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Mathematical Logic
Mathematical logic is a subfield of mathematics exploring the applications of formal logic to mathematics. It bears close connections to metamathematics, the foundations of mathematics, and theoretical computer science.[1] The unifying themes in mathematical logic include the study of the expressive power of formal systems and the deductive power of formal proof systems. Mathematical logic is often divided into the fields of set theory, model theory, recursion theory, and proof theory. These areas share basic results on logic, particularly first-order logic, and definability. In computer science (particularly in the ACM Classification) mathematical logic encompasses additional topics not detailed in this article; see Logic
Logic
in computer science for those. Since its inception, mathematical logic has both contributed to, and has been motivated by, the study of foundations of mathematics
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Begriffsschrift
Begriffsschrift
Begriffsschrift
(German for, roughly, "concept-script") is a book on logic by Gottlob Frege, published in 1879, and the formal system set out in that book. Begriffsschrift
Begriffsschrift
is usually translated as concept writing or concept notation; the full title of the book identifies it as "a formula language, modeled on that of arithmetic, of pure thought." Frege's motivation for developing his formal approach to logic resembled Leibniz's motivation for his calculus ratiocinator (despite that, in the foreword Frege clearly denies that he achieved this aim, and also that his main aim would be constructing an ideal language like Leibniz's, which Frege declares to be a quite hard and idealistic, however not impossible, task)
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ACM Computing Classification System
The ACM Computing
Computing
Classification System (CCS) is a subject classification system for computing devised by the Association for Computing
Computing
Machinery (ACM). The system is comparable to the Mathematics Subject Classification (MSC) in scope, aims, and structure, being used by the various ACM journals to organise subjects by area.Contents1 History 2 Structure 3 See also 4 References 5 External linksHistory[edit] The system has gone through seven revisions, the first version being published in 1964, and revised versions appearing in 1982, 1983, 1987, 1991, 1998, and the now current version in 2012. Structure[edit] The ACM Computing
Computing
Classification System, version 2012, has a revolutionary change in some areas, for example, in "Software" that now is called "Software and its engineering" which has three main subjects:Software organization and properties
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George Peacock
George Peacock
George Peacock
(9 April 1791 – 8 November 1858) was an English mathematician.Contents1 Early life 2 Mathematical career 3 Clerical career 4 Private life 5 Algebraic theory 6 References 7 Sources 8 External linksEarly life[edit] Peacock was born on 9 April 1791 at Thornton Hall, Denton, near Darlington, County Durham.[1] His father, Thomas Peacock, was a priest of the Church of England, incumbent and for 50 years curate of the parish of Denton, where he also kept a school
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Augustus De Morgan
Augustus De Morgan
Augustus De Morgan
(/dɪ ˈmɔːrɡən/;[1] 27 June 1806 – 18 March 1871) was a British mathematician and logician. He formulated De Morgan's laws and introduced the term mathematical induction, making its idea rigorous.[2]Contents1 Biography1.1 Childhood 1.2 University education 1.3 London University 1.4 Family 1.5 Retirement and death2 Mathematical work2.1 Trigonometry
Trigonometry
and Double Algebra 2.2 Formal Logic 2.3 Budget of Paradoxes 2.4 Relations3 Spiritualism 4 Legacy 5 Selected writings 6 See also 7 Notes and references 8 Further reading 9 External linksBiography[edit] Childhood[edit] Augustus De Morgan
Augustus De Morgan
was born in Madurai, India in 1806.[a] His father was Lieut.-Colonel John De Morgan (1772–1816), who held various appointments in the service of the East India Company
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Johann Heinrich Lambert
Johann Heinrich Lambert
Johann Heinrich Lambert
(German: [ˈlambɛʁt], Jean-Henri Lambert in French; 26 August 1728 – 25 September 1777) was a Swiss polymath who made important contributions to the subjects of mathematics, physics (particularly optics), philosophy, astronomy and map projections. Edward Tufte
Edward Tufte
calls him and William Playfair
William Playfair
"The two great inventors of modern graphical designs" (Visual Display of Quantitative Information, p.32).Contents1 Biography 2 Work2.1 Mathematics 2.2 Map projection 2.3 Physics 2.4 Philosophy 2.5 Astronomy 2.6 Logic3 See also 4 Notes 5 References 6 External linksBiography[edit] Lambert was born in 1728 into a Huguenot
Huguenot
family in the city of Mulhouse
Mulhouse
(now in Alsace, France), at that time an exclave of Switzerland
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Logic In Islamic Philosophy
Early Islamic law
Islamic law
placed importance on formulating standards of argument, which gave rise to a "novel approach to logic" (منطق manṭiq "speech, eloquence") in Kalam
Kalam
(Islamic scholasticism)[citation needed] However, with the rise of the
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Logic In India
The development of Indian logic dates back to the anviksiki of Medhatithi Gautama (c. 6th century BCE) the Sanskrit grammar rules of Pāṇini (c. 5th century BCE); the Vaisheshika school's analysis of atomism (c. 6th century BCE to 2nd century BCE); the analysis of inference by Gotama (c. 6th century BC to 2nd century CE), founder of the Nyaya school of Hindu philosophy; and the tetralemma of Nagarjuna (c. 2nd century CE). Indian logic stands as one of the three original traditions of logic, alongside the Greek and the Chinese logic
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Logic In China
Formal logic in China has a special place in the history of logic due to its repression and abandonment—in contrast to the strong ancient adoption and continued development of the study of logic in Europe, India, and the Islamic world.Contents1 Mohist logic 2 The repression of the study of logic 3 Buddhist logic 4 References 5 Bibliography 6 External linksMohist logic[edit]History of science and technology in ChinaInventionsFour Great InventionsDiscoveriesBy subjectMathematics Astronomy Calendar Units of measurement Cartography Geography Printing Ceramics Metallurgy Coinage Chinese Alchemy Traditional medicineherbologyAgricultureSericultureSilk industry ArchitectureClassic gardens BridgesTransportNavigationMilitaryNavyBy eraHan Tang Song Yuan People's RepublicAgriculturev t eIn China, a contemporary of Confucius, Mozi, "Maste
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Rhetoric
Rhetoric
Rhetoric
is the art of discourse, wherein a writer or speaker strives to inform, persuade or motivate particular audiences in specific situations. It can also be in a visual form; as a subject of formal study and a productive civic practice, rhetoric has played a central role in the European tradition.[1] Its best known definition comes from Aristotle, who considers it a counterpart of both logic and politics, and calls it "the faculty of observing in any given case the available means of persuasion."[2] Rhetoric
Rhetoric
typically provides heuristics for understanding, discovering, and developing arguments for particular situations, such as Aristotle's three persuasive audience appeals, logos, pathos, and ethos. The five canons of rhetoric, which trace the traditional tasks in designing a persuasive speech, were first codified in classical Rome: invention, arrangement, style, memory, and delivery
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Constructive Mathematics
In the philosophy of mathematics, constructivism asserts that it is necessary to find (or "construct") a mathematical object to prove that it exists. In standard 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. This proof by contradiction is not constructively valid. 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.[1] 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
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Analysis
Analysis
Analysis
is the process of breaking a complex topic or substance into smaller parts in order to gain a better understanding of it. The technique has been applied in the study of mathematics and logic since before Aristotle
Aristotle
(384–322 B.C.), though analysis as a formal concept is a relatively recent development.[1] The word comes from the Ancient Greek
Ancient Greek
ἀνάλυσις (analysis, "a breaking up", from ana- "up, throughout" and lysis "a loosening").[2] As a formal concept, the method has variously been ascribed to Alhazen,[3] René Descartes
René Descartes
(Discourse on the Method), and Galileo Galilei
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Gerhard Gentzen
Gerhard Karl Erich Gentzen (November 24, 1909 – August 4, 1945) was a German mathematician and logician. He made major contributions to the foundations of mathematics, proof theory, especially on natural deduction and sequent calculus. He died in 1945 after the Second World War, because he was deprived of food after being arrested in Prague.Contents1 Life and career 2 Work 3 Publications3.1 Posthumous4 See also 5 Notes 6 References 7 External linksLife and career[edit] Gentzen was a student of Paul Bernays
Paul Bernays
at the University of Göttingen. Bernays was fired as "non-Aryan" in April 1933 and therefore Hermann Weyl formally acted as his supervisor. Gentzen joined the Sturmabteilung
Sturmabteilung
in November 1933 although he was by no means compelled to do so.[1] Nevertheless he kept in contact with Bernays until the beginning of the Second World War
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Categorical Logic
Categorical logic is the branch of mathematics in which tools and concepts from category theory are applied to the study of mathematical logic. It is also notable for its connections to theoretical computer science. In broad terms, categorical logic represents both syntax and semantics by a category, and an interpretation by a functor. The categorical framework provides a rich conceptual background for logical and type-theoretic constructions. The subject has been recognisable in these terms since around 1970. Overview[edit] There are three important themes in the categorical approach to logic:Categorical semantics Categorical logic introduces the notion of structure valued in a category C with the classical model theoretic notion of a structure appearing in the particular case where C is the category of sets and functions. This notion has proven useful when the set-theoretic notion of a model lacks generality and/or is inconvenient. R.A.G
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Charles Sanders Peirce
CDPT: Commens Dictionary of Peirce's Terms CP x.y: Collected Papers, volume x, paragraph y EP x:y: The Essential Peirce, volume x, page y W x:y Writings of Charles S. Peirce, volume x, page yv t e Charles Sanders Peirce
Charles Sanders Peirce
(/pɜːrs/,[9] like "purse"; September 10, 1839 – April 19, 1914) was an American philosopher, logician, mathematician, and scientist who is sometimes known as "the father of pragmatism". He was educated as a chemist and employed as a scientist for 30 years. Today he is appreciated largely for his contributions to logic, mathematics, philosophy, scientific methodology, and semiotics, and for his founding of pragmatism. An innovator in mathematics, statistics, philosophy, research methodology, and various sciences, Peirce considered himself, first and foremost, a logician. He made major contributions to logic, but logic for him encompassed much of that which is now called epistemology and philosophy of science
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Philosophy
Philosophy
Philosophy
(from Greek φιλοσοφία, philosophia, literally "love of wisdom"[1][2][3][4]) is the study of general and fundamental problems concerning matters such as existence, knowledge, values, reason, mind, and language.[5][6] The term was probably coined by Pythagoras
Pythagoras
(c. 570–495 BCE)
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