Solving Mathematical Problems: A Personal Perspective

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Humans, as well as some other animals, find symmetric patterns to be more beautiful. [189] Mathematically, the symmetries of an object form a group known as the symmetry group. [190] Main articles: Mathematical notation, Language of mathematics, and Glossary of mathematics An explanation of the sigma (Σ) summation notation Calculus, formerly called infinitesimal calculus, was introduced independently and simultaneously by 17th-century mathematicians Newton and Leibniz. [46] It is fundamentally the study of the relationship of variables that depend on each other. Calculus was expanded in the 18th century by Euler with the introduction of the concept of a function and many other results. [47] Presently, "calculus" refers mainly to the elementary part of this theory, and "analysis" is commonly used for advanced parts. Ramana, B. V. (2007). Applied Mathematics. Tata McGraw–Hill Education. p.2.10. ISBN 978-0-07-066753-2 . Retrieved July 30, 2022. The mathematical study of change, motion, growth or decay is calculus. Trefethen, Lloyd N. (2008). "Numerical Analysis". In Gowers, Timothy; Barrow-Green, June; Leader, Imre (eds.). The Princeton Companion to Mathematics (PDF). Princeton University Press. pp.604–615. ISBN 978-0-691-11880-2. Archived (PDF) from the original on March 7, 2023 . Retrieved November 13, 2022.

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In the 19th century, mathematicians discovered non-Euclidean geometries, which do not follow the parallel postulate. By questioning that postulate's truth, this discovery has been viewed as joining Russell's paradox in revealing the foundational crisis of mathematics. This aspect of the crisis was solved by systematizing the axiomatic method, and adopting that the truth of the chosen axioms is not a mathematical problem. [35] [10] In turn, the axiomatic method allows for the study of various geometries obtained either by changing the axioms or by considering properties that do not change under specific transformations of the space. [36] Until the 19th century, the development of mathematics in the West was mainly motivated by the needs of technology and science, and there was no clear distinction between pure and applied mathematics. [109] For example, the natural numbers and arithmetic were introduced for the need of counting, and geometry was motivated by surveying, architecture and astronomy. Later, Isaac Newton introduced infinitesimal calculus for explaining the movement of the planets with his law of gravitation. Moreover, most mathematicians were also scientists, and many scientists were also mathematicians. [110] However, a notable exception occurred with the tradition of pure mathematics in Ancient Greece. [111] Marchuk, Gurii Ivanovich (April 2020). "G I Marchuk's plenary: ICM 1970". MacTutor. School of Mathematics and Statistics, University of St Andrews, Scotland. Archived from the original on November 13, 2022 . Retrieved November 13, 2022.However, many people have rejected or criticized the concept of Homo economicus. [146] [ bettersourceneeded] Economists note that real people usually have limited information and often make poor choices. [146] [ bettersourceneeded] Also, as shown in laboratory experiments, people care about fairness and sometimes altruism, not just personal gain. [146] [ bettersourceneeded] According to critics, mathematization is a veneer that allows for the material's scientific valorization. [ citation needed] Hennig, Christian (2010). "Mathematical Models and Reality: A Constructivist Perspective". Foundations of Science. 15: 29–48. doi: 10.1007/s10699-009-9167-x. S2CID 6229200 . Retrieved November 17, 2022. Creativity and rigor are not the only psychological aspects of the activity of mathematicians. Some mathematicians can see their activity as a game, more specifically as solving puzzles. [181] This aspect of mathematical activity is emphasized in recreational mathematics. Main articles: Mathematical logic and Set theory The Venn diagram is a commonly used method to illustrate the relations between sets. a b c Borel, Armand (1983). "Mathematics: Art and Science". The Mathematical Intelligencer. Springer. 5 (4): 9–17. doi: 10.4171/news/103/8. ISSN 1027-488X.

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Wolfram, Stephan (October 2000). Mathematical notation: Past and future. MathML and Math on the Web: MathML International Conference 2000, Urbana Champaign, USA. Archived from the original on November 16, 2022 . Retrieved November 16, 2022. Main articles: Statistics and Probability theory Whatever the form of a random population distribution (μ), the sampling mean (x̄) tends to a Gaussian distribution and its variance (σ) is given by the central limit theorem of probability theory. [64] Weil, André (2007). Number Theory, An Approach Through History From Hammurapi to Legendre. Birkhäuser Boston. pp.1–3. ISBN 978-0-8176-4571-7 . Retrieved March 19, 2023. This became the foundational crisis of mathematics. [57] It was eventually solved in mainstream mathematics by systematizing the axiomatic method inside a formalized set theory. Roughly speaking, each mathematical object is defined by the set of all similar objects and the properties that these objects must have. [23] For example, in Peano arithmetic, the natural numbers are defined by "zero is a number", "each number has a unique successor", "each number but zero has a unique predecessor", and some rules of reasoning. [58] This mathematical abstraction from reality is embodied in the modern philosophy of formalism, as founded by David Hilbert around 1910. [59]Wilson, Edwin B.; Lewis, Gilbert N. (November 1912). "The Space-Time Manifold of Relativity. The Non-Euclidean Geometry of Mechanics and Electromagnetics". Proceedings of the American Academy of Arts and Sciences. 48 (11): 389–507. doi: 10.2307/20022840. JSTOR 20022840. A famous list of 23 open problems, called " Hilbert's problems", was compiled in 1900 by German mathematician David Hilbert. [208] This list has achieved great celebrity among mathematicians, [209] and, as of 2022 [update], at least thirteen of the problems (depending how some are interpreted) have been solved. [208] Combinatorics, the art of enumerating mathematical objects that satisfy some given constraints. Originally, these objects were elements or subsets of a given set; this has been extended to various objects, which establishes a strong link between combinatorics and other parts of discrete mathematics. For example, discrete geometry includes counting configurations of geometric shapes Algebra is the art of manipulating equations and formulas. Diophantus (3rd century) and al-Khwarizmi (9th century) were the two main precursors of algebra. [38] [39] Diophantus solved some equations involving unknown natural numbers by deducing new relations until he obtained the solution. Al-Khwarizmi introduced systematic methods for transforming equations, such as moving a term from one side of an equation into the other side. The term algebra is derived from the Arabic word al-jabr meaning 'the reunion of broken parts' [40] that he used for naming one of these methods in the title of his main treatise. Here, algebra is taken in its modern sense, which is, roughly speaking, the art of manipulating formulas.

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Hartshorne, Robin (November 11, 2013). Geometry: Euclid and Beyond. Springer New York. pp.9–13. ISBN 978-0-387-22676-7 . Retrieved March 19, 2023. Rouaud, Mathieu (2013). Probability, Statistics and Estimation (PDF). p.10. Archived (PDF) from the original on October 9, 2022. Musielak, Dora (2022). Leonhard Euler and the Foundations of Celestial Mechanics. Springer International Publishing. pp.1–183. ISBN 978-3-031-12322-1 . Retrieved March 19, 2023. In Latin, and in English until around 1700, the term mathematics more commonly meant " astrology" (or sometimes " astronomy") rather than "mathematics"; the meaning gradually changed to its present one from about 1500 to 1800. This change has resulted in several mistranslations: For example, Saint Augustine's warning that Christians should beware of mathematici, meaning "astrologers", is sometimes mistranslated as a condemnation of mathematicians. [15]

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Rao, C.R. (1981). "Foreword". In Arthanari, T.S.; Dodge, Yadolah (eds.). Mathematical programming in statistics. Wiley Series in Probability and Mathematical Statistics. New York: Wiley. pp.vii–viii. ISBN 978-0-471-08073-2. MR 0607328. The unreasonable effectiveness of mathematics is a phenomenon that was named and first made explicit by physicist Eugene Wigner. [7] It is the fact that many mathematical theories (even the "purest") have applications outside their initial object. These applications may be completely outside their initial area of mathematics, and may concern physical phenomena that were completely unknown when the mathematical theory was introduced. [123] Examples of unexpected applications of mathematical theories can be found in many areas of mathematics. Main articles: Mathematical and theoretical biology and Mathematical chemistry The skin of this giant pufferfish exhibits a Turing pattern, which can be modeled by reaction–diffusion systems. Mathematics is used in most sciences for modeling phenomena, which then allows predictions to be made from experimental laws. [99] The independence of mathematical truth from any experimentation implies that the accuracy of such predictions depends only on the adequacy of the model. [100] Inaccurate predictions, rather than being caused by invalid mathematical concepts, imply the need to change the mathematical model used. [101] For example, the perihelion precession of Mercury could only be explained after the emergence of Einstein's general relativity, which replaced Newton's law of gravitation as a better mathematical model. [102]

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Edling, Christofer R. (2002). "Mathematics in Sociology". Annual Review of Sociology. 28 (1): 197–220. doi: 10.1146/annurev.soc.28.110601.140942. ISSN 0360-0572. a b Ferreirós, J. (2007). "Ό Θεὸς Άριθμητίζει: The Rise of Pure Mathematics as Arithmetic with Gauss". In Goldstein, Catherine; Schappacher, Norbert; Schwermer, Joachim (eds.). The Shaping of Arithmetic after C.F. Gauss's Disquisitiones Arithmeticae. Springer Science & Business Media. pp.235–268. ISBN 978-3-540-34720-0. Bishop, Alan (1991). "Environmental activities and mathematical culture". Mathematical Enculturation: A Cultural Perspective on Mathematics Education. Norwell, Massachusetts: Kluwer Academic Publishers. pp.20–59. ISBN 978-0-7923-1270-3 . Retrieved April 5, 2020. In the 6th century BC, Greek mathematics began to emerge as a distinct discipline and some Ancient Greeks such as the Pythagoreans appeared to have considered it a subject in its own right. [76] Around 300 BC, Euclid organized mathematical knowledge by way of postulates and first principles, which evolved into the axiomatic method that is used in mathematics today, consisting of definition, axiom, theorem, and proof. [77] His book, Elements, is widely considered the most successful and influential textbook of all time. [78] The greatest mathematician of antiquity is often held to be Archimedes ( c. 287– c. 212 BC) of Syracuse. [79] He developed formulas for calculating the surface area and volume of solids of revolution and used the method of exhaustion to calculate the area under the arc of a parabola with the summation of an infinite series, in a manner not too dissimilar from modern calculus. [80] Other notable achievements of Greek mathematics are conic sections ( Apollonius of Perga, 3rd century BC), [81] trigonometry ( Hipparchus of Nicaea, 2nd century BC), [82] and the beginnings of algebra (Diophantus, 3rd century AD). [83] The numerals used in the Bakhshali manuscript, dated between the 2nd century BC and the 2nd century ADOre, Øystein (1988). Number Theory and Its History. Courier Corporation. pp.19–24. ISBN 978-0-486-65620-5 . Retrieved November 14, 2022. Game theory (although continuous games are also studied, most common games, such as chess and poker are discrete) a b Geuvers, H. (February 2009). "Proof assistants: History, ideas and future". Sādhanā. 34: 3–4. doi: 10.1007/s12046-009-0001-5. S2CID 14827467. Archived from the original on December 29, 2022 . Retrieved December 29, 2022. Dehaene, Stanislas; Dehaene-Lambertz, Ghislaine; Cohen, Laurent (August 1998). "Abstract representations of numbers in the animal and human brain". Trends in Neurosciences. 21 (8): 355–361. doi: 10.1016/S0166-2236(98)01263-6. PMID 9720604. S2CID 17414557. Corry, Leo (December 6, 2012). Modern Algebra and the Rise of Mathematical Structures. Birkhäuser Basel. pp.247–252. ISBN 978-3-0348-7917-0 . Retrieved March 19, 2023.