1.
A Mathematician's Apology
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A Mathematicians Apology is a 1940 essay by British mathematician G. H. Hardy. It concerns the aesthetics of mathematics with some content. In the books title, Hardy uses the word apology in the sense of a justification or defence. Hardy felt the need to justify his lifes work in mathematics at this time mainly for two reasons, firstly, at age 62, Hardy felt the approach of old age and the decline of his mathematical creativity and skills. By devoting time to writing the Apology, Hardy was admitting that his own time as a mathematician was finished. In his foreword to the 1967 edition of the book, C. P. Snow describes the Apology as a lament for creative powers that used to be. In Hardys words, Exposition, criticism, appreciation, is work for second-rate minds and it is a melancholy experience for a professional mathematician to find himself writing about mathematics. The function of a mathematician is to do something, to new theorems, to add to mathematics. Secondly, at the start of the World War II, Hardy, Hardy was an atheist, and makes his justification not to God but to his fellow man. One of the themes of the book is the beauty that mathematics possesses. For Hardy, the most beautiful mathematics was that which had no applications in the outside world and, in particular. He justifies the pursuit of mathematics with the argument that its very uselessness on the whole meant that it could not be misused to cause harm. On the other hand, Hardy denigrates much of the mathematics as either being trivial, ugly, or dull, and contrasts it with real mathematics. Hardy expounds by commenting about a phrase attributed to Carl Friedrich Gauss that Mathematics is the queen of the sciences, if an application of number theory were to be found, then certainly no one would try to dethrone the queen of mathematics because of that. What Gauss meant, according to Hardy, is that the concepts that constitute number theory are deeper. This view reflects Hardys increasing depression at the wane of his own mathematical powers, for Hardy, real mathematics was essentially a creative activity, rather than an explanatory or expository one. Hardys opinions were influenced by the academic culture of the universities of Cambridge. Some of Hardys examples seem unfortunate in retrospect, for example, he writes, No one has yet discovered any warlike purpose to be served by the theory of numbers or relativity, and it seems unlikely that anyone will do so for many years
2.
International Standard Book Number
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The International Standard Book Number is a unique numeric commercial book identifier. An ISBN is assigned to each edition and variation of a book, for example, an e-book, a paperback and a hardcover edition of the same book would each have a different ISBN. The ISBN is 13 digits long if assigned on or after 1 January 2007, the method of assigning an ISBN is nation-based and varies from country to country, often depending on how large the publishing industry is within a country. The initial ISBN configuration of recognition was generated in 1967 based upon the 9-digit Standard Book Numbering created in 1966, the 10-digit ISBN format was developed by the International Organization for Standardization and was published in 1970 as international standard ISO2108. Occasionally, a book may appear without a printed ISBN if it is printed privately or the author does not follow the usual ISBN procedure, however, this can be rectified later. Another identifier, the International Standard Serial Number, identifies periodical publications such as magazines, the ISBN configuration of recognition was generated in 1967 in the United Kingdom by David Whitaker and in 1968 in the US by Emery Koltay. The 10-digit ISBN format was developed by the International Organization for Standardization and was published in 1970 as international standard ISO2108, the United Kingdom continued to use the 9-digit SBN code until 1974. The ISO on-line facility only refers back to 1978, an SBN may be converted to an ISBN by prefixing the digit 0. For example, the edition of Mr. J. G. Reeder Returns, published by Hodder in 1965, has SBN340013818 -340 indicating the publisher,01381 their serial number. This can be converted to ISBN 0-340-01381-8, the check digit does not need to be re-calculated, since 1 January 2007, ISBNs have contained 13 digits, a format that is compatible with Bookland European Article Number EAN-13s. An ISBN is assigned to each edition and variation of a book, for example, an ebook, a paperback, and a hardcover edition of the same book would each have a different ISBN. The ISBN is 13 digits long if assigned on or after 1 January 2007, a 13-digit ISBN can be separated into its parts, and when this is done it is customary to separate the parts with hyphens or spaces. Separating the parts of a 10-digit ISBN is also done with either hyphens or spaces, figuring out how to correctly separate a given ISBN number is complicated, because most of the parts do not use a fixed number of digits. ISBN issuance is country-specific, in that ISBNs are issued by the ISBN registration agency that is responsible for country or territory regardless of the publication language. Some ISBN registration agencies are based in national libraries or within ministries of culture, in other cases, the ISBN registration service is provided by organisations such as bibliographic data providers that are not government funded. In Canada, ISBNs are issued at no cost with the purpose of encouraging Canadian culture. In the United Kingdom, United States, and some countries, where the service is provided by non-government-funded organisations. Australia, ISBNs are issued by the library services agency Thorpe-Bowker
3.
Alfred North Whitehead
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Alfred North Whitehead OM FRS was an English mathematician and philosopher. In his early career Whitehead wrote primarily on mathematics, logic and his most notable work in these fields is the three-volume Principia Mathematica, which he wrote with former student Bertrand Russell. Beginning in the late 1910s and early 1920s, Whitehead gradually turned his attention from mathematics to philosophy of science and he developed a comprehensive metaphysical system which radically departed from most of western philosophy. Today Whiteheads philosophical works – particularly Process and Reality – are regarded as the texts of process philosophy. For this reason, one of the most promising applications of Whiteheads thought in recent years has been in the area of ecological civilization, cobb, Jr. Alfred North Whitehead was born in Ramsgate, Kent, England, in 1861. His father, Alfred Whitehead, was a minister and schoolmaster of Chatham House Academy, Whitehead himself recalled both of them as being very successful schoolmasters, but that his grandfather was the more extraordinary man. Whiteheads mother was Maria Sarah Whitehead, formerly Maria Sarah Buckmaster, Whitehead was apparently not particularly close with his mother, as he never mentioned her in any of his writings, and there is evidence that Whiteheads wife, Evelyn, had a low opinion of her. Whitehead was educated at Sherborne School, Dorset, then considered one of the best public schools in the country and his childhood was described as over-protected, but when at school he excelled in sports and mathematics and was head prefect of his class. In 1880, Whitehead began attending Trinity College, Cambridge, and his academic advisor was Edward John Routh. He earned his BA from Trinity in 1884, and graduated as fourth wrangler, in 1890, Whitehead married Evelyn Wade, an Irish woman raised in France, they had a daughter, Jessie Whitehead, and two sons, Thomas North Whitehead and Eric Whitehead. Eric Whitehead died in action serving in the Royal Flying Corps during World War I at the age of 19. In 1910, Whitehead resigned his Senior Lectureship in Mathematics at Trinity, toward the end of his time in England, Whitehead turned his attention to philosophy. Though he had no advanced training in philosophy, his work soon became highly regarded. After publishing The Concept of Nature in 1920, he served as president of the Aristotelian Society from 1922 to 1923, in 1924, Henry Osborn Taylor invited the 63-year-old Whitehead to join the faculty at Harvard University as a professor of philosophy. During his time at Harvard, Whitehead produced his most important philosophical contributions, in 1925, he wrote Science and the Modern World, which was immediately hailed as an alternative to the Cartesian dualism that plagued popular science. A few years later, he published his seminal work Process and Reality, the Whiteheads spent the rest of their lives in the United States. Alfred North retired from Harvard in 1937 and remained in Cambridge, the two volume biography of Whitehead by Victor Lowe is the most definitive presentation of the life of Whitehead. However, many details of Whiteheads life remain obscure because he left no Nachlass, additionally, Whitehead was known for his almost fanatical belief in the right to privacy, and for writing very few personal letters of the kind that would help to gain insight on his life
4.
Bertrand Russell
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Bertrand Arthur William Russell, 3rd Earl Russell, OM, FRS was a British philosopher, logician, mathematician, historian, writer, social critic, political activist and Nobel laureate. At various points in his life he considered himself a liberal, a socialist, and a pacifist and he was born in Monmouthshire into one of the most prominent aristocratic families in the United Kingdom. In the early 20th century, Russell led the British revolt against idealism and he is considered one of the founders of analytic philosophy along with his predecessor Gottlob Frege, colleague G. E. Moore, and protégé Ludwig Wittgenstein. He is widely held to be one of the 20th centurys premier logicians, with A. N. Whitehead he wrote Principia Mathematica, an attempt to create a logical basis for mathematics. His philosophical essay On Denoting has been considered a paradigm of philosophy, Russell mostly was a prominent anti-war activist, he championed anti-imperialism. Occasionally, he advocated preventive nuclear war, before the opportunity provided by the monopoly is gone. He went to prison for his pacifism during World War I, in 1950 Russell was awarded the Nobel Prize in Literature in recognition of his varied and significant writings in which he champions humanitarian ideals and freedom of thought. Bertrand Russell was born on 18 May 1872 at Ravenscroft, Trellech, Monmouthshire and his parents, Viscount and Viscountess Amberley, were radical for their times. Lord Amberley consented to his wifes affair with their childrens tutor, both were early advocates of birth control at a time when this was considered scandalous. Lord Amberley was an atheist and his atheism was evident when he asked the philosopher John Stuart Mill to act as Russells secular godfather, Mill died the year after Russells birth, but his writings had a great effect on Russells life. His paternal grandfather, the Earl Russell, had asked twice by Queen Victoria to form a government. The Russells had been prominent in England for several centuries before this, coming to power, Lady Amberley was the daughter of Lord and Lady Stanley of Alderley. Russell often feared the ridicule of his grandmother, one of the campaigners for education of women. Russell had two siblings, brother Frank, and sister Rachel, in June 1874 Russells mother died of diphtheria, followed shortly by Rachels death. In January 1876, his father died of bronchitis following a period of depression. Frank and Bertrand were placed in the care of their staunchly Victorian paternal grandparents and his grandfather, former Prime Minister Earl Russell, died in 1878, and was remembered by Russell as a kindly old man in a wheelchair. His grandmother, the Countess Russell, was the dominant family figure for the rest of Russells childhood, the countess was from a Scottish Presbyterian family, and successfully petitioned the Court of Chancery to set aside a provision in Amberleys will requiring the children to be raised as agnostics. Her favourite Bible verse, Thou shalt not follow a multitude to do evil, the atmosphere at Pembroke Lodge was one of frequent prayer, emotional repression, and formality, Frank reacted to this with open rebellion, but the young Bertrand learned to hide his feelings
5.
Gottlob Frege
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Friedrich Ludwig Gottlob Frege was a German philosopher, logician, and mathematician. Considered a major figure in mathematics, he is responsible for the development of modern logic and he is also understood by many to be the father of analytic philosophy, where he concentrated on the philosophy of language and mathematics. Though largely ignored during his lifetime, Giuseppe Peano and Bertrand Russell introduced his work to generations of logicians. Frege was born in 1848 in Wismar, Mecklenburg-Schwerin and his father Carl Alexander Frege was the co-founder and headmaster of a girls high school until his death. In childhood, Frege encountered philosophies that would guide his future scientific career, Frege studied at a gymnasium in Wismar and graduated in 1869. His teacher Gustav Adolf Leo Sachse, who was a poet, played the most important role in determining Freges future scientific career, Frege matriculated at the University of Jena in the spring of 1869 as a citizen of the North German Confederation. In the four semesters of his studies he attended approximately twenty courses of lectures and his most important teacher was Ernst Karl Abbe. Abbe was more than a teacher to Frege, he was a trusted friend, after Freges graduation, they came into closer correspondence. His other notable university teachers were Christian Philipp Karl Snell, Hermann Karl Julius Traugott Schaeffer, Frege married Margarete Katharina Sophia Anna Lieseberg on 14 March 1887. Though his education and early work focused primarily on geometry. His Begriffsschrift, eine der arithmetischen nachgebildete Formelsprache des reinen Denkens, Halle a/S, the Begriffsschrift broke new ground, including a rigorous treatment of the ideas of functions and variables. Previous logic had dealt with the constants and, or. Freges conceptual notation however can represent such inferences, one of Freges stated purposes was to isolate genuinely logical principles of inference, so that in the proper representation of mathematical proof, one would at no point appeal to intuition. If there was an element, it was to be isolated and represented separately as an axiom, from there on. Already in the 1879 Begriffsschrift important preliminary theorems, for example a generalized form of law of trichotomy, were derived within what Frege understood to be pure logic and this idea was formulated in non-symbolic terms in his The Foundations of Arithmetic. Later, in his Basic Laws of Arithmetic, Frege attempted to derive, by use of his symbolism, most of these axioms were carried over from his Begriffsschrift, though not without some significant changes. The one truly new principle was one he called the Basic Law V, the crucial case of the law may be formulated in modern notation as follows. Let denote the extension of the predicate Fx, i. e. the set of all Fs, then Basic Law V says that the predicates Fx and Gx have the same extension iff ∀x
6.
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
7.
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
8.
Cardinal numbers
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In mathematics, cardinal numbers, or cardinals for short, are a generalization of the natural numbers used to measure the cardinality of sets. The cardinality of a set is a natural number, the number of elements in the set. The transfinite cardinal numbers describe the sizes of infinite sets, cardinality is defined in terms of bijective functions. Two sets have the same cardinality if, and only if, in the case of finite sets, this agrees with the intuitive notion of size. In the case of sets, the behavior is more complex. It is also possible for a subset of an infinite set to have the same cardinality as the original set. There is a sequence of cardinal numbers,0,1,2,3, …, n, …, ℵ0, ℵ1, ℵ2, …, ℵ α, …. This sequence starts with the natural numbers including zero, which are followed by the aleph numbers, the aleph numbers are indexed by ordinal numbers. Under the assumption of the axiom of choice, this transfinite sequence includes every cardinal number, If one rejects that axiom, the situation is more complicated, with additional infinite cardinals that are not alephs. Cardinality is studied for its own sake as part of set theory and it is also a tool used in branches of mathematics including model theory, combinatorics, abstract algebra, and mathematical analysis. In category theory, the numbers form a skeleton of the category of sets. The notion of cardinality, as now understood, was formulated by Georg Cantor, cardinality can be used to compare an aspect of finite sets, e. g. the sets and are not equal, but have the same cardinality, namely three. Cantor applied his concept of bijection to infinite sets, e. g. the set of natural numbers N =, thus, all sets having a bijection with N he called denumerable sets and they all have the same cardinal number. This cardinal number is called ℵ0, aleph-null and he called the cardinal numbers of these infinite sets transfinite cardinal numbers. Cantor proved that any unbounded subset of N has the same cardinality as N and he later proved that the set of all real algebraic numbers is also denumerable. His proof used an argument with nested intervals, but in an 1891 paper he proved the result using his ingenious. The new cardinal number of the set of numbers is called the cardinality of the continuum. His continuum hypothesis is the proposition that c is the same as ℵ1 and this hypothesis has been found to be independent of the standard axioms of mathematical set theory, it can neither be proved nor disproved from the standard assumptions
9.
Ordinal numbers
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In set theory, an ordinal number, or ordinal, is one generalization of the concept of a natural number that is used to describe a way to arrange a collection of objects in order, one after another. Any finite collection of objects can be put in order just by the process of counting, labeling the objects with distinct whole numbers, Ordinal numbers are thus the labels needed to arrange collections of objects in order. An ordinal number is used to describe the type of a well ordered set. Whereas ordinals are useful for ordering the objects in a collection, they are distinct from cardinal numbers, although the distinction between ordinals and cardinals is not always apparent in finite sets, different infinite ordinals can describe the same cardinal. Like other kinds of numbers, ordinals can be added, multiplied, a natural number can be used for two purposes, to describe the size of a set, or to describe the position of an element in a sequence. When restricted to finite sets these two concepts coincide, there is one way to put a finite set into a linear sequence. This is because any set has only one size, there are many nonisomorphic well-orderings of any infinite set. Whereas the notion of number is associated with a set with no particular structure on it. A well-ordered set is an ordered set in which there is no infinite decreasing sequence, equivalently. Ordinals may be used to label the elements of any given well-ordered set and this length is called the order type of the set. Any ordinal is defined by the set of ordinals that precede it, in fact, the most common definition of ordinals identifies each ordinal as the set of ordinals that precede it. For example, the ordinal 42 is the type of the ordinals less than it, i. e. the ordinals from 0 to 41. Conversely, any set of ordinals that is downward-closed—meaning that for any ordinal α in S and any ordinal β < α, β is also in S—is an ordinal. There are infinite ordinals as well, the smallest infinite ordinal is ω, which is the type of the natural numbers. After all of these come ω·2, ω·2+1, ω·2+2, and so on, then ω·3, now the set of ordinals formed in this way must itself have an ordinal associated with it, and that is ω2. Further on, there will be ω3, then ω4, and so on, and ωω, then ωωω, then later ωωωω and this can be continued indefinitely far. The smallest uncountable ordinal is the set of all countable ordinals, in a well-ordered set, every non-empty subset contains a distinct smallest element. Given the axiom of dependent choice, this is equivalent to just saying that the set is ordered and there is no infinite decreasing sequence
10.
Real numbers
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In mathematics, a real number is a value that represents a quantity along a line. The adjective real in this context was introduced in the 17th century by René Descartes, the real numbers include all the rational numbers, such as the integer −5 and the fraction 4/3, and all the irrational numbers, such as √2. Included within the irrationals are the numbers, such as π. Real numbers can be thought of as points on a long line called the number line or real line. Any real number can be determined by a possibly infinite decimal representation, such as that of 8.632, the real line can be thought of as a part of the complex plane, and complex numbers include real numbers. These descriptions of the numbers are not sufficiently rigorous by the modern standards of pure mathematics. All these definitions satisfy the definition and are thus equivalent. The statement that there is no subset of the reals with cardinality greater than ℵ0. Simple fractions were used by the Egyptians around 1000 BC, the Vedic Sulba Sutras in, c.600 BC, around 500 BC, the Greek mathematicians led by Pythagoras realized the need for irrational numbers, in particular the irrationality of the square root of 2. Arabic mathematicians merged the concepts of number and magnitude into a general idea of real numbers. In the 16th century, Simon Stevin created the basis for modern decimal notation, in the 17th century, Descartes introduced the term real to describe roots of a polynomial, distinguishing them from imaginary ones. In the 18th and 19th centuries, there was work on irrational and transcendental numbers. Johann Heinrich Lambert gave the first flawed proof that π cannot be rational, Adrien-Marie Legendre completed the proof, Évariste Galois developed techniques for determining whether a given equation could be solved by radicals, which gave rise to the field of Galois theory. Charles Hermite first proved that e is transcendental, and Ferdinand von Lindemann, lindemanns proof was much simplified by Weierstrass, still further by David Hilbert, and has finally been made elementary by Adolf Hurwitz and Paul Gordan. The development of calculus in the 18th century used the set of real numbers without having defined them cleanly. The first rigorous definition was given by Georg Cantor in 1871, in 1874, he showed that the set of all real numbers is uncountably infinite but the set of all algebraic numbers is countably infinite. Contrary to widely held beliefs, his first method was not his famous diagonal argument, the real number system can be defined axiomatically up to an isomorphism, which is described hereafter. Another possibility is to start from some rigorous axiomatization of Euclidean geometry, from the structuralist point of view all these constructions are on equal footing
11.
Real analysis
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Real analysis is a branch of mathematical analysis dealing with the real numbers and real-valued functions of a real variable. The theorems of real analysis rely intimately upon the structure of the number line. The real number system consists of a set, together with two operations and an order, and is, formally speaking, an ordered quadruple consisting of these objects, there are several ways of formalizing the definition of the real number system. The synthetic approach gives a list of axioms for the numbers as a complete ordered field. Under the usual axioms of set theory, one can show that these axioms are categorical, in the sense there is a model for the axioms. Any one of these models must be constructed, and most of these models are built using the basic properties of the rational number system as an ordered field. These constructions are described in detail in the main article. In addition to these notions, the real numbers, equipped with the absolute value function as a metric. Many important theorems in real analysis remain valid when they are restated as statements involving metric spaces and these theorems are frequently topological in nature, and placing them in the more abstract setting of metric spaces may lead to proofs that are shorter, more natural, or more elegant. The real numbers have several important lattice-theoretic properties that are absent in the complex numbers, most importantly, the real numbers form an ordered field, in which addition and multiplication preserve positivity. Moreover, the ordering of the numbers is total. These order-theoretic properties lead to a number of important results in analysis, such as the monotone convergence theorem, the intermediate value theorem. However, while the results in analysis are stated for real numbers. In particular, many ideas in analysis and operator theory generalize properties of the real numbers – such generalizations include the theories of Riesz spaces. Also, mathematicians consider real and imaginary parts of complex sequences, a sequence is a function whose domain is a countable, totally ordered set, usually taken to be the natural numbers or whole numbers. Occasionally, it is convenient to consider bidirectional sequences indexed by the set of all integers. Of interest in analysis, a real-valued sequence, here indexed by the natural numbers, is a map a, N → R, n ↦ a n. Each a = a n is referred to as a term of the sequence, a sequence that tends to a limit is said to be convergent, otherwise it is divergent
12.
Geometry
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Geometry is a branch of mathematics concerned with questions of shape, size, relative position of figures, and the properties of space. A mathematician who works in the field of geometry is called a geometer, Geometry arose independently in a number of early cultures as a practical way for dealing with lengths, areas, and volumes. Geometry began to see elements of mathematical science emerging in the West as early as the 6th century BC. By the 3rd century BC, geometry was put into a form by Euclid, whose treatment, Euclids Elements. Geometry arose independently in India, with texts providing rules for geometric constructions appearing as early as the 3rd century BC, islamic scientists preserved Greek ideas and expanded on them during the Middle Ages. By the early 17th century, geometry had been put on a solid footing by mathematicians such as René Descartes. Since then, and into modern times, geometry has expanded into non-Euclidean geometry and manifolds, while geometry has evolved significantly throughout the years, there are some general concepts that are more or less fundamental to geometry. These include the concepts of points, lines, planes, surfaces, angles, contemporary geometry has many subfields, Euclidean geometry is geometry in its classical sense. The mandatory educational curriculum of the majority of nations includes the study of points, lines, planes, angles, triangles, congruence, similarity, solid figures, circles, Euclidean geometry also has applications in computer science, crystallography, and various branches of modern mathematics. Differential geometry uses techniques of calculus and linear algebra to problems in geometry. It has applications in physics, including in general relativity, topology is the field concerned with the properties of geometric objects that are unchanged by continuous mappings. In practice, this often means dealing with large-scale properties of spaces, convex geometry investigates convex shapes in the Euclidean space and its more abstract analogues, often using techniques of real analysis. It has close connections to convex analysis, optimization and functional analysis, algebraic geometry studies geometry through the use of multivariate polynomials and other algebraic techniques. It has applications in areas, including cryptography and string theory. Discrete geometry is concerned mainly with questions of relative position of simple objects, such as points. It shares many methods and principles with combinatorics, Geometry has applications to many fields, including art, architecture, physics, as well as to other branches of mathematics. The earliest recorded beginnings of geometry can be traced to ancient Mesopotamia, the earliest known texts on geometry are the Egyptian Rhind Papyrus and Moscow Papyrus, the Babylonian clay tablets such as Plimpton 322. For example, the Moscow Papyrus gives a formula for calculating the volume of a truncated pyramid, later clay tablets demonstrate that Babylonian astronomers implemented trapezoid procedures for computing Jupiters position and motion within time-velocity space
13.
Formalism (mathematics)
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In playing this game one can prove that the Pythagorean theorem is valid because the string representing the Pythagorean theorem can be constructed using only the stated rules. According to formalism, the truths expressed in logic and mathematics are not about numbers, sets, or triangles or any other subject matter — in fact. They are syntactic forms whose shapes and locations have no meaning unless they are given an interpretation, Formalism is associated with rigorous method. In common use, a means the out-turn of the effort towards formalisation of a given limited area. In other words, matters can be formally discussed once captured in a formal system, complete formalisation is in the domain of computer science. Formalism stresses axiomatic proofs using theorems, specifically associated with David Hilbert, a formalist is an individual who belongs to the school of formalism, which is a certain mathematical-philosophical doctrine descending from Hilbert. Formalists are relatively tolerant and inviting to new approaches to logic, non-standard number systems, new set theories, the more games they study, the better. However, in all three of these examples, motivation is drawn from existing mathematical or philosophical concerns, the games are usually not arbitrary. Because of their connection with computer science, this idea is also advocated by mathematical intuitionists and constructivists in the computability tradition. Another version of formalism is known as deductivism. In deductivism, the Pythagorean theorem is not an absolute truth, under deductivism, the same view is held to be true for all other statements of formal logic and mathematics. Thus, formalism need not mean that these deductive sciences are nothing more than meaningless symbolic games and it is usually hoped that there exists some interpretation in which the rules of the game hold. Taking the deductivist view allows the working mathematician to suspend judgement on the philosophical questions. Many formalists would say that in practice, the systems to be studied are suggested by the demands of the particular science. A major early proponent of formalism was David Hilbert, whose program was intended to be a complete, Hilbert aimed to show the consistency of mathematical systems from the assumption that the finitary arithmetic was consistent. The way that Hilbert tried to show that a system was consistent was by formalizing it using a particular language. In order to formalize a system, you must first choose a language in which you can express. This language must include five components, It must include such as x
14.
Sheffer stroke
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It is also called nand or the alternative denial, since it says in effect that at least one of its operands is false. In Boolean algebra and digital electronics it is known as the NAND operation, like its dual, the NOR operator, NAND can be used by itself, without any other logical operator, to constitute a logical formal system. This property makes the NAND gate crucial to modern digital electronics, the NAND operation is a logical operation on two logical values. It produces a value of true, if — and only if — at least one of the propositions is false, the truth table of A NAND B is as follows, The stroke is named after Henry M. Because of self-duality of Boolean algebras, Sheffers axioms are equally valid for either of the NAND or NOR operations in place of the stroke. Sheffer interpreted the stroke as a sign for non-disjunction in his paper, mentioning non-conjunction only in a footnote and it was Jean Nicod who first used the stroke as a sign for non-conjunction in a paper of 1917 and which has since become current practice. Russell and Whitehead used the Sheffer stroke in the 1927 second edition of Principia Mathematica and suggested it as a replacement for the or and not operations of the first edition. Charles Sanders Peirce had discovered the functional completeness of NAND or NOR more than 30 years earlier, using the term ampheck, therefore is a functionally complete set. This can also be realized as follows, All three elements of the complete set can be constructed using only NAND. Thus the set must be complete as well. Hence any formal system including the Sheffer stroke must also include a means of indicating grouping and we shall employ to this effect. We also write p, q, r, … instead of p0, p1, Construction Rule I, For each natural number n, the symbol pn is a well-formed formula, called an atom. Construction Rule II, If X and Y are wffs, then is a wff, closure Rule, Any formulae which cannot be constructed by means of the first two Construction Rules are not wffs. The letters U, V, W, X, and Y are metavariables standing for wffs, then repeat this recursive deconstruction process to each of the subformulae. Eventually the formula should be reduced to its atoms, but if some subformula cannot be so reduced, All wffs of the form are axioms. Instances of, U ⊢ W are inference rules, since the only connective of this logic is |, the symbol | could be discarded altogether, leaving only the parentheses to group the letters. A pair of parentheses must always enclose a pair of wffs, examples of theorems in this simplified notation are. The notation can be simplified further, by letting, = ≡ U for any U and this simplification causes the need to change some rules, More than two letters are allowed within parentheses
15.
Alonzo Church
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Alonzo Church was an American mathematician and logician who made major contributions to mathematical logic and the foundations of theoretical computer science. He is best known for the calculus, Church–Turing thesis, proving the undecidability of the Entscheidungsproblem, Frege–Church ontology. Alonzo Church was born on June 14,1903, in Washington, D. C. where his father, the family later moved to Virginia after his father lost this position because of failing eyesight. With help from his uncle, also named Alonzo Church, he was able to attend the Ridgefield School for Boys in Ridgefield and he stayed at Princeton, earning a Ph. D. in mathematics in three years under Oswald Veblen. He married Mary Julia Kuczinski in 1925 and the couple had three children, Alonzo Church, Jr. Mary Ann and Mildred, after receiving his Ph. D. he taught briefly as an instructor at the University of Chicago and then received a two-year National Research Fellowship. This allowed him to attend Harvard University in 1927–1928 and then both University of Göttingen and University of Amsterdam the following year and he taught philosophy and mathematics at Princeton, 1929–1967, and at the University of California, Los Angeles, 1967–1990. He was a Plenary Speaker at the ICM in 1962 in Stockholm, a deeply religious person, he was a lifelong member of the Presbyterian church. He died in 1995 and was buried in Princeton Cemetery and this is known as Churchs theorem. His proof that Peano arithmetic is undecidable and his articulation of what has come to be known as the Church–Turing thesis. He was the editor of the Journal of Symbolic Logic. His creation of the lambda calculus, the lambda calculus emerged in his 1936 paper showing the unsolvability of the Entscheidungsproblem. This result preceded Alan Turings work on the problem, which also demonstrated the existence of a problem unsolvable by mechanical means. This resulted in the Church–Turing thesis, the lambda calculus influenced the design of the LISP programming language and functional programming languages in general. The Church encoding is named in his honor, many of Churchs doctoral students have led distinguished careers, including C. Anthony Anderson, Peter B. Andrews, George A. Barnard, David Berlinski, William W. Boone, Martin Davis, Alfred L. Foster, Leon Henkin, John G. Kemeny, Stephen C. Kochen, Maurice LAbbé, Isaac Malitz, Gary R. Mar, Michael O. Rabin, Nicholas Rescher, Hartley Rogers, Jr. J. Barkley Rosser, Dana Scott, Raymond Smullyan, a more complete list of Churchs students is available via Mathematics Genealogy Project. Alonzo Church, Introduction to Mathematical Logic Alonzo Church, The Calculi of Lambda-Conversion Alonzo Church, A Bibliography of Symbolic Logic, Introduction to the Collected Works of Alonzo Church, MIT Press, not yet published. In memoriam, Alonzo Church, The Bulletin of Symbolic Logic, wade, Nicholas, Alonzo Church,92, Theorist of the Limits of Mathematics, The New York Times, September 5,1995, p. B6
16.
Ernst Zermelo
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Ernst Friedrich Ferdinand Zermelo was a German logician and mathematician, whose work has major implications for the foundations of mathematics. He is known for his role in developing Zermelo–Fraenkel axiomatic set theory, Ernst Zermelo graduated from Berlins Luisenstädtisches Gymnasium in 1889. He then studied mathematics, physics and philosophy at the universities of Berlin, Halle and he finished his doctorate in 1894 at the University of Berlin, awarded for a dissertation on the calculus of variations. Zermelo remained at the University of Berlin, where he was appointed assistant to Planck, in 1897, Zermelo went to Göttingen, at that time the leading centre for mathematical research in the world, where he completed his habilitation thesis in 1899. In 1910, Zermelo left Göttingen upon being appointed to the chair of mathematics at Zurich University and he was appointed to an honorary chair at Freiburg im Breisgau in 1926, which he resigned in 1935 because he disapproved of Adolf Hitlers regime. At the end of World War II and at his request, Zermelo began to work on the problems of set theory under Hilberts influence and in 1902 published his first work concerning the addition of transfinite cardinals. By that time he had discovered the so-called Russell paradox. In 1904, he succeeded in taking the first step suggested by Hilbert towards the continuum hypothesis when he proved the well-ordering theorem and this result brought fame to Zermelo, who was appointed Professor in Göttingen, in 1905. Zermelo began to set theory in 1905, in 1908. See the article on Zermelo set theory for an outline of this paper, together with the original axioms, in 1922, Adolf Fraenkel and Thoralf Skolem independently improved Zermelos axiom system. The resulting 8 axiom system, now called Zermelo-Fraenkel axioms, is now the most commonly used system for set theory. Proposed in 1931, the Zermelos navigation problem is an optimal control problem. The problem deals with a boat navigating on a body of water, the boat is capable of a certain maximum speed, and we want to derive the best possible control to reach D in the least possible time. Without considering external forces such as current and wind, the control is for the boat to always head towards D. Its path then is a segment from O to D. With consideration of current and wind, if the force applied to the boat is non-zero the control for no current. Zermelo, Ernst, Ebbinghaus, Heinz-Dieter, Fraser, Craig G. Kanamori, Akihiro, from Frege to Gödel, A Source Book in Mathematical Logic, 1879–1931. Proof that every set can be well-ordered, 139−41, a new proof of the possibility of well-ordering, 183–98
17.
Predicate logic
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First-order logic – also known as first-order predicate calculus and predicate logic – is a collection of formal systems used in mathematics, philosophy, linguistics, and computer science. This distinguishes it from propositional logic, which does not use quantifiers, Sometimes theory is understood in a more formal sense, which is just a set of sentences in first-order logic. In first-order theories, predicates are associated with sets. In interpreted higher-order theories, predicates may be interpreted as sets of sets, There are many deductive systems for first-order logic which are both sound and complete. Although the logical relation is only semidecidable, much progress has been made in automated theorem proving in first-order logic. First-order logic also satisfies several metalogical theorems that make it amenable to analysis in proof theory, such as the Löwenheim–Skolem theorem, first-order logic is the standard for the formalization of mathematics into axioms and is studied in the foundations of mathematics. Peano arithmetic and Zermelo–Fraenkel set theory are axiomatizations of number theory and set theory, respectively, no first-order theory, however, has the strength to uniquely describe a structure with an infinite domain, such as the natural numbers or the real line. Axioms systems that do fully describe these two structures can be obtained in stronger logics such as second-order logic, for a history of first-order logic and how it came to dominate formal logic, see José Ferreirós. While propositional logic deals with simple declarative propositions, first-order logic additionally covers predicates, a predicate takes an entity or entities in the domain of discourse as input and outputs either True or False. Consider the two sentences Socrates is a philosopher and Plato is a philosopher, in propositional logic, these sentences are viewed as being unrelated and might be denoted, for example, by variables such as p and q. The predicate is a philosopher occurs in both sentences, which have a structure of a is a philosopher. The variable a is instantiated as Socrates in the first sentence and is instantiated as Plato in the second sentence, while first-order logic allows for the use of predicates, such as is a philosopher in this example, propositional logic does not. Relationships between predicates can be stated using logical connectives, consider, for example, the first-order formula if a is a philosopher, then a is a scholar. This formula is a statement with a is a philosopher as its hypothesis. The truth of this depends on which object is denoted by a. Quantifiers can be applied to variables in a formula, the variable a in the previous formula can be universally quantified, for instance, with the first-order sentence For every a, if a is a philosopher, then a is a scholar. The universal quantifier for every in this sentence expresses the idea that the if a is a philosopher. The negation of the sentence For every a, if a is a philosopher, then a is a scholar is logically equivalent to the sentence There exists a such that a is a philosopher and a is not a scholar
18.
Truth table
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In particular, truth tables can be used to show whether a propositional expression is true for all legitimate input values, that is, logically valid. A truth table has one column for each variable. Each row of the table contains one possible configuration of the input variables. See the examples below for further clarification, the Com row indicates whether an operator, op, is commutative - P op Q = Q op P. The L id row shows the operators left identities if it has any - values I such that I op Q = Q, the R id row shows the operators right identities if it has any - values I such that P op I = P. The four combinations of values for p, q, are read by row from the table above. The output function for p, q combination, can be read, by row. Key, The following table is oriented by column, rather than by row, There are four columns rather than four rows, to display the four combinations of p, q, as input. P, T T F Fq, T F T F There are 16 rows in this key, the output row for ↚ is thus 2, F F T F and the 16-row key is Logical operators can also be visualized using Venn diagrams. Logical conjunction is an operation on two values, typically the values of two propositions, that produces a value of true if both of its operands are true. The truth table for p AND q is as follows, In ordinary language terms, for all other assignments of logical values to p and to q the conjunction p ∧ q is false. The truth table for p OR q is as follows, Stated in English, if p, then p ∨ q is p, otherwise p ∨ q is q. The truth table associated with the conditional if p then q. Logical equality is an operation on two values, typically the values of two propositions, that produces a value of true if both operands are false or both operands are true. The truth table for p XNOR q is as follows, So p EQ q is true if p and q have the truth value. Exclusive disjunction is an operation on two values, typically the values of two propositions, that produces a value of true if one but not both of its operands is true. The truth table for p XOR q is as follows, For two propositions, XOR can also be written as ∨. The logical NAND is an operation on two values, typically the values of two propositions, that produces a value of false if both of its operands are true
19.
Syntax
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In linguistics, syntax is the set of rules, principles, and processes that govern the structure of sentences in a given language, specifically word order. The term syntax is used to refer to the study of such principles and processes. The goal of many syntacticians is to discover the syntactic rules common to all languages, in mathematics, syntax refers to the rules governing the behavior of mathematical systems, such as formal languages used in logic. The word syntax comes from Ancient Greek, σύνταξις coordination, which consists of σύν syn, together, and τάξις táxis, a basic feature of a languages syntax is the sequence in which the subject, verb, and object usually appear in sentences. Over 85% of languages usually place the subject first, either in the sequence SVO or the sequence SOV, the other possible sequences are VSO, VOS, OVS, and OSV, the last three of which are rare. In the West, the school of thought came to be known as traditional grammar began with the work of Dionysius Thrax. For centuries, work in syntax was dominated by a known as grammaire générale. This system took as its premise the assumption that language is a direct reflection of thought processes and therefore there is a single. It became apparent that there was no such thing as the most natural way to express a thought, the Port-Royal grammar modeled the study of syntax upon that of logic. Syntactic categories were identified with logical ones, and all sentences were analyzed in terms of Subject – Copula – Predicate, initially, this view was adopted even by the early comparative linguists such as Franz Bopp. The central role of syntax within theoretical linguistics became clear only in the 20th century, there are a number of theoretical approaches to the discipline of syntax. One school of thought, founded in the works of Derek Bickerton, sees syntax as a branch of biology, other linguists take a more Platonistic view, since they regard syntax to be the study of an abstract formal system. Yet others consider syntax a taxonomical device to reach broad generalizations across languages, the hypothesis of generative grammar is that language is a structure of the human mind. The goal of grammar is to make a complete model of this inner language. This model could be used to all human language and to predict the grammaticality of any given utterance. This approach to language was pioneered by Noam Chomsky, most generative theories assume that syntax is based upon the constituent structure of sentences. Generative grammars are among the theories that focus primarily on the form of a sentence and this complex category is notated as instead of V. NP\S is read as a category that searches to the left for an NP and outputs a sentence. The category of verb is defined as an element that requires two NPs to form a sentence
20.
Carnap
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Rudolf Carnap was a German-born philosopher who was active in Europe before 1935 and in the United States thereafter. He was a member of the Vienna Circle and an advocate of logical positivism. He is considered one of the giants among twentieth-century philosophers, Carnaps father had risen from the status of a poor ribbon-weaver to become the owner of a ribbon-making factory. His mother came from stock, her father was an educational reformer and her oldest brother was the archaeologist Wilhelm Dörpfeld. As a ten-year-old, Carnap accompanied his uncle on an expedition to Greece, Carnap was raised in a religious family, but later became an atheist. He began his education at the Barmen Gymnasium. From 1910 to 1914, he attended the University of Jena, but he also studied carefully Kants Critique of Pure Reason during a course taught by Bruno Bauch, and was one of very few students to attend Gottlob Freges courses in mathematical logic. While Carnap held moral and political opposition to World War I, after three years of service, he was given permission to study physics at the University of Berlin, 1917–18, where Albert Einstein was a newly appointed professor. Carnap then attended the University of Jena, where he wrote a thesis defining a theory of space. The physics department said it was too philosophical, and Bruno Bauch of the department said it was pure physics. Carnap then wrote another thesis in 1921, with Bauchs supervision, on the theory of space in a more orthodox Kantian style, in it he makes the clear distinction between formal, physical and perceptual spaces. Freges course exposed him to Bertrand Russells work on logic and philosophy and he accepted the effort to surpass traditional philosophy with logical innovations that inform the sciences. In 1924 and 1925, he attended seminars led by Edmund Husserl, the founder of phenomenology, Carnap discovered a kindred spirit when he met Hans Reichenbach at a 1923 conference. Reichenbach introduced Carnap to Moritz Schlick, a professor at the University of Vienna who offered Carnap a position in his department, when Wittgenstein visited Vienna, Carnap would meet with him. He wrote the 1929 manifesto of the Circle, and initiated the philosophy journal Erkenntnis, the formal system of the Aufbau was grounded in a single primitive dyadic predicate, which is satisfied if two individuals resemble each other. The Aufbau was greatly influenced by Principia Mathematica, and warrants comparison with the mereotopological metaphysics A. N. Whitehead developed over 1916–29 and it appears, however, that Carnap soon became somewhat disenchanted with this book. In particular, he did not authorize an English translation until 1967, pseudoproblems in Philosophy asserted that many philosophical questions were meaningless, i. e. the way they were posed amounted to an abuse of language. An operational implication of this opinion was taken to be the elimination of metaphysics from responsible human discourse and this is the statement for which Carnap was best known for many years
21.
Axiom of choice
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In mathematics, the axiom of choice, or AC, is an axiom of set theory equivalent to the statement that the Cartesian product of a collection of non-empty sets is non-empty. It states that for every indexed family i ∈ I of nonempty sets there exists an indexed family i ∈ I of elements such that x i ∈ S i for every i ∈ I. The axiom of choice was formulated in 1904 by Ernst Zermelo in order to formalize his proof of the well-ordering theorem. Informally put, the axiom of choice says that any collection of bins, each containing at least one object. One motivation for use is that a number of generally accepted mathematical results, such as Tychonoffs theorem. Contemporary set theorists also study axioms that are not compatible with the axiom of choice, the axiom of choice is avoided in some varieties of constructive mathematics, although there are varieties of constructive mathematics in which the axiom of choice is embraced. A choice function is a function f, defined on a collection X of nonempty sets, each choice function on a collection X of nonempty sets is an element of the Cartesian product of the sets in X. The axiom of choice asserts the existence of elements, it is therefore equivalent to, Given any family of nonempty sets. In this article and other discussions of the Axiom of Choice the following abbreviations are common, ZF – Zermelo–Fraenkel set theory omitting the Axiom of Choice. ZFC – Zermelo–Fraenkel set theory, extended to include the Axiom of Choice, There are many other equivalent statements of the axiom of choice. These are equivalent in the sense that, in the presence of basic axioms of set theory. One variation avoids the use of functions by, in effect. Given any set X of pairwise disjoint non-empty sets, there exists at least one set C that contains one element in common with each of the sets in X. This guarantees for any partition of a set X the existence of a subset C of X containing exactly one element from each part of the partition. Another equivalent axiom only considers collections X that are essentially powersets of other sets, For any set A, authors who use this formulation often speak of the choice function on A, but be advised that this is a slightly different notion of choice function. With this alternate notion of function, the axiom of choice can be compactly stated as Every set has a choice function. Which is equivalent to For any set A there is a function f such that for any non-empty subset B of A, f lies in B. The negation of the axiom can thus be expressed as, There is a set A such that for all functions f, however, that particular case is a theorem of Zermelo–Fraenkel set theory without the axiom of choice, it is easily proved by mathematical induction
22.
Frank P. Ramsey
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Frank Plumpton Ramsey was a British philosopher, mathematician and economist who died at the age of 26. Like Wittgenstein, he was a member of the Cambridge Apostles, Ramsey was born on 22 February 1903 in Cambridge where his father Arthur Stanley Ramsey, also a mathematician, was President of Magdalene College. His mother was Mary Agnes Stanley and he was the eldest of two brothers and two sisters, and his brother Michael Ramsey, the only one of the four siblings who was to remain Christian, later became Archbishop of Canterbury. He entered Winchester College in 1915 and later returned to Cambridge to study mathematics at Trinity College, while studying mathematics at Trinity College, Ramsey became a student to John Maynard Keynes, and an active member in the Apostles, a Cambridge discussion group. In 1923, he received his bachelors degree in mathematics, passing his examinations with the result of first class with distinction, easy-going, simple and modest, Ramsey had many interests besides his scientific work. After about an hour I said ‘Margaret will you fuck with me and he, like many of his contemporaries, including his Viennese flatmate and fellow Apostle Lionel Penrose, was intellectually interested in psychoanalysis. Ramseys analyst was Theodor Reik, a disciple of Freud, as one of the justifications for undertaking the therapy, he asserted in a letter to his mother that unconscious impulses might even affect the work of a mathematician. While in Vienna, he visited Wittgenstein in Puchberg, was befriended by the Wittgenstein family, neills experimental school four hours from Vienna at Sonntagsberg. In the summer of 1924, he continued his analysis by joining Reik at Dobbiaco, Ramsey returned to England in October 1924, with John Maynard Keyness support he became a fellow of Kings College, Cambridge. Ramsey married Lettice Baker in September 1925, the taking place in a Register Office since Ramsey was, as his wife described him. Despite his atheism, Ramsey was quite tolerant towards his brother when the decided to become a priest in the Church of England. In 1926 he became a university lecturer in mathematics and later a Director of Studies in Mathematics at Kings College, the Vienna Circle manifesto lists three of his publications in a bibliography of closely related authors. When I. A. Richards and C. K. Ogden, according to Richards, he mastered the language in almost hardly over a week, although other sources show he took before that one year of German in school. Ramsey was then able, at the age of 19, to make the first draft of the translation of the German text of Wittgensteins Tractatus Logico Philosophicus, for two weeks Ramsey discussed the difficulties he was facing in understanding the Tractatus. Wittgenstein made some corrections to the English translation in Ramseys copy and some annotations, Ramsey and John Maynard Keynes cooperated to try to bring Ludwig Wittgenstein back to Cambridge. Once Wittgenstein had returned to Cambridge, Ramsey became his nominal supervisor, Wittgenstein submitted the Tractatus Logico-Philosophicus as his doctoral thesis. G. E. Moore and Bertrand Russell acted as examiners, later, the three of them arranged financial aid for Wittgenstein to help him continue his research work. The contributions of Ramsey to these conversations were acknowledged by both Sraffa and Wittgenstein in their later work, suffering from chronic liver problems, Ramsey developed jaundice after an abdominal operation and died on 19 January 1930 at Guys Hospital in London at the age of 26
23.
Logical truth
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Logical truth is one of the most fundamental concepts in logic, and there are different theories on its nature. A logical truth is a statement which is true, and remains true under all reinterpretations of its components other than its logical constants and it is a type of analytic statement. All of philosophical logic can be thought of as providing accounts of the nature of logical truth, Logical truths are truths which are considered to be necessarily true. This is to say that they are considered to be such that they could not be untrue and it must be true in every sense of intuition, practices, and bodies of beliefs. However, it is not universally agreed that there are any statements which are necessarily true, a logical truth is considered by some philosophers to be a statement which is true in all possible worlds. This is contrasted with facts which are true in this world, as it has historically unfolded, later, with the rise of formal logic a logical truth was considered to be a statement which is true under all possible interpretations. Empiricists commonly respond to this objection by arguing that logical truths, are analytic, Logical truths, being analytic statements, do not contain any information about any matters of fact. Other than logical truths, there is also a class of analytic statements. The characteristic of such a statement is that it can be turned into a logical truth by substituting synonyms for synonyms salva veritate, can be turned into No unmarried man is married. By substituting unmarried man for its synonym bachelor, in his essay, Two Dogmas of Empiricism, the philosopher W. V. O. Quine called into question the distinction between analytic and synthetic statements, in his conclusion, Quine rejects that logical truths are necessary truths. Instead he posits that the truth-value of any statement can be changed, including logical truths, considering different interpretations of the same statement leads to the notion of truth value. The simplest approach to truth values means that the statement may be true in one case, in one sense of the term tautology, it is any type of formula or proposition which turns out to be true under any possible interpretation of its terms. This is synonymous to logical truth, however, the term tautology is also commonly used to refer to what could more specifically be called truth-functional tautologies. Not all logical truths are tautologies of such a kind, Logical constants, including logical connectives and quantifiers, can all be reduced conceptually to logical truth. For instance, two statements or more are logically incompatible if, and only if their conjunction is logically false, one statement logically implies another when it is logically incompatible with the negation of the other. A statement is true if, and only if its opposite is logically false. The opposite statements must contradict one another, in this way all logical connectives can be expressed in terms of preserving logical truth
24.
Inconsistency
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In classical deductive logic, a consistent theory is one that does not contain a contradiction. The lack of contradiction can be defined in either semantic or syntactic terms, the semantic definition states that a theory is consistent if and only if it has a model, i. e. there exists an interpretation under which all formulas in the theory are true. This is the 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 and only if there is no formula ϕ such that both ϕ and its negation ¬ ϕ are elements of the set T. Let A be set of closed sentences and ⟨ A ⟩ the set of closed sentences provable from A under some formal deductive system, the set of axioms A is consistent when ⟨ A ⟩ is. If there exists a system for which these semantic and syntactic definitions are equivalent for any theory formulated in a particular deductive logic. Stronger logics, such as logic, are not complete. A consistency proof is a proof that a particular theory is consistent. The early development of mathematical theory was driven by the desire to provide finitary consistency proofs for all of mathematics as part of Hilberts program. Hilberts program was impacted by incompleteness theorems, which showed that sufficiently strong proof theories cannot prove their own consistency. Although consistency can be proved by means of theory, it is often done in a purely syntactical way. The cut-elimination implies the consistency of the calculus, since there is obviously no cut-free proof of falsity, in theories of arithmetic, such as Peano arithmetic, there is an intricate relationship between the consistency of the theory and its completeness. A theory is complete if, for every formula φ in its language, Presburger arithmetic is an axiom system for the natural numbers under addition. It is both consistent and complete, Gödels incompleteness theorems show that any sufficiently strong recursively enumerable theory of arithmetic cannot be both complete and consistent. Gödels theorem applies to the theories of Peano arithmetic and Primitive recursive arithmetic, moreover, Gödels second incompleteness theorem shows that the consistency of sufficiently strong recursively enumerable theories of arithmetic can be tested in a particular way. Thus the consistency of a strong, recursively enumerable, consistent theory of arithmetic can never be proven in that system itself. The same result is true for recursively enumerable theories that can describe a strong enough fragment of arithmetic—including set theories such as Zermelo–Fraenkel set theory and these set theories cannot prove their own Gödel sentence—provided that they are consistent, which is generally believed. Because consistency of ZF is not provable in ZF, the weaker notion relative consistency is interesting in set theory
25.
Propositional logic
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Logical connectives are found in natural languages. In English for example, some examples are and, or, not”, the following is an example of a very simple inference within the scope of propositional logic, Premise 1, If its raining then its cloudy. Both premises and the conclusion are propositions, the premises are taken for granted and then with the application of modus ponens the conclusion follows. Not only that, but they will also correspond with any other inference of this form, Propositional logic may be studied through a formal system in which formulas of a formal language may be interpreted to represent propositions. A system of rules and axioms allows certain formulas to be derived. These derived formulas are called theorems and may be interpreted to be true propositions, a constructed sequence of such formulas is known as a derivation or proof and the last formula of the sequence is the theorem. The derivation may be interpreted as proof of the represented by the theorem. When a formal system is used to represent formal logic, only statement letters are represented directly, usually in truth-functional propositional logic, formulas are interpreted as having either a truth value of true or a truth value of false. Truth-functional propositional logic and systems isomorphic to it, are considered to be zeroth-order logic, although propositional logic had been hinted by earlier philosophers, it was developed into a formal logic by Chrysippus in the 3rd century BC and expanded by his successor Stoics. The logic was focused on propositions and this advancement was different from the traditional syllogistic logic which was focused on terms. However, later in antiquity, the propositional logic developed by the Stoics was no longer understood, consequently, the system was essentially reinvented by Peter Abelard in the 12th century. Propositional logic was eventually refined using symbolic logic, the 17th/18th-century mathematician Gottfried Leibniz has been credited with being the founder of symbolic logic for his work with the calculus ratiocinator. Although his work was the first of its kind, it was unknown to the larger logical community, consequently, many of the advances achieved by Leibniz were recreated by logicians like George Boole and Augustus De Morgan completely independent of Leibniz. Just as propositional logic can be considered an advancement from the earlier syllogistic logic, one author describes predicate logic as combining the distinctive features of syllogistic logic and propositional logic. Consequently, predicate logic ushered in a new era in history, however, advances in propositional logic were still made after Frege, including Natural Deduction. Natural deduction was invented by Gerhard Gentzen and Jan Łukasiewicz, Truth-Trees were invented by Evert Willem Beth. The invention of truth-tables, however, is of controversial attribution, within works by Frege and Bertrand Russell, are ideas influential to the invention of truth tables. The actual tabular structure, itself, is credited to either Ludwig Wittgenstein or Emil Post
26.
Catch-22 (logic)
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A catch-22 is a paradoxical situation from which an individual cannot escape because of contradictory rules. The term was coined by Joseph Heller, who used it in his 1961 novel Catch-22, an example would be, How am I supposed to gain experience if Im constantly turned down for not having any. Catch-22s often result from rules, regulations, or procedures that an individual is subject to but has no control over because to fight the rule is to accept it. Another example is a situation in which someone is in need of something that can only be had by not being in need of it. One connotation of the term is that the creators of the situation have created arbitrary rules in order to justify. Joseph Heller coined the term in his 1961 novel Catch-22, which describes absurd bureaucratic constraints on soldiers in World War II and this phrase also means a dilemma or difficult circumstance from which there is no escape because of mutually conflicting or dependent conditions. Different formulations of Catch-22 appear throughout the novel, the term is applied to various loopholes and quirks of the military system, always with the implication that rules are inaccessible to and slanted against those lower in the hierarchy. In chapter 6, Yossarian is told that Catch-22 requires him to do anything his commanding officer tells him to do, besides referring to an unsolvable logical dilemma, Catch-22 is invoked to explain or justify the military bureaucracy. For example, in the first chapter, it requires Yossarian to sign his name to letters that he censors while he is confined to a hospital bed. One clause mentioned in chapter 10 closes a loophole in promotions, through courts-martial for going AWOL, he would be busted in rank back to private, but Catch-22 limited the number of times he could do this before being sent to the stockade. At another point in the book, a prostitute explains to Yossarian that she marry him because he is crazy. She considers any man crazy who would marry a woman who is not a virgin, captain Black asks Milo, Youre not against Catch-22, are you. In chapter 40, Catch-22 forces Colonels Korn and Cathcart to promote Yossarian to Major and they fear that if they do not, others will refuse to fly, just as Yossarian did. Heller originally wanted to call the phrase by other numbers, but he, the number has no particular significance, it was chosen more or less for euphony. The title was originally Catch-18, but Heller changed it after the popular Mila 18 was published a short time beforehand, the term catch-22 has filtered into common usage in the English language. In a 1975 interview, Heller said the term would not translate well into other languages, james E. Combs and Dan D. Nimmo suggest that the idea of a catch-22 has gained popular currency because so many people in modern society are exposed to frustrating bureaucratic logic. They write, Everyone, then, who deals with organizations understands the bureaucratic logic of Catch-22, along with George Orwells doublethink, Catch-22 has become one of the best-recognized ways to describe the predicament of being trapped by contradictory rules. A significant type of definition of medicine has been termed a catch-22
27.
Tractatus Logico-Philosophicus
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The Tractatus Logico-Philosophicus is the only book-length philosophical work published by the Austrian philosopher Ludwig Wittgenstein in his lifetime. G. E. Moore originally suggested the works Latin title as homage to the Tractatus Theologico-Politicus by Baruch Spinoza. Wittgenstein wrote the notes for the Tractatus while he was a soldier during World War I and completed it when a prisoner of war at Como and it was first published in German in 1921 as Logisch-Philosophische Abhandlung. The Tractatus was influential chiefly amongst the logical positivists of the Vienna Circle, such as Rudolf Carnap, Bertrand Russells article The Philosophy of Logical Atomism is presented as a working out of ideas that he had learned from Wittgenstein. The Tractatus employs a notoriously austere and succinct literary style, the work contains almost no arguments as such, but rather consists of declarative statements that are meant to be self-evident. The statements are hierarchically numbered, with seven basic propositions at the primary level, Wittgensteins later works, notably the posthumously published Philosophical Investigations, criticised many of the ideas in the Tractatus. There are seven main propositions in the text and these are, The world is everything that is the case. What is the case is the existence of states of affairs, a logical picture of facts is a thought. A thought is a proposition with a sense, a proposition is a truth-function of elementary propositions. The general form of a proposition is the form of a truth function. This is the form of a proposition. Whereof one cannot speak, thereof one must be silent, the first chapter is very brief,1 The world is all that is the case. 1.1 The world is the totality of facts, not of things,1.11 The world is determined by the facts, and by their being all the facts. 1.12 For the totality of facts determines what is the case,1.13 The facts in logical space are the world. 1.2 The world divides into facts,1.21 Each item can be the case or not the case while everything else remains the same. This along with the beginning of two can be taken to be the relevant parts of Wittgensteins metaphysical view that he use to support his picture theory of language. These sections concern Wittgensteins view that the sensible, changing world we perceive does not consist of substance, Proposition two begins with a discussion of objects, form and substance. 2 What is the case—a fact—is the existence of atomic facts,2.01 An atomic fact is a combination of objects
28.
Wittgenstein
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Ludwig Josef Johann Wittgenstein was an Austrian-British philosopher who worked primarily in logic, the philosophy of mathematics, the philosophy of mind, and the philosophy of language. From 1929 to 1947, Wittgenstein taught at the University of Cambridge, during his lifetime he published just one slim book, the 75-page Tractatus Logico-Philosophicus, one article, one book review and a childrens dictionary. His voluminous manuscripts were edited and published posthumously, Philosophical Investigations appeared as a book in 1953, and has since come to be recognised as one of the most important works of philosophy in the twentieth century. His teacher Bertrand Russell described Wittgenstein as the most perfect example I have ever known of genius as traditionally conceived, passionate, profound, intense, born in Vienna into one of Europes richest families, he inherited a large fortune from his father in 1913. Three of his brothers committed suicide, with Wittgenstein contemplating it too and he described philosophy as the only work that gives me real satisfaction. His philosophy is divided into an early period, exemplified by the Tractatus. The later Wittgenstein rejected many of the assumptions of the Tractatus, ludwigs grandmother Fanny was a first cousin of the famous violinist Joseph Joachim. They had 11 children—among them Wittgensteins father, Karl Otto Clemens Wittgenstein became an industrial tycoon, and by the late 1880s was one of the richest men in Europe, with an effective monopoly on Austrias steel cartel. Thanks to Karl, the Wittgensteins became the second wealthiest family in Austria-Hungary, however, their wealth diminished due to post-1918 hyperinflation and subsequently during the Great Depression, although even as late as 1938 they owned 13 mansions in Vienna alone. Wittgensteins mother was Leopoldine Maria Josefa Kalmus, known among friends as Poldi and her father was a Bohemian Jew and her mother was Austrian-Slovene Catholic—she was Wittgensteins only non-Jewish grandparent. She was an aunt of the Nobel Prize laureate Friedrich Hayek on her maternal side, Wittgenstein was born at 8,30 pm on 26 April 1889 in the so-called Wittgenstein Palace at Alleegasse 16, now the Argentinierstrasse, near the Karlskirche. Karl and Poldi had nine children in all, the children were baptized as Catholics, received formal Catholic instruction, and raised in an exceptionally intense environment. The family was at the center of Viennas cultural life, Bruno Walter described the life at the Wittgensteins palace as an atmosphere of humanity. Karl was a patron of the arts, commissioning works by Auguste Rodin and financing the citys exhibition hall and art gallery. Gustav Klimt painted Wittgensteins sister for her portrait, and Johannes Brahms. For Wittgenstein, who highly valued precision and discipline, contemporary music was never considered acceptable at all, music, he said to his friend Drury in 1930, came to a full stop with Brahms, and even in Brahms I can begin to hear the noise of machinery. He also learnt to play the clarinet in his thirties, a fragment of music, composed by Wittgenstein, was discovered in one of his 1931 notebooks, by Michael Nedo, Director of the Wittgenstein Institute in Cambridge. Three of the five brothers would commit suicide
29.
Axiom of replacement
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In set theory, the axiom schema of replacement is a schema of axioms in Zermelo–Fraenkel set theory that asserts that the image of any set under any definable mapping is also a set. It is necessary for the construction of certain infinite sets in ZF, the axiom schema is motivated by the idea that whether a class is a set depends only on the cardinality of the class, not on the rank of its elements. Thus, if one class is small enough to be a set, and there is a surjection from that class to a second class, the axiom states that the second class is also a set. However, because ZFC only speaks of sets, not proper classes, the schema is stated only for definable surjections, suppose P is a definable binary relation such that for every set x there is a unique set y such that P holds. There is a definable function F P, where F P = Y if. Consider the class B defined such for every set y, y ∈ B if, B is called the image of A under F P, and denoted F P or. The axiom schema of replacement states that if F is a class function, as above. This can be seen as a principle of smallness, the states that if A is small enough to be a set. It is implied by the axiom of limitation of size. In the formal language of set theory, the schema is, ∀ w 1, …, w n ∀ A The axiom schema of collection is closely related to. While replacement says that the image itself is a set, collection merely says that some superclass of the image is a set, in other words, the resulting set, B, is not required to be minimal. This version of collection also lacks the requirement on ϕ. Suppose that the variables of ϕ are among w 1, …, w n, x, y. Then the axiom schema is, ∀ w 1, …, w n That is, the relation defined by ϕ is not required to be a function--some x ∈ A may correspond to many y s B. In this case, the image set B whose existence is asserted must contain at least one such y for x in the original set. However, the schema as stated requires that, if an element x of A is associated with at least one set y. The resulting axiom schema is also called the axiom schema of boundedness, the axiom schema of collection is equivalent to the axiom schema of replacement over the remainder of the ZF axioms. However, this is not so in the absence of the Power Set Axiom or constructive counterpart of ZF, the ordinal number ω·2 = ω + ω is the first ordinal that cannot be constructed without replacement
30.
Von Neumann ordinal
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In set theory, an ordinal number, or ordinal, is one generalization of the concept of a natural number that is used to describe a way to arrange a collection of objects in order, one after another. Any finite collection of objects can be put in order just by the process of counting, labeling the objects with distinct whole numbers, Ordinal numbers are thus the labels needed to arrange collections of objects in order. An ordinal number is used to describe the type of a well ordered set. Whereas ordinals are useful for ordering the objects in a collection, they are distinct from cardinal numbers, although the distinction between ordinals and cardinals is not always apparent in finite sets, different infinite ordinals can describe the same cardinal. Like other kinds of numbers, ordinals can be added, multiplied, a natural number can be used for two purposes, to describe the size of a set, or to describe the position of an element in a sequence. When restricted to finite sets these two concepts coincide, there is one way to put a finite set into a linear sequence. This is because any set has only one size, there are many nonisomorphic well-orderings of any infinite set. Whereas the notion of number is associated with a set with no particular structure on it. A well-ordered set is an ordered set in which there is no infinite decreasing sequence, equivalently. Ordinals may be used to label the elements of any given well-ordered set and this length is called the order type of the set. Any ordinal is defined by the set of ordinals that precede it, in fact, the most common definition of ordinals identifies each ordinal as the set of ordinals that precede it. For example, the ordinal 42 is the type of the ordinals less than it, i. e. the ordinals from 0 to 41. Conversely, any set of ordinals that is downward-closed—meaning that for any ordinal α in S and any ordinal β < α, β is also in S—is an ordinal. There are infinite ordinals as well, the smallest infinite ordinal is ω, which is the type of the natural numbers. After all of these come ω·2, ω·2+1, ω·2+2, and so on, then ω·3, now the set of ordinals formed in this way must itself have an ordinal associated with it, and that is ω2. Further on, there will be ω3, then ω4, and so on, and ωω, then ωωω, then later ωωωω and this can be continued indefinitely far. The smallest uncountable ordinal is the set of all countable ordinals, in a well-ordered set, every non-empty subset contains a distinct smallest element. Given the axiom of dependent choice, this is equivalent to just saying that the set is ordered and there is no infinite decreasing sequence
31.
Axiomatic 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
32.
Begriffsschrift
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Begriffsschrift is a book on logic by Gottlob Frege, published in 1879, and the formal system set out in that book. Begriffsschrift is usually translated as concept writing or concept notation, the title of the book identifies it as a formula language, modeled on that of arithmetic. Freges motivation for developing his formal approach to logic resembled Leibnizs motivation for his calculus ratiocinator, Frege went on to employ his logical calculus in his research on the foundations of mathematics, carried out over the next quarter century. The calculus contains the first appearance of quantified variables, and is essentially classical bivalent second-order logic with identity and it is bivalent in that sentences or formulas denote either True or False, second order because it includes relation variables in addition to object variables and allows quantification over both. The modifier with identity specifies that the language includes the identity relation, Frege presents his calculus using idiosyncratic two-dimensional notation, connectives and quantifiers are written using lines connecting formulas, rather than the symbols ¬, ∧, and ∀ in use today. For example, that judgement B materially implies judgement A, i. e, let signify that the third of those possibilities does not obtain, but one of the three others does. So if we negate, that means the third possibility is valid, i. e. we negate A, Frege declared nine of his propositions to be axioms, and justified them by arguing informally that, given their intended meanings, they express self-evident truths. – govern material implication, – negation, and identity, expresses Leibnizs indiscernibility of identicals, and asserts that identity is a reflexive relation. This rule is much harder to articulate precisely than the two preceding rules, and Frege invokes it in ways that are not obviously legitimate. The main results of the chapter, titled Parts from a general series theory. Frege applied the results from the Begriffsschrifft, including those on the ancestral of a relation, thus, if we take xRy to be the relation y = x +1, then 0R*y is the predicate y is a natural number. Says that if x, y, and z are natural numbers, then one of the following must hold, x < y, x = y and this is the so-called law of trichotomy. For a careful recent study of how the Begriffsschrift was reviewed in the German mathematical literature, some reviewers, especially Ernst Schröder, were on the whole favorable. Some vestige of Freges notation survives in the turnstile symbol ⊢ derived from his Urteilsstrich │, Frege used these symbols in the Begriffsschrift in the unified form ├─ for declaring that a proposition is true. In his later Grundgesetze he revises slightly his interpretation of the ├─ symbol, in Begriffsschrift the Definitionsdoppelstrich │├─ indicates that a proposition is a definition. Furthermore, the negation sign ¬ can be read as a combination of the horizontal Inhaltsstrich with a vertical negation stroke and this negation symbol was reintroduced by Arend Heyting in 1930 to distinguish intuitionistic from classical negation. It also appears in Gerhard Gentzens doctoral dissertation, in the Tractatus Logico Philosophicus, Ludwig Wittgenstein pays homage to Frege by employing the term Begriffsschrift as a synonym for logical formalism. Freges 1892 essay, Sense and Reference, recants some of the conclusions of the Begriffsschrifft about identity, ancestral relation Freges propositional calculus Gottlob Frege
33.
Boolean algebra (logic)
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In mathematics and mathematical logic, Boolean algebra is the branch of algebra in which the values of the variables are the truth values true and false, usually denoted 1 and 0 respectively. It is thus a formalism for describing logical relations in the way that ordinary algebra describes numeric relations. Boolean algebra was introduced by George Boole in his first book The Mathematical Analysis of Logic, according to Huntington, the term Boolean algebra was first suggested by Sheffer in 1913. Boolean algebra has been fundamental in the development of digital electronics and it is also used in set theory and statistics. Booles algebra predated the modern developments in algebra and mathematical logic. In an abstract setting, Boolean algebra was perfected in the late 19th century by Jevons, Schröder, Huntington, in fact, M. H. Stone proved in 1936 that every Boolean algebra is isomorphic to a field of sets. Shannon already had at his disposal the abstract mathematical apparatus, thus he cast his switching algebra as the two-element Boolean algebra, in circuit engineering settings today, there is little need to consider other Boolean algebras, thus switching algebra and Boolean algebra are often used interchangeably. Efficient implementation of Boolean functions is a problem in the design of combinational logic circuits. Logic sentences that can be expressed in classical propositional calculus have an equivalent expression in Boolean algebra, thus, Boolean logic is sometimes used to denote propositional calculus performed in this way. Boolean algebra is not sufficient to capture logic formulas using quantifiers, the closely related model of computation known as a Boolean circuit relates time complexity to circuit complexity. Whereas in elementary algebra expressions denote mainly numbers, in Boolean algebra they denote the truth values false and these values are represented with the bits, namely 0 and 1. Addition and multiplication then play the Boolean roles of XOR and AND respectively, Boolean algebra also deals with functions which have their values in the set. A sequence of bits is a commonly used such function, another common example is the subsets of a set E, to a subset F of E is associated the indicator function that takes the value 1 on F and 0 outside F. The most general example is the elements of a Boolean algebra, as with elementary algebra, the purely equational part of the theory may be developed without considering explicit values for the variables. The basic operations of Boolean calculus are as follows, AND, denoted x∧y, satisfies x∧y =1 if x = y =1 and x∧y =0 otherwise. OR, denoted x∨y, satisfies x∨y =0 if x = y =0, NOT, denoted ¬x, satisfies ¬x =0 if x =1 and ¬x =1 if x =0. Alternatively the values of x∧y, x∨y, and ¬x can be expressed by tabulating their values with truth tables as follows, the first operation, x → y, or Cxy, is called material implication. If x is then the value of x → y is taken to be that of y
34.
Stephen Kleene
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Stephen Cole Kleene /ˈkleɪniː/ KLAY-nee was an American mathematician. Kleenes work grounds the study of functions are computable. A number of concepts are named after him, Kleene hierarchy, Kleene algebra, the Kleene star, Kleenes recursion theorem. He also invented regular expressions, and made significant contributions to the foundations of mathematical intuitionism, although his last name is commonly pronounced /ˈkliːniː/ KLEE-nee or /ˈkliːn/ kleen, Kleene himself pronounced it /ˈkleɪniː/ KLAY-nee. His son, Ken Kleene, wrote, As far as I am aware this pronunciation is incorrect in all known languages, I believe that this novel pronunciation was invented by my father. Kleene was awarded the BA degree from Amherst College in 1930 and he was awarded the Ph. D. in mathematics from Princeton University in 1934. His thesis, entitled A Theory of Positive Integers in Formal Logic, was supervised by Alonzo Church, in the 1930s, he did important work on Churchs lambda calculus. In 1935, he joined the department at the University of Wisconsin–Madison. After two years as an instructor, he was appointed assistant professor in 1937, while a visiting scholar at the Institute for Advanced Study in Princeton, 1939–40, he laid the foundation for recursion theory, an area that would be his lifelong research interest. In 1941, he returned to Amherst College, where he spent one year as a professor of mathematics. During World War II, Kleene was a lieutenant commander in the United States Navy. He was an instructor of navigation at the U. S. Naval Reserves Midshipmens School in New York, in 1946, Kleene returned to Wisconsin, becoming a full professor in 1948 and the Cyrus C. MacDuffee professor of mathematics in 1964 and he was chair of the Department of Mathematics and Computer Science, 1962–63, and Dean of the College of Letters and Science from 1969 to 1974. The latter appointment he took on despite the considerable student unrest of the day and he retired from the University of Wisconsin in 1979. In 1999 the mathematics library at the University of Wisconsin was renamed in his honor, Kleenes teaching at Wisconsin resulted in three texts in mathematical logic, Kleene and Kleene and Vesley, often cited and still in print. Kleene wrote alternative proofs to the Gödels incompleteness theorems that enhanced their status and made them easier to teach. Kleene and Vesley is the classic American introduction to intuitionist logic, Kleene served as president of the Association for Symbolic Logic, 1956–58, and of the International Union of History and Philosophy of Science,1961. In 1990, he was awarded the National Medal of Science, the importance of Kleenes work led to the saying that Kleeneness is next to Gödelness
35.
Stephen Cole Kleene
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Stephen Cole Kleene /ˈkleɪniː/ KLAY-nee was an American mathematician. Kleenes work grounds the study of functions are computable. A number of concepts are named after him, Kleene hierarchy, Kleene algebra, the Kleene star, Kleenes recursion theorem. He also invented regular expressions, and made significant contributions to the foundations of mathematical intuitionism, although his last name is commonly pronounced /ˈkliːniː/ KLEE-nee or /ˈkliːn/ kleen, Kleene himself pronounced it /ˈkleɪniː/ KLAY-nee. His son, Ken Kleene, wrote, As far as I am aware this pronunciation is incorrect in all known languages, I believe that this novel pronunciation was invented by my father. Kleene was awarded the BA degree from Amherst College in 1930 and he was awarded the Ph. D. in mathematics from Princeton University in 1934. His thesis, entitled A Theory of Positive Integers in Formal Logic, was supervised by Alonzo Church, in the 1930s, he did important work on Churchs lambda calculus. In 1935, he joined the department at the University of Wisconsin–Madison. After two years as an instructor, he was appointed assistant professor in 1937, while a visiting scholar at the Institute for Advanced Study in Princeton, 1939–40, he laid the foundation for recursion theory, an area that would be his lifelong research interest. In 1941, he returned to Amherst College, where he spent one year as a professor of mathematics. During World War II, Kleene was a lieutenant commander in the United States Navy. He was an instructor of navigation at the U. S. Naval Reserves Midshipmens School in New York, in 1946, Kleene returned to Wisconsin, becoming a full professor in 1948 and the Cyrus C. MacDuffee professor of mathematics in 1964 and he was chair of the Department of Mathematics and Computer Science, 1962–63, and Dean of the College of Letters and Science from 1969 to 1974. The latter appointment he took on despite the considerable student unrest of the day and he retired from the University of Wisconsin in 1979. In 1999 the mathematics library at the University of Wisconsin was renamed in his honor, Kleenes teaching at Wisconsin resulted in three texts in mathematical logic, Kleene and Kleene and Vesley, often cited and still in print. Kleene wrote alternative proofs to the Gödels incompleteness theorems that enhanced their status and made them easier to teach. Kleene and Vesley is the classic American introduction to intuitionist logic, Kleene served as president of the Association for Symbolic Logic, 1956–58, and of the International Union of History and Philosophy of Science,1961. In 1990, he was awarded the National Medal of Science, the importance of Kleenes work led to the saying that Kleeneness is next to Gödelness
36.
Ivor Grattan-Guinness
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Ivor Owen Grattan-Guinness was a historian of mathematics and logic. Grattan-Guinness was born in Bakewell, England, his father was a mathematics teacher and he gained his bachelor degree as a Mathematics Scholar at Wadham College, Oxford, and an MSc in Mathematical Logic and the Philosophy of Science at the London School of Economics in 1966. He gained both the doctorate in 1969, and higher doctorate in 1978, in the History of Science at the University of London and he was Emeritus Professor of the History of Mathematics and Logic at Middlesex University, and a Visiting Research Associate at the London School of Economics. He was awarded the Kenneth O, in 2010, he was elected an Honorary Member of the Bertrand Russell Society. Grattan-Guinness spent much of his career at Middlesex University and he was a fellow at the Institute for Advanced Study in Princeton, New Jersey, United States, and a member of the International Academy of the History of Science. From 1974 to 1981, Grattan-Guinness was editor of the history of science journal Annals of Science, in 1979 he founded the journal History and Philosophy of Logic, and edited it until 1992. He was an editor of Historia Mathematica for twenty years from its inception in 1974. He also acted as editor to the editions of the writings of C. S. Peirce and Bertrand Russell. He was a member of the Executive Committee of the International Commission on the History of Mathematics from 1977 to 1993, Grattan-Guinness gave over 570 invited lectures to organisations and societies, or to conferences and congresses, in over 20 countries around the world. These lectures include tours undertaken in Australia, New Zealand, Italy, South Africa, from 1986 to 1988, Grattan-Guinness was the President of the British Society for the History of Mathematics, and for 1992 the Vice-President. In 1991, he was elected a member of the Académie Internationale dHistoire des Sciences. He was the Associate Editor for mathematicians and statisticians for the Oxford Dictionary of National Biography, Grattan-Guinness took an interest in the phenomenon of coincidence and has written on it for the Society for Psychical Research. He died of heart failure on 12 December 2014, aged 73 and he was especially interested in characterising how past thinkers, far removed from us in time, view their findings differently from the way we see them now. He has emphasised the importance of ignorance as a notion in this task. He did extensive research with original sources both published and unpublished, thanks to his reading and spoken knowledge of the main European languages, the Development of the Foundations of Mathematical Analysis from Euler to Riemann. Dear Russell—Dear Jourdain, a Commentary on Russells Logic, Based on His Correspondence with Philip Jourdain, from the Calculus to Set Theory, 1630–1910, An Introductory History. Psychical Research, A Guide to Its History, Principles & Practices - in celebration of 100 years of the Society for Psychical Research, Aquarian Press, convolutions in French Mathematics, 1800–1840 in 3 Vols. The Rainbow of Mathematics, A History of the Mathematical Sciences, from the Calculus to Set Theory 1630–1910, An Introductory History
37.
Ludwig Wittgenstein
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Ludwig Josef Johann Wittgenstein was an Austrian-British philosopher who worked primarily in logic, the philosophy of mathematics, the philosophy of mind, and the philosophy of language. From 1929 to 1947, Wittgenstein taught at the University of Cambridge, during his lifetime he published just one slim book, the 75-page Tractatus Logico-Philosophicus, one article, one book review and a childrens dictionary. His voluminous manuscripts were edited and published posthumously, Philosophical Investigations appeared as a book in 1953, and has since come to be recognised as one of the most important works of philosophy in the twentieth century. His teacher Bertrand Russell described Wittgenstein as the most perfect example I have ever known of genius as traditionally conceived, passionate, profound, intense, born in Vienna into one of Europes richest families, he inherited a large fortune from his father in 1913. Three of his brothers committed suicide, with Wittgenstein contemplating it too and he described philosophy as the only work that gives me real satisfaction. His philosophy is divided into an early period, exemplified by the Tractatus. The later Wittgenstein rejected many of the assumptions of the Tractatus, ludwigs grandmother Fanny was a first cousin of the famous violinist Joseph Joachim. They had 11 children—among them Wittgensteins father, Karl Otto Clemens Wittgenstein became an industrial tycoon, and by the late 1880s was one of the richest men in Europe, with an effective monopoly on Austrias steel cartel. Thanks to Karl, the Wittgensteins became the second wealthiest family in Austria-Hungary, however, their wealth diminished due to post-1918 hyperinflation and subsequently during the Great Depression, although even as late as 1938 they owned 13 mansions in Vienna alone. Wittgensteins mother was Leopoldine Maria Josefa Kalmus, known among friends as Poldi and her father was a Bohemian Jew and her mother was Austrian-Slovene Catholic—she was Wittgensteins only non-Jewish grandparent. She was an aunt of the Nobel Prize laureate Friedrich Hayek on her maternal side, Wittgenstein was born at 8,30 pm on 26 April 1889 in the so-called Wittgenstein Palace at Alleegasse 16, now the Argentinierstrasse, near the Karlskirche. Karl and Poldi had nine children in all, the children were baptized as Catholics, received formal Catholic instruction, and raised in an exceptionally intense environment. The family was at the center of Viennas cultural life, Bruno Walter described the life at the Wittgensteins palace as an atmosphere of humanity. Karl was a patron of the arts, commissioning works by Auguste Rodin and financing the citys exhibition hall and art gallery. Gustav Klimt painted Wittgensteins sister for her portrait, and Johannes Brahms. For Wittgenstein, who highly valued precision and discipline, contemporary music was never considered acceptable at all, music, he said to his friend Drury in 1930, came to a full stop with Brahms, and even in Brahms I can begin to hear the noise of machinery. He also learnt to play the clarinet in his thirties, a fragment of music, composed by Wittgenstein, was discovered in one of his 1931 notebooks, by Michael Nedo, Director of the Wittgenstein Institute in Cambridge. Three of the five brothers would commit suicide
38.
Metamath
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While the large database of proved theorems follows conventional ZFC set theory, the Metamath language is a metalanguage, suitable for developing a wide variety of formal systems. The set of symbols that can be used for constructing formulas is declared using $c and $v statements, for example, axioms and rules of inference are specified with $a statements along with $ for block scoping, for example, $ a1 $a |- $. $ The metamath program can convert statements to more conventional TeX notation, for example, note the inclusion of the proof in the $p statement. This substitution is just the replacement of a variable with an expression. Even if Metamath is used for mathematical proof checking, its algorithm is so general we can extend the field of its usage. In fact Metamath could be used with every sort of formal systems, in contrast, it is largely incompatible with logical systems which uses other things than formulas and inference rules. The original natural deduction system, which uses a stack, is an example of a system that cannot be implemented with Metamath. In the case of natural deduction however it is possible to append the stack to the formulas so that Metamaths requirements are met, what makes Metamath so generic is its substitution algorithm. This algorithm makes no assumption about the logic and only checks the substitutions of variables are correctly done. So here is an example of how this algorithm works. Steps 1 and 2 of the theorem 2p2e4 in set. mm are depicted left, lets explain how Metamath uses its substitution algorithm to check that step 2 is the logical consequence of step 1 when you use the theorem opreq2i. It is the conclusion of the theorem opreq2i, the theorem opreq2i states that if A = B, then =. This theorem would never appear under this form in a textbook but its literate formulation is banal. To check the proof Metamath attempts to unify = with =, there is only one way to do so, unifying C with 2, F with +, A with 2 and B with. So now Metamath uses the premise of opreq2i and this premise states that A = B. As a consequence of its previous computation, Metamath knows that A should be substituted by 2 and B by, the premise A = B becomes 2 = and thus step 1 is therefore generated. In its turn step 1 is unified with df-2, df-2 is the definition of the number 2 and states that 2 =. Here the unification is simply a matter of constants and is straightforward, so the verification is finished and these two steps of the proof of 2p2e4 are correct
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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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Outline of logic
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The following outline is provided as an overview of and topical guide to logic, Logic is the formal science of using reason and is considered a branch of both philosophy and mathematics. Logic investigates and classifies the structure of statements and arguments, both through the study of systems of inference and through the study of arguments in natural language. One of the aims of logic is to identify the correct and incorrect inferences, logicians study the criteria for the evaluation of arguments. By accident or design, fallacies may exploit emotional triggers in the listener or interlocutor, or take advantage of relationships between people. Fallacious arguments are often structured using rhetorical patterns that obscure any logical argument, fallacies can be used to win arguments regardless of the merits. There are dozens of types of fallacies, Formal logic – Mathematical logic, symbolic logic and formal logic are largely, if not completely synonymous. The essential feature of field is the use of formal languages to express the ideas whose logical validity is being studied. Axiom Deductive system Formal proof Formal system Formal theorem Syntactic consequence Syntax Transformation rules Model theory – The study of interpretation of formal systems, the field has grown to include the study of generalized computability and definability. The answers to these questions have led to a theory that is still being actively researched. D. In The Dictionary of the History of Ideas, Logic test Test your logic skills Logic Self-Taught, A Workbook
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History of logic
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The history of logic deals with the study of the development of the science of valid inference. Formal logics developed in ancient times in China, India, Greek methods, particularly Aristotelian logic as found in the Organon, found wide application and acceptance in Western science and mathematics for millennia. The Stoics, especially Chrysippus, began the development of predicate logic, christian and Islamic philosophers such as Boethius and William of Ockham further developed Aristotles logic in the Middle Ages, reaching a high point in the mid-fourteenth century. The period between the fourteenth century and the beginning of the century saw largely decline and neglect. Empirical methods ruled the day, as evidenced by Sir Francis Bacons Novum Organon of 1620, valid reasoning has been employed in all periods of human history. However, logic studies the principles of reasoning, inference. It is probable that the idea of demonstrating a conclusion first arose in connection with geometry, the ancient Egyptians discovered geometry, including the formula for the volume of a truncated pyramid. Ancient Babylon was also skilled in mathematics, while the ancient Egyptians empirically discovered some truths of geometry, the great achievement of the ancient Greeks was to replace empirical methods by demonstrative proof. Both Thales and Pythagoras of the Pre-Socratic philosophers seem aware of geometrys methods, fragments of early proofs are preserved in the works of Plato and Aristotle, and the idea of a deductive system was probably known in the Pythagorean school and the Platonic Academy. The proofs of Euclid of Alexandria are a paradigm of Greek geometry, the three basic principles of geometry are as follows, Certain propositions must be accepted as true without demonstration, such a proposition is known as an axiom of geometry. Every proposition that is not an axiom of geometry must be demonstrated as following from the axioms of geometry, the proof must be formal, that is, the derivation of the proposition must be independent of the particular subject matter in question. Further evidence that early Greek thinkers were concerned with the principles of reasoning is found in the fragment called dissoi logoi and this is part of a protracted debate about truth and falsity. Thales was said to have had a sacrifice in celebration of discovering Thales Theorem just as Pythagoras had the Pythagorean Theorem, Indian and Babylonian mathematicians knew his theorem for special cases before he proved it. It is believed that Thales learned that an angle inscribed in a semicircle is a right angle during his travels to Babylon, before 520 BC, on one of his visits to Egypt or Greece, Pythagoras might have met the c.54 years older Thales. The systematic study of proof seems to have begun with the school of Pythagoras in the sixth century BC. Indeed, the Pythagoreans, believing all was number, are the first philosophers to emphasize rather than matter. He is known for his obscure sayings and this logos holds always but humans always prove unable to understand it, both before hearing it and when they have first heard it. But other people fail to notice what they do when awake, in contrast to Heraclitus, Parmenides held that all is one and nothing changes
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Logic in computer science
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The ACM–IEEE Symposium on Logic in Computer Science is an annual academic conference on the theory and practice of computer science in relation to mathematical logic. Extended versions of selected papers of each years conference appear in renowned international journals such as Logical Methods in Computer Science, since the first installment in 1988, the cover page of the conference proceedings has featured an artwork entitled Irrational Tiling by Logical Quantifiers, by Alvy Ray Smith. Since 1995, each year the Kleene award is given to the best student paper, in addition, since 2006, the LICS Test-of-Time Award is given annually to one among the twenty-year-old LICS papers that have best met the test of time. Each year, since 2006, the LICS Test-of-Time Award recognizes those articles from LICS proceedings 20 years earlier, Kleene, is given for the best student paper. The list of computer science conferences contains other academic conferences in computer science
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Metamathematics
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Metamathematics is the study of mathematics itself using mathematical methods. This study produces metatheories, which are mathematical theories about other mathematical theories, emphasis on metamathematics owes itself to David Hilberts attempt to secure the foundations of mathematics in the early part of the 20th Century. Metamathematics provides a mathematical technique for investigating a great variety of foundation problems for mathematics. An important feature of metamathematics is its emphasis on differentiating between reasoning from inside a system and from outside a system, an informal illustration of this is categorizing the proposition 2+2=4 as belonging to mathematics while categorizing the proposition 2+2=4 is valid as belonging to metamathematics. Something similar can be said around the well-known Russells paradox, Metamathematics was intimately connected to mathematical logic, so that the early histories of the two fields, during the late 19th and early 20th centuries, largely overlap. More recently, mathematical logic has often included the study of new pure mathematics, such as set theory, recursion theory and pure model theory, serious metamathematical reflection began with the work of Gottlob Frege, especially his Begriffsschrift. David Hilbert was the first to invoke the term metamathematics with regularity, in his hands, it meant something akin to contemporary proof theory, in which finitary methods are used to study various axiomatized mathematical theorems. Today, metalogic and metamathematics are largely synonymous with each other, the discovery of hyperbolic geometry had important philosophical consequences for Metamathematics. Before its discovery there was just one geometry and mathematics, the idea that another geometry existed was considered improbable, the uproar of the Boeotians came and went, and gave an impetus to metamathematics and great improvements in mathematical rigour, analytical philosophy and logic. Begriffsschrift is a book on logic by Gottlob Frege, published in 1879, Begriffsschrift is usually translated as concept writing or concept notation, the full title of the book identifies it as a formula language, modeled on that of arithmetic, of pure thought. Freges motivation for developing his formal approach to logic resembled Leibnizs motivation for his calculus ratiocinator, Frege went on to employ his logical calculus in his research on the foundations of mathematics, carried out over the next quarter century. As such, this project is of great importance in the history of mathematics and philosophy. One of the inspirations and motivations for PM was the earlier work of Gottlob Frege on logic. PM sought to avoid this problem by ruling out the creation of arbitrary sets. This was achieved by replacing the notion of a set with notion of a hierarchy of sets of different types. Contemporary mathematics, however, avoids paradoxes such as Russells in less unwieldy ways, gödels completeness theorem is a fundamental theorem in mathematical logic that establishes a correspondence between semantic truth and syntactic provability in first-order logic. It makes a link between model theory that deals with what is true in different models, and proof theory that studies what can be formally proven in particular formal systems. More formally, the theorem says that if a formula is logically valid then there is a finite deduction of the formula
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A priori and a posteriori
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The Latin phrases a priori and a posteriori are philosophical terms of art popularized by Immanuel Kants Critique of Pure Reason, one of the most influential works in the history of philosophy. These terms are used with respect to reasoning to distinguish necessary conclusions from first premises from conclusions based on sense observation, a posteriori knowledge or justification is dependent on experience or empirical evidence, as with most aspects of science and personal knowledge. There are many points of view on two types of knowledge, and their relationship gives rise to one of the oldest problems in modern philosophy. The terms a priori and a posteriori are primarily used as adjectives to modify the noun knowledge, however, a priori is sometimes used to modify other nouns, such as truth. Philosophers also may use apriority and aprioricity as nouns to refer to the quality of being a priori, although definitions and use of the terms have varied in the history of philosophy, they have consistently labeled two separate epistemological notions. See also the related distinctions, deductive/inductive, analytic/synthetic, necessary/contingent, the intuitive distinction between a priori and a posteriori knowledge is best seen in examples. A priori Consider the proposition, If George V reigned at least four days and this is something that one knows a priori, because it expresses a statement that one can derive by reason alone. A posteriori Compare this with the proposition expressed by the sentence and this is something that one must come to know a posteriori, because it expresses an empirical fact unknowable by reason alone. Several philosophers reacting to Kant sought to explain a priori knowledge without appealing to, as Paul Boghossian explains and that has never been described in satisfactory terms. One theory, popular among the positivists of the early 20th century, is what Boghossian calls the analytic explanation of the a priori. The distinction between analytic and synthetic propositions was first introduced by Kant, in short, proponents of this explanation claimed to have reduced a dubious metaphysical faculty of pure reason to a legitimate linguistic notion of analyticity. However, the explanation of a priori knowledge has undergone several criticisms. Most notably, Quine argued that the distinction is illegitimate. Quine states, But for all its a priori reasonableness, a boundary between analytic and synthetic statements simply has not been drawn and that there is such a distinction to be drawn at all is an unempirical dogma of empiricists, a metaphysical article of faith. While the soundness of Quines critique is highly disputed, it had an effect on the project of explaining the a priori in terms of the analytic. The metaphysical distinction between necessary and contingent truths has also related to a priori and a posteriori knowledge. A proposition that is true is one whose negation is self-contradictory. Consider the proposition that all bachelors are unmarried and its negation, the proposition that some bachelors are married, is incoherent, because the concept of being unmarried is part of the concept of being a bachelor
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Definition
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A definition is a statement of the meaning of a term. Definitions can be classified into two categories, intensional definitions and extensional definitions. Another important category of definitions is the class of ostensive definitions, a term may have many different senses and multiple meanings, and thus require multiple definitions. In mathematics, a definition is used to give a meaning to a new term. Definitions and axioms are the basis on all of mathematics is constructed. In modern usage, a definition is something, typically expressed in words, the word or group of words that is to be defined is called the definiendum, and the word, group of words, or action that defines it is called the definiens. In the definition An elephant is a large gray animal native to Asia and Africa, the elephant is the definiendum. Note that the definiens is not the meaning of the word defined, there are many sub-types of definitions, often specific to a given field of knowledge or study. An intensional definition, also called a connotative definition, specifies the necessary, any definition that attempts to set out the essence of something, such as that by genus and differentia, is an intensional definition. An extensional definition, also called a denotative definition, of a concept or term specifies its extension and it is a list naming every object that is a member of a specific set. An extensional definition would be the list of wrath, greed, sloth, pride, lust, envy, a genus–differentia definition is a type of intensional definition that takes a large category and narrows it down to a smaller category by a distinguishing characteristic. The differentia, The portion of the new definition that is not provided by the genus, for example, consider the following genus-differentia definitions, a triangle, A plane figure that has three straight bounding sides. A quadrilateral, A plane figure that has four straight bounding sides and those definitions can be expressed as a genus and two differentiae. It is possible to have two different genus-differentia definitions that describe the same term, especially when the term describes the overlap of two large categories, for instance, both of these genus-differentia definitions of square are equally acceptable, a square, a rectangle that is a rhombus. A square, a rhombus that is a rectangle, thus, a square is a member of both the genus rectangle and the genus rhombus. One important form of the definition is ostensive definition. This gives the meaning of a term by pointing, in the case of an individual, to the thing itself, or in the case of a class, to examples of the right kind. So one can explain who Alice is by pointing her out to another, or what a rabbit is by pointing at several, the process of ostensive definition itself was critically appraised by Ludwig Wittgenstein
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Logical consequence
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Logical consequence is a fundamental concept in logic, which describes the relationship between statements that holds true when one statement logically follows from one or more statements. A valid logical argument is one in which the conclusions are entailed by the premises, the philosophical analysis of logical consequence involves the questions, In what sense does a conclusion follow from its premises. And What does it mean for a conclusion to be a consequence of premises, All of philosophical logic is meant to provide accounts of the nature of logical consequence and the nature of logical truth. Logical consequence is necessary and formal, by way of examples that explain with formal proof and models of interpretation. A sentence is said to be a consequence of a set of sentences, for a given language, if and only if. The most widely prevailing view on how to best account for logical consequence is to appeal to formality and this is to say that whether statements follow from one another logically depends on the structure or logical form of the statements without regard to the contents of that form. Syntactic accounts of logical consequence rely on schemes using inference rules, for instance, we can express the logical form of a valid argument as, All A are B. All C are A. Therefore, all C are B and this argument is formally valid, because every instance of arguments constructed using this scheme are valid. This is in contrast to an argument like Fred is Mikes brothers son, if you know that Q follows logically from P no information about the possible interpretations of P or Q will affect that knowledge. Our knowledge that Q is a consequence of P cannot be influenced by empirical knowledge. Deductively valid arguments can be known to be so without recourse to experience, however, formality alone does not guarantee that logical consequence is not influenced by empirical knowledge. So the a property of logical consequence is considered to be independent of formality. The two prevailing techniques for providing accounts of logical consequence involve expressing the concept in terms of proofs, the study of the syntactic consequence is called proof theory whereas the study of semantic consequence is called model theory. A formula A is a syntactic consequence within some formal system F S of a set Γ of formulas if there is a proof in F S of A from the set Γ. Γ ⊢ F S A Syntactic consequence does not depend on any interpretation of the formal system, or, in other words, the set of the interpretations that make all members of Γ true is a subset of the set of the interpretations that make A true. Modal accounts of logical consequence are variations on the basic idea, Γ ⊢ A is true if and only if it is necessary that if all of the elements of Γ are true. Alternatively, Γ ⊢ A is true if and only if it is impossible for all of the elements of Γ to be true, such accounts are called modal because they appeal to the modal notions of logical necessity and logical possibility. Consider the modal account in terms of the argument given as an example above, the conclusion is a logical consequence of the premises because we cant imagine a possible world where all frogs are green, Kermit is a frog, and Kermit is not green