1.
Philosophy of mathematics
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The philosophy of mathematics is the branch of philosophy that studies the philosophical assumptions, foundations, and implications of mathematics. The aim of the philosophy of mathematics is to provide an account of the nature and methodology of mathematics, the logical and structural nature of mathematics itself makes this study both broad and unique among its philosophical counterparts. The terms philosophy of mathematics and mathematical philosophy are frequently used interchangeably, the latter, however, may be used to refer to several other areas of study. Another refers to the philosophy of an individual practitioner or a like-minded community of practicing mathematicians. Recurrent themes include, What is the role of Mankind in developing mathematics, what are the sources of mathematical subject matter. What is the status of mathematical entities. What does it mean to refer to a mathematical object, what is the character of a mathematical proposition. What is the relation between logic and mathematics, what is the role of hermeneutics in mathematics. What kinds of play a role in mathematics. What are the objectives of mathematical inquiry, what gives mathematics its hold on experience. What are the human traits behind mathematics, what is the source and nature of mathematical truth. What is the relationship between the world of mathematics and the material universe. The origin of mathematics is subject to argument, whether the birth of mathematics was a random happening or induced by necessity duly contingent upon other subjects, say for example physics, is still a matter of prolific debates. Many thinkers have contributed their ideas concerning the nature of mathematics, there are traditions of mathematical philosophy in both Western philosophy and Eastern philosophy. Greek philosophy on mathematics was strongly influenced by their study of geometry, for example, at one time, the Greeks held the opinion that 1 was not a number, but rather a unit of arbitrary length. A number was defined as a multitude, therefore,3, for example, represented a certain multitude of units, and was thus not truly a number. At another point, an argument was made that 2 was not a number. These earlier Greek ideas of numbers were later upended by the discovery of the irrationality of the root of two
2.
Mathematics
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Mathematics is the study of topics such as quantity, structure, space, and change. There is a range of views among mathematicians and philosophers as to the exact scope, Mathematicians seek out patterns and use them to formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proof, when mathematical structures are good models of real phenomena, then mathematical reasoning can provide insight or predictions about nature. Through the use of abstraction and logic, mathematics developed from counting, calculation, measurement, practical mathematics has been a human activity from as far back as written records exist. The research required to solve mathematical problems can take years or even centuries of sustained inquiry, rigorous arguments first appeared in Greek mathematics, most notably in Euclids Elements. Galileo Galilei said, The universe cannot be read until we have learned the language and it is written in mathematical language, and the letters are triangles, circles and other geometrical figures, without which means it is humanly impossible to comprehend a single word. Without these, one is wandering about in a dark labyrinth, carl Friedrich Gauss referred to mathematics as the Queen of the Sciences. Benjamin Peirce called mathematics the science that draws necessary conclusions, David Hilbert said of mathematics, We are not speaking here of arbitrariness in any sense. Mathematics is not like a game whose tasks are determined by arbitrarily stipulated rules, rather, it is a conceptual system possessing internal necessity that can only be so and by no means otherwise. Albert Einstein stated that as far as the laws of mathematics refer to reality, they are not certain, Mathematics is essential in many fields, including natural science, engineering, medicine, finance and the social sciences. Applied mathematics has led to entirely new mathematical disciplines, such as statistics, Mathematicians also engage in pure mathematics, or mathematics for its own sake, without having any application in mind. There is no clear line separating pure and applied mathematics, the history of mathematics can be seen as an ever-increasing series of abstractions. The earliest uses of mathematics were in trading, land measurement, painting and weaving patterns, in Babylonian mathematics elementary arithmetic first appears in the archaeological record. Numeracy pre-dated writing and numeral systems have many and diverse. Between 600 and 300 BC the Ancient Greeks began a study of mathematics in its own right with Greek mathematics. Mathematics has since been extended, and there has been a fruitful interaction between mathematics and science, to the benefit of both. Mathematical discoveries continue to be made today, the overwhelming majority of works in this ocean contain new mathematical theorems and their proofs. The word máthēma is derived from μανθάνω, while the modern Greek equivalent is μαθαίνω, in Greece, the word for mathematics came to have the narrower and more technical meaning mathematical study even in Classical times
3.
Luitzen Egbertus Jan Brouwer
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He was the founder of the mathematical philosophy of intuitionism. Early in his career, Brouwer proved a number of theorems that were in the field of topology. The main results were his fixed point theorem, the invariance of degree. The most popular of the three among mathematicians is the first one called the Brouwer Fixed Point Theorem and it is a simple corollary to the second, about the topological invariance of degree, and this one is the most popular among algebraic topologists. The third is perhaps the hardest, in 1912, at age 31, he was elected a member of the Royal Netherlands Academy of Arts and Sciences. As a variety of mathematics, intuitionism is essentially a philosophy of the foundations of mathematics. It is sometimes and rather simplistically characterized by saying that its adherents refuse to use the law of excluded middle in mathematical reasoning, Brouwer was a member of the Significs group. It formed part of the history of semiotics—the study of symbols—around Victoria. The original meaning of his intuitionism probably can not be completely disentangled from the milieu of that group. In 1905, at the age of 24, Brouwer expressed his philosophy of life in a short tract Life, Art, arthur Schopenhauer had a formative influence on Brouwer, not least because he insisted that all concepts be fundamentally based on sense intuitions. All interwoven with some kind of pessimism and mystical attitude to life which is not mathematics and it was then that Brouwer felt free to return to his revolutionary project which he was now calling intuitionism. He was combative for a young man and he was involved in a very public and eventually demeaning controversy in the later 1920s with Hilbert over editorial policy at Mathematische Annalen, at that time a leading learned journal. He became relatively isolated, the development of intuitionism at its source was taken up by his student Arend Heyting and he was killed in 1966 at the age of 85, struck by a vehicle while crossing the street in front of his house. Jean van Heijenoort,1967 3rd printing 1976 with corrections, A Source Book in Mathematical Logic, harvard University Press, Cambridge MA, ISBN 0-674-32449-8 pbk. The original papers are prefaced with valuable commentary, L. E. J. Brouwer, On the significance of the principle of excluded middle in mathematics, especially in function theory. With two Addenda and corrigenda, 334-45, a. N. Kolmogorov, On the principle of excluded middle, pp. 414–437. Kolmogorov supports most of Brouwers results but disputes a few, he discusses the ramifications of intuitionism with respect to transfinite judgements, L. E. J. Brouwer, On the domains of definition of functions. Brouwers intuitionistic treatment of the continuum, with an extended commentary, david Hilbert, The foundations of mathematics, 464-801927
4.
Intuition (philosophy)
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Intuition is the ability to acquire knowledge without proof, evidence, or conscious reasoning, or without understanding how the knowledge was acquired. The word intuition comes from the Latin verb intueri translated as consider or from late middle English word intuit, both Eastern and Western philosophers have studied the concept in great detail. Philosophy of mind deals with the concept of intuition, there are philosophers who contend that this concept is often confused with other concepts such as truth, belief, and meaning in philosophical discussion. In the East intuition is mostly intertwined with religion and spirituality, in Hinduism various attempts have been made to interpret the Vedic and other esoteric texts. However with knowledge by identity which we currently only gives the awareness of human existence can be extended further to outside of ourselves resulting in intuitive knowledge. He finds that this process seems to be a decent, is actual a circle of progress. As a lower faculty is being pushed to take up as much from a way of working. Advaita vedanta takes intuition to be an experience through which one can come in contact with, in Zen Buddhism various techniques have been developed to help develop ones intuitive capability, such as kó-an – the resolving of which leads to states of minor enlightenment. In parts of Zen Buddhism intuition is deemed a state between the Universal mind and ones individual, discriminating mind. In Islam there are scholars with varied interpretation of intuition. While Ibn Sīnā finds the ability of having intuition as a prophetic capacity terms it as a knowledge obtained without intentionally acquiring it and he finds regular knowledge is based on imitation while intuitive knowledge as based on intellectual certitude. In the West, intuition does not appear as a field of study. In his book Republic he tries to define intuition as a capacity of human reason to comprehend the true nature of reality. In his discussion with Meno & Phaedo, he describes intuition as a pre-existing knowledge residing in the soul of eternity, and he provides an example of mathematical truths, and posits that they are not arrived at by reason. He argues that these truths are accessed using an already present in a dormant form. This concept by Plato is also referred to as anamnesis. The study was continued by his followers. In his book Meditations on first philosophy, Descartes refers to an intuition as a pre-existing knowledge gained through rational reasoning or discovering truth through contemplation and this definition is commonly referred to as rational intuition
5.
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
6.
Intuitionistic logic
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In particular, systems of intuitionistic logic do not include the law of the excluded middle and double negation elimination, which are fundamental inference rules in classical logic. Formalized intuitionistic logic was developed by Arend Heyting to provide a formal basis for Brouwers programme of intuitionism. From a proof-theoretic perspective, Heyting’s calculus is a restriction of classical logic in which the law of excluded middle and double negation elimination have been removed. Excluded middle and double negation elimination can still be proved for some propositions on a case by case basis, however, several systems of semantics for intuitionistic logic have been studied. One semantics mirrors classical Boolean-valued semantics but uses Heyting algebras in place of Boolean algebras and these, however, are technical means for studying Heyting’s deductive system rather than formalizations of Brouwer’s original informal semantic intuitions. Yet such semantics persistently induce logics properly stronger than Heyting’s logic, some authors have argued that this might be an indication of inadequacy of Heyting’s calculus itself, deeming the latter incomplete as a constructive logic. In the semantics of classical logic, propositional formulae are assigned values from the two-element set. This is referred to as the law of excluded middle, because it excludes the possibility of any truth value besides true or false. In contrast, propositional formulae in intuitionistic logic are not assigned a truth value and are only considered true when we have direct evidence, hence proof. Operations in intuitionistic logic therefore preserve justification, with respect to evidence and provability, Intuitionistic logic is one of the set of approaches of constructivism in mathematics. The use of constructivist logics in general has been a topic among mathematicians. A very common objection to their use is the lack of two central rules of classical logic, the law of excluded middle and double negation elimination. To prohibit existence statements and the principle of excluded middle is tantamount to relinquishing the science of mathematics altogether, despite the serious challenges presented by the inability to utilize the valuable rules of excluded middle and double negation elimination, intuitionistic logic has practical use. One reason for this is that its restrictions produce proofs that have the existence property, one reason that this particular aspect of intuitionistic logic is so valuable is that it enables practitioners to utilize a wide range of computerized tools, known as proof assistants. These tools assist their users in the verification of large-scale proofs, one example of a proof which was impossible to formally verify before the advent of these tools is the famous proof of the four color theorem. That proof was controversial for some time, but it was verified using Coq. The syntax of formulas of intuitionistic logic is similar to propositional logic or first-order logic, however, intuitionistic connectives are not definable in terms of each other in the same way as in classical logic, hence their choice matters. In intuitionistic propositional logic it is customary to use →, ∧, ∨, ⊥ as the basic connectives, in intuitionistic first-order logic both quantifiers ∃, ∀ are needed
7.
Mind
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The mind is a set of cognitive faculties including consciousness, perception, thinking, judgement, and memory. It is usually defined as the faculty of an entitys reasoning and it holds the power of imagination, recognition, and appreciation, and is responsible for processing feelings and emotions, resulting in attitudes and actions. One open question regarding the nature of the mind is the mind–body problem, pre-scientific viewpoints included dualism and idealism, which considered the mind somehow non-physical. Modern views center around physicalism and functionalism, which hold that the mind is identical with the brain or reducible to physical phenomena such as neuronal activity. Another question concerns which types of beings are capable of having minds, the concept of mind is understood in many different ways by many different cultural and religious traditions. Some see mind as a property exclusive to humans whereas others ascribe properties of mind to non-living entities, to animals, important philosophers of mind include Plato, Descartes, Leibniz, Searle, Dennett, Fodor, Nagel, and Chalmers. Psychologists such as Freud and James, and computer scientists such as Turing, the original meaning of Old English gemynd was the faculty of memory, not of thought in general. Hence call to mind, come to mind, keep in mind, to have mind of, the word retains this sense in Scotland. Old English had other words to mind, such as hyge mind. The meaning of memory is shared with Old Norse, which has munr, the word is originally from a PIE verbal root *men-, meaning to think, remember, whence also Latin mens mind, Sanskrit manas mind and Greek μένος mind, courage, anger. The generalization of mind to all mental faculties, thought, volition, feeling and memory. The attributes that make up the mind is debated, some psychologists argue that only the higher intellectual functions constitute mind, particularly reason and memory. In this view the emotions — love, hate, fear, joy — are more primitive or subjective in nature and should be seen as different from the mind as such. Others argue that rational and emotional states cannot be so separated, that they are of the same nature and origin. In popular usage, mind is frequently synonymous with thought, the conversation with ourselves that we carry on inside our heads. Thus we make up our minds, change our minds or are of two minds about something, one of the key attributes of the mind in this sense is that it is a private sphere to which no one but the owner has access. No one else can know our mind and they can only interpret what we consciously or unconsciously communicate. Broadly speaking, mental faculties are the functions of the mind
8.
Negation
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Negation is thus a unary logical connective. It may be applied as an operation on propositions, truth values, in classical logic, negation is normally identified with the truth function that takes truth to falsity and vice versa. In intuitionistic logic, according to the Brouwer–Heyting–Kolmogorov interpretation, the negation of a proposition p is the proposition whose proofs are the refutations of p. Classical negation is an operation on one logical value, typically the value of a proposition, that produces a value of true when its operand is false and a value of false when its operand is true. So, if statement A is true, then ¬A would therefore be false, the truth table of ¬p is as follows, Classical negation can be defined in terms of other logical operations. For example, ¬p can be defined as p → F, conversely, one can define F as p & ¬p for any proposition p, where & is logical conjunction. The idea here is that any contradiction is false, while these ideas work in both classical and intuitionistic logic, they do not work in paraconsistent logic, where contradictions are not necessarily false. But in classical logic, we get an identity, p → q can be defined as ¬p ∨ q. Algebraically, classical negation corresponds to complementation in a Boolean algebra and these algebras provide a semantics for classical and intuitionistic logic respectively. The negation of a proposition p is notated in different ways in various contexts of discussion and fields of application. Among these variants are the following, In set theory \ is also used to indicate not member of, U \ A is the set of all members of U that are not members of A. No matter how it is notated or symbolized, the negation ¬p / −p can be read as it is not the case p, not that p. Within a system of logic, double negation, that is. In intuitionistic logic, a proposition implies its double negation but not conversely and this marks one important difference between classical and intuitionistic negation. Algebraically, classical negation is called an involution of period two and this result is known as Glivenkos theorem. De Morgans laws provide a way of distributing negation over disjunction and conjunction, ¬ ≡, in Boolean algebra, a linear function is one such that, If there exists a0, a1. An ∈ such that f = a0 ⊕ ⊕, another way to express this is that each variable always makes a difference in the truth-value of the operation or it never makes a difference. Negation is a logical operator
9.
Logical disjunction
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In logic and mathematics, or is the truth-functional operator of disjunction, also known as alternation, the or of a set of operands is true if and only if one or more of its operands is true. The logical connective that represents this operator is written as ∨ or +. A or B is true if A is true, or if B is true, or if both A and B are true. In logic, or by means the inclusive or, distinguished from an exclusive or. An operand of a disjunction is called a disjunct, related concepts in other fields are, In natural language, the coordinating conjunction or. In programming languages, the short-circuit or control structure, or is usually expressed with an infix operator, in mathematics and logic, ∨, in electronics, +, and in most programming languages, |, ||, or or. In Jan Łukasiewiczs prefix notation for logic, the operator is A, logical disjunction is an operation on two logical values, typically the values of two propositions, that has a value of false if and only if both of its operands are false. More generally, a disjunction is a formula that can have one or more literals separated only by ors. A single literal is often considered to be a degenerate disjunction, the disjunctive identity is false, which is to say that the or of an expression with false has the same value as the original expression. In keeping with the concept of truth, when disjunction is defined as an operator or function of arbitrary arity. Falsehood-preserving, The interpretation under which all variables are assigned a value of false produces a truth value of false as a result of disjunction. The mathematical symbol for logical disjunction varies in the literature, in addition to the word or, and the formula Apq, the symbol ∨, deriving from the Latin word vel is commonly used for disjunction. For example, A ∨ B is read as A or B, such a disjunction is false if both A and B are false. In all other cases it is true, all of the following are disjunctions, A ∨ B ¬ A ∨ B A ∨ ¬ B ∨ ¬ C ∨ D ∨ ¬ E. The corresponding operation in set theory is the set-theoretic union, operators corresponding to logical disjunction exist in most programming languages. Disjunction is often used for bitwise operations, for example, x = x | 0b00000001 will force the final bit to 1 while leaving other bits unchanged. Logical disjunction is usually short-circuited, that is, if the first operand evaluates to true then the second operand is not evaluated, the logical disjunction operator thus usually constitutes a sequence point. In a parallel language, it is possible to both sides, they are evaluated in parallel, and if one terminates with value true
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Law of excluded middle
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In logic, the law of excluded middle is the third of the three classic laws of thought. It states that for any proposition, either that proposition is true, the law is also known as the law of the excluded third, in Latin principium tertii exclusi. Another Latin designation for this law is tertium non datur, no third is given, the principle was stated as a theorem of propositional logic by Russell and Whitehead in Principia Mathematica as, ∗2 ⋅11. The principle should not be confused with the principle of bivalence. The principle of excluded middle, along with its complement, the law of contradiction, are correlates of the law of identity, some systems of logic have different but analogous laws. For some finite n-valued logics, there is a law called the law of excluded n+1th. If negation is cyclic and ∨ is a max operator, then the law can be expressed in the language by. It is easy to check that the sentence must receive at least one of the n truth values, Other systems reject the law entirely. For example, if P is the proposition, Socrates is mortal, then the law of excluded middle holds that the logical disjunction, Either Socrates is mortal, or it is not the case that Socrates is mortal. is true by virtue of its form alone. That is, the position, that Socrates is neither mortal nor not-mortal, is excluded by logic. An example of an argument that depends on the law of excluded middle follows and we seek to prove that there exist two irrational numbers a and b such that a b is rational. It is known that 2 is irrational, clearly this number is either rational or irrational. If it is rational, the proof is complete, and a =2 and b =2, but if 22 is irrational, then let a =22 and b =2. Then a b =2 =2 =22 =2, in the above argument, the assertion this number is either rational or irrational invokes the law of excluded middle. An intuitionist, for example, would not accept this argument without further support for that statement and this might come in the form of a proof that the number in question is in fact irrational, or a finite algorithm that could determine whether the number is rational. By non-constructive Davis means that a proof that actually are mathematic entities satisfying certain conditions would have to provide a method to exhibit explicitly the entities in question. For example, to prove there exists an n such that P, under both the classical and the intuitionistic logic, by reductio ad absurdum this gives not for all n, not P. The classical logic allows this result to be transformed into there exists an n such that P, indeed, David Hilbert and Luitzen E. J. Brouwer both give examples of the law of excluded middle extended to the infinite
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De Morgan's laws
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In propositional logic and boolean algebra, De Morgans laws are a pair of transformation rules that are both valid rules of inference. They are named after Augustus De Morgan, a 19th-century British mathematician, the rules allow the expression of conjunctions and disjunctions purely in terms of each other via negation. Applications of the rules include simplification of logical expressions in computer programs, De Morgans laws are an example of a more general concept of mathematical duality. The negation of conjunction rule may be written in sequent notation, the negation of disjunction rule may be written as, ¬ ⊢. De Morgans laws are shown in the compact form above, with negation of the output on the left. A clearer form for substitution can be stated as, ≡ ¬, ≡ ¬ and this emphasizes the need to invert both the inputs and the output, as well as change the operator, when doing a substitution. In set notation, De Morgans laws can be remembered using the mnemonic break the line, De Morgan’s laws commonly apply to text searching using Boolean operators AND, OR, and NOT. Consider a set of documents containing the words “cars” and “trucks”, Document 3, Contains both “cars” and “trucks”. Document 4, Contains neither “cars” nor “trucks”, to evaluate Search A, clearly the search “” will hit on Documents 1,2, and 3. So the negation of that search will hit everything else, which is Document 4, evaluating Search B, the search “” will hit on documents that do not contain “cars”, which is Documents 2 and 4. Similarly the search “” will hit on Documents 1 and 4, applying the AND operator to these two searches will hit on the documents that are common to these two searches, which is Document 4. A similar evaluation can be applied to show that the two searches will return the same set of documents, Search C, NOT, Search D. The laws are named after Augustus De Morgan, who introduced a version of the laws to classical propositional logic. De Morgans formulation was influenced by algebraization of logic undertaken by George Boole, nevertheless, a similar observation was made by Aristotle, and was known to Greek and Medieval logicians. For example, in the 14th century, William of Ockham wrote down the words that would result by reading the laws out, jean Buridan, in his Summulae de Dialectica, also describes rules of conversion that follow the lines of De Morgans laws. Still, De Morgan is given credit for stating the laws in the terms of formal logic. De Morgans laws can be proved easily, and may seem trivial. Nonetheless, these laws are helpful in making inferences in proofs
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Truth
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Truth is most often used to mean being in accord with fact or reality, or fidelity to an original or standard. Truth may also often be used in modern contexts to refer to an idea of truth to self, the commonly understood opposite of truth is falsehood, which, correspondingly, can also take on a logical, factual, or ethical meaning. The concept of truth is discussed and debated in several contexts, including philosophy, art, Some philosophers view the concept of truth as basic, and unable to be explained in any terms that are more easily understood than the concept of truth itself. Commonly, truth is viewed as the correspondence of language or thought to an independent reality, other philosophers take this common meaning to be secondary and derivative. On this view, the conception of truth as correctness is a derivation from the concepts original essence. Various theories and views of truth continue to be debated among scholars, philosophers, language and words are a means by which humans convey information to one another and the method used to determine what is a truth is termed a criterion of truth. The English word truth is derived from Old English tríewþ, tréowþ, trýwþ, Middle English trewþe, cognate to Old High German triuwida, like troth, it is a -th nominalisation of the adjective true. Old Norse trú, faith, word of honour, religious faith, thus, truth involves both the quality of faithfulness, fidelity, loyalty, sincerity, veracity, and that of agreement with fact or reality, in Anglo-Saxon expressed by sōþ. All Germanic languages besides English have introduced a distinction between truth fidelity and truth factuality. To express factuality, North Germanic opted for nouns derived from sanna to assert, affirm, while continental West Germanic opted for continuations of wâra faith, trust, pact. Romance languages use terms following the Latin veritas, while the Greek aletheia, Russian pravda, each presents perspectives that are widely shared by published scholars. However, the theories are not universally accepted. More recently developed deflationary or minimalist theories of truth have emerged as competitors to the substantive theories. Minimalist reasoning centres around the notion that the application of a term like true to a statement does not assert anything significant about it, for instance, anything about its nature. Minimalist reasoning realises truth as a label utilised in general discourse to express agreement, to stress claims, correspondence theories emphasise that true beliefs and true statements correspond to the actual state of affairs. This type of theory stresses a relationship between thoughts or statements on one hand, and things or objects on the other and it is a traditional model tracing its origins to ancient Greek philosophers such as Socrates, Plato, and Aristotle. This class of theories holds that the truth or the falsity of a representation is determined in principle entirely by how it relates to things, Aquinas also restated the theory as, A judgment is said to be true when it conforms to the external reality. Many modern theorists have stated that this ideal cannot be achieved without analysing additional factors, for example, language plays a role in that all languages have words to represent concepts that are virtually undefined in other languages
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Natural number
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In mathematics, the natural numbers are those used for counting and ordering. In common language, words used for counting are cardinal numbers, texts that exclude zero from the natural numbers sometimes refer to the natural numbers together with zero as the whole numbers, but in other writings, that term is used instead for the integers. These chains of extensions make the natural numbers canonically embedded in the number systems. Properties of the numbers, such as divisibility and the distribution of prime numbers, are studied in number theory. Problems concerning counting and ordering, such as partitioning and enumerations, are studied in combinatorics, the most primitive method of representing a natural number is to put down a mark for each object. Later, a set of objects could be tested for equality, excess or shortage, by striking out a mark, the first major advance in abstraction was the use of numerals to represent numbers. This allowed systems to be developed for recording large numbers, the ancient Egyptians developed a powerful system of numerals with distinct hieroglyphs for 1,10, and all the powers of 10 up to over 1 million. A stone carving from Karnak, dating from around 1500 BC and now at the Louvre in Paris, depicts 276 as 2 hundreds,7 tens, and 6 ones, and similarly for the number 4,622. A much later advance was the development of the idea that 0 can be considered as a number, with its own numeral. The use of a 0 digit in place-value notation dates back as early as 700 BC by the Babylonians, the Olmec and Maya civilizations used 0 as a separate number as early as the 1st century BC, but this usage did not spread beyond Mesoamerica. The use of a numeral 0 in modern times originated with the Indian mathematician Brahmagupta in 628, the first systematic study of numbers as abstractions is usually credited to the Greek philosophers Pythagoras and Archimedes. Some Greek mathematicians treated the number 1 differently than larger numbers, independent studies also occurred at around the same time in India, China, and Mesoamerica. In 19th century Europe, there was mathematical and philosophical discussion about the nature of the natural numbers. A school of Naturalism stated that the numbers were a direct consequence of the human psyche. Henri Poincaré was one of its advocates, as was Leopold Kronecker who summarized God made the integers, in opposition to the Naturalists, the constructivists saw a need to improve the logical rigor in the foundations of mathematics. In the 1860s, Hermann Grassmann suggested a recursive definition for natural numbers thus stating they were not really natural, later, two classes of such formal definitions were constructed, later, they were shown to be equivalent in most practical applications. The second class of definitions was introduced by Giuseppe Peano and is now called Peano arithmetic and it is based on an axiomatization of the properties of ordinal numbers, each natural number has a successor and every non-zero natural number has a unique predecessor. Peano arithmetic is equiconsistent with several systems of set theory
14.
Alexander Esenin-Volpin
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Alexander Sergeyevich Esenin-Volpin was a prominent Russian-American poet and mathematician. Alexander Volpin was born on May 12,1924 in the Soviet Union and his mother, Nadezhda Volpin, was a poet and translator from French and English. His father was Sergei Yesenin, a celebrated Russian poet, who never knew his son, Alexander and his mother moved from Leningrad to Moscow in 1933. His first psychiatric imprisonments took place in 1949 for anti-Soviet poetry, in 1959 for smuggling abroad samizdat, including his Свободный философский трактат, Esenin-Volpin graduated from Moscow State University with a “candidate” dissertation in the spring of 1949. After graduation, Volpin was sent to the Ukrainian city of Chernovtsy to teach mathematics at the state university. Less than a month after his arrival in Chernovtsy he was arrested by the MGB, sent on a back to Moscow. He was charged with systematically conducting anti-Soviet agitation, writing anti-Soviet poems, apprehensive about the prospect of prison and labor camp, Volpin faked a suicide attempt in order to initiate a psychiatric evaluation. Psychiatrists at Moscows Serbsky Institute declared Volpin mentally incompetent, and in October 1949 he was transferred to the Leningrad Psychiatric Prison Hospital for an indefinite stay. A year later he was released from the prison hospital. In Karagada, he found employment as a teacher of evening, in 1953, after the death of Joseph Stalin, Volpin was released due to a general amnesty. Soon he became a mathematician specializing in the fields of ultrafinitism and intuitionism. The meeting was attended by about 200 people, many of whom turned out to be KGB operatives, the slogans read, Требуем гласности суда над Синявским и Даниэлем and Уважайте советскую конституцию. In the following years, Esenin-Volpin became an important voice in the human movement in the Soviet Union. He was one of the first Soviet dissidents who took on a legalist strategy of dissent and he proclaimed that it is possible and necessary to defend human rights by strictly observing the law, and in turn demand that the authorities observe the formally guaranteed rights. Esenin-Volpin was again hospitalized in February 1968 as one of those protesting most strongly against the trial of Alexander Ginzburg, after his 1968 psychiatric confinement,99 Soviet mathematicians sent a letter to the Soviet authorities asking for his release. This fact became public and the Voice of America conducted a broadcast on the topic, vladimir Bukovsky was quoted as saying that Volpins diagnosis was pathological honesty. In 1968, Esenin-Volpin circulated his famous Памятка для тех, кому предстоят допросы widely used by fellow dissidents, in 1969, he signed the first Appeal to The UN Committee for Human Rights, drafted by the Initiative Group for the Defense of Human Rights in the USSR. In 1970, Volpin joined the Committee on Human Rights in the USSR and worked with Yuri Orlov, Andrei Sakharov, in May 1972, he emigrated to the United States, but his Soviet citizenship was not revoked as was customary at the time
15.
Georg Cantor
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Georg Ferdinand Ludwig Philipp Cantor was a German mathematician. He invented set theory, which has become a theory in mathematics. In fact, Cantors method of proof of this theorem implies the existence of an infinity of infinities and he defined the cardinal and ordinal numbers and their arithmetic. Cantors work is of great philosophical interest, a fact of which he was well aware, E. J. Brouwer, while Ludwig Wittgenstein raised philosophical objections. Cantor, a devout Lutheran, believed the theory had been communicated to him by God, Kronecker objected to Cantors proofs that the algebraic numbers are countable, and that the transcendental numbers are uncountable, results now included in a standard mathematics curriculum. The harsh criticism has been matched by later accolades, in 1904, the Royal Society awarded Cantor its Sylvester Medal, the highest honor it can confer for work in mathematics. David Hilbert defended it from its critics by declaring, From his paradise that Cantor with us unfolded, we hold our breath in awe, knowing, we shall not be expelled. Georg Cantor was born in the merchant colony in Saint Petersburg, Russia. Georg, the oldest of six children, was regarded as an outstanding violinist and his grandfather Franz Böhm was a well-known musician and soloist in a Russian imperial orchestra. In 1860, Cantor graduated with distinction from the Realschule in Darmstadt, his skills in mathematics. In 1862, Cantor entered the Swiss Federal Polytechnic and he spent the summer of 1866 at the University of Göttingen, then and later a center for mathematical research. Cantor submitted his dissertation on number theory at the University of Berlin in 1867, after teaching briefly in a Berlin girls school, Cantor took up a position at the University of Halle, where he spent his entire career. He was awarded the habilitation for his thesis, also on number theory. In 1874, Cantor married Vally Guttmann and they had six children, the last born in 1886. Cantor was able to support a family despite modest academic pay, during his honeymoon in the Harz mountains, Cantor spent much time in mathematical discussions with Richard Dedekind, whom he had met two years earlier while on Swiss holiday. Cantor was promoted to Extraordinary Professor in 1872 and made full Professor in 1879, however, his work encountered too much opposition for that to be possible. Worse yet, Kronecker, a figure within the mathematical community and Cantors former professor. Cantor came to believe that Kroneckers stance would make it impossible for him ever to leave Halle, in 1881, Cantors Halle colleague Eduard Heine died, creating a vacant chair
16.
Leopold Kronecker
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Leopold Kronecker was a German mathematician who worked on number theory, algebra and logic. He criticized Cantors work on set theory, and was quoted by Weber as having said, Die ganzen Zahlen hat der liebe Gott gemacht, Kronecker was a student and lifelong friend of Ernst Kummer. Leopold Kronecker was born on 7 December 1823 in Liegnitz, Prussia in a wealthy Jewish family, Kronecker then went to the Liegnitz Gymnasium where he was interested in a wide range of topics including science, history and philosophy, while also practicing gymnastics and swimming. At the gymnasium he was taught by Ernst Kummer, who noticed and encouraged the boys interest in mathematics, in 1841 Kronecker became a student at the University of Berlin where his interest did not immediately focus on mathematics, but rather spread over several subjects including astronomy and philosophy. He spent the summer of 1843 at the University of Bonn studying astronomy, back in Berlin, Kronecker studied mathematics with Peter Gustav Lejeune Dirichlet and in 1845 defended his dissertation in algebraic number theory written under Dirichlets supervision. After obtaining his degree, Kronecker did not follow his interest in research on a career path. He went back to his hometown to manage a large farming estate built up by his mothers uncle, in 1848 he married his cousin Fanny Prausnitzer, and the couple had six children. For several years Kronecker focused on business, and although he continued to study mathematics as a hobby and corresponded with Kummer, in 1853 he wrote a memoir on the algebraic solvability of equations extending the work of Évariste Galois on the theory of equations. Due to his business activity, Kronecker was financially comfortable, Dirichlet, whose wife Rebecka came from the wealthy Mendelssohn family, had introduced Kronecker to the Berlin elite. He became a friend of Karl Weierstrass, who had recently joined the university. Over the following years Kronecker published numerous papers resulting from his previous years independent research, as a result of this published research, he was elected a member of the Berlin Academy in 1861. Although he held no official university position, Kronecker had the right as a member of the Academy to hold classes at the University of Berlin and he decided to do so, starting in 1862. In 1866, when Riemann died, Kronecker was offered the chair at the University of Göttingen. Only in 1883, when Kummer retired from the University, was Kronecker invited to succeed him, Kronecker was the supervisor of Kurt Hensel, Adolf Kneser, Mathias Lerch, and Franz Mertens, amongst others. Kronecker died on 29 December 1891 in Berlin, several months after the death of his wife, in the last year of his life, he converted to Christianity. He is buried in the Alter St Matthäus Kirchhof cemetery in Berlin-Schöneberg, an important part of Kroneckers research focused on number theory and algebra. In an 1853 paper on the theory of equations and Galois theory he formulated the Kronecker–Weber theorem and he also introduced the structure theorem for finitely-generated abelian groups. Kronecker studied elliptic functions and conjectured his liebster Jugendtraum, a generalization that was put forward by Hilbert in a modified form as his twelfth problem
17.
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
18.
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
19.
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
20.
David Hilbert
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David Hilbert was a German mathematician. He is recognized as one of the most influential and universal mathematicians of the 19th, Hilbert discovered and developed a broad range of fundamental ideas in many areas, including invariant theory and the axiomatization of geometry. He also formulated the theory of Hilbert spaces, one of the foundations of functional analysis, Hilbert adopted and warmly defended Georg Cantors set theory and transfinite numbers. A famous example of his leadership in mathematics is his 1900 presentation of a collection of problems set the course for much of the mathematical research of the 20th century. Hilbert and his students contributed significantly to establishing rigor and developed important tools used in mathematical physics. Hilbert is known as one of the founders of theory and mathematical logic. In late 1872, Hilbert entered the Friedrichskolleg Gymnasium, but, after a period, he transferred to. Upon graduation, in autumn 1880, Hilbert enrolled at the University of Königsberg, in early 1882, Hermann Minkowski, returned to Königsberg and entered the university. Hilbert knew his luck when he saw it, in spite of his fathers disapproval, he soon became friends with the shy, gifted Minkowski. In 1884, Adolf Hurwitz arrived from Göttingen as an Extraordinarius, Hilbert obtained his doctorate in 1885, with a dissertation, written under Ferdinand von Lindemann, titled Über invariante Eigenschaften spezieller binärer Formen, insbesondere der Kugelfunktionen. Hilbert remained at the University of Königsberg as a Privatdozent from 1886 to 1895, in 1895, as a result of intervention on his behalf by Felix Klein, he obtained the position of Professor of Mathematics at the University of Göttingen. During the Klein and Hilbert years, Göttingen became the preeminent institution in the mathematical world and he remained there for the rest of his life. Among Hilberts students were Hermann Weyl, chess champion Emanuel Lasker, Ernst Zermelo, john von Neumann was his assistant. At the University of Göttingen, Hilbert was surrounded by a circle of some of the most important mathematicians of the 20th century, such as Emmy Noether. Between 1902 and 1939 Hilbert was editor of the Mathematische Annalen, good, he did not have enough imagination to become a mathematician. Hilbert lived to see the Nazis purge many of the prominent faculty members at University of Göttingen in 1933 and those forced out included Hermann Weyl, Emmy Noether and Edmund Landau. One who had to leave Germany, Paul Bernays, had collaborated with Hilbert in mathematical logic and this was a sequel to the Hilbert-Ackermann book Principles of Mathematical Logic from 1928. Hermann Weyls successor was Helmut Hasse, about a year later, Hilbert attended a banquet and was seated next to the new Minister of Education, Bernhard Rust
21.
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
22.
Alan Turing
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Alan Mathison Turing OBE FRS was an English computer scientist, mathematician, logician, cryptanalyst and theoretical biologist. Turing is widely considered to be the father of computer science. During the Second World War, Turing worked for the Government Code and Cypher School at Bletchley Park, for a time he led Hut 8, the section responsible for German naval cryptanalysis. After the war, he worked at the National Physical Laboratory and he wrote a paper on the chemical basis of morphogenesis, and predicted oscillating chemical reactions such as the Belousov–Zhabotinsky reaction, first observed in the 1960s. Turing was prosecuted in 1952 for homosexual acts, when by the Labouchere Amendment and he accepted chemical castration treatment, with DES, as an alternative to prison. Turing died in 1954,16 days before his 42nd birthday, an inquest determined his death as suicide, but it has been noted that the known evidence is also consistent with accidental poisoning. In 2009, following an Internet campaign, British Prime Minister Gordon Brown made a public apology on behalf of the British government for the appalling way he was treated. Queen Elizabeth II granted him a pardon in 2013. The Alan Turing law is now a term for a 2017 law in the United Kingdom that retroactively pardons men cautioned or convicted under historical legislation that outlawed homosexual acts. Turings father was the son of a clergyman, the Rev. John Robert Turing, from a Scottish family of merchants that had based in the Netherlands. Turings mother, Julius wife, was Ethel Sara, daughter of Edward Waller Stoney, the Stoneys were a Protestant Anglo-Irish gentry family from both County Tipperary and County Longford, while Ethel herself had spent much of her childhood in County Clare. Julius work with the ICS brought the family to British India and he had an elder brother, John. At Hastings, Turing stayed at Baston Lodge, Upper Maze Hill, St Leonards-on-Sea, very early in life, Turing showed signs of the genius that he was later to display prominently. His parents purchased a house in Guildford in 1927, and Turing lived there during school holidays, the location is also marked with a blue plaque. Turings parents enrolled him at St Michaels, a day school at 20 Charles Road, St Leonards-on-Sea, the headmistress recognised his talent early on, as did many of his subsequent educators. From January 1922 to 1926, Turing was educated at Hazelhurst Preparatory School, in 1926, at the age of 13, he went on to Sherborne School, an independent school in the market town of Sherborne in Dorset. Turings natural inclination towards mathematics and science did not earn him respect from some of the teachers at Sherborne and his headmaster wrote to his parents, I hope he will not fall between two stools. If he is to stay at school, he must aim at becoming educated
23.
Arend Heyting
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Arend Heyting was a Dutch mathematician and logician. He was a student of Luitzen Egbertus Jan Brouwer at the University of Amsterdam, Heyting gave the first formal development of intuitionistic logic in order to codify Brouwers way of doing mathematics. The inclusion of Brouwers name in the Brouwer–Heyting–Kolmogorov interpretation is largely honorific, in 1942 he became member of the Royal Netherlands Academy of Arts and Sciences. Heyting was born in Amsterdam, Netherlands, and died in Lugano, die formalen Regeln der intuitionistischen Logik. 3 parts, In, Sitzungsberichte der preußischen Akademie der Wissenschaften, Heyting, A. Untersuchungen der intuitionistische Algebra. 2,36 pp. Heyting, A. Intuitionism, axioms for intuitionistic plane affine geometry. With special reference to geometry and physics, proceedings of an International Symposium held at the Univ. of Calif. Berkeley, Dec.26, 1957–Jan 4,1958 pp. 160–173 Studies in Logic,1962 Logic, Methodology and Philosophy of Science pp. 194–197 Stanford Univ. Bibliotheca Mathematica, Vol. V. Interscience Publishers John Wiley & Sons, new York, P. Noordhoff N. V. Groningen, North-Holland Publishing Co. Second revised edition North-Holland Publishing Co, at the occasion of the Brouwer memorial lecture given by Prof. A. Robinson on the 26th April 1973. Heyting, A. Mathematische Grundlagenforschung, Intuitionismus, Beweistheorie, bibliotheca Mathematica, V. Wolters-Noordhoff Scientific Publications, Ltd. Heyting algebra Heyting arithmetic OConnor, John J. Robertson, Edmund F. Arend Heyting, MacTutor History of Mathematics archive, Arend Heyting at the Mathematics Genealogy Project
24.
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
25.
Benjamin Peirce
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Benjamin Peirce was an American mathematician who taught at Harvard University for approximately 50 years. He made contributions to celestial mechanics, statistics, number theory, algebra, and he was the son of Benjamin Peirce, later librarian of Harvard, and Lydia Ropes Nichols Peirce. After graduating from Harvard, he remained as a tutor, and was appointed professor of mathematics in 1831. He added astronomy to his portfolio in 1842, and remained as Harvard professor until his death. In addition, he was instrumental in the development of Harvards science curriculum, served as the college librarian, Benjamin Peirce is often regarded as the earliest American scientist whose research was recognized as world class. He was an apologist for slavery, opining that it should be condoned if it was used to allow an elite to pursue scientific enquiry, in number theory, he proved there is no odd perfect number with fewer than four prime factors. In algebra, he was notable for the study of associative algebras and he first introduced the terms idempotent and nilpotent in 1870 to describe elements of these algebras, and he also introduced the Peirce decomposition. In the philosophy of mathematics, he known for the statement that Mathematics is the science that draws necessary conclusions. Peirces definition of mathematics was credited by his son, Charles Sanders Peirce, like George Boole, Peirce believed that mathematics could be used to study logic. These ideas were developed by Charles Sanders Peirce, who noted that logic also includes the study of faulty reasoning. In contrast, the later logicist program of Gottlob Frege and Bertrand Russell attempted to base mathematics on logic, Peirce proposed what came to be known as Peirces Criterion for the statistical treatment of outliers, that is, of apparently extreme observations. His ideas were developed by Charles Sanders Peirce. Peirce was a witness in the Howland will forgery trial. Their analysis of the questioned signature showed that it resembled another particular handwriting example so closely that the chances of such a match were statistically extremely remote and he was devoutly religious, though he seldom published his theological thoughts. Peirce credited God as shaping nature in ways that account for the efficacy of pure mathematics in describing empirical phenomena, Peirce viewed mathematics as study of Gods work by Gods creatures, according to an encyclopedia. He married Sarah Hunt Mills, the daughter of U. S, the lunar crater Peirce is named for Peirce. Post-doctoral positions in Harvard Universitys mathematics department are named in his honor as Benjamin Peirce Fellows, the United States Coast Survey ship USCS Benjamin Peirce, in commission from 1855 to 1868, was named for him. An Elementary Treatise on Plane and Spherical Trigonometry, Boston, James Munroe, google Eprints of successive editions 1840–1862
26.
Constructive logic
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In particular, systems of intuitionistic logic do not include the law of the excluded middle and double negation elimination, which are fundamental inference rules in classical logic. Formalized intuitionistic logic was developed by Arend Heyting to provide a formal basis for Brouwers programme of intuitionism. From a proof-theoretic perspective, Heyting’s calculus is a restriction of classical logic in which the law of excluded middle and double negation elimination have been removed. Excluded middle and double negation elimination can still be proved for some propositions on a case by case basis, however, several systems of semantics for intuitionistic logic have been studied. One semantics mirrors classical Boolean-valued semantics but uses Heyting algebras in place of Boolean algebras and these, however, are technical means for studying Heyting’s deductive system rather than formalizations of Brouwer’s original informal semantic intuitions. Yet such semantics persistently induce logics properly stronger than Heyting’s logic, some authors have argued that this might be an indication of inadequacy of Heyting’s calculus itself, deeming the latter incomplete as a constructive logic. In the semantics of classical logic, propositional formulae are assigned values from the two-element set. This is referred to as the law of excluded middle, because it excludes the possibility of any truth value besides true or false. In contrast, propositional formulae in intuitionistic logic are not assigned a truth value and are only considered true when we have direct evidence, hence proof. Operations in intuitionistic logic therefore preserve justification, with respect to evidence and provability, Intuitionistic logic is one of the set of approaches of constructivism in mathematics. The use of constructivist logics in general has been a topic among mathematicians. A very common objection to their use is the lack of two central rules of classical logic, the law of excluded middle and double negation elimination. To prohibit existence statements and the principle of excluded middle is tantamount to relinquishing the science of mathematics altogether, despite the serious challenges presented by the inability to utilize the valuable rules of excluded middle and double negation elimination, intuitionistic logic has practical use. One reason for this is that its restrictions produce proofs that have the existence property, one reason that this particular aspect of intuitionistic logic is so valuable is that it enables practitioners to utilize a wide range of computerized tools, known as proof assistants. These tools assist their users in the verification of large-scale proofs, one example of a proof which was impossible to formally verify before the advent of these tools is the famous proof of the four color theorem. That proof was controversial for some time, but it was verified using Coq. The syntax of formulas of intuitionistic logic is similar to propositional logic or first-order logic, however, intuitionistic connectives are not definable in terms of each other in the same way as in classical logic, hence their choice matters. In intuitionistic propositional logic it is customary to use →, ∧, ∨, ⊥ as the basic connectives, in intuitionistic first-order logic both quantifiers ∃, ∀ are needed
27.
Fuzzy Logic
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Fuzzy logic is a form of many-valued logic in which the truth values of variables may be any real number between 0 and 1. By contrast, in Boolean logic, the values of variables may only be the integer values 0 or 1. Fuzzy logic has been employed to handle the concept of partial truth, furthermore, when linguistic variables are used, these degrees may be managed by specific functions. The term fuzzy logic was introduced with the 1965 proposal of set theory by Lotfi Zadeh. Fuzzy logic had however been studied since the 1920s, as infinite-valued logic—notably by Łukasiewicz, Fuzzy logic has been applied to many fields, from control theory to artificial intelligence. Classical logic only permits conclusions which are true or false. However, there are also propositions with variable answers, such as one might find when asking a group of people to identify a color. In such instances, the truth appears as the result of reasoning from inexact or partial knowledge in which the answers are mapped on a spectrum. Humans and animals often operate using fuzzy evaluations in many everyday situations, instead the person instinctively applies quick fuzzy estimates, based upon previous experience, to determine what output values of force, direction and vertical angle to use to make the toss. Take, for example, the concepts of empty and full, the meaning of each of them can be represented by a certain fuzzy set. The concept of emptiness would be subjective and thus would depend on the observer or designer, a basic application might characterize various sub-ranges of a continuous variable. For instance, a measurement for anti-lock brakes might have several separate membership functions defining particular temperature ranges needed to control the brakes properly. Each function maps the temperature value to a truth value in the 0 to 1 range. These truth values can then be used to determine how the brakes should be controlled, in this image, the meanings of the expressions cold, warm, and hot are represented by functions mapping a temperature scale. A point on that scale has three truth values—one for each of the three functions, the vertical line in the image represents a particular temperature that the three arrows gauge. Since the red arrow points to zero, this temperature may be interpreted as not hot, the orange arrow may describe it as slightly warm and the blue arrow fairly cold. While variables in mathematics usually take numerical values, in fuzzy logic applications non-numeric values are used to facilitate the expression of rules. A linguistic variable such as age may accept values such as young, because natural languages do not always contain enough value terms to express a fuzzy value scale, it is common practice to modify linguistic values with adjectives or adverbs
28.
Intuition (knowledge)
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Intuition is the ability to acquire knowledge without proof, evidence, or conscious reasoning, or without understanding how the knowledge was acquired. The word intuition comes from the Latin verb intueri translated as consider or from late middle English word intuit, both Eastern and Western philosophers have studied the concept in great detail. Philosophy of mind deals with the concept of intuition, there are philosophers who contend that this concept is often confused with other concepts such as truth, belief, and meaning in philosophical discussion. In the East intuition is mostly intertwined with religion and spirituality, in Hinduism various attempts have been made to interpret the Vedic and other esoteric texts. However with knowledge by identity which we currently only gives the awareness of human existence can be extended further to outside of ourselves resulting in intuitive knowledge. He finds that this process seems to be a decent, is actual a circle of progress. As a lower faculty is being pushed to take up as much from a way of working. Advaita vedanta takes intuition to be an experience through which one can come in contact with, in Zen Buddhism various techniques have been developed to help develop ones intuitive capability, such as kó-an – the resolving of which leads to states of minor enlightenment. In parts of Zen Buddhism intuition is deemed a state between the Universal mind and ones individual, discriminating mind. In Islam there are scholars with varied interpretation of intuition. While Ibn Sīnā finds the ability of having intuition as a prophetic capacity terms it as a knowledge obtained without intentionally acquiring it and he finds regular knowledge is based on imitation while intuitive knowledge as based on intellectual certitude. In the West, intuition does not appear as a field of study. In his book Republic he tries to define intuition as a capacity of human reason to comprehend the true nature of reality. In his discussion with Meno & Phaedo, he describes intuition as a pre-existing knowledge residing in the soul of eternity, and he provides an example of mathematical truths, and posits that they are not arrived at by reason. He argues that these truths are accessed using an already present in a dormant form. This concept by Plato is also referred to as anamnesis. The study was continued by his followers. In his book Meditations on first philosophy, Descartes refers to an intuition as a pre-existing knowledge gained through rational reasoning or discovering truth through contemplation and this definition is commonly referred to as rational intuition
29.
Topos theory
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In mathematics, a topos is a category that behaves like the category of sheaves of sets on a topological space. Topoi behave much like the category of sets and possess a notion of localization, the Grothendieck topoi find applications in algebraic geometry, the more general elementary topoi are used in logic. Since the introduction of sheaves into mathematics in the 1940s a major theme has been to study a space by studying sheaves on a space and this idea was expounded by Alexander Grothendieck by introducing the notion of a topos. An important example of this idea to date is the étale topos of a scheme. A theorem of Giraud states that the following are equivalent, There is a small category D, C is the category of sheaves on a Grothendieck site. A category with these properties is called a topos, here Presh denotes the category of contravariant functors from D to the category of sets, such a contravariant functor is frequently called a presheaf. Girauds axioms for a category C are, C has a set of generators. Furthermore, colimits commute with fiber products, in other words, the fiber product of X and Y over their sum is the initial object in C. All equivalence relations in C are effective, the last axiom needs the most explanation. If X is an object of C, an equivalence relation R on X is a map R→X×X in C such that for any object Y in C, since C has colimits we may form the coequalizer of the two maps R→X, call this X/R. The equivalence relation is effective if the canonical map R → X × X / R X is an isomorphism, girauds theorem already gives sheaves on sites as a complete list of examples. Note, however, that nonequivalent sites often give rise to equivalent topoi, as indicated in the introduction, sheaves on ordinary topological spaces motivate many of the basic definitions and results of topos theory. The category of sets is an important special case, it plays the role of a point in topos theory, indeed, a set may be thought of as a sheaf on a point. More exotic examples, and the raison dêtre of topos theory, to a scheme and even a stack one may associate an étale topos, an fppf topos, a Nisnevich topos. Topos theory is, in sense, a generalization of classical point-set topology. One should therefore expect to see old and new instances of pathological behavior, for instance, there is an example due to Pierre Deligne of a nontrivial topos that has no points. If X and Y are topoi, a morphism u. Note that u∗ automatically preserves colimits by virtue of having a right adjoint, by Freyds adjoint functor theorem, to give a geometric morphism X → Y is to give a functor u∗, Y → X that preserves finite limits and all small colimits
30.
Fuzzy logic
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Fuzzy logic is a form of many-valued logic in which the truth values of variables may be any real number between 0 and 1. By contrast, in Boolean logic, the values of variables may only be the integer values 0 or 1. Fuzzy logic has been employed to handle the concept of partial truth, furthermore, when linguistic variables are used, these degrees may be managed by specific functions. The term fuzzy logic was introduced with the 1965 proposal of set theory by Lotfi Zadeh. Fuzzy logic had however been studied since the 1920s, as infinite-valued logic—notably by Łukasiewicz, Fuzzy logic has been applied to many fields, from control theory to artificial intelligence. Classical logic only permits conclusions which are true or false. However, there are also propositions with variable answers, such as one might find when asking a group of people to identify a color. In such instances, the truth appears as the result of reasoning from inexact or partial knowledge in which the answers are mapped on a spectrum. Humans and animals often operate using fuzzy evaluations in many everyday situations, instead the person instinctively applies quick fuzzy estimates, based upon previous experience, to determine what output values of force, direction and vertical angle to use to make the toss. Take, for example, the concepts of empty and full, the meaning of each of them can be represented by a certain fuzzy set. The concept of emptiness would be subjective and thus would depend on the observer or designer, a basic application might characterize various sub-ranges of a continuous variable. For instance, a measurement for anti-lock brakes might have several separate membership functions defining particular temperature ranges needed to control the brakes properly. Each function maps the temperature value to a truth value in the 0 to 1 range. These truth values can then be used to determine how the brakes should be controlled, in this image, the meanings of the expressions cold, warm, and hot are represented by functions mapping a temperature scale. A point on that scale has three truth values—one for each of the three functions, the vertical line in the image represents a particular temperature that the three arrows gauge. Since the red arrow points to zero, this temperature may be interpreted as not hot, the orange arrow may describe it as slightly warm and the blue arrow fairly cold. While variables in mathematics usually take numerical values, in fuzzy logic applications non-numeric values are used to facilitate the expression of rules. A linguistic variable such as age may accept values such as young, because natural languages do not always contain enough value terms to express a fuzzy value scale, it is common practice to modify linguistic values with adjectives or adverbs