1.
Paradox
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A paradox is a statement that, despite apparently sound reasoning from true premises, leads to a self-contradictory or a logically unacceptable conclusion. A paradox involves contradictory yet interrelated elements that exist simultaneously and persist over time, some logical paradoxes are known to be invalid arguments but are still valuable in promoting critical thinking. Some paradoxes have revealed errors in definitions assumed to be rigorous, others, such as Currys paradox, are not yet resolved. Examples outside logic include the Ship of Theseus from philosophy, paradoxes can also take the form of images or other media. Escher featured perspective-based paradoxes in many of his drawings, with walls that are regarded as floors from other points of view, and staircases that appear to climb endlessly. In common usage, the word often refers to statements that may be both true and false i. e. ironic or unexpected, such as the paradox that standing is more tiring than walking. Common themes in paradoxes include self-reference, infinite regress, circular definitions, patrick Hughes outlines three laws of the paradox, Self-reference An example is This statement is false, a form of the liar paradox. The statement is referring to itself, another example of self-reference is the question of whether the barber shaves himself in the barber paradox. One more example would be Is the answer to this question No, contradiction This statement is false, the statement cannot be false and true at the same time. Another example of contradiction is if a man talking to a genie wishes that wishes couldnt come true, vicious circularity, or infinite regress This statement is false, if the statement is true, then the statement is false, thereby making the statement true. Another example of vicious circularity is the group of statements. Other paradoxes involve false statements or half-truths and the biased assumptions. This form is common in howlers, for example, consider a situation in which a father and his son are driving down the road. The car crashes into a tree and the father is killed, the boy is rushed to the nearest hospital where he is prepared for emergency surgery. On entering the suite, the surgeon says, I cant operate on this boy. The apparent paradox is caused by a hasty generalization, for if the surgeon is the boys father, the paradox is resolved if it is revealed that the surgeon is a woman — the boys mother. Paradoxes which are not based on a hidden error generally occur at the fringes of context or language, paradoxes that arise from apparently intelligible uses of language are often of interest to logicians and philosophers. Russells paradox, which shows that the notion of the set of all sets that do not contain themselves leads to a contradiction, was instrumental in the development of modern logic

2.
Mathematics
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Mathematics is the study of topics such as quantity, structure, space, and change. There is a range of views among mathematicians and philosophers as to the exact scope, Mathematicians seek out patterns and use them to formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proof, when mathematical structures are good models of real phenomena, then mathematical reasoning can provide insight or predictions about nature. Through the use of abstraction and logic, mathematics developed from counting, calculation, measurement, practical mathematics has been a human activity from as far back as written records exist. The research required to solve mathematical problems can take years or even centuries of sustained inquiry, rigorous arguments first appeared in Greek mathematics, most notably in Euclids Elements. Galileo Galilei said, The universe cannot be read until we have learned the language and it is written in mathematical language, and the letters are triangles, circles and other geometrical figures, without which means it is humanly impossible to comprehend a single word. Without these, one is wandering about in a dark labyrinth, carl Friedrich Gauss referred to mathematics as the Queen of the Sciences. Benjamin Peirce called mathematics the science that draws necessary conclusions, David Hilbert said of mathematics, We are not speaking here of arbitrariness in any sense. Mathematics is not like a game whose tasks are determined by arbitrarily stipulated rules, rather, it is a conceptual system possessing internal necessity that can only be so and by no means otherwise. Albert Einstein stated that as far as the laws of mathematics refer to reality, they are not certain, Mathematics is essential in many fields, including natural science, engineering, medicine, finance and the social sciences. Applied mathematics has led to entirely new mathematical disciplines, such as statistics, Mathematicians also engage in pure mathematics, or mathematics for its own sake, without having any application in mind. There is no clear line separating pure and applied mathematics, the history of mathematics can be seen as an ever-increasing series of abstractions. The earliest uses of mathematics were in trading, land measurement, painting and weaving patterns, in Babylonian mathematics elementary arithmetic first appears in the archaeological record. Numeracy pre-dated writing and numeral systems have many and diverse. Between 600 and 300 BC the Ancient Greeks began a study of mathematics in its own right with Greek mathematics. Mathematics has since been extended, and there has been a fruitful interaction between mathematics and science, to the benefit of both. Mathematical discoveries continue to be made today, the overwhelming majority of works in this ocean contain new mathematical theorems and their proofs. The word máthēma is derived from μανθάνω, while the modern Greek equivalent is μαθαίνω, in Greece, the word for mathematics came to have the narrower and more technical meaning mathematical study even in Classical times

3.
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

4.
0.999...
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In mathematics, the repeating decimal 0. 999… denotes a real number that can be shown to be the number one. In other words, the symbols 0. 999… and 1 represent the same number, more generally, every nonzero terminating decimal has an equal twin representation with infinitely many trailing 9s. The terminating decimal representation is preferred, contributing to the misconception that it is the only representation. The same phenomenon occurs in all other bases or in any representation of the real numbers. The equality of 0. 999… and 1 is closely related to the absence of nonzero infinitesimals in the number system. Some alternative number systems, such as the hyperreals, do contain nonzero infinitesimals, the equality 0. 999… =1 has long been accepted by mathematicians and is part of general mathematical education. Nonetheless, some find it sufficiently counterintuitive that they question or reject it. Such skepticism is common enough that the difficulty of convincing them of the validity of this identity has been the subject of studies in mathematics education. Algebraic proofs showing that 0. 999… represents the number 1 use concepts such as fractions, long division, however, these proofs are not rigorous as they do not include a careful analytic definition of 0. 999…. One reason that infinite decimals are an extension of finite decimals is to represent fractions. Using long division, a division of integers like 1⁄9 becomes a recurring decimal,0. 111…. This decimal yields a quick proof for 0. 999… =1, If 0. 999… is to be consistent, it must equal 9⁄9 =1. 0.333 … =390.888 … =890.999 … =99 =1 When a number in decimal notation is multiplied by 10, the digits do not change but each digit moves one place to the left. Thus 10 ×0. 999… equals 9. 999…, which is 9 greater than the original number, in introductory arithmetic, such proofs help explain why 0. 999… =1 but 0. 333… <0.34. In introductory algebra, the proofs help explain why the method of converting between fractions and repeating decimals works. Once a representation scheme is defined, it can be used to justify the rules of decimal arithmetic used in the above proofs. Moreover, one can demonstrate that the decimals 0. 999… and 1. 000… both represent the same real number, it is built into the definition. Since the question of 0. 999… does not affect the development of mathematics

5.
Berry paradox
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The Berry paradox is a self-referential paradox arising from an expression like the smallest positive integer not definable in fewer than twelve words. Consider the expression, The smallest positive integer not definable in under sixty letters, since there are only twenty-six letters, 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 many positive integers, this means that there are positive integers that cannot be defined by phrases of under sixty letters. This is the integer to which the expression refers. This is a paradox, there must be an integer defined by this expression, the Berry paradox as formulated above arises because of systematic ambiguity in the word definable. In other formulations of the Berry paradox, such as one that instead reads. not nameable in less, the term nameable is also one that has this systematic ambiguity. Terms of this kind give rise to vicious circle fallacies, other terms with this type of ambiguity are, satisfiable, true, false, function, property, class, relation, cardinal, and ordinal. To resolve one of these means to pinpoint exactly where our use of language went wrong. This family of paradoxes can be resolved by incorporating stratifications of meaning in language, terms with systematic ambiguity may be written with subscripts denoting that one level of meaning is considered a higher priority than another in their interpretation. The number not nameable0 in less than eleven words may be nameable1 in less than eleven words under this scheme. Using programs or proofs of bounded lengths, it is possible to construct an analogue of the Berry expression in a formal mathematical language, though the formal analogue does not lead to a logical contradiction, it does prove certain impossibility results. George Boolos built on a version of Berrys paradox to prove Gödels Incompleteness Theorem in a new. Then the proposition m is the first number not definable in less than k symbols can be formalized and it is not possible in general to unambiguously define what is the minimal number of symbols required to describe a given string. Some long strings can be described exactly using fewer symbols than those required by their full representation, the complexity of a given string is then defined as the minimal length that a description requires in order to refer to the full representation of that string. The Kolmogorov complexity is defined using formal languages, or Turing machines which avoids ambiguities about which string results from a given description and it can be proven that the Kolmogorov complexity is not computable. Busy beaver Chaitins incompleteness theorem Definable number Hilbert–Bernays paradox Interesting number paradox List of paradoxes Richards paradox Bennett, Boolos, George A new proof of the Gödel Incompleteness Theorem, Notices of the American Mathematical Society 36, 388–90,676. Reprinted in his Logic, Logic, and Logic, Chaitin, Gregory, Transcript of lecture given 27 October 1993 at the University of New Mexico Chaitin, Gregory The Berry Paradox. The False Assumption Underlying Berrys Paradox, Journal of Symbolic Logic 53, Russell, Bertrand Les paradoxes de la logique, Revue de métaphysique et de morale 14, 627–650 Russell, Bertrand, Whitehead, Alfred N. Principia Mathematica

6.
Braess's paradox
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Braess paradox or Braesss paradox is a proposed explanation for a seeming improvement to a road network being able to impede traffic through it. The paradox may have analogues in electrical grids and biological systems. It has been suggested that in theory, the improvement of a network could be accomplished by removing certain parts of it. Dietrich Braess, a mathematician at Ruhr University, Germany, noticed the flow in a network could be impeded by adding a new road. More formally, the idea behind Braess discovery is that the Nash equilibrium may not equate with the best overall flow through a network. The paradox is stated as follows, For each point of a network, let there be given the number of cars starting from it. Under these conditions one wishes to estimate the distribution of traffic flow, whether one street is preferable to another depends not only on the quality of the road, but also on the density of the flow. If every driver takes the path that looks most favorable to him, furthermore, it is indicated by an example that an extension of the road network may cause a redistribution of the traffic that results in longer individual running times. Adding extra capacity to a network when the moving entities selfishly choose their route can in some cases reduce overall performance and that is because the Nash equilibrium of such a system is not necessarily optimal. The network change induces a new structure which leads to a prisoners dilemma. In a Nash equilibrium, drivers have no incentive to change their routes, while the system is not in a Nash equilibrium, individual drivers are able to improve their respective travel times by changing the routes they take. In the case of Braess paradox, drivers continue to switch until they reach Nash equilibrium despite the reduction in overall performance. If the latency functions are linear, adding an edge can never make total travel time at equilibrium worse by a factor of more than 4/3. As a corollary, they obtain that Braess paradox is about as likely to occur as not occur, their result applies to rather than planned networks. In 1968, Dietrich Braess showed that the extension of the network may cause a redistribution of the traffic that results in longer individual running times. This paradox has a counterpart in case of a reduction of the road network, in Seoul, South Korea, a speeding up in traffic around the city was seen when a motorway was removed as part of the Cheonggyecheon restoration project. In Stuttgart, Germany, after investments into the network in 1969. In 1990 the temporary closing of 42nd Street in New York City for Earth Day reduced the amount of congestion in the area, in 2009, New York experimented with closures of Broadway at Times Square and Herald Square, which resulted in improved traffic flow and permanent pedestrian plazas