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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

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Hans Freudenthal
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Hans Freudenthal was a German-born Dutch mathematician. He made substantial contributions to topology and also took an interest in literature, philosophy, history. Freudenthal was born in Luckenwalde, Brandenburg, on 17 September 1905 and he was interested in both mathematics and literature as a child, and studied mathematics at the University of Berlin beginning in 1923. He met Brouwer in 1927, when Brouwer came to Berlin to give a lecture and he completed his thesis work with Heinz Hopf at Berlin, defended a thesis on the ends of topological groups in 1930, and was officially awarded a degree in October 1931. After defending his thesis in 1930, he moved to Amsterdam to take up a position as assistant to Brouwer, in this pre-war period in Amsterdam, he was promoted to lecturer at the University of Amsterdam, and married his wife, Suus Lutter, a Dutch teacher. Although he was a German Jew, Freudenthals position in the Netherlands insulated him from the laws that had been passed in Germany beginning with the Nazi rise to power in 1933. However, in 1940 the Germans invaded the Netherlands, following which Freudenthal was suspended from duties at the University of Amsterdam by the Nazis. In 1943 Freudenthal was sent to a camp in the village of Havelte in the Netherlands. During this period Freudenthal occupied his time in literary pursuits, including winning first prize under a name in a novel-writing contest. He served as the 8th president of the International Commission on Mathematical Instruction from 1967 to 1970, in 1972 he founded and became editor-in-chief of the journal Geometriae Dedicata. He retired from his professorship in 1975 and from his editorship in 1981. He died in Utrecht in 1990, sitting on a bench in a park where he took a morning walk. In his thesis work, published as an article in 1931. Ends remain of great importance in topological group theory, Freudenthals motivating application, in 1936, while working with Brouwer, Freudenthal proved the Freudenthal spectral theorem on the existence of uniform approximations by simple functions in Riesz spaces. The Freudenthal magic square is a construction in Lie algebra developed by Freudenthal in the 1950s and 1960s, later in his life, Freudenthal focused on elementary mathematics education. In the 1970s, his single-handed intervention prevented the Netherlands from following the trend of new math. He was also a fervent critic of one of the first international school achievement studies, Freudenthal published the Impossible Puzzle, a mathematical puzzle that appears to lack sufficient information for a solution, in 1969. He also designed a constructed language, Lincos, to make communication with extraterrestrial intelligence

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Measure (mathematics)
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In mathematical analysis, a measure on a set is a systematic way to assign a number to each suitable subset of that set, intuitively interpreted as its size. In this sense, a measure is a generalization of the concepts of length, area, for instance, the Lebesgue measure of the interval in the real numbers is its length in the everyday sense of the word – specifically,1. Technically, a measure is a function that assigns a real number or +∞ to subsets of a set X. It must further be countably additive, the measure of a subset that can be decomposed into a finite number of smaller disjoint subsets, is the sum of the measures of the smaller subsets. In general, if one wants to associate a consistent size to each subset of a set while satisfying the other axioms of a measure. This problem was resolved by defining measure only on a sub-collection of all subsets, the so-called measurable subsets and this means that countable unions, countable intersections and complements of measurable subsets are measurable. Non-measurable sets in a Euclidean space, on which the Lebesgue measure cannot be defined consistently, are complicated in the sense of being badly mixed up with their complement. Indeed, their existence is a consequence of the axiom of choice. Measure theory was developed in stages during the late 19th and early 20th centuries by Émile Borel, Henri Lebesgue, Johann Radon. The main applications of measures are in the foundations of the Lebesgue integral, in Andrey Kolmogorovs axiomatisation of probability theory, probability theory considers measures that assign to the whole set the size 1, and considers measurable subsets to be events whose probability is given by the measure. Ergodic theory considers measures that are invariant under, or arise naturally from, let X be a set and Σ a σ-algebra over X. A function μ from Σ to the real number line is called a measure if it satisfies the following properties, Non-negativity. Countable additivity, For all countable collections i =1 ∞ of pairwise disjoint sets in Σ, μ = ∑ k =1 ∞ μ One may require that at least one set E has finite measure. Then the empty set automatically has measure zero because of countable additivity, because μ = μ = μ + μ + μ + …, which implies that μ =0. If only the second and third conditions of the definition of measure above are met, the pair is called a measurable space, the members of Σ are called measurable sets. If and are two spaces, then a function f, X → Y is called measurable if for every Y-measurable set B ∈ Σ Y. See also Measurable function#Caveat about another setup, a triple is called a measure space. A probability measure is a measure with total measure one – i. e, a probability space is a measure space with a probability measure