1.
Penrose triangle
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The Penrose triangle, also known as the Penrose tribar, or the impossible tribar, is an impossible object. It was first created by the Swedish artist Oscar Reutersvärd in 1934, the psychiatrist Lionel Penrose and his mathematician son Roger Penrose independently devised and popularized it in the 1950s, describing it as impossibility in its purest form. It is featured prominently in the works of artist M. C, Escher, whose earlier depictions of impossible objects partly inspired it. The tribar appears to be an object, made of three straight beams of square cross-section which meet pairwise at right angles at the vertices of the triangle they form. The beams may be broken, forming cubes or cuboids and this combination of properties cannot be realized by any 3-dimensional object in ordinary Euclidean space. Such an object can exist in certain Euclidean 3-manifolds, there also exist 3-dimensional solid shapes each of which, when viewed from a certain angle, appears the same as the 2-dimensional depiction of the Penrose triangle on this page. The term Penrose triangle can refer to the 2-dimensional depiction or the object itself. Eschers lithograph Waterfall depicts a watercourse that flows in a zigzag along the sides of two elongated Penrose triangles, so that it ends up two stories higher than it began. The resulting waterfall, forming the sides of both triangles, drives a water wheel. Escher helpfully points out that in order to keep the wheel turning some water must occasionally be added to compensate for evaporation, if a line is traced around the Penrose triangle, a 3-loop Möbius strip is formed. Although the tribar is named one of the objects, there exist many more that fit into the same category. Other impossible objects include the devils fork, the dancing elephant, three hares Penrose steps Penrose square root law An article about impossible triangle sculpture in Perth Escher for Real constructions
2.
Russell's paradox
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According to naive set theory, any definable collection is a set. Let R be the set of all sets that are not members of themselves. Symbolically, Let R =, then R ∈ R ⟺ R ∉ R In 1908, two ways of avoiding the paradox were proposed, Russells type theory and the Zermelo set theory, the first constructed axiomatic set theory. Zermelos axioms went well beyond Gottlob Freges axioms of extensionality and unlimited set abstraction, Let us call a set abnormal if it is a member of itself, and normal otherwise. For example, take the set of all squares in the plane and that set is not itself a square in the plane, and therefore is not a member of the set of all squares in the plane. On the other hand, if we take the set that contains all non-. Now we consider the set of all sets, R. This leads to the conclusion that R is neither normal nor abnormal, then by existential instantiation and universal instantiation we have y ∈ y ⟺ y ∉ y a contradiction. Modifications to this axiomatic theory proposed in the 1920s by Abraham Fraenkel, Thoralf Skolem and this theory became widely accepted once Zermelos axiom of choice ceased to be controversial, and ZFC has remained the canonical axiomatic set theory down to the present day. ZFC does not assume that, for property, there is a set of all things satisfying that property. Rather, it asserts that any set X, any subset of X definable using first-order logic exists. The object R discussed above cannot be constructed in this fashion, in some extensions of ZFC, objects like R are called proper classes. ZFC is silent about types, although the hierarchy has a notion of layers that resemble types. Zermelo himself never accepted Skolems formulation of ZFC using the language of first-order logic and this 2nd order ZFC preferred by Zermelo, including axiom of foundation, allowed a rich cumulative hierarchy. Ferreirós writes that Zermelos layers are essentially the same as the types in the versions of simple TT offered by Gödel. One can describe the cumulative hierarchy into which Zermelo developed his models as the universe of a cumulative TT in which types are allowed. Thus, simple TT and ZFC could now be regarded as systems that talk essentially about the same intended objects, the main difference is that TT relies on a strong higher-order logic, while Zermelo employed second-order logic, and ZFC can also be given a first-order formulation. The first-order description of the hierarchy is much weaker, as is shown by the existence of denumerable models
3.
Set theory
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Set theory is a branch of mathematical logic that studies sets, which informally are collections of objects. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics, the language of set theory can be used in the definitions of nearly all mathematical objects. The modern study of set theory was initiated by Georg Cantor, Set theory is commonly employed as a foundational system for mathematics, particularly in the form of Zermelo–Fraenkel set theory with the axiom of choice. Beyond its foundational role, set theory is a branch of mathematics in its own right, contemporary research into set theory includes a diverse collection of topics, ranging from the structure of the real number line to the study of the consistency of large cardinals. Mathematical topics typically emerge and evolve through interactions among many researchers, Set theory, however, was founded by a single paper in 1874 by Georg Cantor, On a Property of the Collection of All Real Algebraic Numbers. Since the 5th century BC, beginning with Greek mathematician Zeno of Elea in the West and early Indian mathematicians in the East, especially notable is the work of Bernard Bolzano in the first half of the 19th century. Modern understanding of infinity began in 1867–71, with Cantors work on number theory, an 1872 meeting between Cantor and Richard Dedekind influenced Cantors thinking and culminated in Cantors 1874 paper. Cantors work initially polarized the mathematicians of his day, while Karl Weierstrass and Dedekind supported Cantor, Leopold Kronecker, now seen as a founder of mathematical constructivism, did not. This utility of set theory led to the article Mengenlehre contributed in 1898 by Arthur Schoenflies to Kleins encyclopedia, in 1899 Cantor had himself posed the question What is the cardinal number of the set of all sets. Russell used his paradox as a theme in his 1903 review of continental mathematics in his The Principles of Mathematics, in 1906 English readers gained the book Theory of Sets of Points by William Henry Young and his wife Grace Chisholm Young, published by Cambridge University Press. The momentum of set theory was such that debate on the paradoxes did not lead to its abandonment, the work of Zermelo in 1908 and Abraham Fraenkel in 1922 resulted in the set of axioms ZFC, which became the most commonly used set of axioms for set theory. The work of such as Henri Lebesgue demonstrated the great mathematical utility of set theory. Set theory is used as a foundational system, although in some areas category theory is thought to be a preferred foundation. Set theory begins with a binary relation between an object o and a set A. If o is a member of A, the notation o ∈ A is used, since sets are objects, the membership relation can relate sets as well. A derived binary relation between two sets is the relation, also called set inclusion. If all the members of set A are also members of set B, then A is a subset of B, for example, is a subset of, and so is but is not. As insinuated from this definition, a set is a subset of itself, for cases where this possibility is unsuitable or would make sense to be rejected, the term proper subset is defined
4.
Property (philosophy)
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In philosophy, mathematics, and logic, a property is a characteristic of an object, a red object is said to have the property of redness. The property may be considered a form of object in its own right, a property however differs from individual objects in that it may be instantiated, and often in more than one thing. Understanding how different individual entities can in some sense have some of the properties is the basis of the problem of universals. The terms attribute and quality have similar meanings, in classical Aristotelian terminology, a property is one of the predicables. It is a quality of a species, but a quality which is nevertheless characteristically present in members of that species. For example, ability to laugh may be considered a characteristic of human beings. However, laughter is not a quality of the species human. A determinable property is one that can get more specific, for example, color is a determinable property because it can be restricted to redness, blueness, etc. A determinate property is one that become more specific. This distinction may be useful in dealing with issues of identity, in other words, it is the view that non-physical, mental properties inhere in some physical substances. e. The set, p is its indicator function and it may be objected that this defines merely the extension of a property, and says nothing about what causes the property to hold for exactly those values. The ontological fact that something has a property is represented in language by applying a predicate to a subject. However, taking any grammatical predicate whatsoever to be a property, or to have a property, leads to certain difficulties, such as Russells paradox. Other predicates, such as is an individual, or has some properties are uninformative or vacuous, there is some resistance to regarding such so-called Cambridge properties as legitimate. An intrinsic property is a property that an object or a thing has of itself, independently of other things, an extrinsic property is a property that depends on a things relationship with other things. A relation is considered to be a more general case of a property. Relations are true of several particulars, or shared amongst them, holds between two individuals, who would occupy the two ellipses. Relations can be expressed by N-place predicates, where N is greater than 1 and it is widely accepted that there are at least some apparent relational properties which are merely derived from non-relational properties
5.
Curry's paradox
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The paradox is named after the logician Haskell Curry. The paradox may be expressed in language and in various mathematical settings, including certain forms of set theory, lambda calculus. It has also been called Löbs paradox after Martin Hugo Löb, claims of the form if A, then B are called conditional claims. Currys paradox uses a kind of self-referential conditional sentence, as demonstrated in this example, If this sentence is true. Even though Germany does not border China, the example sentence certainly is a natural-language sentence, the paradox follows from this analysis. The analysis consists of two steps, first, common natural-language proof techniques can be used to prove that the example sentence is true. Second, the truth of the sentence can be used to prove that Germany borders China. Because Germany does not border China, this suggests there has been an error in one of the proofs. The claim Germany borders China could be replaced by any other claim, because the proof uses only well-accepted methods of deduction, and because none of these methods appears to be incorrect, this situation is paradoxical. The following analysis is used to show that the sentence If this sentence is true, the quoted sentence is of the form If A, then B where A refers to the sentence itself and B refers to Germany borders China. Such sentences are called conditional sentences, and the method for proving them is called conditional proof. To apply this method, the first step is to assume for the sake of argument that the hypothesis is true, the goal is then to show that the conclusion can be proven from that assumption. Therefore, for the purpose of the proof, assume A, because A refers to the overall sentence, this means that assuming A is the same as assuming If A, then B. Therefore, in assuming A, we have assumed both A and If A, then B, from these, we can obtain B by an application of the standard deduction rule modus ponens. Therefore, A implies B, and we have proved If this sentence is true, therefore Germany borders China, should be true—but it actually is false. The paradox is caused by the fact that the rules used in the proof are standard rules that are used in many other settings, thus. If we assume the sentence is true, then Germany must border China, but Germany does not border China, so the sentence is false. So we assume the sentence is false, then Germany may not border China, since Germany in fact does not border China, the sentence is true
6.
M. C. Escher
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Maurits Cornelis Escher, or commonly M. C. Escher, was a Dutch graphic artist who made mathematically inspired woodcuts, lithographs, early in his career, he drew inspiration from nature, making studies of insects, landscapes, and plants such as lichens, all of which he used as details in his artworks. Apart from being used in a variety of papers, his work has appeared on the covers of many books. He was one of the inspirations of Douglas Hofstadters 1979 book Gödel, Escher. Maurits Cornelis Escher was born on 17 June 1898 in Leeuwarden, Friesland and he was the youngest son of the civil engineer George Arnold Escher and his second wife, Sara Gleichman. In 1903, the moved to Arnhem, where he attended primary and secondary school until 1918. Known to his friends and family as Mauk, he was a sickly child, although he excelled at drawing, his grades were generally poor. He also took carpentry and piano lessons until he was thirteen years old, in 1918, he went to the Technical College of Delft. From 1919 to 1922, Escher attended the Haarlem School of Architecture and Decorative Arts, learning drawing and he briefly studied architecture, but he failed a number of subjects and switched to decorative arts, studying under the graphic artist Samuel Jessurun de Mesquita. In 1922, an important year of his life, Escher traveled through Italy, visiting Florence, San Gimignano, Volterra, Siena, in the same year he traveled through Spain, visiting Madrid, Toledo, and Granada. He was impressed by the Italian countryside, and in Granada by the Moorish architecture of the fourteenth-century Alhambra, Escher returned to Italy, and lived in Rome from 1923 to 1935. While in Italy, Escher met Jetta Umiker – a Swiss woman, the couple settled in Rome where their first son, Giorgio Arnaldo Escher, named after his grandfather, was born. Escher and Jetta later had two sons, Arthur and Jan. He travelled frequently, visiting Viterbo in 1926, the Abruzzi in 1927 and 1929, Corsica in 1928 and 1933, Calabria in 1930, the townscapes and landscapes of these places feature prominently in his artworks. In May and June 1936, Escher travelled back to Spain, revisiting the Alhambra, the sketches he made in the Alhambra formed a major source for his work from that time on. He also studied the architecture of the Mezquita, the Moorish mosque of Cordoba and this turned out to be the last of his long study journeys, after 1937, his artworks were created in his studio rather than in the field. All the same, even his work already shows his interest in the nature of space, the unusual, perspective. In 1935, the climate in Italy became unacceptable to Escher
7.
Perspective (graphical)
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Perspective in the graphic arts is an approximate representation, on a flat surface, of an image as it is seen by the eye. If viewed from the spot as the windowpane was painted. Each painted object in the scene is thus a flat, scaled down version of the object on the side of the window. All perspective drawings assume the viewer is a distance away from the drawing. Objects are scaled relative to that viewer, an object is often not scaled evenly, a circle often appears as an ellipse and a square can appear as a trapezoid. This distortion is referred to as foreshortening, Perspective drawings have a horizon line, which is often implied. This line, directly opposite the viewers eye, represents objects infinitely far away and they have shrunk, in the distance, to the infinitesimal thickness of a line. It is analogous to the Earths horizon, any perspective representation of a scene that includes parallel lines has one or more vanishing points in a perspective drawing. A one-point perspective drawing means that the drawing has a vanishing point, usually directly opposite the viewers eye. All lines parallel with the line of sight recede to the horizon towards this vanishing point. This is the standard receding railroad tracks phenomenon, a two-point drawing would have lines parallel to two different angles. Any number of vanishing points are possible in a drawing, one for each set of lines that are at an angle relative to the plane of the drawing. Perspectives consisting of parallel lines are observed most often when drawing architecture. In contrast, natural scenes often do not have any sets of parallel lines, the only method to indicate the relative position of elements in the composition was by overlapping, of which much use is made in works like the Parthenon Marbles. Chinese artists made use of perspective from the first or second century until the 18th century. It is not certain how they came to use the technique, some authorities suggest that the Chinese acquired the technique from India, oblique projection is also seen in Japanese art, such as in the Ukiyo-e paintings of Torii Kiyonaga. This was detailed within Aristotles Poetics as skenographia, using flat panels on a stage to give the illusion of depth, the philosophers Anaxagoras and Democritus worked out geometric theories of perspective for use with skenographia. Alcibiades had paintings in his house designed using skenographia, so this art was not confined merely to the stage, Euclids Optics introduced a mathematical theory of perspective, but there is some debate over the extent to which Euclids perspective coincides with the modern mathematical definition
8.
Irony
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Irony, in its broadest sense, is a rhetorical device, literary technique, or event in which what appears, on the surface, to be the case, differs radically from what is actually the case. Irony may be divided into such as verbal, dramatic. Verbal, dramatic, and situational irony are often used for emphasis in the assertion of a truth, other forms, as identified by historian Connop Thirlwall, include dialectic and practical irony. Also, Eric Partridge, in Usage and Abusage, writes that Irony consists in stating the contrary of what is meant, the use of irony may require the concept of a double audience. The term is used as a synonym for incongruous and applied to every trivial oddity in situations where there is no double audience. The American Heritage Dictionarys secondary meaning for irony, incongruity between what might be expected and what actually occurs and this sense, however, is not synonymous with incongruous but merely a definition of dramatic or situational irony. It is often included in definitions of irony not only that incongruity is present, according to the Encyclopædia Britannica, The term irony has its roots in the Greek comic character Eiron, a clever underdog who by his wit repeatedly triumphs over the boastful character Alazon. The Socratic irony of the Platonic dialogues derives from this comic origin. e, the word came into English as a figure of speech in the 16th century as similar to the French ironie. It derives from the Latin ironia and ultimately from the Greek εἰρωνεία eirōneía, meaning dissimulation, Romantic irony, A self-aware and self-critical form of fiction. Cosmic irony, A contrast between the absolute and the relative, the general and the individual, which Hegel expressed by the phrase, when the audience knows that the character will die before morning. It is most often used when the author causes a character to speak or act erroneously, in tragic irony, the audience knows the character is making a mistake, even as the character is making it. Verbal irony is distinguished from situational irony and dramatic irony in that it is produced intentionally by speakers, for instance, if a man exclaims, Im not upset. But reveals an upset emotional state through his voice while truly trying to claim hes not upset, but if the same speaker said the same words and intended to communicate that he was upset by claiming he was not, the utterance would be verbal irony. This distinction illustrates an important aspect of verbal irony—speakers communicate implied propositions that are contradictory to the propositions contained in the words themselves. Were left in no doubt as to whos ambitious and whos honourable, the literal truth of whats written clashes with the perceived truth of whats meant to revealing effect, which is irony in a nutshell. Ironic similes are a form of verbal irony where a speaker intends to communicate the opposite of what they mean, a fair amount of confusion has surrounded the issue of the relationship between verbal irony and sarcasm. Fowlers A Dictionary of Modern English Usage states, Sarcasm does not necessarily involve irony and irony has often no touch of sarcasm and this suggests that the two concepts are linked but may be considered separately. 2 a, a mode of satirical wit depending for its effect on bitter, caustic, the psychologist Martin, in The Psychology of Humour, is quite clear that irony is where the literal meaning is opposite to the intended and sarcasm is aggressive humor that pokes fun
9.
Self-reference
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Self-reference occurs in natural or formal languages when a sentence, idea or formula refers to itself. The reference may be expressed either directly—through some intermediate sentence or formula—or by means of some encoding. In philosophy, it refers to the ability of a subject to speak of or refer to itself, to have the kind of thought expressed by the first person nominative singular pronoun. Self-reference is studied and has applications in mathematics, philosophy, computer programming, self-referential statements are sometimes paradoxical, and can also be considered recursive. Self-reference occurs in literature and film when an author refers to his or her own work in the context of the work itself and this is closely related to the concepts of breaking the fourth wall and meta-reference, which often involve self-reference. The surrealist painter René Magritte is famous for his self-referential works, in computer science, self-reference occurs in reflection, where a program can read or modify its own instructions like any other data. Numerous programming languages support reflection to some extent with varying degrees of expressiveness, additionally, self-reference is seen in recursion, where a code structure refers back to itself during computation. A word that describes itself is called an autological word, a sentence which inventories its own letters and punctuation marks is called an autogram. There is a case of meta-sentence in which the content of the sentence in the metalanguage. Such a sentence is referring to itself, however some meta-sentences of this type can lead to paradoxes. Can be considered to be a self-referential meta-sentence which is obviously true, however This sentence is false is a meta-sentence which leads to a self-referential paradox. Hofstadters law, which specifies that It always takes longer than you expect, fumblerules are a list of rules of good grammar and writing, demonstrated through sentences that violate those very rules, such as Avoid cliches like the plague and Dont use no double negatives. The term was coined in a published list of rules by William Safire. Bartlett, Steven J. Reflexivity, A Source-book in Self-reference, rePub, Erasmus University Hofstadter, D. R. Gödel, Escher, Bach, an Eternal Golden Braid
10.
Infinite regress
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An infinite regress in a series of propositions arises if the truth of proposition P1 requires the support of proposition P2, the truth of proposition P2 requires the support of proposition P3. The truth of proposition Pn−1 requires the support of proposition Pn, distinction is made between infinite regresses that are vicious and those that are not. Others think there is, but that all truths are demonstrable, neither doctrine is either true or a necessary deduction from the premises. Our own doctrine is not all knowledge is demonstrative, on the contrary. Such, then, is our doctrine, and in addition we maintain that besides scientific knowledge there is its original source which enables us to recognize the definitions. Infinite regress is akin to the phenomenon wherein an infinite sequence of receding images is formed between two parallel facing mirrors. A similar process occurs in optical feedback
11.
Circular definition
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A circular definition is one that uses the term being defined as a part of the definition or assumes a prior understanding of the term being defined. There are several kinds of circular definition, and several ways of characterising the term, such definitions lead to a need for additional information that motivated someone to look at the definition in the first place and, thus, violate the principle of providing new or useful information. Here are some examples, Suppose we define oak as a tree which has catkins and grows from an acorn, to someone who does not know which trees are oaks, nor which nuts are acorns, the definition is inadequate. If someone wants to know what a phone is, telling them that it is a phone that is cellular will not be especially illuminating. Much more helpful would be to explain the concept of a cell in the context of telecommunications, similarly, defining dialectical materialism as materialism that involves dialectic is unhelpful. Consequently, when constructing systems of definitions, authors should use good practices that avoid producing viciously circular definitions, in many learners dictionaries, circular definitions are greatly reduced by writing definitions using only the words in a constrained defining vocabulary. The circular definition object, a thing → thing, an entity → entity, the classic genus-difference dictionary definition is in terms of nearest kind and specific differences. This genus-difference description may be involved in producing circular definitions of part and kind relationships, for example, rake, tine, a sharp spike at the end of a rake. In practice, an approach is often taken in considering the effects of circularity in dictionary definitions. From a linguistic point of view, some intuitively circular definitions in the derivation of words can easily be shown to be non-circular, for example, sometimes a definition like musicality, the quality or state of being musical is said to be circular. But strictly speaking, the condition the term being defined as a part of the definition or assumes a prior understanding of the term being defined is false in this case, a definition chain which ends is not circular. In each case, the terms to be defined and the terms in the definition are different, formal approaches to characterizing circular definitions are found in logic, mathematics and in computer science. A branch of mathematics called non-well-founded set theory allows for the construction of circular sets, circular sets are good for modelling cycles and, despite the fields name, this area of mathematics is well founded. Computer science allows for procedures to be defined by using recursion, such definitions are not circular as long as they terminate. Dictionary entries are given as examples of apparent circular definitions. Thus, the example of a definition of oak given above is not completely useless, even if acorn and catkin are defined in terms of oak, in that it supplies additional concepts in the definition. While a dictionary might produce a circle among the terms, oak, catkin, in another case it might produce a true circle. Taken as a whole, dictionaries are circular because each and every word is defined in terms of words that are contained within the dictionary
12.
Abstraction
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An abstraction is the product of this process — a concept that acts as a super-categorical noun for all subordinate concepts, and connects any related concepts as a group, field, or category. Conceptual abstractions may be formed by filtering the information content of a concept or an observable phenomenon, in a type–token distinction, a type is more abstract than its tokens. Abstraction in its use is a material process, discussed in the themes below. Its development is likely to have been connected with the development of human language. Abstraction involves induction of ideas or the synthesis of particular facts into one theory about something. It is the opposite of specification, which is the analysis or breaking-down of an idea or abstraction into concrete facts. Thales believed that everything in the universe comes from one main substance and he deduced or specified from a general idea, everything is water, to the specific forms of water such as ice, snow, fog, and rivers. Modern scientists can use the opposite approach of abstraction, or going from particular facts collected into one general idea. This conceptual scheme emphasizes the inherent equality of both constituent and abstract data, thus avoiding problems arising from the distinction between abstract and concrete, in this sense the process of abstraction entails the identification of similarities between objects, and the process of associating these objects with an abstraction. For example, picture 1 below illustrates the concrete relationship Cat sits on Mat, for example, graph 1 below expresses the abstraction agent sits on location. This conceptual scheme entails no specific hierarchical taxonomy, only a progressive exclusion of detail, things that do not exist at any particular place and time are often considered abstract. By contrast, instances, or members, of such a thing might exist in many different places and times. Those abstract things are said to be multiply instantiated, in the sense of picture 1, picture 2. It is not sufficient, however, to abstract ideas as those that can be instantiated. Although the concepts cat and telephone are abstractions, they are not abstract in the sense of the objects in graph 1 below, perhaps confusingly, some philosophies refer to tropes as abstract particulars — e. g. the particular redness of a particular apple is an abstract particular. This is similar to qualia and sumbebekos, karl Marxs writing on the commodity abstraction recognizes a parallel process. The state as both concept and material practice exemplifies the two sides of this process of abstraction, conceptually, the current concept of the state is an abstraction from the much more concrete early-modern use as the standing or status of the prince, his visible estates. At the same time, materially, the practice of statehood is now constitutively and materially more abstract than at the time when princes ruled as the embodiment of extended power and that difference accounts for the ontological usefulness of the word abstract
13.
Patrick Hughes (artist)
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Patrick Hughes is a British artist working in London. He is the creator of reverspective, an illusion on a 3-dimensional surface where the parts of the picture which seem farthest away are actually physically the nearest. Patrick Hughes was born in Birmingham, went to school in Hull, later he taught at the Leeds College of Art before becoming an independent artist. He has three sons by his first wife, Rennie Paterson, and was married to the author Molly Parkin. Hughes lives above his studio near Old Street, London, with his third wife and he has been represented by Angela Flowers for more than forty years. In July 2011, Hughes celebrated Fifty Years in Showbusiness with two exhibitions, a retrospective at Flowers East, and current works in Flowers Cork Street and he explored visual oxymorons and paradoxes. His fascination with the illusion of perspective began with works like Infinity, Three Doors, later prints like Leaf Art and paintings like Realistic Paint expressed similar interests with colour. His first reverse perspective or reverspective was Sticking Out Room, which was a room for the Institute of Contemporary Arts in 1970. His work entitled Paradoxymoron is currently exhibited in the British Library and it shows a set of library book stacks, which appear to move in an extremely disconcerting way as the viewers eyes move. The two lighter rectangles which appear to be in the distance at the end of the buildings are the flat tops, Hughes reverspective is the subject of scientific papers on the psychology of perception, by Nicholas Wade and Thomas Papathomas of Rutgers Universitys Laboratory of Vision Research. Hughes has written four books investigating themes that parallel his art and his latest is Paradoxymoron, Foolish Wisdom in Words and Pictures, published in 2011. He has written for The Observer, The Guardian, and ICA Magazine, among others, on art, artists, a collection of his writings, Left to Write, was published by Momentum in 2008. The third edition of John Slyces Patrick Hughes, Perverspective was published in 2011, the Leeds-based surrealist Anthony Earnshaw was a friend and inspiration. Exhibition Catalogues Patrick Hughes, Deeperspective London, Patrick Hughes, The Prints in Between 1983 –1995 London. Monographs Ades, D. Kemp, M, Papathomas, T. et al, reverspective Ltd, London Hughes, P. Paradoxymoron, Foolish Wisdom in Words and Pictures. Slyce, J. Patrick Hughes, Perverspective Momentum, London, official web site Flowers East gallery Patrick Hughes work in the Tate gallery
14.
Liar paradox
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In philosophy and logic, the classical liar paradox or liars paradox is the statement of a liar who states that they are lying, for instance, declaring that I am lying or everything I say is false. If they are indeed lying, they are telling the truth, in this sentence is a lie the paradox is strengthened in order to make it amenable to more rigorous logical analysis. It is still called the liar paradox although abstraction is made precisely from the liar themself. Trying to assign to this statement, the liar, a classical binary truth value leads to a contradiction. If this sentence is false is true, then the sentence is false, but if the states that it is false, and it is false, then it must be true. The Epimenides paradox has been suggested as an example of the liar paradox, the semi-mythical seer Epimenides, a Cretan, reportedly stated that All Cretans are liars. However, Epimenides statement that all Cretans are liars can be resolved as false and it is precisely in order to avoid uncertainties deriving from the human factor and from fuzzy concepts that modern logicians proposed a strengthened liar such as the sentence this sentence is false. The paradoxs name translates as pseudómenos lógos in Ancient Greek, one version of the liar paradox is attributed to the Greek philosopher Eubulides of Miletus who lived in the 4th century BC. Eubulides reportedly asked, A man says that he is lying, is what he says true or false. The paradox was discussed by St. Jerome in a sermon, I said in my alarm. Is David telling the truth or is he lying, if it is true that every man is a liar, and Davids statement, Every man is a liar is true, then David also is lying, he, too, is a man. But if he, too, is lying, his statement, Every man is a liar, whatever way you turn the proposition, the conclusion is a contradiction. Since David himself is a man, it follows that he also is lying, the Indian grammarian-philosopher Bhartrhari was well aware of a liar paradox which he formulated as everything I am saying is false. He analyzes this paradox together with the paradox of unsignifiability and explores the boundary between statements that are unproblematic in daily life and paradoxes, in early Islamic tradition liar paradox was discussed for at least five centuries starting from late 9th century apparently without being influenced by any other tradition. Naṣīr al-Dīn al-Ṭūsī could have been the first logician to identify the liar paradox as self-referential, the problem of the liar paradox is that it seems to show that common beliefs about truth and falsity actually lead to a contradiction. Sentences can be constructed that cannot consistently be assigned a value even though they are completely in accord with grammar. The simplest version of the paradox is the sentence, This statement is false, if is true, then This statement is false is true. The hypothesis that is true leads to the conclusion that is false, if is false, then This statement is false is false
15.
Barber
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A barber is a person whose occupation is mainly to cut, dress, groom, style and shave mens and boys hair. A barbers place of work is known as a shop or a barbers. Barber shops are also places of social interaction and public discourse, in some instances, barbershops are also public forums. They are the locations of open debates, voicing public concerns and they were also influential in helping shape male identity. In previous times, barbers performed surgery and dentistry. With the development of safety razors and the prevalence of beards, in English-speaking cultures. In modern times, the barber is used both as a professional title and to refer to hairdressers who specialize in mens hair. Historically, all hairdressers were considered barbers, in the 20th century, the profession of cosmetology branched off from barbering, and today hairdressers may be licensed as either barbers or cosmetologists. Barbers differ with respect to where they work, which services they are licensed to provide, part of this terminology difference depends on the regulations in a given location. Different states in the US vary on their labor and licensing laws, for example, in Maryland, a cosmetologist cannot use a straight razor, strictly reserved for barbers. In Australia, the term for a barber is hairdresser, barber is only a popular title for mens hairdressers. Most would work in a hairdressing salon, the barbers trade has a long history, razors have been found among relics of the Bronze Age in Egypt. In ancient Egyptian culture, barbers were highly respected individuals, priests and men of medicine are the earliest recorded examples of barbers. Men in Ancient Greece would have their beards, hair, and fingernails trimmed and styled by the κουρεύς, in an agora which also served as a gathering for debates. Barbering was introduced to Rome by the Greek colonies in Sicily in 296 BC, starting from the Middle Ages, barbers often served as surgeons and dentists. In addition to haircutting, hairdressing, and shaving, barbers performed surgery, bloodletting and leeching, fire cupping, enemas, the barber pole, featuring red and white spiraling stripes, symbolized different aspects of the craft. Barbers received higher pay than surgeons until surgeons were entered into British warships during naval wars, some of the duties of the barber included neck manipulation, cleansing of ears and scalp, draining of boils, fistula and lancing of cysts with wicks. Barbershops were influential at the turn of the 19th century as African American businesses that helped to develop African American culture and economy
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Barber paradox
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The barber paradox is a puzzle derived from Russells paradox. It was used by Bertrand Russell himself as an illustration of the paradox and it shows that an apparently plausible scenario is logically impossible. Specifically, it describes a barber who is defined such that he shaves himself and does not shave himself. The barber is the one who shaves all those, and those only, the question is, does the barber shave himself. Answering this question results in a contradiction, the barber cannot shave himself as he only shaves those who do not shave themselves. As such, if he shaves himself he ceases to be a barber, conversely, if the barber does not shave himself, then he fits into the group of people who would be shaved by the barber, and thus, as the barber, he must shave himself. Despite its popular name, however, the paradox is not really a paradox in the true sense of this word. A man who shaves exactly those men who do not shave themselves simply cannot and does not exist and this paradox is often attributed to Bertrand Russell. It was suggested to him as a form of Russells paradox. However, Russell denied that the Barbers paradox was an instance of his own and this sentence is unsatisfiable because of the universal quantifier. The universal quantifier y will include every element in the domain. So when the x is assigned to y, the sentence can be rewritten to shaves ↔ ¬ shaves. In Prolog, one aspect of the paradox can be expressed by a self-referencing clause. If we apply the stratification test known from Datalog, the predicate shaves is exposed as unstratifiable since it is defined recursively over its negation, in his book Alice in Puzzleland, the logician Raymond Smullyan had the character Humpty Dumpty explain the apparent paradox to Alice. A paradox is supposed to arise from plausible and apparently consistent statements, the paradox is also mentioned several times in David Foster Wallaces first novel, The Broom of the System, as well as The Information by James Gleick. If the paradox is altered so there may be multiple barbers in the town, then the paradox may or may not be resolved. If the initial state that every man in town must keep himself clean-shaven, either by Shaving himself. Each barber can be shaved by another barber, however, if the initial rules describe the responsibilities of the barbers rather than the towns residents in general, then the paradox remains
17.
Virtuous circle and vicious circle
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The terms virtuous circle and vicious circle refer to complex chains of events which reinforce themselves through a feedback loop. A virtuous circle has favorable results, while a circle has detrimental results. Both circles are complexes of events with no tendency towards equilibrium, both systems of events have feedback loops in which each iteration of the cycle reinforces the previous one. These cycles will continue in the direction of their momentum until an external factor intervenes, the prefix hyper- is sometimes used to describe these cycles if they are extreme. The best-known example of a circle is hyperinflation. The contemporary subprime mortgage crisis is a complex of vicious circles, a specific example is the circle related to housing. As housing prices decline, more homeowners go underwater, when the value of a home drops below the mortgage on it. This provides an incentive to walk away from the home, increasing defaults and this, in turn, lowers housing values further from over-supply, reinforcing the cycle. The foreclosures reduce the cash flowing into banks and the value of mortgage-backed securities widely held by banks, banks incur losses and require additional funds, also called “recapitalization”. If banks are not capitalized sufficiently to lend, economic activity slows and unemployment increases, economist Nouriel Roubini described the vicious circles within and across the housing market and financial markets during interviews with Charlie Rose in September and October 2008. Other examples include the poverty cycle, sharecropping, and the intensification of drought, in climate change science, feedback loops involve positive feedbacks and negative feedbacks that respectively serve to intensify or dampen the effects of global warming. And Heskett, J. Breaking the cycle of failure in services, Sloan Management Review,31, spring 1991, pp.17 –28. Http, //william-king. www. drexel. edu/top/prin/txt/gro/gro21b. html – An introduction to 20th century virtuous circle theory, china, A Stabilizing or Deflationary Influence in East Asia. The Problem of Conflicted Virtue – In this link the author is using virtue in the sense of an outcome that conflicts with long term regional growth. Http, //www. bioenergycentres. com/resources. shtml – an essay in 3 parts by Polarity Therapist Andrew Harry, Bath, defining The Virtuous Cycle as an integrated healing process
18.
Howler (error)
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A howler is a glaring blunder, typically an amusing one. The Oxford English Dictionary defines howler,3.3 slang, something crying, clamant, or excessive, spec. A glaring blunder, esp. in an examination, etc. eric Partridges Dictionary of Slang and Unconventional English says, the 1951 edition of Partridge defined it in part as. A glaring blunder, from before 1890, literally something that howls or cries for notice, or perhaps. By way of contracting howling blunder, another common interpretation of this usage is that a howler is a mistake fit to make one howl with laughter. All over the world, probably in all languages, there are many informal terms for blunders. Boner is another colloquialism that means much the same as howler in the context of this article, for one thing, boner is not traditionally used as a general intensifier or for specifically describing an accident or the like, as howler and howling are. Assorted other terms have much longer histories and some of them are not regarded as slang, for example, Bull and Blunder have long been used in similar senses, each with its own overtones and assorted extraneous meanings. Bulls and Blunders, an American book published in the 1890s, uses the word howler only once, in the passage, graham, of Annerley, has received a prize from the University Correspondent for the best collection of schoolboy howlers. Although he did not otherwise use the word himself, the author did not define a term so familiar on both sides of the Atlantic even at that time. Mathematicians sometimes speak of howlers, mainly in the form of an error which leads innocently, terms related to howlers and fallacies include sophism, in which an error is wilfully concealed, whether for didactic purposes or for entertainment. In one sense the converse of either a howler or a sophism is a mathematical paradox, however, in the terminology of Willard V. O. Quine, that would be a veridical paradox, whereas sophisms and fallacies would be falsidical paradoxes. As such, a howler could be a blunder in any field of knowledge. In the short story by Eden Philpotts Doctor Dunstons Howler, the howler in question was not even verbal, it was flogging the wrong boy, with disastrous consequences. Conversely, on inspection of many examples of bulls and howlers, they simply may be the products of unfortunate wording, punctuation, schoolboy howlers in particular sometimes amount to what Richard Feynman called Perfectly reasonable deviations from the beaten track. Such specimens may variously be based on mondegreens, or they might be derived from misunderstandings of fact by the elders, teachers or communities, not all howlers originate with the pupil. As illustrated, terms such as howler need not specify the discipline in which the blunder was perpetrated, howlers have little special application to any particular field, except perhaps education. Most collections refer mainly to the schoolboy howler, politicians howler, epitaph howler, judicial howler, sometimes such lampooning is fairly good-humoured, sometimes it is deliberately used as a political weapon
19.
Surgery
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An act of performing surgery may be called a surgical procedure, operation, or simply surgery. In this context, the verb means to perform surgery. The adjective surgical means pertaining to surgery, e. g. surgical instruments or surgical nurse, the patient or subject on which the surgery is performed can be a person or an animal. A surgeon is a person who practices surgery and an assistant is a person who practices surgical assistance. A surgical team is made up of surgeon, surgeons assistant, anesthesia provider, circulating nurse, Surgery usually spans minutes to hours, but it is typically not an ongoing or periodic type of treatment. The term surgery can also refer to the place where surgery is performed, or simply the office of a physician, dentist, Surgery is a technology consisting of a physical intervention on tissues. As a general rule, a procedure is considered surgical when it involves cutting of a patients tissues or closure of a previously sustained wound, surgical procedures are commonly categorized by urgency, type of procedure, body system involved, degree of invasiveness, and special instrumentation. Based on timing, Elective surgery is done to correct a condition, and is carried out at the patients request, subject to the surgeons. A semi-elective surgery is one that must be done to avoid permanent disability or death, emergency surgery is surgery which must be done promptly to save life, limb, or functional capacity. Based on purpose, Exploratory surgery is performed to aid or confirm a diagnosis, therapeutic surgery treats a previously diagnosed condition. Cosmetic surgery is done to improve the appearance of an otherwise normal structure. By type of procedure, Amputation involves cutting off a part, usually a limb or digit. Resection is the removal of all or part of an organ or part of the body. Replantation involves reattaching a severed body part, reconstructive surgery involves reconstruction of an injured, mutilated, or deformed part of the body. Excision is the cutting out or removal of an organ, tissue, transplant surgery is the replacement of an organ or body part by insertion of another from different human into the patient. Removing an organ or body part from a human or animal for use in transplant is also a type of surgery. By body part, When surgery is performed on one system or structure. Examples include cardiac surgery, gastrointestinal surgery, and orthopedic surgery, by contrast, an open surgical procedure such as a laparotomy requires a large incision to access the area of interest
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Language
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Language is the ability to acquire and use complex systems of communication, particularly the human ability to do so, and a language is any specific example of such a system. The scientific study of language is called linguistics, questions concerning the philosophy of language, such as whether words can represent experience, have been debated since Gorgias and Plato in Ancient Greece. Thinkers such as Rousseau have argued that language originated from emotions while others like Kant have held that it originated from rational and logical thought, 20th-century philosophers such as Wittgenstein argued that philosophy is really the study of language. Major figures in linguistics include Ferdinand de Saussure and Noam Chomsky, estimates of the number of languages in the world vary between 5,000 and 7,000. However, any precise estimate depends on an arbitrary distinction between languages and dialects. Natural languages are spoken or signed, but any language can be encoded into secondary media using auditory, visual, or tactile stimuli – for example, in whistling, signed and this is because human language is modality-independent. All languages rely on the process of semiosis to relate signs to particular meanings, human language has the properties of productivity and displacement, and relies entirely on social convention and learning. Its complex structure affords a wider range of expressions than any known system of animal communication. Language is processed in different locations in the human brain. Humans acquire language through interaction in early childhood, and children generally speak fluently when they are approximately three years old. The use of language is deeply entrenched in human culture, a group of languages that descend from a common ancestor is known as a language family. The languages of the Dravidian family that are mostly in Southern India include Tamil. Academic consensus holds that between 50% and 90% of languages spoken at the beginning of the 21st century will probably have become extinct by the year 2100. The English word language derives ultimately from Proto-Indo-European *dn̥ǵʰwéh₂s tongue, speech, language through Latin lingua, language, tongue, and Old French language. The word is used to refer to codes, ciphers. Unlike conventional human languages, a language in this sense is a system of signs for encoding and decoding information. This article specifically concerns the properties of human language as it is studied in the discipline of linguistics. As an object of study, language has two primary meanings, an abstract concept, and a specific linguistic system, e. g. French
21.
Logic
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Logic, originally meaning the word or what is spoken, is generally held to consist of the systematic study of the form of arguments. A valid argument is one where there is a relation of logical support between the assumptions of the argument and its conclusion. Historically, logic has been studied in philosophy and mathematics, and recently logic has been studied in science, linguistics, psychology. The concept of form is central to logic. The validity of an argument is determined by its logical form, traditional Aristotelian syllogistic logic and modern symbolic logic are examples of formal logic. Informal logic is the study of natural language arguments, the study of fallacies is an important branch of informal logic. Since much informal argument is not strictly speaking deductive, on some conceptions of logic, formal logic is the study of inference with purely formal content. An inference possesses a purely formal content if it can be expressed as an application of a wholly abstract rule, that is. The works of Aristotle contain the earliest known study of logic. Modern formal logic follows and expands on Aristotle, in many definitions of logic, logical inference and inference with purely formal content are the same. This does not render the notion of informal logic vacuous, because no formal logic captures all of the nuances of natural language, Symbolic logic is the study of symbolic abstractions that capture the formal features of logical inference. Symbolic logic is divided into two main branches, propositional logic and predicate logic. Mathematical logic is an extension of logic into other areas, in particular to the study of model theory, proof theory, set theory. Logic is generally considered formal when it analyzes and represents the form of any valid argument type, the form of an argument is displayed by representing its sentences in the formal grammar and symbolism of a logical language to make its content usable in formal inference. Simply put, formalising simply means translating English sentences into the language of logic and this is called showing the logical form of the argument. It is necessary because indicative sentences of ordinary language show a variety of form. Second, certain parts of the sentence must be replaced with schematic letters, thus, for example, the expression all Ps are Qs shows the logical form common to the sentences all men are mortals, all cats are carnivores, all Greeks are philosophers, and so on. The schema can further be condensed into the formula A, where the letter A indicates the judgement all - are -, the importance of form was recognised from ancient times
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Philosopher
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A philosopher is someone who practices philosophy, which involves rational inquiry into areas that are outside of either theology or science. The term philosopher comes from the Ancient Greek φιλόσοφος meaning lover of wisdom, the coining of the term has been attributed to the Greek thinker Pythagoras. Typically, these brands of philosophy are Hellenistic ones and those who most arduously commit themselves to this lifestyle may be considered philosophers. The separation of philosophy and science from theology began in Greece during the 6th century BC, thales, an astronomer and mathematician, was considered by Aristotle to be the first philosopher of the Greek tradition. While Pythagoras coined the word, the first known elaboration on the topic was conducted by Plato, in his Symposium, he concludes that Love is that which lacks the object it seeks. Therefore, the philosopher is one who seeks wisdom, if he attains wisdom, therefore, the philosopher in antiquity was one who lives in the constant pursuit of wisdom, and living in accordance to that wisdom. Disagreements arose as to what living philosophically entailed and these disagreements gave rise to different Hellenistic schools of philosophy. In consequence, the ancient philosopher thought in a tradition, as the ancient world became schism by philosophical debate, the competition lay in living in manner that would transform his whole way of living in the world. Philosophy is a discipline which can easily carry away the individual in analyzing the universe. The second is the change through the Medieval era. With the rise of Christianity, the way of life was adopted by its theology. Thus, philosophy was divided between a way of life and the conceptual, logical, physical and metaphysical materials to justify that way of life, philosophy was then the servant to theology. The third is the sociological need with the development of the university, the modern university requires professionals to teach. Maintaining itself requires teaching future professionals to replace the current faculty, therefore, the discipline degrades into a technical language reserved for specialists, completely eschewing its original conception as a way of life. In the fourth century, the word began to designate a man or woman who led a monastic life. Gregory of Nyssa, for example, describes how his sister Macrina persuaded their mother to forsake the distractions of life for a life of philosophy. Later during the Middle Ages, persons who engaged with alchemy was called a philosopher - thus, many philosophers still emerged from the Classical tradition, as saw their philosophy as a way of life. Among the most notable are René Descartes, Baruch Spinoza, Nicolas Malebranche, with the rise of the university, the modern conception of philosophy became more prominent
23.
Set (mathematics)
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In mathematics, a set is a well-defined collection of distinct objects, considered as an object in its own right. For example, the numbers 2,4, and 6 are distinct objects when considered separately, Sets are one of the most fundamental concepts in mathematics. Developed at the end of the 19th century, set theory is now a part of mathematics. In mathematics education, elementary topics such as Venn diagrams are taught at a young age, the German word Menge, rendered as set in English, was coined by Bernard Bolzano in his work The Paradoxes of the Infinite. A set is a collection of distinct objects. The objects that make up a set can be anything, numbers, people, letters of the alphabet, other sets, Sets are conventionally denoted with capital letters. Sets A and B are equal if and only if they have precisely the same elements. Cantors definition turned out to be inadequate, instead, the notion of a set is taken as a notion in axiomatic set theory. There are two ways of describing, or specifying the members of, a set, one way is by intensional definition, using a rule or semantic description, A is the set whose members are the first four positive integers. B is the set of colors of the French flag, the second way is by extension – that is, listing each member of the set. An extensional definition is denoted by enclosing the list of members in curly brackets, one often has the choice of specifying a set either intensionally or extensionally. In the examples above, for instance, A = C and B = D, there are two important points to note about sets. First, in a definition, a set member can be listed two or more times, for example. However, per extensionality, two definitions of sets which differ only in one of the definitions lists set members multiple times, define, in fact. Hence, the set is identical to the set. The second important point is that the order in which the elements of a set are listed is irrelevant and we can illustrate these two important points with an example, = =. For sets with many elements, the enumeration of members can be abbreviated, for instance, the set of the first thousand positive integers may be specified extensionally as, where the ellipsis indicates that the list continues in the obvious way. Ellipses may also be used where sets have infinitely many members, thus the set of positive even numbers can be written as
24.
Thought experiment
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A thought experiment considers some hypothesis, theory, or principle for the purpose of thinking through its consequences. Given the structure of the experiment, it may not be possible to perform it, perhaps the key experiment in the history of modern science is Galileos demonstration that falling objects must fall at the same rate regardless of their masses. The experiment is described by Galileo in Discorsi e dimostrazioni matematiche thus, do you not agree with me in this opinion. Hence the heavier body moves with less speed than the lighter, thus you see how, from your assumption that the heavier body moves more rapidly than the lighter one, I infer that the heavier body moves more slowly. Although the extract does not convey the elegance and power of the demonstration terribly well, it is clear that it is a thought experiment, instead, many philosophers prefer to consider Thought Experiments to be merely the use of a hypothetical scenario to help understand the way things are. Thought experiments have used in a variety of fields, including philosophy, law, physics. In philosophy, they have used at least since classical antiquity. In law, they were well-known to Roman lawyers quoted in the Digest, in physics and other sciences, notable thought experiments date from the 19th and especially the 20th century, but examples can be found at least as early as Galileo. Johann Witt-Hansen established that Hans Christian Ørsted was the first to use the Latin-German mixed term Gedankenexperiment circa 1812, Ørsted was also the first to use its entirely German equivalent, Gedankenversuch, in 1820. The English term thought experiment was coined from Machs Gedankenexperiment, prior to its emergence, the activity of posing hypothetical questions that employed subjunctive reasoning had existed for a very long time. However, people had no way of categorizing it or speaking about it and this helps to explain the extremely wide and diverse range of the application of the term thought experiment once it had been introduced into English. In physics and other sciences many thought experiments date from the 19th and especially the 20th Century, in Galileo’s thought experiment, for example, the rearrangement of empirical experience consists in the original idea of combining bodies of different weight. Thought experiments have used in philosophy, physics, and other fields. In law, the hypothetical is frequently used for such experiments. Regardless of their goal, all thought experiments display a patterned way of thinking that is designed to allow us to explain, predict and control events in a better. However, they may make those theories themselves irrelevant, and could create new problems that are just as difficult. Ensure the avoidance of past failures Scientists tend to use thought experiments as imaginary, in these cases, the result of the proxy experiment will often be so clear that there will be no need to conduct a physical experiment at all. Scientists also use thought experiments when particular physical experiments are impossible to conduct, such as Einsteins thought experiment of chasing a light beam, leading to Special Relativity
25.
Time travel
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Some ancient myths depict skipping forward in time. The Buddhist Pāli Canon mentions the relativity of time, in the period of our century, one hundred years, only a single day, twenty four hours would have passed for them. The Japanese tale of Urashima Tarō, first described in the Nihongi tells of a fisherman named Urashima Taro who visits an undersea palace. After three days, he returns home to his village and finds himself 300 years in the future, where he has forgotten, his house is in ruins. Early science-fiction stories feature characters who sleep for years and awaken in a changed society. Among them LAn 2440, rêve sil en fût jamais by Louis-Sébastien Mercier, Rip Van Winkle by Washington Irving, Looking Backward by Edward Bellamy, prolonged sleep, like the more familiar time machine, is used as a means of time travel in these stories. Decades-long and centuries-long sleep is featured in ancient myths. Like forward time travel, backward time travel has an uncertain origin, the narrator rides to ancient Greece on a hippogriff, meets Aristotle, and goes on a voyage with Alexander the Great before returning to the 19th century. While the narrator waits under a tree for a coach to take him out of Newcastle and he encounters the Venerable Bede in a monastery and explains to him the developments of the coming centuries. However, the story never makes it clear whether these events are real or a dream, charles Dickenss A Christmas Carol has early depictions of time travel in both directions, as the protagonist, Ebenezer Scrooge, is transported to Christmases past and future. However, these might be interpreted as visions rather than as time travel because Scrooge experiences the time periods as a rather than as a participant. A clearer example of time travel is found in the popular 1861 book Paris avant les hommes by the French botanist and geologist Pierre Boitard. This may have been the first story to feature an alternate history created as a result of time travel. Early time machines One of the first stories to feature time travel by means of a machine is The Clock that Went Backward by Edward Page Mitchell, however, the mechanism borders on fantasy. An unusual clock, when wound, runs backwards and transports people nearby back in time, the author does not explain the origin or properties of the clock. Enrique Gaspar y Rimbaus El Anacronópete may have been the first story to feature a vessel engineered to travel through time, H. G. Wellss The Time Machine popularized the concept of time travel by mechanical means. In technical papers, physicists generally avoid the language of moving or traveling through time. Movement normally refers only to a change in position as the time coordinate is varied
26.
Butterfly effect
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The butterfly effect is the concept that small causes can have large effects. Initially, it was used with weather prediction but later the term became a used in. A very small change in initial conditions had created a different outcome. The idea that small causes may have effects in general. The butterfly effect is exhibited by very simple systems, in The Vocation of Man, Fichte says that you could not remove a single grain of sand from its place without thereby. Changing something throughout all parts of the immeasurable whole, Chaos theory and the sensitive dependence on initial conditions were described in the literature in a particular case of the three-body problem by Henri Poincaré in 1890. He later proposed that such phenomena could be common, for example, in 1898, Jacques Hadamard noted general divergence of trajectories in spaces of negative curvature. Pierre Duhem discussed the general significance of this in 1908. In 1961, Lorenz was running a computer model to redo a weather prediction from the middle of the previous run as a shortcut. He entered the initial condition 0.506 from the printout instead of entering the full precision 0.506127 value, the result was a completely different weather scenario. In 1963 Lorenz published a study of this effect in a highly cited. Elsewhere he stated, One meteorologist remarked that if the theory were correct, the controversy has not yet been settled, but the most recent evidence seems to favor the sea gulls. Following suggestions from colleagues, in speeches and papers Lorenz used the more poetic butterfly. The flapping wing represents a change in the initial condition of the system. The butterfly effect presents a challenge to prediction, since initial conditions for a system such as the weather can never be known to complete accuracy. This problem motivated the development of ensemble forecasting, in which a number of forecasts are made from perturbed initial conditions, some scientists have since argued that the weather system is not as sensitive to initial conditions as previously believed. David Orrell argues that the contributor to weather forecast error is model error. Recurrence, the return of a system towards its initial conditions
27.
Willard Van Orman Quine
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Willard Van Orman Quine was an American philosopher and logician in the analytic tradition, recognized as one of the most influential philosophers of the twentieth century. He filled the Edgar Pierce Chair of Philosophy at Harvard from 1956 to 1978, a 2009 poll conducted among analytic philosophers named Quine as the fifth most important philosopher of the past two centuries. This led to his famous quip that philosophy of science is philosophy enough, in philosophy of mathematics, he and his Harvard colleague Hilary Putnam developed the Quine–Putnam indispensability thesis, an argument for the reality of mathematical entities. According to his autobiography, The Time of My Life, Quine grew up in Akron, Ohio and his father, Cloyd R. was a manufacturing entrepreneur and his mother, Harriett E. was a schoolteacher and later a housewife. He received his B. A. in mathematics from Oberlin College in 1930 and his thesis supervisor was Alfred North Whitehead. He was then appointed a Harvard Junior Fellow, which excused him from having to teach for four years. During the academic year 1932–33, he travelled in Europe thanks to a Sheldon fellowship, meeting Polish logicians and members of the Vienna Circle and it was through Quines good offices that Tarski was invited to attend the September 1939 Unity of Science Congress in Cambridge. To attend that Congress, Tarski sailed for the USA on the last ship to leave Danzig before the Third Reich invaded Poland, Tarski survived the war and worked another 44 years in the USA. For the academic year 1964–1965, Quine was a fellow on the faculty in the Center for Advanced Studies at Wesleyan University, in 1980 Quine received an honorary doctorate from the Faculty of Humanities at Uppsala University, Sweden. Quine was an atheist when he was a teenager and he had four children by two marriages. Guitarist Robert Quine was his nephew, Quine was politically conservative, but the bulk of his writing was in technical areas of philosophy removed from direct political issues. Quines Ph. D. thesis and early publications were on formal logic, only after World War II did he, by virtue of seminal papers on ontology, epistemology and language, emerge as a major philosopher. By the 1960s, he had worked out his naturalized epistemology whose aim was to answer all questions of knowledge and meaning using the methods. Quine roundly rejected the notion that there should be a first philosophy and these views are intrinsic to his naturalism. Quine could lecture in French, Spanish, Portuguese and German, like the logical positivists, Quine evinced little interest in the philosophical canon, only once did he teach a course in the history of philosophy, on Hume. This distinction was central to logical positivism, like other Analytic philosophers before him, Quine accepted the definition of analytic as true in virtue of meaning alone. Unlike them, however, he concluded that ultimately the definition was circular, in other words, Quine accepted that analytic statements are those that are true by definition, then argued that the notion of truth by definition was unsatisfactory. Quines chief objection to analyticity is with the notion of synonymy, the objection to synonymy hinges upon the problem of collateral information
28.
The Pirates of Penzance
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The Pirates of Penzance, or, The Slave of Duty is a comic opera in two acts, with music by Arthur Sullivan and libretto by W. S. Gilbert. The operas official premiere was at the Fifth Avenue Theatre in New York City on 31 December 1879, where the show was well received by both audiences and critics. Its London debut was on 3 April 1880, at the Opera Comique, the story concerns Frederic, who, having completed his 21st year, is released from his apprenticeship to a band of tender-hearted pirates. He meets Mabel, the daughter of Major-General Stanley, and the two people fall instantly in love. Frederic soon learns, however, that he was born on the 29th of February and his indenture specifies that he remain apprenticed to the pirates until his twenty-first birthday, meaning that he must serve for another 63 years. Bound by his own sense of duty, Frederics only solace is that Mabel agrees to wait for him faithfully, Pirates was the fifth Gilbert and Sullivan collaboration and introduced the much-parodied Major-Generals Song. The opera was performed for over a century by the DOyly Carte Opera Company in Britain and by other opera companies. Pirates remains popular today, taking its place along with The Mikado, Pinafore as one of the most frequently played Gilbert and Sullivan operas. The Pirates of Penzance was the only Gilbert and Sullivan opera to have its premiere in the United States. At the time, American law offered no protection to foreigners. After the pairs previous opera, H. M. S, however, Gilbert, Sullivan, and their producer, Richard DOyly Carte, failed in their efforts, over the next decade, to control the American performance copyrights to Pirates and their other operas. Fiction and plays about pirates were ubiquitous in the 19th century, walter Scotts The Pirate and James Fenimore Coopers The Red Rover were key sources for the romanticised, dashing pirate image and the idea of repentant pirates. Both Gilbert and Sullivan had parodied these ideas early in their careers, Sullivan had written a comic opera called The Contrabandista, in 1867, about a hapless British tourist who is captured by bandits and forced to become their chief. Gilbert had written several works that involved pirates or bandits. In Gilberts 1876 opera Princess Toto, the character is eager to be captured by a brigand chief. Gilbert had translated Jacques Offenbachs operetta Les brigands, in 1871, as in Les brigands, The Pirates of Penzance absurdly treats stealing as a professional career path, with apprentices and tools of the trade such as the crowbar and life preserver. While Pinafore was running strongly at the Opera Comique in London, Gilbert was eager to get started on his and Sullivans next opera and he re-used several elements of his 1870 one-act piece, Our Island Home, which had introduced a pirate chief, Captain Bang. Bang was mistakenly apprenticed to a band as a child by his deaf nursemaid
29.
Leap day
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February 29, also known as leap day or leap year day, is a date added to most years that are divisible by 4, such as 2008,2012,2016,2020, and 2024. A leap day is added in various solar calendars, including the Gregorian calendar standard in most of the world, lunisolar calendars instead add a leap or intercalary month. In the Gregorian calendar, years that are divisible by 100, thus,1700,1800, and 1900 did not contain a leap day, neither will 2100,2200, and 2300. Conversely,1600 and 2000 did and 2400 will, Years containing a leap day are called leap years. Years not containing a leap day are called common years, February 29 is the 60th day of the Gregorian calendar, in such a year, with 306 days remaining until the end of the year. In the Chinese calendar, this day will only occur in years of the monkey, dragon, a leap day is observed because a complete revolution around the Sun takes approximately 6 hours longer than 365 days. A leap day compensates for this lag, realigning the calendar with the Earths position in the Solar System, otherwise, although most modern calendar years have 365 days, a complete revolution around the Sun takes approximately 365 days and 6 hours. An extra 24 hours thus accumulates every four years, requiring that a calendar day be added to align the calendar with the Suns apparent position. A solar year is slightly shorter than 365 days and 6 hours. The currently accepted figure is 365 days,5 hours,48 minutes,45 seconds. Adding a calendar day every four years, therefore, results in an excess of around 44 minutes for those four years, to compensate for this, three days are removed every 400 years. The Gregorian calendar reform implements this adjustment by making an exception to the rule that there is a leap year every four years. Instead, a year divisible by 100 is not a year unless that year is also divisible by 400. This means that the years 1600,2000, and 2400 are leap years, while the years 1700,1800,1900,2100,2200,2300, the Gregorian calendar repeats itself every 400 years, which is exactly 20,871 weeks including 97 leap days. Over this period, February 29 falls on Sunday, Tuesday, and Thursday 13 times each,14 times each on Friday and Saturday, the calendar of the Roman king Numa Pompilius had only 355 days which meant that it would quickly become unsynchronized with the solar year. An earlier Roman solution to this problem was to lengthen the calendar periodically by adding days to February. February consisted of two parts, each with an odd number of days, the first part ended with the Terminalia on the 23rd, which was considered the end of the religious year, and the five remaining days formed the second part. To keep the calendar year roughly aligned with the solar year, the second part of February was incorporated in the intercalary month as its last five days, with no change either in their dates or the festivals observed on them
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Arrow's impossibility theorem
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The theorem is often cited in discussions of voting theory as it is further interpreted by the Gibbard–Satterthwaite theorem. The theorem is named after economist Kenneth Arrow, who demonstrated the theorem in his thesis and popularized it in his 1951 book Social Choice. The original paper was titled A Difficulty in the Concept of Social Welfare, if every voters preference between X and Y remains unchanged, then the groups preference between X and Y will also remain unchanged. There is no dictator, no single voter possesses the power to determine the groups preference. Cardinal voting systems are not covered by the theorem, as they convey more information than rank orders, the theorem can also be sidestepped by weakening the notion of independence. The axiomatic approach Arrow adopted can treat all conceivable rules within one unified framework, in that sense, the approach is qualitatively different from the earlier one in voting theory, in which rules were investigated one by one. One can therefore say that the paradigm of social choice theory started from this theorem. The practical consequences of the theorem are debatable, Arrow has said Most systems are not going to work all of the time. All I proved is that all can work badly at times, the framework for Arrows theorem assumes that we need to extract a preference order on a given set of options. Each individual in the society gives an order of preferences on the set of outcomes. We are searching for a voting system, called a social welfare function. It cannot simply mimic the preferences of a single voter, unrestricted domain, or universality For any set of individual voter preferences, the social welfare function should yield a unique and complete ranking of societal choices. Thus, Independence of irrelevant alternatives The social preference between x and y should depend only on the preferences between x and y. More generally, changes in individuals rankings of irrelevant alternatives should have no impact on the ranking of the subset. For example, the introduction of a candidate to a two-candidate election should not affect the outcome of the election unless the third candidate wins. An individual should not be able to hurt an option by ranking it higher, non-imposition, or citizen sovereignty Every possible societal preference order should be achievable by some set of individual preference orders. This means that the social function is surjective, It has an unrestricted target space. This, again, is a demand that the social welfare function will be sensitive to the preference profile
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Monty Hall problem
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The Monty Hall problem is a brain teaser, in the form of a probability puzzle, loosely based on the American television game show Lets Make a Deal and named after its original host, Monty Hall. The problem was posed in a letter by Steve Selvin to the American Statistician in 1975. 1, and the host, who knows whats behind the doors, opens another door and he then says to you, Do you want to pick door No.2. Is it to advantage to switch your choice. Vos Savants response was that the contestant should switch to the other door, under the standard assumptions, contestants who switch have a 2/3 chance of winning the car, while contestants who stick to their initial choice have only a 1/3 chance. The given probabilities depend on assumptions about how the host. Another insight is that switching doors is a different action than choosing between the two remaining doors at random, as the first action uses the information and the latter does not. Other possible behaviors than the one described can reveal different additional information, or none at all, many readers of vos Savants column refused to believe switching is beneficial despite her explanation. After the problem appeared in Parade, approximately 10,000 readers, including nearly 1,000 with PhDs, wrote to the magazine, even when given explanations, simulations, and formal mathematical proofs, many people still do not accept that switching is the best strategy. Paul Erdős, one of the most prolific mathematicians in history, the problem is a paradox of the veridical type, because the correct result is so counterintuitive it can seem absurd, but is nevertheless demonstrably true. The Monty Hall problem is closely related to the earlier Three Prisoners problem. Steve Selvin wrote a letter to the American Statistician in 1975 describing a problem based on the game show Lets Make a Deal. 1, and the host, who knows whats behind the doors, opens another door and he then says to you, Do you want to pick door No.2. Is it to advantage to switch your choice. The behavior of the host is key to the 2/3 solution, ambiguities in the Parade version do not explicitly define the protocol of the host. The host must always open a door to reveal a goat, the host must always offer the chance to switch between the originally chosen door and the remaining closed door. When any of these assumptions is varied, it can change the probability of winning by switching doors as detailed in the section below. It is also presumed that the car is initially hidden behind a random door
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Mathematical fallacy
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In mathematics, certain kinds of mistaken proof are often exhibited, and sometimes collected, as illustrations of a concept of mathematical fallacy. For example, the reason validity fails may be a division by zero that is hidden by algebraic notation. There is a quality of the mathematical fallacy, as typically presented, it leads not only to an absurd result. Therefore, these fallacies, for reasons, usually take the form of spurious proofs of obvious contradictions. The latter applies normally to a form of argument that is not a rule of logic. Beyond pedagogy, the resolution of a fallacy can lead to insights into a subject. Pseudaria, an ancient lost book of false proofs, is attributed to Euclid, mathematical fallacies exist in many branches of mathematics. Well-known fallacies also exist in elementary Euclidean geometry and calculus, examples exist of mathematically correct results derived by incorrect lines of reasoning. Such an argument, however true the conclusion, is invalid and is commonly known as a howler. Consider for instance the calculation,1664 =16 /6 /4 =14, although the conclusion 1664 =14 is correct, there is a fallacious, invalid cancellation in the middle step. Another classical example of a howler is proving the Cayley-Hamilton theorem by simply substituting the scalar variables of the characteristic polynomial by the matrix, bogus proofs, calculations, or derivations constructed to produce a correct result in spite of incorrect logic or operations were termed howlers by Maxwell. Outside the field of mathematics the term howler has various meanings, the division-by-zero fallacy has many variants. The following example uses division by zero to prove that 2 =1, the fallacy is in line 5, the progression from line 4 to line 5 involves division by a − b, which is zero since a equals b. Since division by zero is undefined, the argument is invalid, many functions do not have a unique inverse. For instance squaring a number gives a value, but there are two possible square roots of a positive number. The square root of the square of −2 is 2, calculus as the mathematical study of infinitesimal change and limits can lead to mathematical fallacies if the properties of integrals and differentials are ignored. For instance, a use of integration by parts can be used to give a false proof that 0 =1. The problem is that antiderivatives are only defined up to a constant, the error really comes to light when we introduce arbitrary integration limits a and b
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Division by zero
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In mathematics, division by zero is division where the divisor is zero. Such a division can be expressed as a/0 where a is the dividend. In ordinary arithmetic, the expression has no meaning, as there is no number which, multiplied by 0, gives a, and so division by zero is undefined. Since any number multiplied by zero is zero, the expression 0/0 also has no defined value, in computing, a program error may result from an attempt to divide by zero. When division is explained at the elementary level, it is often considered as splitting a set of objects into equal parts. As an example, consider having ten cookies, and these cookies are to be distributed equally to five people at a table, each person would receive 105 =2 cookies. Similarly, if there are ten cookies, and only one person at the table, so, for dividing by zero, what is the number of cookies that each person receives when 10 cookies are evenly distributed amongst 0 people at a table. Certain words can be pinpointed in the question to highlight the problem, the problem with this question is the when. There is no way to evenly distribute 10 cookies to nobody, in mathematical jargon, a set of 10 items cannot be partitioned into 0 subsets. So 100, at least in elementary arithmetic, is said to be either meaningless, similar problems occur if one has 0 cookies and 0 people, but this time the problem is in the phrase the number. A partition is possible, but since the partition has 0 parts, vacuously every set in our partition has a number of elements, be it 0,2,5. If there are, say,5 cookies and 2 people, in any integer partition of a 5-set into 2 parts, one of the parts of the partition will have more elements than the other. But the problem with 5 cookies and 2 people can be solved by cutting one cookie in half, the problem with 5 cookies and 0 people cannot be solved in any way that preserves the meaning of divides. Another way of looking at division by zero is that division can always be checked using multiplication. Considering the 10/0 example above, setting x = 10/0, if x equals ten divided by zero, then x times zero equals ten, but there is no x that, when multiplied by zero, gives ten. If instead of x=10/0 we have x=0/0, then every x satisfies the question what number x, multiplied by zero, the Brahmasphutasiddhanta of Brahmagupta is the earliest known text to treat zero as a number in its own right and to define operations involving zero. The author could not explain division by zero in his texts, according to Brahmagupta, A positive or negative number when divided by zero is a fraction with the zero as denominator. Zero divided by a negative or positive number is zero or is expressed as a fraction with zero as numerator
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Zeno's paradoxes
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It is usually assumed, based on Platos Parmenides, that Zeno took on the project of creating these paradoxes because other philosophers had created paradoxes against Parmenides view. Plato has Socrates claim that Zeno and Parmenides were essentially arguing exactly the same point, some of Zenos nine surviving paradoxes are essentially equivalent to one another. Aristotle offered a refutation of some of them, three of the strongest and most famous—that of Achilles and the tortoise, the Dichotomy argument, and that of an arrow in flight—are presented in detail below. Zenos arguments are perhaps the first examples of a method of proof called reductio ad absurdum also known as proof by contradiction and they are also credited as a source of the dialectic method used by Socrates. Some mathematicians and historians, such as Carl Boyer, hold that Zenos paradoxes are simply mathematical problems, some philosophers, however, say that Zenos paradoxes and their variations remain relevant metaphysical problems. The origins of the paradoxes are somewhat unclear, Diogenes Laertius, a fourth source for information about Zeno and his teachings, citing Favorinus, says that Zenos teacher Parmenides was the first to introduce the Achilles and the tortoise paradox. But in a passage, Laertius attributes the origin of the paradox to Zeno. In a race, the quickest runner can never overtake the slowest, since the pursuer must first reach the point whence the pursued started, so that the slower must always hold a lead. – as recounted by Aristotle, Physics VI,9, 239b15 In the paradox of Achilles, Achilles allows the tortoise a head start of 100 meters, for example. If we suppose that each racer starts running at constant speed, then after some finite time, Achilles will have run 100 meters. During this time, the tortoise has run a shorter distance. Thus, whenever Achilles reaches somewhere the tortoise has been, he still has farther to go, therefore, because there are an infinite number of points Achilles must reach where the tortoise has already been, he can never overtake the tortoise. That which is in locomotion must arrive at the stage before it arrives at the goal. – as recounted by Aristotle, Physics VI,9. Before he can get there, he must get halfway there, before he can get halfway there, he must get a quarter of the way there. Before traveling a quarter, he must travel one-eighth, before an eighth, one-sixteenth, the resulting sequence can be represented as, This description requires one to complete an infinite number of tasks, which Zeno maintains is an impossibility. This sequence also presents a problem in that it contains no first distance to run, for any possible first distance could be divided in half. Hence, the trip cannot even begin, the paradoxical conclusion then would be that travel over any finite distance can neither be completed nor begun, and so all motion must be an illusion. An alternative conclusion, proposed by Henri Bergson, is that motion is not actually divisible and this argument is called the Dichotomy because it involves repeatedly splitting a distance into two parts
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