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Self-referential
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 also refers to the ability of a subject to speak of or refer to itself, that is, to have the kind of thought expressed by the first person nominative singular pronoun "I" in English. Self-reference is studied and has applications in mathematics, philosophy, computer programming, second-order cybernetics, and linguistics, as well as in humor. Self-referential statements are sometimes paradoxical, and can also be considered recursive. In logic, mathematics and computing In classical philosophy, paradoxes were created by self-referential concepts such as the omnipotence paradox of asking if it was possible for a being to exist so powerful that it could create a stone that it could not lift. The Epimenides paradox, 'All Cretans a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Self-referential Humor
Self-referential humor, also known as self-reflexive humor, self-aware humor, or meta humor, is a type of comedic expression that—either directed toward some other subject, or openly directed toward itself—is self-referential in some way, intentionally alluding to the very person who is expressing the humor in a comedic fashion, or to some specific aspect of that same comedic expression. Self-referential humor expressed discreetly and surrealistically is a form of bathos. In general, self-referential humor often uses hypocrisy, oxymoron, or paradox to create a contradictory or otherwise absurd situation that is humorous to the audience. History Old Comedy of Classical Athens is held to be the first—in the extant sources—form of self-referential comedy. Aristophanes, whose plays form the only remaining fragments of Old Comedy, used fantastical plots, grotesque and inhuman masks and status reversals of characters to slander prominent politicians and court his audience' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Berry's Paradox
The Berry paradox is a self-referential paradox arising from an expression like "The smallest positive integer not definable in under sixty letters" (a phrase with fifty-seven letters). Bertrand Russell, the first to discuss the paradox in print, attributed it to G. G. Berry (1867–1928), a junior librarian at Oxford's Bodleian Library. Russell called Berry "the only person in Oxford who understood mathematical logic". The paradox was called "Richard's paradox" by Jean-Yves Girard. Overview Consider the expression: :"The smallest positive integer not definable in under sixty letters." Since there are only twenty-six letters in the English alphabet, there are finitely many phrases of under sixty letters, and hence finitely many positive integers that are defined by phrases of under sixty letters. Since there are infinitely many positive integers, this means that there are positive integers that cannot be defined by phrases of under sixty letters. If there are positive integers ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gödel's Incompleteness Theorems
Gödel's incompleteness theorems are two theorems of mathematical logic that are concerned with the limits of in formal axiomatic theories. These results, published by Kurt Gödel in 1931, are important both in mathematical logic and in the philosophy of mathematics. The theorems are widely, but not universally, interpreted as showing that Hilbert's program to find a complete and consistent set of axioms for all mathematics is impossible. The first incompleteness theorem states that no consistent system of axioms whose theorems can be listed by an effective procedure (i.e., an algorithm) is capable of proving all truths about the arithmetic of natural numbers. For any such consistent formal system, there will always be statements about natural numbers that are true, but that are unprovable within the system. The second incompleteness theorem, an extension of the first, shows that the system cannot demonstrate its own consistency. Employing a diagonal argument, Gödel's in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ouroboros
The ouroboros or uroboros () is an ancient symbol depicting a serpent or dragon eating its own tail. The ouroboros entered Western tradition via ancient Egyptian iconography and the Greek magical tradition. It was adopted as a symbol in Gnosticism and Hermeticism and most notably in alchemy. The term derives , from ''oura'' 'tail' plus ''-boros'' '-eating'. The ''ouroboros'' is often interpreted as a symbol for eternal cyclic renewal or a cycle of life, death, and rebirth; the snake’s skin-sloughing symbolizes the transmigration of souls. The snake biting its own tail is a fertility symbol in some religions: the tail is a phallic symbol and the mouth is a yonic or womb-like symbol. Some snakes, such as rat snakes, have been known to consume themselves. One captive snake attempted to consume itself twice, dying in the second attempt. Another wild rat snake was found having swallowed about two-thirds of its body. Historical representations Ancient Egypt One of the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Second-order Cybernetics
Second-order cybernetics, also known as the cybernetics of cybernetics, is the recursive application of cybernetics to itself and the reflexive practice of cybernetics according to such a critique. It is cybernetics where "the role of the observer is appreciated and acknowledged rather than disguised, as had become traditional in western science". Glanville, R. (2002). "Second order cybernetics." In F. Parra-Luna (ed.), Systems science and cybernetics. In ''Encyclopaedia of Life Support Systems'' (EOLSS). OxfordEoLSS Second-order cybernetics was developed between the late 1960s and mid 1970s by Heinz von Foerster and others, with key inspiration coming from Margaret Mead. Foerster referred to it as "the control of control and the communication of communication" and differentiated first order cybernetics as "the cybernetics of observed systems" and second-order cybernetics as "the cybernetics of observing systems". Foerster, Heinz von, ed. ''Cybernetics of Cybernetics: Or, the Contro ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Recursion
Recursion (adjective: ''recursive'') occurs when a thing is defined in terms of itself or of its type. Recursion is used in a variety of disciplines ranging from linguistics to logic. The most common application of recursion is in mathematics and computer science, where a function being defined is applied within its own definition. While this apparently defines an infinite number of instances (function values), it is often done in such a way that no infinite loop or infinite chain of references ("crock recursion") can occur. Formal definitions In mathematics and computer science, a class of objects or methods exhibits recursive behavior when it can be defined by two properties: * A simple ''base case'' (or cases) — a terminating scenario that does not use recursion to produce an answer * A ''recursive step'' — a set of rules that reduces all successive cases toward the base case. For example, the following is a recursive definition of a person's ''ancestor''. One's an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
The Halting Problem
In computability theory, the halting problem is the problem of determining, from a description of an arbitrary computer program and an input, whether the program will finish running, or continue to run forever. Alan Turing proved in 1936 that a general algorithm to solve the halting problem for all possible program–input pairs cannot exist. For any program that might determine whether programs halt, a "pathological" program , called with some input, can pass its own source and its input to ''f'' and then specifically do the opposite of what ''f'' predicts ''g'' will do. No ''f'' can exist that handles this case. A key part of the proof is a mathematical definition of a computer and program, which is known as a Turing machine; the halting problem is '' undecidable'' over Turing machines. It is one of the first cases of decision problems proven to be unsolvable. This proof is significant to practical computing efforts, defining a class of applications which no programming invent ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Recursion
Recursion (adjective: ''recursive'') occurs when a thing is defined in terms of itself or of its type. Recursion is used in a variety of disciplines ranging from linguistics to logic. The most common application of recursion is in mathematics and computer science, where a function being defined is applied within its own definition. While this apparently defines an infinite number of instances (function values), it is often done in such a way that no infinite loop or infinite chain of references ("crock recursion") can occur. Formal definitions In mathematics and computer science, a class of objects or methods exhibits recursive behavior when it can be defined by two properties: * A simple ''base case'' (or cases) — a terminating scenario that does not use recursion to produce an answer * A ''recursive step'' — a set of rules that reduces all successive cases toward the base case. For example, the following is a recursive definition of a person's ''ancestor''. One's an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Omnipotence Paradox
The omnipotence paradox is a family of paradoxes that arise with some understandings of the term ''omnipotent''. The paradox arises, for example, if one assumes that an omnipotent being has no limits and is capable of realizing any outcome, even a logically contradictory one such as creating a square circle. Atheological arguments based on the omnipotence paradox are sometimes described as evidence for countering theism. Other possible resolutions to the paradox hinge on the definition of omnipotence applied and the nature of God regarding this application and whether omnipotence is directed toward God himself or outward toward his external surroundings. The omnipotence paradox has medieval origins, dating at least to the 10th century, when the Saadia Gaon responded to the question of whether God's omnipotence extended to logical absurdities. It was later addressed by AverroësAverroës, ''Tahafut al-Tahafut (The Incoherence of the Incoherence)'' trans. Simon Van Den Bergh, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Computer Programming
Computer programming is the process of performing a particular computation (or more generally, accomplishing a specific computing result), usually by designing and building an executable computer program. Programming involves tasks such as analysis, generating algorithms, profiling algorithms' accuracy and resource consumption, and the implementation of algorithms (usually in a chosen programming language, commonly referred to as coding). The source code of a program is written in one or more languages that are intelligible to programmers, rather than machine code, which is directly executed by the central processing unit. The purpose of programming is to find a sequence of instructions that will automate the performance of a task (which can be as complex as an operating system) on a computer, often for solving a given problem. Proficient programming thus usually requires expertise in several different subjects, including knowledge of the application domain, specialized algorit ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Consistency
In classical deductive logic, a consistent theory is one that does not lead to a logical contradiction. The lack of contradiction can be defined in either semantic or syntactic terms. The semantic definition states that a theory is consistent if it has a model, i.e., there exists an interpretation under which all formulas in the theory are true. This is the sense used in traditional Aristotelian logic, although in contemporary mathematical logic the term ''satisfiable'' is used instead. The syntactic definition states a theory T is consistent if there is no formula \varphi such that both \varphi and its negation \lnot\varphi are elements of the set of consequences of T. Let A be a set of closed sentences (informally "axioms") and \langle A\rangle the set of closed sentences provable from A under some (specified, possibly implicitly) formal deductive system. The set of axioms A is consistent when \varphi, \lnot \varphi \in \langle A \rangle for no formula \varphi. If there e ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Computer Science
Computer science is the study of computation, automation, and information. Computer science spans theoretical disciplines (such as algorithms, theory of computation, information theory, and automation) to practical disciplines (including the design and implementation of hardware and software). Computer science is generally considered an area of academic research and distinct from computer programming. Algorithms and data structures are central to computer science. The theory of computation concerns abstract models of computation and general classes of problems that can be solved using them. The fields of cryptography and computer security involve studying the means for secure communication and for preventing security vulnerabilities. Computer graphics and computational geometry address the generation of images. Programming language theory considers different ways to describe computational processes, and database theory concerns the management of repositories o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
