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Paul Lorenzen
Paul Lorenzen (March 24, 1915 – October 1, 1994) was a German philosopher and mathematician, founder of the Erlangen School (with Wilhelm Kamlah) and inventor of game semantics (with Kuno Lorenz). Biography Lorenzen studied at the University of Göttingen until he earned his PhD there in 1938 under Helmut Hasse with a thesis titled ''Zur Abstrakten Begründung der multiplikativen Idealtheorie''. In 1939, he became an assistant to Wolfgang Krull at the University of Bonn. His main work was on the foundations of mathematics—proof theory. He created and modified constructive mathematics. Lorenzen taught at Stanford, the University of Texas, and Boston University in the USA. He was John Locke Lecturer in 1967/1968. Theory Lorenzen came in 1962 to University of Erlangen (South Germany) and founded the Erlangen School of epistemological constructivism there. He wrote with Wilhelm Kamlah the famous book ''Logical Propaedeutic'' ("Logische Propädeutik") and worked on game seman ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kiel
Kiel () is the capital and most populous city in the northern Germany, German state of Schleswig-Holstein, with a population of 246,243 (2021). Kiel lies approximately north of Hamburg. Due to its geographic location in the southeast of the Jutland peninsula on the southwestern shore of the Baltic Sea, Kiel has become one of Germany's major maritime centres, known for a variety of international sailing events, including the annual Kiel Week, which is the biggest sailing event in the world. Kiel is also known for the Kiel mutiny, Kiel Mutiny, when sailors refused to board their vessels in protest against Germany's further participation in World War I, resulting in the abdication of the Wilhelm II, German Emperor, Kaiser and the formation of the Weimar Republic. The Olympic sailing competitions of the 1936 Summer Olympics, 1936 and the 1972 Summer Olympics#Venues, 1972 Summer Olympics were held in the Bay of Kiel. Kiel has also been one of the traditional homes of the German Nav ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stanford
Stanford University, officially Leland Stanford Junior University, is a private research university in Stanford, California. The campus occupies , among the largest in the United States, and enrolls over 17,000 students. Stanford is considered among the most prestigious universities in the world. Stanford was founded in 1885 by Leland and Jane Stanford in memory of their only child, Leland Stanford Jr., who had died of typhoid fever at age 15 the previous year. Leland Stanford was a U.S. senator and former governor of California who made his fortune as a railroad tycoon. The school admitted its first students on October 1, 1891, as a coeducational and non-denominational institution. Stanford University struggled financially after the death of Leland Stanford in 1893 and again after much of the campus was damaged by the 1906 San Francisco earthquake. Following World War II, provost of Stanford Frederick Terman inspired and supported faculty and graduates' entrepreneurialism ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Classical Analysis
Analysis is the branch of mathematics dealing with continuous functions, limits, and related theories, such as differentiation, integration, measure, infinite sequences, series, and analytic functions. These theories are usually studied in the context of real and complex numbers and functions. Analysis evolved from calculus, which involves the elementary concepts and techniques of analysis. Analysis may be distinguished from geometry; however, it can be applied to any space of mathematical objects that has a definition of nearness (a topological space) or specific distances between objects (a metric space). History Ancient Mathematical analysis formally developed in the 17th century during the Scientific Revolution, but many of its ideas can be traced back to earlier mathematicians. Early results in analysis were implicitly present in the early days of ancient Greek mathematics. For instance, an infinite geometric sum is implicit in Zeno's paradox of the dichotomy. (St ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Impredicativity
In mathematics, logic and philosophy of mathematics, something that is impredicative is a self-referencing definition. Roughly speaking, a definition is impredicative if it invokes (mentions or quantifies over) the set being defined, or (more commonly) another set that contains the thing being defined. There is no generally accepted precise definition of what it means to be predicative or impredicative. Authors have given different but related definitions. The opposite of impredicativity is predicativity, which essentially entails building stratified (or ramified) theories where quantification over lower levels results in variables of some new type, distinguished from the lower types that the variable ranges over. A prototypical example is intuitionistic type theory, which retains ramification so as to discard impredicativity. Russell's paradox is a famous example of an impredicative construction—namely the set of all sets that do not contain themselves. The paradox is that su ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hermann Weyl
Hermann Klaus Hugo Weyl, (; 9 November 1885 – 8 December 1955) was a German mathematician, theoretical physicist and philosopher. Although much of his working life was spent in Zürich, Switzerland, and then Princeton, New Jersey, he is associated with the University of Göttingen tradition of mathematics, represented by Carl Friedrich Gauss, David Hilbert and Hermann Minkowski. His research has had major significance for theoretical physics as well as purely mathematical disciplines such as number theory. He was one of the most influential mathematicians of the twentieth century, and an important member of the Institute for Advanced Study during its early years. Weyl contributed to an exceptionally wide range of mathematical fields, including works on space, time, matter, philosophy, logic, symmetry and the history of mathematics. He was one of the first to conceive of combining general relativity with the laws of electromagnetism. Freeman Dyson wrote that Weyl alone bore ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Constructive Analysis
In mathematics, constructive analysis is mathematical analysis done according to some principles of constructive mathematics. This contrasts with ''classical analysis'', which (in this context) simply means analysis done according to the (more common) principles of classical mathematics. Generally speaking, constructive analysis can reproduce theorems of classical analysis, but only in application to separable spaces; also, some theorems may need to be approached by approximations. Furthermore, many classical theorems can be stated in ways that are logically equivalent according to classical logic, but not all of these forms will be valid in constructive analysis, which uses intuitionistic logic. Examples The intermediate value theorem For a simple example, consider the intermediate value theorem (IVT). In classical analysis, IVT implies that, given any continuous function ''f'' from a closed interval 'a'',''b''to the real line ''R'', if ''f''(''a'') is negative while ''f''(''b ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Constructive Type Theory
Intuitionistic type theory (also known as constructive type theory, or Martin-Löf type theory) is a type theory and an alternative foundation of mathematics. Intuitionistic type theory was created by Per Martin-Löf, a Swedish mathematician and philosopher, who first published it in 1972. There are multiple versions of the type theory: Martin-Löf proposed both intensional and extensional variants of the theory and early impredicative versions, shown to be inconsistent by Girard's paradox, gave way to predicative versions. However, all versions keep the core design of constructive logic using dependent types. Design Martin-Löf designed the type theory on the principles of mathematical constructivism. Constructivism requires any existence proof to contain a "witness". So, any proof of "there exists a prime greater than 1000" must identify a specific number that is both prime and greater than 1000. Intuitionistic type theory accomplished this design goal by internalizing the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Constructive Logic
Intuitionistic logic, sometimes more generally called constructive logic, refers to systems of symbolic logic that differ from the systems used for classical logic by more closely mirroring the notion of constructive proof. In particular, systems of intuitionistic logic do not assume the law of the excluded middle and double negation elimination, which are fundamental inference rules in classical logic. Formalized intuitionistic logic was originally developed by Arend Heyting to provide a formal basis for L. E. J. Brouwer's programme of intuitionism. From a proof-theoretic perspective, Heyting’s calculus is a restriction of classical logic in which the law of excluded middle and double negation elimination have been removed. Excluded middle and double negation elimination can still be proved for some propositions on a case by case basis, however, but do not hold universally as they do with classical logic. The standard explanation of intuitionistic logic is the BHK interpretation ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Peter Janich
Peter Janich (4 January 1942 – 4 September 2016) was a professor of philosophy at the University of Marburg. He was born in Munich. Janich studied physics, philosophy and psychology at the Universities of Erlangen and Hamburg. He attained a doctorate in philosophy in 1969 and during 1969/70 was a guest lecturer at the University of Texas at Austin. From 1973 to 1980 he was a professor for philosophy of science of the accurate sciences at the University of Konstanz. Since 1980 he has held the chair for systematic philosophy with an emphasis on theoretical philosophy at the Philipps University of Marburg. Janich is the joint founder and representative of the approach of methodical culturalism, a development of the methodical constructivism of the Erlangen School of constructivism. He developed the concept of protophysics with Paul Lorenzen and Rüdiger Inhetveen. Janich died of cancer at the age of 74 on 4 September 2016 in Rauschenberg Milton Ernest "Robert" Rauschenberg (Oct ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Epistemological Constructivism
Constructivism is a view in the philosophy of science that maintains that scientific knowledge is constructed by the scientific community, which seeks to measure and construct models of the natural world. According to the constructivist, natural science, therefore, consists of mental constructs that aim to explain sensory experience and measurements. According to constructivists, the world is independent of human minds, but knowledge of the world is always a human and social construction. Constructivism opposes the philosophy of objectivism, embracing the belief that a human can come to know the truth about the natural world not mediated by scientific approximations with different degrees of validity and accuracy. According to constructivists, there is no single valid methodology in science but rather a diversity of useful methods. Etymology The term originates from psychology, education, and social constructivism. The expression "constructivist epistemology" was first used b ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Erlangen School Of Constructivism
Erlangen (; East Franconian: ''Erlang'', Bavarian: ''Erlanga'') is a Middle Franconian city in Bavaria, Germany. It is the seat of the administrative district Erlangen-Höchstadt (former administrative district Erlangen), and with 116,062 inhabitants (as of 30 March 2022), it is the smallest of the eight major cities (''Großstadt'') in Bavaria. The number of inhabitants exceeded the threshold of 100,000 in 1974, making Erlangen a major city according to the statistical definition officially used in Germany. Together with Nuremberg, Fürth, and Schwabach, Erlangen forms one of the three metropolises in Bavaria. With the surrounding area, these cities form the European Metropolitan Region of Nuremberg, one of 11 metropolitan areas in Germany. The cities of Nuremberg, Fürth, and Erlangen also form a triangle on a map, which represents the heartland of the Nuremberg conurbation. An element of the city that goes back a long way in history, but is still noticeable, is the settlemen ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
University Of Erlangen
A university () is an institution of higher (or tertiary) education and research which awards academic degrees in several academic disciplines. Universities typically offer both undergraduate and postgraduate programs. In the United States, the designation is reserved for colleges that have a graduate school. The word ''university'' is derived from the Latin ''universitas magistrorum et scholarium'', which roughly means "community of teachers and scholars". The first universities were created in Europe by Catholic Church monks. The University of Bologna (''Università di Bologna''), founded in 1088, is the first university in the sense of: *Being a high degree-awarding institute. *Having independence from the ecclesiastic schools, although conducted by both clergy and non-clergy. *Using the word ''universitas'' (which was coined at its foundation). *Issuing secular and non-secular degrees: grammar, rhetoric, logic, theology, canon law, notarial law.Hunt Janin: "The university ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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