Results for 'david hilbert'

Hilbert System

Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. In logic, especially mathematical logic, a Hilbert system, sometimes called Hilbert calculus or Hilbert-Ackermann system, is a type of system of formal deduction attributed to Gottlob Frege and David Hilbert. These deductive systems are most often studied for first-order logic, but are of interest for other logics as well. Most variants of Hilbert systems take a characteristic tack in the way they balance a trade-off between logical axioms and rules of inference. Hilbert systems can be characterised by the choice of a large number of schemes of logical axioms and a small set of rules of inference. The most commonly studied Hilbert systems have either just one rule of inference -modus ponens, for propositional logics- or two - with generalisation, to handle predicate logics, as well- and several infinite axiom schemes. Hilbert systems for propositional modal logics, sometimes called Hilbert-Lewis systems, are generally axiomatised with two additional rules, the necessitation rule and the uniform substitution rule.

Calculabilité

Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. La théorie de la calculabilité (appelée aussi parfois théorie de la récursion) est une branche de la logique mathématique et de l'informatique théorique. Alors que la notion intuitive de fonction calculable est aussi vieille que les mathématiques (voir l'article Histoire des mathématiques), la formalisation de ces notions a commencé dans la décennie 1930 afin de répondre à des problèmes fondamentaux de logique mathématique, dont celui énoncé par David Hilbert et appelé Entscheidung Problem ou Problème de la décision. La calculabilité cherche d'une part à identifier la classe des fonctions qui peuvent être calculées à l'aide d'un algorithme et d'autre part à appliquer ces concepts à des questions fondamentales des mathématiques. Une bonne appréhension de ce qui est calculable et de ce qui ne l'est pas permet de voir les limites des problèmes que peuvent résoudre les ordinateurs.

Problèmes de Hilbert

Théorie des Ensembles

Brouwer-Hilbert Controversy

Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. In a foundational controversy in twentieth century mathematics, L. E. J. Brouwer, a supporter of intuitionism, opposed David Hilbert, the founder of formalism. Brouwer in effect founded the mathematical philosophy of intuitionism as a challenge to the then-prevailing formalism of David Hilbert and his collaborators Paul Bernays, Wilhelm Ackermann, John von Neumann and others. As a variety of constructive mathematics, intuitionism is essentially a philosophy of the foundations of mathematics. It is sometimes and rather simplistically characterized by saying that its adherents refuse to use the law of excluded middle in mathematical reasoning. In the later 1920s, Brouwer became involved in a public and demeaning controversy with Hilbert over editorial policy at Mathematische Annalen, at that time a leading learned journal. He became relatively isolated; the development of intuitionism at its source was taken up by his student Arend Heyting.

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