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All Horses Are The Same Colour
All horses are the same color is a falsidical paradox that arises from a flawed use of mathematical induction to prove the statement ''All horses are the same color''. There is no actual contradiction, as these arguments have a crucial flaw that makes them incorrect. This example was originally raised by George Pólya in a 1954 book in different terms: "Are any numbers equal?" or "Any girls have eyes of the same color", as an exercise in mathematical induction. It has also been restated as "All cows have the same color".Thomas VanDrunen, ''Discrete Mathematics and Functional Programming'', Franklin, Beedle and Associates, 2012, Section "Induction Gone Awry" The "horses" version of the paradox was presented in 1961 in a satirical article by Joel E. Cohen. It was stated as a lemma (mathematics), lemma, which in particular allowed the author to "prove" that Alexander the Great did not exist, and he had an infinite number of limbs. The argument The argument is proof by induction. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Falsidical Paradox
A paradox is a logically self-contradictory statement or a statement that runs contrary to one's expectation. It is a statement that, despite apparently valid reasoning from true premises, leads to a seemingly self-contradictory or a logically unacceptable conclusion. A paradox usually involves contradictory-yet-interrelated elements that exist simultaneously and persist over time. They result in "persistent contradiction between interdependent elements" leading to a lasting "unity of opposites". In logic, many paradoxes exist that are known to be Validity (logic), invalid arguments, yet are nevertheless valuable in promoting critical thinking, while other paradoxes have revealed errors in definitions that were assumed to be rigorous, and have caused axioms of mathematics and logic to be re-examined. One example is Russell's paradox, which questions whether a "list of all lists that do not contain themselves" would include itself, and showed that attempts to found set theory on t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematical Induction
Mathematical induction is a method for proving that a statement ''P''(''n'') is true for every natural number ''n'', that is, that the infinitely many cases ''P''(0), ''P''(1), ''P''(2), ''P''(3), ... all hold. Informal metaphors help to explain this technique, such as falling dominoes or climbing a ladder: A proof by induction consists of two cases. The first, the base case, proves the statement for ''n'' = 0 without assuming any knowledge of other cases. The second case, the induction step, proves that ''if'' the statement holds for any given case ''n'' = ''k'', ''then'' it must also hold for the next case ''n'' = ''k'' + 1. These two steps establish that the statement holds for every natural number ''n''. The base case does not necessarily begin with ''n'' = 0, but often with ''n'' = 1, and possibly with any fixed natural number ''n'' = ''N'', establishing the truth of the statement for all natu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Horse
The horse (''Equus ferus caballus'') is a domesticated, one-toed, hoofed mammal. It belongs to the taxonomic family Equidae and is one of two extant subspecies of ''Equus ferus''. The horse has evolved over the past 45 to 55 million years from a small multi-toed creature, ''Eohippus'', into the large, single-toed animal of today. Humans began domesticating horses around 4000 BCE, and their domestication is believed to have been widespread by 3000 BCE. Horses in the subspecies ''caballus'' are domesticated, although some domesticated populations live in the wild as feral horses. These feral populations are not true wild horses, as this term is used to describe horses that have never been domesticated. There is an extensive, specialized vocabulary used to describe equine-related concepts, covering everything from anatomy to life stages, size, colors, markings, breeds, locomotion, and behavior. Horses are adapted to run, allowing them to quickly escape predators, and po ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
George Pólya
George Pólya (; hu, Pólya György, ; December 13, 1887 – September 7, 1985) was a Hungarian mathematician. He was a professor of mathematics from 1914 to 1940 at ETH Zürich and from 1940 to 1953 at Stanford University. He made fundamental contributions to combinatorics, number theory, numerical analysis and probability theory. He is also noted for his work in heuristics and mathematics education. He has been described as one of The Martians, an informal category which included one of his most famous students at ETH Zurich, John Von Neumann. Life and works Pólya was born in Budapest, Austria-Hungary, to Anna Deutsch and Jakab Pólya, Hungarian Jews who had converted to Christianity in 1886. Although his parents were religious and he was baptized into the Catholic Church upon birth, George eventually grew up to be an agnostic. He was a professor of mathematics from 1914 to 1940 at ETH Zürich in Switzerland and from 1940 to 1953 at Stanford University. He remained a Pr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Joel E , a community in the United States
{{disambiguation, hn, geo ...
Joel or Yoel is a name meaning "Yahweh Is God" and may refer to: * Joel (given name), origin of the name including a list of people with the first name. * Joel (surname), a surname * Joel (footballer, born 1904), Joel de Oliveira Monteiro, Brazilian football goalkeeper * Joel (footballer, born 1980), Joel Bertoti Padilha, Brazilian football centre-back * Joel (prophet), a prophet of ancient Israel ** Book of Joel, a book in the Jewish Tanakh, and in the Christian Bible, ascribed to the prophet * Joel, Georgia, a community in the United States * Joel, Wisconsin The Town of Clayton is located in Polk County, Wisconsin, Polk County, Wisconsin, United States. The population was 571 at the 2000 census. The Clayton (village), Wisconsin, Village of Clayton and the unincorporated communities of Joel and Richard ... [...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] |
Alexander The Great
Alexander III of Macedon ( grc, wikt:Ἀλέξανδρος, Ἀλέξανδρος, Alexandros; 20/21 July 356 BC – 10/11 June 323 BC), commonly known as Alexander the Great, was a king of the Ancient Greece, ancient Greek kingdom of Macedonia (ancient kingdom), Macedon. He succeeded his father Philip II of Macedon, Philip II to the throne in 336 BC at the age of 20, and spent most of his ruling years conducting a lengthy military campaign throughout Western Asia and ancient Egypt, Egypt. By the age of thirty, he had created one of the List of largest empires, largest empires in history, stretching from Greece to northwestern Historical India, India. He was undefeated in battle and is widely considered to be one of history's greatest and most successful military commanders. Until the age of 16, Alexander was tutored by Aristotle. In 335 BC, shortly after his assumption of kingship over Macedon, he Alexander's Balkan campaign, campaigned in the Balkans and reasserted control ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Worm Runner's Digest
The ''Worm Runner's Digest'' (''W.R.D.'') was created in 1959 by biologist James V. McConnell after his experiments with memory transfer in planarian worms generated a torrent of mail enquiries. The ''W.R.D.'' published both satirical articles, such as "A Stress Analysis of a Strapless Evening Gown", and scientific papers, the most famous of which, "Memory transfer through cannibalism in planaria", was a result of McConnell's RNA memory transfer experiments with planarian worms and was later published in the ''Journal of Neuropsychiatry''. The title for the W.R.D., McConnell explained, was an extension of the psychological jargon that terms psychologists who work with rats "rat runners" and those who work with insects "bug runners." After complaints that the satirical articles and the scientific publications were not distinguishable, the satirical articles were printed upside down in the back half of the ''W.R.D.'' along with a topsy turvy back cover. In 1966, the title was chan ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Horses Induction2
The horse (''Equus ferus caballus'') is a domesticated, one-toed, hoofed mammal. It belongs to the taxonomic family Equidae and is one of two extant subspecies of ''Equus ferus''. The horse has evolved over the past 45 to 55 million years from a small multi-toed creature, ''Eohippus'', into the large, single-toed animal of today. Humans began domesticating horses around 4000 BCE, and their domestication is believed to have been widespread by 3000 BCE. Horses in the subspecies ''caballus'' are domesticated, although some domesticated populations live in the wild as feral horses. These feral populations are not true wild horses, as this term is used to describe horses that have never been domesticated. There is an extensive, specialized vocabulary used to describe equine-related concepts, covering everything from anatomy to life stages, size, colors, markings, breeds, locomotion, and behavior. Horses are adapted to run, allowing them to quickly escape predators, and po ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Proof By Induction
Mathematical induction is a method for mathematical proof, proving that a statement ''P''(''n'') is true for every natural number ''n'', that is, that the infinitely many cases ''P''(0), ''P''(1), ''P''(2), ''P''(3), ... all hold. Informal metaphors help to explain this technique, such as falling dominoes or climbing a ladder: A proof by induction consists of two cases. The first, the base case, proves the statement for ''n'' = 0 without assuming any knowledge of other cases. The second case, the induction step, proves that ''if'' the statement holds for any given case ''n'' = ''k'', ''then'' it must also hold for the next case ''n'' = ''k'' + 1. These two steps establish that the statement holds for every natural number ''n''. The base case does not necessarily begin with ''n'' = 0, but often with ''n'' = 1, and possibly with any fixed natural number ''n'' = ''N'', establishing the truth of the sta ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Subset
In mathematics, Set (mathematics), set ''A'' is a subset of a set ''B'' if all Element (mathematics), elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are unequal, then ''A'' is a proper subset of ''B''. The relationship of one set being a subset of another is called inclusion (or sometimes containment). ''A'' is a subset of ''B'' may also be expressed as ''B'' includes (or contains) ''A'' or ''A'' is included (or contained) in ''B''. A ''k''-subset is a subset with ''k'' elements. The subset relation defines a partial order on sets. In fact, the subsets of a given set form a Boolean algebra (structure), Boolean algebra under the subset relation, in which the join and meet are given by Intersection (set theory), intersection and Union (set theory), union, and the subset relation itself is the Inclusion (Boolean algebra), Boolean inclusion relation. Definition If ''A'' and ''B'' are sets and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Unexpected Hanging Paradox
The unexpected hanging paradox or surprise test paradox is a paradox about a person's expectations about the timing of a future event which they are told will occur at an unexpected time. The paradox is variously applied to a prisoner's hanging or a surprise school test. It was first introduced to the public in Martin Gardner's March 1963 Mathematical Games column in ''Scientific American'' magazine. There is no consensus on its precise nature and consequently a canonical resolution has not been agreed on. Logical analyses focus on "truth values", for example by identifying it as paradox of self-reference. Epistemological studies of the paradox instead focus on issues relating to ''knowledge''; for example, one interpretation reduces it to Moore's paradox. Some regard it as a "significant problem" for philosophy. Description The paradox has been described as follows: Other versions of the paradox replace the death sentence with a surprise fire drill, examination, pop quiz, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
