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Definitions Of Mathematics
Mathematics has no generally accepted definition. Different schools of thought, particularly in philosophy, have put forth radically different definitions. All are controversial. Early definitions Aristotle defined mathematics as: The science of quantity. In Aristotle's classification of the sciences, discrete quantities were studied by arithmetic, continuous quantities by geometry.James Franklin, "Aristotelian Realism" in ''Philosophy of Mathematics", ed. A.D. Irvinep. 104 Elsevier (2009). Aristotle also thought that quantity alone does not distinguish mathematics from sciences like physics; in his view, abstraction and studying quantity as a property "separable in thought" from real instances set mathematics apart. Auguste Comte's definition tried to explain the role of mathematics in coordinating phenomena in all other fields: The science of indirect measurement. Auguste Comte 1851 The "indirectness" in Comte's definition refers to determining quantities that cannot be m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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Mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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Deductive Reasoning
Deductive reasoning is the process of drawing valid inferences. An inference is valid if its conclusion follows logically from its premises, meaning that it is impossible for the premises to be true and the conclusion to be false. For example, the inference from the premises "all men are mortal" and " Socrates is a man" to the conclusion "Socrates is mortal" is deductively valid. An argument is ''sound'' if it is valid ''and'' all its premises are true. One approach defines deduction in terms of the intentions of the author: they have to intend for the premises to offer deductive support to the conclusion. With the help of this modification, it is possible to distinguish valid from invalid deductive reasoning: it is invalid if the author's belief about the deductive support is false, but even invalid deductive reasoning is a form of deductive reasoning. Deductive logic studies under what conditions an argument is valid. According to the semantic approach, an argument is valid ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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Encyclopædia Britannica
The is a general knowledge, general-knowledge English-language encyclopaedia. It has been published by Encyclopædia Britannica, Inc. since 1768, although the company has changed ownership seven times. The 2010 version of the 15th edition, which spans 32 volumes and 32,640 pages, was the last printed edition. Since 2016, it has been published exclusively as an online encyclopedia, online encyclopaedia. Printed for 244 years, the ''Britannica'' was the longest-running in-print encyclopaedia in the English language. It was first published between 1768 and 1771 in Edinburgh, Scotland, in three volumes. The encyclopaedia grew in size; the second edition was 10 volumes, and by its fourth edition (1801–1810), it had expanded to 20 volumes. Its rising stature as a scholarly work helped recruit eminent contributors, and the 9th (1875–1889) and Encyclopædia Britannica Eleventh Edition, 11th editions (1911) are landmark encyclopaedias for scholarship and literary ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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The American Heritage Dictionary Of The English Language
''The American Heritage Dictionary of the English Language'' (''AHD'') is a dictionary of American English published by HarperCollins. It is currently in its fifth edition (since 2011). Before HarperCollins acquired certain business lines from Houghton Mifflin Harcourt in 2022, the family of American Heritage dictionaries had long been published by Houghton Mifflin Harcourt and its predecessor Houghton Mifflin. The first edition appeared in 1969, an outgrowth of the editorial effort for Houghton Mifflin's ''American Heritage'' brand of history books and journals. The dictionary's creation was spurred by the controversy during the 1960s over the perceived permissiveness of the ''Webster's Third New International Dictionary'' (1961). A college dictionary followed several years later. The main dictionary became the flagship title as the brand grew into a family of various dictionaries, a dictionary-thesaurus combination, and a usage (language), usage guide. History James Parton ( ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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Oxford English Dictionary
The ''Oxford English Dictionary'' (''OED'') is the principal historical dictionary of the English language, published by Oxford University Press (OUP), a University of Oxford publishing house. The dictionary, which published its first edition in 1884, traces the historical development of the English language, providing a comprehensive resource to scholars and academic researchers, and provides ongoing descriptions of English language usage in its variations around the world. In 1857, work first began on the dictionary, though the first edition was not published until 1884. It began to be published in unbound Serial (literature), fascicles as work continued on the project, under the name of ''A New English Dictionary on Historical Principles; Founded Mainly on the Materials Collected by The Philological Society''. In 1895, the title ''The Oxford English Dictionary'' was first used unofficially on the covers of the series, and in 1928 the full dictionary was republished in 10 b ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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Abstraction (mathematics)
Abstraction in mathematics is the process of extracting the underlying structures, patterns or properties of a mathematical concept, removing any dependence on real world objects with which it might originally have been connected, and generalizing it so that it has wider applications or matching among other abstract descriptions of equivalent phenomena. In other words, to be abstract is to remove context and application. Two of the most highly abstract areas of modern mathematics are category theory and model theory. Description Many areas of mathematics began with the study of real world problems, before the underlying rules and concepts were identified and defined as abstract structures. For example, geometry has its origins in the calculation of distances and areas in the real world, and algebra started with methods of solving problems in arithmetic. Abstraction is an ongoing process in mathematics and the historical development of many mathematical topics exhibits a progres ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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Walter Warwick Sawyer
Walter Warwick Sawyer (or W. W. Sawyer; April 5, 1911 – February 15, 2008) was a mathematician, mathematics educator and author, who taught on several continents. Life and career Walter Warwick Sawyer was born in St. Ives, Hunts, England on April 5, 1911. He attended Highgate School in London. He was an undergraduate at St. John's College, Cambridge, obtaining a BA in 1933 and specializing in quantum theory and relativity. He was an assistant lecturer in mathematics from 1933 to 1937 at University College, Dundee and from 1937 to 1944 at University of Manchester. In 1940 he met Betty ilda Elizabeth Crowtherand within two weeks, they were married. In 1943 their one child, daughter Anne Elizabeth, was born. 1943 was also the year that Sawyer's publishing career began with the book Mathematician's Delight published by Pelican Books, the non-fiction imprint of Penguin Books founded by Allen Lane and V. K. Krishna Menon. From 1945 to 1947, he was the head of mathematics at Lei ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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Haskell Curry
Haskell Brooks Curry ( ; September 12, 1900 – September 1, 1982) was an American mathematician, logician and computer scientist. Curry is best known for his work in combinatory logic, whose initial concept is based on a paper by Moses Schönfinkel, for which Curry did much of the development. Curry is also known for Curry's paradox and the Curry–Howard correspondence. Named for him are three programming languages: Haskell, Brook, and Curry, and the concept of ''currying'', a method to transform functions, used in mathematics and computer science. Life Curry was born on in Millis, Massachusetts, to Samuel Silas Curry and Anna Baright Curry, who ran a school for elocution. He entered Harvard University in 1916 to study medicine but switched to mathematics before graduating in 1920. After two years of graduate work in electrical engineering at Massachusetts Institute of Technology (MIT), he returned to Harvard to study physics, earning a Master of Arts (M.A.) in 1924. Cur ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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Intuitionistic 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 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 interpre ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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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] |
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Uta Merzbach
Uta Caecilia Merzbach (February 9, 1933 – June 27, 2017) was a German-American historian of mathematics who became the first curator of mathematical instruments at the Smithsonian Institution. Early life Merzbach was born in Berlin, where her mother was a philologist and her father was an economist who worked for the Reich Association of Jews in Germany during World War II. The Nazi government closed the association in June 1943; they arrested the family, along with other leading members of the association, and sent them to the Theresienstadt concentration camp on August 4, 1943. The Merzbachs survived the war and the camp, and after living for a year in a refugee camp in Deggendorf they moved to Georgetown, Texas in 1946, where her father found a faculty position at Southwestern University. Education After high school in Brownwood, Texas, Merzbach entered Southwestern, but transferred after two years to the University of Texas at Austin, where she graduated in 1952 with ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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Categorical Proposition
In logic, a categorical proposition, or categorical statement, is a proposition that asserts or denies that all or some of the members of one category (the ''subject term'') are included in another (the ''predicate term''). The study of arguments using categorical statements (i.e., syllogisms) forms an important branch of deductive reasoning that began with the Ancient Greeks. The Ancient Greeks such as Aristotle identified four primary distinct types of categorical proposition and gave them standard forms (now often called ''A'', ''E'', ''I'', and ''O''). If, abstractly, the subject category is named ''S'' and the predicate category is named ''P'', the four standard forms are: * All ''S'' are ''P''. (''A'' form) * No ''S'' are ''P''. (''E'' form) * Some ''S'' are ''P''. (''I'' form) * Some ''S'' are not ''P''. (''O'' form) A large number of sentences may be translated into one of these canonical forms while retaining all or most of the original meaning of the sentence. Greek inves ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |