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Constructive Logic
Constructive logic is a family of logics where proofs must be constructive (i.e., proving something means one must build or exhibit it, not just argue it “must exist” abstractly). No “non-constructive” proofs are allowed (like the classic proof by contradiction without a witness). The main constructive logics are the following: 1. Intuitionistic logic Founder: L. E. J. Brouwer (1908, philosophy) formalized by A. Heyting (1930) and A. N. Kolmogorov (1932) Key Idea: Truth = having a proof. One cannot assert “P or not P” unless one can prove P or prove \neg \neg P. Features: * No law of excluded middle (P \lor \neg P is not generally valid). * No double negation elimination (\neg \neg\ P \to P is not valid generally). * Implication is constructive: a proof of P \to Q is a method turning any proof of P into a proof of Q. Used in: type theory, constructive mathematics. 2. Modal logics for constructive reasoning Founder(s): * K F. Gödel (1933) showed that intui ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Arend Heyting
__NOTOC__ Arend Heyting (; 9 May 1898 – 9 July 1980) was a Dutch mathematician and logician. Biography Heyting was a student of Luitzen Egbertus Jan Brouwer at the University of Amsterdam, and did much to put intuitionistic logic on a footing where it could become part of mathematical logic. Heyting gave the first formal development of intuitionistic logic in order to codify Brouwer's way of doing mathematics. The inclusion of Brouwer's name in the Brouwer–Heyting–Kolmogorov interpretation is largely honorific, as Brouwer was opposed in principle to the formalisation of certain intuitionistic principles (and went as far as calling Heyting's work a "sterile exercise"). In 1942 he became a member of the Royal Netherlands Academy of Arts and Sciences. Heyting was born in Amsterdam, Netherlands, and died in Lugano Lugano ( , , ; ) is a city and municipality within the Lugano District in the canton of Ticino, Switzerland. It is the largest city in both Ticino and th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Agda (programming Language)
Agda is a dependently typed functional programming language originally developed by Ulf Norell at Chalmers University of Technology with implementation described in his PhD thesis. The original Agda system was developed at Chalmers by Catarina Coquand in 1999. The current version, originally named Agda 2, is a full rewrite, which should be considered a new language that shares a name and tradition. Agda is also a proof assistant based on the ''propositions-as-types'' paradigm (Curry–Howard correspondence), but unlike Rocq, has no separate ''tactics'' language, and proofs are written in a functional programming style. The language has ordinary programming constructs such as data types, pattern matching, records, let expressions and modules, and a Haskell-like syntax. The system has Emacs, Atom, and VS Code interfaces but can also be run in batch processing mode from a command-line interface. Agda is based on Zhaohui Luo's unified theory of dependent types (UTT), a type ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Constructivism (mathematics)
In the philosophy of mathematics, constructivism asserts that it is necessary to find (or "construct") a specific example of a mathematical object in order to prove that an example exists. Contrastingly, in classical mathematics, one can prove the existence of a mathematical object without "finding" that object explicitly, by assuming its non-existence and then deriving a contradiction from that assumption. Such a proof by contradiction might be called non-constructive, and a constructivist might reject it. The constructive viewpoint involves a verificational interpretation of the existential quantifier, which is at odds with its classical interpretation. There are many forms of constructivism. These include the program of intuitionism founded by Brouwer, the finitism of Hilbert and Bernays, the constructive recursive mathematics of Shanin and Markov, and Bishop's program of constructive analysis. Constructivism also includes the study of constructive set theories such a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Non-classical Logic
Non-classical logics (and sometimes alternative logics or non-Aristotelian logics) are formal systems that differ in a significant way from standard logical systems such as propositional and predicate logic. There are several ways in which this is commonly the case, including by way of extensions, deviations, and variations. The aim of these departures is to make it possible to construct different models of logical consequence and logical truth. Philosophical logic is understood to encompass and focus on non-classical logics, although the term has other meanings as well. In addition, some parts of theoretical computer science can be thought of as using non-classical reasoning, although this varies according to the subject area. For example, the basic boolean functions (e.g. AND, OR, NOT, etc) in computer science are very much classical in nature, as is clearly the case given that they can be fully described by classical truth tables. However, in contrast, some computeriz ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Logic In Computer Science
Logic in computer science covers the overlap between the field of logic and that of computer science. The topic can essentially be divided into three main areas: * Theoretical foundations and analysis * Use of computer technology to aid logicians * Use of concepts from logic for computer applications Theoretical foundations and analysis Logic plays a fundamental role in computer science. Some of the key areas of logic that are particularly significant are computability theory (formerly called recursion theory), modal logic and category theory. The theory of computation is based on concepts defined by logicians and mathematicians such as Alonzo Church and Alan Turing. Church first showed the existence of Undecidable problem, algorithmically unsolvable problems using his notion of lambda-definability. Turing gave the first compelling analysis of what can be called a mechanical procedure and Kurt Gödel asserted that he found Turing's analysis "perfect.". In addition some other ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Compositio Mathematica
''Compositio Mathematica'' is a monthly peer-reviewed mathematics journal established by L.E.J. Brouwer in 1935. It is owned by the Foundation Compositio Mathematica, and since 2004 it has been published on behalf of the Foundation by the London Mathematical Society in partnership with Cambridge University Press. According to the ''Journal Citation Reports'', the journal has a 2020 2-year impact factor of 1.456 and a 2020 5-year impact factor of 1.696. The editors-in-chief are Fabrizio Andreatta, David Holmes, Bruno Klingler, and Éric Vasserot. Early history The journal was established by L. E. J. Brouwer in response to his dismissal from ''Mathematische Annalen'' in 1928. An announcement of the new journal was made in a 1934 issue of the ''American Mathematical Monthly''. In 1940, the publication of the journal was suspended due to the German occupation of the Netherlands Despite Dutch neutrality, Nazi Germany German invasion of the Netherlands, invaded the Netherlands ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Bulletin Of The American Mathematical Society
The ''Bulletin of the American Mathematical Society'' is a quarterly mathematical journal published by the American Mathematical Society. Scope It publishes surveys on contemporary research topics, written at a level accessible to non-experts. It also publishes, by invitation only, book reviews and short ''Mathematical Perspectives'' articles. History It began as the ''Bulletin of the New York Mathematical Society'' and underwent a name change when the society became national. The Bulletin's function has changed over the years; its original function was to serve as a research journal for its members. Indexing The Bulletin is indexed in Mathematical Reviews, Science Citation Index, ISI Alerting Services, CompuMath Citation Index, and Current Contents/Physical, Chemical & Earth Sciences. See also *'' Journal of the American Mathematical Society'' *'' Memoirs of the American Mathematical Society'' *'' Notices of the American Mathematical Society'' *'' Proceedings of the Ame ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
De Gruyter
Walter de Gruyter GmbH, known as De Gruyter (), is a German scholarly publishing house specializing in academic literature. History The roots of the company go back to 1749 when Frederick the Great granted the Königliche Realschule in Berlin the royal privilege to open a bookstore and "to publish good and useful books". In 1800, the store was taken over by Georg Reimer (1776–1842), operating as the ''Reimer'sche Buchhandlung'' from 1817, while the school's press eventually became the ''Georg Reimer Verlag''. From 1816, Reimer used a representative palace at Wilhelmstraße 73 in Berlin for his family and the publishing house, whereby the wings contained his print shop and press. The building later served as the Palace of the Reich President. Born in Ruhrort in 1862, Walter de Gruyter took a position with Reimer Verlag in 1894. By 1897, at the age of 35, he had become sole proprietor of the hundred-year-old company then known for publishing the works of German romantic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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. The Grothendieck topoi find applications in algebraic geometry, and 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 formalizing the heuristic. An important example of this progra ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Realizability
In mathematical logic, realizability is a collection of methods in proof theory used to study constructive proofs and extract additional information from them. Formulas from a formal theory are "realized" by objects, known as "realizers", in a way that knowledge of the realizer gives knowledge about the truth of the formula. There are many variations of realizability; exactly which class of formulas is studied and which objects are realizers differ from one variation to another. Realizability can be seen as a formalization of the Brouwer–Heyting–Kolmogorov (BHK) interpretation of intuitionistic logic. In realizability the notion of "proof" (which is left undefined in the BHK interpretation) is replaced with a formal notion of "realizer". Most variants of realizability begin with a theorem that any statement that is provable in the formal system being studied is realizable. The realizer, however, usually gives more information about the formula than a formal proof would directly ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Constructive Set Theory
Constructivism may refer to: Art and architecture * Constructivism (art), an early 20th-century artistic movement that extols art as a practice for social purposes * Constructivist architecture, an architectural movement in the Soviet Union in the 1920s and 1930s * British Constructivists, a group of British artists who were active between 1951 and 1955. Education * Constructivism (philosophy of education), a theory about the nature of learning that focuses on how humans make meaning from their experiences * Constructivism in science education * Constructivist teaching methods, based on constructivist learning theory Mathematics * Constructivism (philosophy of mathematics), a logic for founding mathematics that accepts only objects that can be effectively constructed * Constructivist set theory * Constructivist type theory Philosophy * Constructivism (philosophy of mathematics), a philosophical view that asserts the necessity of constructing a mathematical object to p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Jean-Yves Girard
Jean-Yves Girard (; born 1947) is a French logician working in proof theory. He is a research director (emeritus) at the mathematical institute of University of Aix-Marseille, at Luminy. Biography Jean-Yves Girard is an alumnus of the École normale supérieure de Saint-Cloud. He made a name for himself in the 1970s with his proof of strong normalization in a system of second-order logic called System F. This result gave a new proof of Takeuti's conjecture, which was proven a few years earlier by William W. Tait, Motō Takahashi and Dag Prawitz. For this purpose, he introduced the notion of "reducibility candidate" ("candidat de réducibilité"). He is also credited with the discovery of Girard's paradox, linear logic, the geometry of interaction, ludics, and (satirically) the mustard watch. He obtained the CNRS Silver Medal in 1983 and is a member of the French Academy of Sciences. Bibliography * * * * Jean-Yves Girard (2011). ''The Blind Spot: Lectures on ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
