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Corollaries
In mathematics and logic, a corollary ( , ) is a theorem of less importance which can be readily deduced from a previous, more notable statement. A corollary could, for instance, be a proposition which is incidentally proved while proving another proposition; it might also be used more casually to refer to something which naturally or incidentally accompanies something else (e.g., violence as a corollary of revolutionary social changes). Overview In mathematics, a corollary is a theorem connected by a short proof to an existing theorem. The use of the term ''corollary'', rather than ''proposition'' or ''theorem'', is intrinsically subjective. More formally, proposition ''B'' is a corollary of proposition ''A'', if ''B'' can be readily deduced from ''A'' or is self-evident from its proof. In many cases, a corollary corresponds to a special case of a larger theorem, which makes the theorem easier to use and apply, even though its importance is generally considered to be secondary t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Theorems
In mathematics, a theorem is a statement that has been proved, or can be proved. The ''proof'' of a theorem is a logical argument that uses the inference rules of a deductive system to establish that the theorem is a logical consequence of the axioms and previously proved theorems. In the mainstream of mathematics, the axioms and the inference rules are commonly left implicit, and, in this case, they are almost always those of Zermelo–Fraenkel set theory with the axiom of choice, or of a less powerful theory, such as Peano arithmetic. A notable exception is Wiles's proof of Fermat's Last Theorem, which involves the Grothendieck universes whose existence requires the addition of a new axiom to the set theory. Generally, an assertion that is explicitly called a theorem is a proved result that is not an immediate consequence of other known theorems. Moreover, many authors qualify as ''theorems'' only the most important results, and use the terms ''lemma'', ''proposition'' and ' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Theorem
In mathematics, a theorem is a statement that has been proved, or can be proved. The ''proof'' of a theorem is a logical argument that uses the inference rules of a deductive system to establish that the theorem is a logical consequence of the axioms and previously proved theorems. In the mainstream of mathematics, the axioms and the inference rules are commonly left implicit, and, in this case, they are almost always those of Zermelo–Fraenkel set theory with the axiom of choice, or of a less powerful theory, such as Peano arithmetic. A notable exception is Wiles's proof of Fermat's Last Theorem, which involves the Grothendieck universes whose existence requires the addition of a new axiom to the set theory. Generally, an assertion that is explicitly called a theorem is a proved result that is not an immediate consequence of other known theorems. Moreover, many authors qualify as ''theorems'' only the most important results, and use the terms ''lemma'', ''proposition'' and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lemma (mathematics)
In mathematics, informal logic and argument mapping, a lemma (plural lemmas or lemmata) is a generally minor, proven proposition which is used as a stepping stone to a larger result. For that reason, it is also known as a "helping theorem" or an "auxiliary theorem". In many cases, a lemma derives its importance from the theorem it aims to prove; however, a lemma can also turn out to be more important than originally thought. The word "lemma" derives from the Ancient Greek ("anything which is received", such as a gift, profit, or a bribe). Comparison with theorem There is no formal distinction between a lemma and a theorem, only one of intention (see Theorem terminology). However, a lemma can be considered a minor result whose sole purpose is to help prove a more substantial theorem – a step in the direction of proof. Well-known lemmas A good stepping stone can lead to many others. Some powerful results in mathematics are known as lemmas, first named for their originally min ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematical Terminology
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Roosevelt Corollary
In the history of United States foreign policy, the Roosevelt Corollary was an addition to the Monroe Doctrine articulated by President Theodore Roosevelt in his State of the Union address in 1904 after the Venezuelan crisis of 1902–1903. The corollary states that the United States could intervene in the internal affairs of Latin American countries if they committed flagrant and chronic wrongdoings. Roosevelt tied his policy to the Monroe Doctrine, and it was also consistent with his foreign policy included in his Big Stick Diplomacy. Roosevelt stated that in keeping with the Monroe Doctrine, the United States was justified in exercising "international police power" to put an end to chronic unrest or wrongdoing in the Western Hemisphere. President Herbert Hoover in 1930 endorsed the Clark Memorandum that repudiated the Roosevelt Corollary in favor of what was later called the Good Neighbor policy. Background The Roosevelt Corollary was articulated in the aftermath of t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Monroe Doctrine
The Monroe Doctrine was a United States foreign policy position that opposed European colonialism in the Western Hemisphere. It held that any intervention in the political affairs of the Americas by foreign powers was a potentially hostile act against the United States. The doctrine was central to American foreign policy for much of the 19th and early 20th centuries. President James Monroe first articulated the doctrine on December 2, 1823, during his seventh annual State of the Union Address to Congress (though it would not be named after him until 1850). At the time, nearly all Spanish colonies in the Americas had either achieved or were close to independence. Monroe asserted that the New World and the Old World were to remain distinctly separate spheres of influence, and thus further efforts by European powers to control or influence sovereign states in the region would be viewed as a threat to U.S. security. In turn, the United States would recognize and not interfere with ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lodge Corollary
The Lodge Corollary was a corollary to the Monroe Doctrine. Proposed by Henry Cabot Lodge and ratified by the U.S. Senate in 1912, it forbade any foreign power or foreign interest of any kind from acquiring sufficient territory in the Western Hemisphere as to put that government in "practical power of control." As Lodge argued, the corollary reaffirmed the basic right of nations to provide for their safety and extended the principles behind the Monroe Doctrine beyond colonialism to include corporate territorial acquisitions as well. The proposal was a reaction to negotiations between a Japanese syndicate and Mexico for the purchase of a considerable portion of Baja California including a harbor considered to be of strategic value, Magdalena Bay Magdalena Bay ( es, Bahía Magdalena) is a long bay in Comondú Municipality along the western coast of the Mexican state of Baja California Sur. It is protected from the Pacific Ocean by the unpopulated sandy barrier islands of Isla Ma ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Proposition
In logic and linguistics, a proposition is the meaning of a declarative sentence. In philosophy, " meaning" is understood to be a non-linguistic entity which is shared by all sentences with the same meaning. Equivalently, a proposition is the non-linguistic bearer of truth or falsity which makes any sentence that expresses it either true or false. While the term "proposition" may sometimes be used in everyday language to refer to a linguistic statement which can be either true or false, the technical philosophical term, which differs from the mathematical usage, refers exclusively to the non-linguistic meaning behind the statement. The term is often used very broadly and can also refer to various related concepts, both in the history of philosophy and in contemporary analytic philosophy. It can generally be used to refer to some or all of the following: The primary bearers of truth values (such as "true" and "false"); the objects of belief and other propositional attitudes (i.e ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Porism
A porism is a mathematical proposition or corollary. It has been used to refer to a direct consequence of a Mathematical proof, proof, analogous to how a corollary refers to a direct consequence of a theorem. In modern usage, it is a relationship that holds for an infinite range of values but only if a certain condition is assumed, such as Steiner chain, Steiner's porism. The term originates from three books of Euclid that have been lost. A proposition may not have been proven, so a porism may not be a theorem or true. Origins The book that talks about porisms first is Euclid's ''Porisms''. What is known of it is in Pappus of Alexandria's ''Collection'', who mentions it along with other geometrical treatises, and gives several lemma (mathematics), lemmas necessary for understanding it. Pappus states: :The porisms of all classes are neither theorems nor problems, but occupy a position intermediate between the two, so that their enunciations can be stated either as theorems or probl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Logic
Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the science of deductively valid inferences or of logical truths. It is a formal science investigating how conclusions follow from premises in a topic-neutral way. When used as a countable noun, the term "a logic" refers to a logical formal system that articulates a proof system. Formal logic contrasts with informal logic, which is associated with informal fallacies, critical thinking, and argumentation theory. While there is no general agreement on how formal and informal logic are to be distinguished, one prominent approach associates their difference with whether the studied arguments are expressed in formal or informal languages. Logic plays a central role in multiple fields, such as philosophy, mathematics, computer science, and linguistics. Logic studies arguments, which consist of a set of premises together with a conclusion. Premises and conclusions are usually un ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Charles Sanders Peirce Bibliography
This Charles Sanders Peirce bibliography consolidates numerous references to the writings of Charles Sanders Peirce, including letters, manuscripts, publications, and . For an extensive chronological list of Peirce's works (titled in English), see the (Chronological Overview) on the (Writings) page for Charles Sanders Peirce. Abbreviations Click on abbreviation in order to jump down this page to the relevant edition information. Click on the abbreviation appearing with that edition information in order to return here. Main editions (posthumous) Other Primary literature Bibliographies and microfilms Other bibliographies of primary literature * Burks, Arthur W. (1958). "Bibliography of the Works of Charles Sanders Peirce." CP 8:260–321. * Cohen, Morris R. (1916). "Charles S. Peirce and a Tentative Bibliography of His Published Writings." ''The Journal of Philosophy, Psychology, and Scientific Methods'' 13(26):726–37. *Fisch, Max H. (1964). "A First Supplement ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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