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Strange Loop
A strange loop is a cyclic structure that goes through several levels in a hierarchical system. It arises when, by moving only upwards or downwards through the system, one finds oneself back where one started. Strange loops may involve self-reference and paradox. The concept of a strange loop was proposed and extensively discussed by Douglas Hofstadter in ''Gödel, Escher, Bach'', and is further elaborated in Hofstadter's book '' I Am a Strange Loop'', published in 2007. A tangled hierarchy is a hierarchical consciousness system in which a strange loop appears. Definitions A strange loop is a hierarchy of levels, each of which is linked to at least one other by some type of relationship. A strange loop hierarchy is "tangled" (Hofstadter refers to this as a " heterarchy"), in that there is no well defined highest or lowest level; moving through the levels, one eventually returns to the starting point, i.e., the original level. Examples of strange loops that Hofstadter offers i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cycle (graph Theory)
In graph theory, a cycle in a graph is a non-empty trail in which only the first and last vertices are equal. A directed cycle in a directed graph is a non-empty directed trail in which only the first and last vertices are equal. A graph without cycles is called an ''acyclic graph''. A directed graph without directed cycles is called a ''directed acyclic graph''. A connected graph without cycles is called a '' tree''. Definitions Circuit and cycle * A circuit is a non-empty trail in which the first and last vertices are equal (''closed trail''). : Let be a graph. A circuit is a non-empty trail with a vertex sequence . * A cycle or simple circuit is a circuit in which only the first and last vertices are equal. Directed circuit and directed cycle * A directed circuit is a non-empty directed trail in which the first and last vertices are equal (''closed directed trail''). : Let be a directed graph. A directed circuit is a non-empty directed trail with a vertex sequ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
This Statement Is False
In philosophy and logic, the classical liar paradox or liar's paradox or antinomy of the liar is the statement of a liar that they are lying: for instance, declaring that "I am lying". If the liar is indeed lying, then the liar is telling the truth, which means the liar just lied. In "this sentence is a lie" the paradox is strengthened in order to make it amenable to more rigorous logical analysis. It is still generally called the "liar paradox" although abstraction is made precisely from the liar making the statement. Trying to assign to this statement, the strengthened liar, a classical binary truth value leads to a contradiction. If "this sentence is false" is true, then it is false, but the sentence states that it is false, and if it is false, then it must be true, and so on. History The Epimenides paradox (circa 600 BC) has been suggested as an example of the liar paradox, but they are not logically equivalent. The semi-mythical seer Epimenides, a Cretan, reportedly stated ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ascending And Descending
''Ascending and Descending'' is a lithograph print by the Dutch artist M. C. Escher first printed in March 1960. The original print measures . The lithograph depicts a large building roofed by a never-ending staircase. Two lines of identically dressed men appear on the staircase, one line ascending while the other descends. Two figures sit apart from the people on the endless staircase: one in a secluded courtyard, the other on a lower set of stairs. While most two-dimensional artists use relative proportions to create an illusion of depth, Escher here and elsewhere uses conflicting proportions to create the visual paradox. ''Ascending and Descending'' was influenced by, and is an artistic implementation of, the Penrose stairs, an impossible object; Lionel Penrose had first published his concept in the February 1958 issue of the ''British Journal of Psychology''. Escher developed the theme further in his print '' Waterfall'', which appeared in 1961. The two concentric process ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Drawing Hands
''Drawing Hands'' is a lithograph by the Dutch artist M. C. Escher first printed in January 1948. It depicts a sheet of paper, out of which two hands rise, in the paradoxical act of drawing one another into existence. This is one of the most obvious examples of Escher's common use of paradox. It is referenced in the book ''Gödel, Escher, Bach'', by Douglas Hofstadter, who calls it an example of a strange loop. It is used in ''Structure and Interpretation of Computer Programs'' by Harold Abelson and Gerald Jay Sussman as an allegory for the eval and apply In mathematics and computer science, apply is a function that applies a function to arguments. It is central to programming languages derived from lambda calculus, such as LISP and Scheme, and also in functional languages. It has a role in th ... functions of programming language interpreters in computer science, which feed each other. ''Drawing Hands'' has been referenced and copied many times by artists in dif ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Waterfall (M
A waterfall is a point in a river or stream where water flows over a vertical drop or a series of steep drops. Waterfalls also occur where meltwater drops over the edge of a tabular iceberg or ice shelf. Waterfalls can be formed in several ways, but the most common method of formation is that a river courses over a top layer of resistant bedrock before falling on to softer rock, which erodes faster, leading to an increasingly high fall. Waterfalls have been studied for their impact on species living in and around them. Humans have had a distinct relationship with waterfalls for years, travelling to see them, exploring and naming them. They can present formidable barriers to navigation along rivers. Waterfalls are religious sites in many cultures. Since the 18th century they have received increased attention as tourist destinations, sources of hydropower, andparticularly since the mid-20th centuryas subjects of research. Definition and terminology A waterfall is generally d ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Johann Sebastian Bach
Johann Sebastian Bach (28 July 1750) was a German composer and musician of the late Baroque period. He is known for his orchestral music such as the '' Brandenburg Concertos''; instrumental compositions such as the Cello Suites; keyboard works such as the '' Goldberg Variations'' and '' The Well-Tempered Clavier''; organ works such as the '' Schubler Chorales'' and the Toccata and Fugue in D minor; and vocal music such as the '' St Matthew Passion'' and the Mass in B minor. Since the 19th-century Bach revival he has been generally regarded as one of the greatest composers in the history of Western music. The Bach family already counted several composers when Johann Sebastian was born as the last child of a city musician in Eisenach. After being orphaned at the age of 10, he lived for five years with his eldest brother Johann Christoph, after which he continued his musical education in Lüneburg. From 1703 he was back in Thuringia, working as a musician for Protest ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Peano Arithmetic
In mathematical logic, the Peano axioms, also known as the Dedekind–Peano axioms or the Peano postulates, are axioms for the natural numbers presented by the 19th century Italian mathematician Giuseppe Peano. These axioms have been used nearly unchanged in a number of metamathematical investigations, including research into fundamental questions of whether number theory is consistent and complete. The need to formalize arithmetic was not well appreciated until the work of Hermann Grassmann, who showed in the 1860s that many facts in arithmetic could be derived from more basic facts about the successor operation and induction. In 1881, Charles Sanders Peirce provided an axiomatization of natural-number arithmetic. In 1888, Richard Dedekind proposed another axiomatization of natural-number arithmetic, and in 1889, Peano published a simplified version of them as a collection of axioms in his book, ''The principles of arithmetic presented by a new method'' ( la, Arithm ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Typographical Number Theory
Typographical Number Theory (TNT) is a formal axiomatic system describing the natural numbers that appears in Douglas Hofstadter's book '' Gödel, Escher, Bach''. It is an implementation of Peano arithmetic that Hofstadter uses to help explain Gödel's incompleteness theorems. Like any system implementing the Peano axioms, TNT is capable of referring to itself (it is self-referential). Numerals TNT does not use a distinct symbol for each natural number. Instead it makes use of a simple, uniform way of giving a compound symbol to each natural number: : The symbol S can be interpreted as "the successor of", or "the number after". Since this is, however, a number theory, such interpretations are useful, but not strict. It cannot be said that because four is the successor of three that four is SSSS0, but rather that since three is the successor of two, which is the successor of one, which is the successor of zero, which has been described as 0, four can be "proved" to be SSS ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Leon Henkin
Leon Albert Henkin (April 19, 1921, Brooklyn, New York - November 1, 2006, Oakland, California) was an American logician, whose works played a strong role in the development of logic, particularly in the theory of types. He was an active scholar at the University of California, Berkeley, where he made great contributions as a researcher, teacher, as well as in administrative positions. At this university he directed, together with Alfred Tarski, the Group in Logic and the Methodology of Science',Manzano, María; Alonso, Enrique (2014). «Leon Henkin». In Manzano et al., María, ed. ''The Life and Work of Leon Henkin''. Springer International Publishing. pp. 3-22. . doi:10.1007/978-3-319-09719-0_11. from which many important logicians and philosophers emerged. He had a strong sense of social commitment and was a passionate defensor of his pacifist and progressive ideas. He took part in many social projects aimed at teaching mathematics, as well as projects aimed at supporting ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Neurons
A neuron, neurone, or nerve cell is an electrically excitable cell that communicates with other cells via specialized connections called synapses. The neuron is the main component of nervous tissue in all animals except sponges and placozoa. Non-animals like plants and fungi do not have nerve cells. Neurons are typically classified into three types based on their function. Sensory neurons respond to stimuli such as touch, sound, or light that affect the cells of the sensory organs, and they send signals to the spinal cord or brain. Motor neurons receive signals from the brain and spinal cord to control everything from muscle contractions to glandular output. Interneurons connect neurons to other neurons within the same region of the brain or spinal cord. When multiple neurons are connected together, they form what is called a neural circuit. A typical neuron consists of a cell body (soma), dendrites, and a single axon. The soma is a compact structure, and the axon an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Self-consciousness
Self-consciousness is a heightened sense of awareness of oneself. It is not to be confused with consciousness in the sense of qualia. Historically, "self-consciousness" was synonymous with "self-awareness", referring to a state of awareness that one exists and that one has consciousness. While "self-conscious" and "self-aware" are still sometimes used interchangeably, particularly in philosophy, self-consciousness is now also commonly used to refer to a preoccupation with oneself, especially with how others might perceive one's appearance or one's actions. An unpleasant feeling of self-consciousness may occur when one realizes that one is being watched or observed, the feeling that "everyone is looking" at oneself. Some people are habitually more self-conscious than others. Unpleasant feelings of self-consciousness are sometimes associated with shyness or paranoia. Impairment When feeling self-conscious, one becomes aware of even the smallest of one's own actions. Such awareness ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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