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Mathematical Theory
A mathematical theory is a mathematical model of a branch of mathematics that is based on a set of axioms. It can also simultaneously be a body of knowledge (e.g., based on known axioms and definitions), and so in this sense can refer to an area of mathematical research within the established framework. Explanatory depth is one of the most significant theoretical virtues in mathematics. For example, set theory has the ability to systematize and explain number theory and geometry/analysis. Despite the widely logical necessity (and self-evidence) of arithmetic truths such as 1<3, 2+2=4, 6-1=5, and so on, a theory that just postulates an infinite blizzard of such truths would be inadequate. Rather an adequate theory is one in which such truths are derived from explanatorily prior axioms, such as the Peano Axioms or set theoretic axioms, which lie at the foundation of ZFC axiomatic set theory. The singular accomplishment of axiomatic set theory is its ability to give a foundation fo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematical Model
A mathematical model is a description of a system using mathematical concepts and language. The process of developing a mathematical model is termed mathematical modeling. Mathematical models are used in the natural sciences (such as physics, biology, earth science, chemistry) and engineering disciplines (such as computer science, electrical engineering), as well as in non-physical systems such as the social sciences (such as economics, psychology, sociology, political science). The use of mathematical models to solve problems in business or military operations is a large part of the field of operations research. Mathematical models are also used in music, linguistics, and philosophy (for example, intensively in analytic philosophy). A model may help to explain a system and to study the effects of different components, and to make predictions about behavior. Elements of a mathematical model Mathematical models can take many forms, including dynamical systems, statisti ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Axioms
An axiom, postulate, or assumption is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Ancient Greek word (), meaning 'that which is thought worthy or fit' or 'that which commends itself as evident'. The term has subtle differences in definition when used in the context of different fields of study. As defined in classic philosophy, an axiom is a statement that is so evident or well-established, that it is accepted without controversy or question. As used in modern logic, an axiom is a premise or starting point for reasoning. As used in mathematics, the term ''axiom'' is used in two related but distinguishable senses: "logical axioms" and "non-logical axioms". Logical axioms are usually statements that are taken to be true within the system of logic they define and are often shown in symbolic form (e.g., (''A'' and ''B'') implies ''A''), while non-logical axioms (e.g., ) are actually s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Body Of Knowledge
A body of knowledge (BOK or BoK) is the complete set of concepts, terms and activities that make up a professional domain, as defined by the relevant learned society or professional association.Oliver, G.R. (2012). ''Foundations of the Assumed Business Operations and Strategy Body of Knowledge (BOSBOK): An Outline of Shareable Knowledge'', p. 3. It is a type of knowledge representation by any knowledge organization. Several definitions of BOK have been developed, for example: * "Structured knowledge that is used by members of a discipline to guide their practice or work." "The prescribed aggregation of knowledge in a particular area an individual is expected to have mastered to be considered or certified as a practitioner." (BOK-def). Waite's pragmatic view is also worth noting (Ören 2005): "BOK is a stepping stone to unifying community" (Waite 2004). * The systematic collection of activities and outcomes in terms of their values, constructs, models, principles and instantiat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
List Of Mathematical Theories
This is a list of mathematical theories. {{columns-list, colwidth=20em, *Algebraic K-theory * Almgren–Pitts min-max theory * Approximation theory *Asymptotic theory *Automata theory *Bifurcation theory *Braid theory *Brill–Noether theory * Catastrophe theory * Category theory * Chaos theory *Character theory *Choquet theory * Class field theory *Coding theory *Cohomology theory * Complexity theory * Computation theory *Control theory * Deformation theory * Dimension theory * Distribution theory *Elimination theory *Extremal graph theory * Field theory *Galois theory * Game theory *Graph theory *Grothendieck's Galois theory *Group theory * Hodge theory * Homology theory * Homotopy theory * Information theory *Invariant theory *K-theory * Knot theory * L-theory *Local class field theory *M-theory *Matrix theory * Measure theory * Model theory *Morse theory *Module theory * Network theory *Nevanlinna theory *Number theory *Obstruction theory *Operator theory *Optimization theory ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Theorem
In mathematics, a theorem is a statement that has been proved, or can be proved. The ''proof'' of a theorem is a logical argument that uses the inference rules of a deductive system to establish that the theorem is a logical consequence of the axioms and previously proved theorems. In the mainstream of mathematics, the axioms and the inference rules are commonly left implicit, and, in this case, they are almost always those of Zermelo–Fraenkel set theory with the axiom of choice, or of a less powerful theory, such as Peano arithmetic. A notable exception is Wiles's proof of Fermat's Last Theorem, which involves the Grothendieck universes whose existence requires the addition of a new axiom to the set theory. Generally, an assertion that is explicitly called a theorem is a proved result that is not an immediate consequence of other known theorems. Moreover, many authors qualify as ''theorems'' only the most important results, and use the terms ''lemma'', ''proposition'' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Theory (mathematical Logic)
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, after which an element \phi\in T of a deductively closed theory T is then called a theorem of the theory. In many deductive systems there is usually a subset \Sigma \subseteq T that is called "the set of axioms" of the theory T, in which case the deductive system is also called an "axiomatic system". By definition, every axiom is automatically a theorem. A first-order theory is a set of first-order sentences (theorems) recursively obtained by the inference rules of the system applied to the set of axioms. General theories (as expressed in formal language) When defining theories for foundational purposes, additional care must be taken, as normal set-theoretic language may not be appropriate. The construction of a theory begins by specifying a definite non-empty ''conceptual class'' \mathcal, the eleme ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Unifying Theories In Mathematics
There have been several attempts in history to reach a unified theory of mathematics. Some of the most respected mathematicians in the academia have expressed views that the whole subject should be fitted into one theory. Historical perspective The process of unification might be seen as helping to define what constitutes mathematics as a discipline. For example, mechanics and mathematical analysis were commonly combined into one subject during the 18th century, united by the differential equation concept; while algebra and geometry were considered largely distinct. Now we consider analysis, algebra, and geometry, but not mechanics, as parts of mathematics because they are primarily deductive formal sciences, while mechanics like physics must proceed from observation. There is no major loss of content, with analytical mechanics in the old sense now expressed in terms of symplectic topology, based on the newer theory of manifolds. Mathematical theories The term ''theory'' is u ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Università Di Genova
The University of Genoa, known also with the acronym UniGe ( it, Università di Genova), is one of the largest universities in Italy. It is located in the city of Genoa and regional Metropolitan City of Genoa, on the Italian Riviera in the Liguria region of northwestern Italy. The original university was founded in 1481. According to Microsoft Academic Search 2016 rankings, the University of Genoa has high-ranking positions among the European universities in multiple computer science fields: * in machine learning and pattern recognition the University of Genoa is the best scientific institution in Italy and is ranked 36th in Europe; * in computer vision the University of Genoa is the best scientific institution in Italy and is ranked 34th in Europe; * in computer graphics the University of Genoa is ranked 2nd institution in Italy and 35th in Europe. The University of Genoa has a strong collaboration with the Italian Institute of Technology (IIT), since its foundation in 2005. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Formal Theories
Formal, formality, informal or informality imply the complying with, or not complying with, some set of requirements ( forms, in Ancient Greek). They may refer to: Dress code and events * Formal wear, attire for formal events * Semi-formal attire, attire for semi-formal events * Informal attire, more controlled attire than casual but less than formal * Formal (university), official university dinner, ball or other event * School formal, official school dinner, ball or other event Logic and mathematics *Formal logic, or mathematical logic ** Informal logic, the complement, whose definition and scope is contentious *Formal fallacy, reasoning of invalid structure ** Informal fallacy, the complement *Informal mathematics, also called naïve mathematics *Formal cause, Aristotle's intrinsic, determining cause * Formal power series, a generalization of power series without requiring convergence, used in combinatorics *Formal calculation, a calculation which is systematic, but withou ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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