of composition because it is inconsistent with the possibility of gunk. Sider's argument also applies to a simpler view than van Inwagen's: mereological ''nihilism'', the view that only material simples exist. If nihilism is necess ... [...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] |
Meronomy
A meronomy or partonomy is a type of hierarchy that deals with part–whole relationships, in contrast to a taxonomy whose categorisation is based on discrete sets. Accordingly, the unit of meronomical classification is meron, while the unit of taxonomical classification is taxon. These conceptual structures are used in linguistics and computer science, with applications in biology. The part–whole relationship is sometimes referred to as ''HAS-A'', and corresponds to object composition in object-oriented programming. The study of meronomy is known as ''mereology'', and in linguistics a ''meronym'' is the name given to a constituent part of, a substance of, or a member of something. "X" is a meronym of "Y" if an X is a part of a Y. Example *Cars have parts: engine, headlight, wheel **Engines have parts: crankcase, carburetor **Headlights have parts: headlight bulb, reflector In knowledge representation In formal terms, in the context of knowledge representation and ontologi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematical Logic
Mathematical logic is the study of logic, formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory. Research in mathematical logic commonly addresses the mathematical properties of formal systems of logic such as their expressive or deductive power. However, it can also include uses of logic to characterize correct mathematical reasoning or to establish foundations of mathematics. Since its inception, mathematical logic has both contributed to and been motivated by the study of foundations of mathematics. This study began in the late 19th century with the development of axiomatic frameworks for geometry, arithmetic, and Mathematical analysis, analysis. In the early 20th century it was shaped by David Hilbert's Hilbert's program, program to prove the consistency of foundational theories. Results of Kurt Gödel, Gerhard Gentzen, and others provided partial resolution to the program, and clarified the issues involved in pr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Formal Ontology
In philosophy, the term formal ontology is used to refer to an ontology defined by axioms in a formal language with the goal to provide an unbiased (domain- and application-independent) view on reality, which can help the modeler of domain- or application-specific ontologies (information science) to avoid possibly erroneous ontological assumptions encountered in modeling large-scale ontologies. By maintaining an independent view on reality a formal ( upper level) ontology gains the following properties: *indefinite expandability: *:the ontology remains consistent with increasing content. *content and context independence: *:any kind of 'concept' can find its place. *accommodate different levels of granularity. Historical background Theories on how to conceptualize reality date back as far as Plato and Aristotle. The term 'formal ontology' itself was coined by Edmund Husserl in the second edition of his '' Logical Investigations'' (1900–01), where it refers to an ontological co ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Set (mathematics)
A set is the mathematical model for a collection of different things; a set contains '' elements'' or ''members'', which can be mathematical objects of any kind: numbers, symbols, points in space, lines, other geometrical shapes, variables, or even other sets. The set with no element is the empty set; a set with a single element is a singleton. A set may have a finite number of elements or be an infinite set. Two sets are equal if they have precisely the same elements. Sets are ubiquitous in modern mathematics. Indeed, set theory, more specifically Zermelo–Fraenkel set theory, has been the standard way to provide rigorous foundations for all branches of mathematics since the first half of the 20th century. History The concept of a set emerged in mathematics at the end of the 19th century. The German word for set, ''Menge'', was coined by Bernard Bolzano in his work ''Paradoxes of the Infinite''. Georg Cantor, one of the founders of set theory, gave the following defin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
General Systems Theory
Systems theory is the interdisciplinary study of systems, i.e. cohesive groups of interrelated, interdependent components that can be natural or human-made. Every system has causal boundaries, is influenced by its context, defined by its structure, function and role, and expressed through its relations with other systems. A system is "more than the sum of its parts" by expressing synergy or emergent behavior. Changing one component of a system may affect other components or the whole system. It may be possible to predict these changes in patterns of behavior. For systems that learn and adapt, the growth and the degree of adaptation depend upon how well the system is engaged with its environment and other contexts influencing its organization. Some systems support other systems, maintaining the other system to prevent failure. The goals of systems theory are to model a system's dynamics, constraints, conditions, and relations; and to elucidate principles (such as purpose, measure ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Joseph Goguen
__NOTOC__ Joseph Amadee Goguen ( ; June 28, 1941 – July 3, 2006) was an American computer scientist. He was professor of Computer Science at the University of California and University of Oxford, and held research positions at IBM and SRI International. In the 1960s, along with Lotfi Zadeh, Goguen was one of the earliest researchers in fuzzy logic and made profound contributions to fuzzy set theory. In the 1970s Goguen's work was one of the earliest approaches to the algebraic characterisation of abstract data types and he originated and helped develop the OBJ family of programming languages. He was author of ''A Categorical Manifesto'' and founderBurstall R., "My friend Joseph Goguen", in ''Goguen Festschrift'', K. Futatsugi et al. (Eds.), Lecture Notes in Computer Science 4060, Springer, pp. 25–30 (2006). and Editor-in-Chief of the ''Journal of Consciousness Studies''. His development of institution theory impacted the field of universal logic. Standard implication in p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Steve Vickers (computer Scientist)
Steve Vickers (born c. 1953) is a British mathematician and computer scientist. In the early 1980s, he wrote ROM firmware and manuals for three home computers, the ZX81, ZX Spectrum, and Jupiter Ace. The latter was produced by Jupiter Cantab, a short-lived company Vickers formed together with Richard Altwasser, after the two had left Sinclair Research. Since the late 1980s, Vickers has been an academic in the field of geometric logic, writing over 30 papers in scholarly journals on mathematical aspects of computer science. His book ''Topology via Logic'' has been influential over a range of fields (extending even to theoretical physics, where Christopher Isham of Imperial College London has cited Vickers as an early influence on his work on topoi and quantum gravity). In October 2018, he retired as senior lecturer at the University of Birmingham. As announced on his university homepage, he continues to supervise PhD students at the university and focus on his research. Educatio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Category Theory
Category theory is a general theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Nowadays, category theory is used in almost all areas of mathematics, and in some areas of computer science. In particular, many constructions of new mathematical objects from previous ones, that appear similarly in several contexts are conveniently expressed and unified in terms of categories. Examples include quotient spaces, direct products, completion, and duality. A category is formed by two sorts of objects: the objects of the category, and the morphisms, which relate two objects called the ''source'' and the ''target'' of the morphism. One often says that a morphism is an ''arrow'' that ''maps'' its source to its target. Morphisms can be ''composed'' if the target of the first morphism equals the source of the second one, and morphism compos ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Topos
In mathematics, a topos (, ; plural topoi or , or toposes) is a category that behaves like the category of sheaves of sets on a topological space (or more generally: on a site). Topoi behave much like the category of sets and possess a notion of localization; they are a direct generalization of point-set topology. The Grothendieck topoi find applications in algebraic geometry; the more general elementary topoi are used in logic. The mathematical field that studies topoi is called topos theory. Grothendieck topos (topos in geometry) Since the introduction of sheaves into mathematics in the 1940s, a major theme has been to study a space by studying sheaves on a space. This idea was expounded by Alexander Grothendieck by introducing the notion of a "topos". The main utility of this notion is in the abundance of situations in mathematics where topological heuristics are very effective, but an honest topological space is lacking; it is sometimes possible to find a topos formaliz ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sheaf Theory
In mathematics, a sheaf is a tool for systematically tracking data (such as sets, abelian groups, rings) attached to the open sets of a topological space and defined locally with regard to them. For example, for each open set, the data could be the ring of continuous functions defined on that open set. Such data is well behaved in that it can be restricted to smaller open sets, and also the data assigned to an open set is equivalent to all collections of compatible data assigned to collections of smaller open sets covering the original open set (intuitively, every piece of data is the sum of its parts). The field of mathematics that studies sheaves is called sheaf theory. Sheaves are understood conceptually as general and abstract objects. Their correct definition is rather technical. They are specifically defined as sheaves of sets or as sheaves of rings, for example, depending on the type of data assigned to the open sets. There are also maps (or morphisms) from one ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
