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Glossari

 

Presentació

Espai Hilbert

Biblioteca Digital

Glossari

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Aquest glossari és una mostra dels termes més representatius de David Hilbert. Les definicions han estat extretes dels recursos següents:

 

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Algebraic invariants

Invariant theory is a branch of abstract algebra dealing with actions of groups on algebraic varieties, such as vector spaces, from the point of view of their effect on functions. Classically, the theory dealt with the question of explicit description of polynomial functions that do not change, or are invariant, under the transformations from a given linear group. For example, if we consider the action of the special linear group SLn on the space of n by n matrices by left multiplication, then the determinant is an invariant of this action because the determinant of A X equals the determinant of X, when A is in SLn.

Citació: " Invariant theory" A: Wikipedia. Wikimedia Foundation, 2011. [en línia]. [Consulta: 19 setembre 2017]. Disponible a: <https://en.wikipedia.org/wiki/Invariant_theory>

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Axiomatisation

In mathematics, an axiomatic system is any set of axioms from which some or all axioms can be used in conjunction to logically derive theorems. A theory consists of an axiomatic system and all its derived theorems. An axiomatic system that is completely described is a special kind of formal system. A formal theory typically means an axiomatic system, for example formulated within model theory. A formal proof is a complete rendition of a mathematical proof within a formal system.

Citació: "Axiomatic system" A: Wikipedia. Wikimedia Foundation, 2011. [en línia]. [Consulta: 19 setembre 2017]. Disponible a: <https://en.wikipedia.org/wiki/Biostatistics>

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Consistency of axioms

In classical deductive logic, a consistent theory is one that does not contain a contradiction.The lack of contradiction can be defined in either semantic or syntactic terms. The semantic definition states that a theory is consistent if and only if it has a model, i.e., there exists an interpretation under which all formulas in the theory are true. This is the sense used in traditional Aristotelian logic, although in contemporary mathematical logic the term satisfiable is used instead.

Citació: "Consistency" A: Wikipedia. Wikimedia Foundation, 2011. [en línia]. [Consulta: 19 setembre 2017]. Disponible a: <https://en.wikipedia.org/wiki/Consistency>

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Entscheidungsproblem

In mathematics and computer science, the Entscheidungsproblem (pronounced [ɛntˈʃaɪ̯dʊŋspʁoˌbleːm], German for "decision problem") is a challenge posed by David Hilbert in 1928.The problem asks for an algorithm that takes as input a statement of a first-order logic (possibly with a finite number of axioms beyond the usual axioms of first-order logic) and answers "Yes" or "No" according to whether the statement is universally valid, i.e., valid in every structure satisfying the axioms. By the completeness theorem of first-order logic, a statement is universally valid if and only if it can be deduced from the axioms, so the Entscheidungsproblem can also be viewed as asking for an algorithm to decide whether a given statement is provable from the axioms using the rules of logic.

Citació: "Entscheidungsproblem" A: Wikipedia. Wikimedia Foundation, 2011. [en línia]. [Consulta: 19 setembre 2017]. Disponible a:
<https://en.wikipedia.org/wiki/Entscheidungsproblem>

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Formalism

In foundations of mathematics, philosophy of mathematics, and philosophy of logic, formalism is a theory that holds that statements of mathematics and logic can be considered to be statements about the consequences of certain string manipulation rules.

Citació: "Formalism (philosophy of mathematics)" A: Wikipedia. Wikimedia Foundation, 2011. [en línia]. [Consulta: 19 setembre 2017]. Disponible a: <https://en.wikipedia.org/wiki/Formalism_(philosophy_of_mathematics)>

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Foundations of geometry

Foundations of geometry is the study of geometries as axiomatic systems. There are several sets of axioms which give rise to Euclidean geometry or to non-Euclidean geometries. These are fundamental to the study and of historical importance, but there are a great many modern geometries that are not Euclidean which can be studied from this viewpoint. The term axiomatic geometry can be applied to any geometry that is developed from an axiom system, but is often used to mean Euclidean geometry studied from this point of view. The completeness and independence of general axiomatic systems are important mathematical considerations, but there are also issues to do with the teaching of geometry which come into play.

Citació: "Foundations of geometry" A: Wikipedia. Wikimedia Foundation, 2011. [en línia]. [Consulta: 19 setembre 2017]. Disponible a: <https://en.wikipedia.org/wiki/Foundations_of_geometry>

 


Foundations of mathematics

Foundations of mathematics is the study of the philosophical and logical and/or algorithmic basis of mathematics, or, in a broader sense, the mathematical investigation of what underlies the philosophical theories concerning the nature of mathematics. In this latter sense, the distinction between foundations of mathematics and philosophy of mathematics turns out to be quite vague. Foundations of mathematics can be conceived as the study of the basic mathematical concepts (number, geometrical figure, set, function, etc.) and how they form hierarchies of more complex structures and concepts, especially the fundamentally important structures that form the language of mathematics (formulas, theories and their models giving a meaning to formulas, definitions, proofs, algorithms, etc.) also called metamathematical concepts, with an eye to the philosophical aspects and the unity of mathematics. The search for foundations of mathematics is a central question of the philosophy of mathematics; the abstract nature of mathematical objects presents special philosophical challenges.

Citació: "Foundations of mathematics" A: Wikipedia. Wikimedia Foundation, 2011. [en línia]. [Consulta: 19 setembre 2017]. Disponible a: <https://en.wikipedia.org/wiki/Foundations_of_mathematics>

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Hilbert's problems

Hilbert's problems are twenty-three problems in mathematics published by German mathematician David Hilbert in 1900. The problems were all unsolved at the time, and several of them were very influential for 20th-century mathematics. Hilbert presented ten of the problems (1, 2, 6, 7, 8, 13, 16, 19, 21 and 22) at the Paris conference of the International Congress of Mathematicians, speaking on August 8 in the Sorbonne. The complete list of 23 problems was published later, most notably in English translation in 1902 by Mary Frances Winston Newson in the Bulletin of the American Mathematical Society.

Citació: "Hilbert's problems" A: Wikipedia. Wikimedia Foundation, 2011. [en línia]. [Consulta: 19 setembre 2017]. Disponible a: <https://en.wikipedia.org/wiki/Hilbert%27s_problems>

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Integral equations

In mathematics, an integral equation is an equation in which an unknown function appears under an integral sign.

There is a close connection between differential and integral equations, and some problems may be formulated either way. See, for example, Green's function, Fredholm theory, and Maxwell's equations.

Citació: "Integral equations " A: Wikipedia. Wikimedia Foundation, 2011. [en línia]. [Consulta: 19 setembre 2017]. Disponible a: <https://en.wikipedia.org/wiki/Integral_equation>

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Mathematical physics

Mathematical physics refers to development of mathematical methods for application to problems in physics. The Journal of Mathematical Physics defines the field as "the application of mathematics to problems in physics and the development of mathematical methods suitable for such applications and for the formulation of physical theories". It is a branch of applied mathematics, but deals with physical problems.

Citació: "Mathematical physics" A: Wikipedia. Wikimedia Foundation, 2011. [en línia]. [Consulta: 19 setembre 2017]. Disponible a: <https://en.wikipedia.org/wiki/Mathematical_physics>

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Nullstellensatz

Hilbert's Nullstellensatz (German for "theorem of zeros," or more literally, "zero-locus-theorem"—see Satz) is a theorem that establishes a fundamental relationship between geometry and algebra. This relationship is the basis of algebraic geometry, a branch of mathematics. It relates algebraic sets to ideals in polynomial rings over algebraically closed fields. This relationship was discovered by David Hilbert who proved the Nullstellensatz and several other important related theorems named after him (like Hilbert's basis theorem).

Citació: "Nullstellensatz" A: Wikipedia. Wikimedia Foundation, 2011. [en línia]. [Consulta: 19 setembre 2017]. Disponible a: <https://en.wikipedia.org/wiki/Hilbert%27s_Nullstellensatz>

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Relativity field equations

The Einstein–Hilbert action (also referred to as Hilbert action) in general relativity is the action that yields the Einstein field equations through the principle of least action

Citació: "Einstein–Hilbert action" A: Wikipedia. Wikimedia Foundation, 2011. [en línia]. [Consulta: 19 setembre 2017]. Disponible a: <https://en.wikipedia.org/wiki/Einstein%E2%80%93Hilbert_action>

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Variational calculus

Calculus of variations is a field of mathematical analysis that deals with maximizing or minimizing functionals, which are mappings from a set of functions to the real numbers. Functionals are often expressed as definite integrals involving functions and their derivatives. The interest is in extremal functions that make the functional attain a maximum or minimum value – or stationary functions – those where the rate of change of the functional is zero.

Citació: "Calculus of variations" A: Wikipedia. Wikimedia Foundation, 2011. [en línia]. [Consulta: 19 setembre 2017]. Disponible a: <https://en.wikipedia.org/wiki/Mathematical_physics>

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