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{{Short description|Area of knowledge}}{{Redirect-several|Mathematics|Maths|Math}}{{pp|small=yes}}{{pp-move}}{{Use American English|date=August 2022}}{{Use mdy dates|date=May 2023}}{{CS1 config|mode=cs1}}{{Math topics TOC}}Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory,WEB,weblink Mathematics (noun), Oxford English Dictionary, Oxford University Press, January 17, 2024, The science of space, number, quantity, and arrangement, whose methods involve logical reasoning and usually the use of symbolic notation, and which includes geometry, arithmetic, algebra, and analysis., algebra,BOOK, Kneebone, G. T., 1963, Traditional Logic, 4, Mathematical Logic and the Foundations of Mathematics: An Introductory Survey, D. Van Nostard Company, 62019535, 0150021, 792731, 118005003, Mathematics ... is simply the study of abstract structures, or formal patterns of connectedness., geometry, and analysis,BOOK, LaTorre, Donald R., Kenelly, John W., Reed, Iris B., Carpenter, Laurel R., Harris, Cynthia R., Biggers, Sherry, 2008, Models and Functions, 2, Calculus Concepts: An Applied Approach to the Mathematics of Change, 4th, Houghton Mifflin Company, 978-0-618-78983-2, 2006935429, 125397884, Calculus is the study of change—how things change and how quickly they change., respectively. There is no general consensus among mathematicians about a common definition for their academic discipline.Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature or{{emdash}}in modern mathematics{{emdash}}entities that are stipulated to have certain properties, called axioms. A proof consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, and{{emdash}}in case of abstraction from nature{{emdash}}some basic properties that are considered true starting points of the theory under consideration.BOOK, Hipólito, Inês Viegas, Kanzian, Christian, Mitterer, Josef, Josef Mitterer, Neges, Katharina, August 9–15, 2015, Abstract Cognition and the Nature of Mathematical Proof, 132–134, Realismus – Relativismus – Konstruktivismus: Beiträge des 38. Internationalen Wittgenstein Symposiums, Realism – Relativism – Constructivism: Contributions of the 38th International Wittgenstein Symposium, 23, de, en, Austrian Ludwig Wittgenstein Society, Kirchberg am Wechsel, Austria, 1022-3398, 236026294,weblink live,weblink November 7, 2022, January 17, 2024, (at ResearchGate {{open access}} {{Webarchive|url= |date=November 5, 2022}})Mathematics is essential in the natural sciences, engineering, medicine, finance, computer science, and the social sciences. Although mathematics is extensively used for modeling phenomena, the fundamental truths of mathematics are independent from any scientific experimentation. Some areas of mathematics, such as statistics and game theory, are developed in close correlation with their applications and are often grouped under applied mathematics. Other areas are developed independently from any application (and are therefore called pure mathematics), but often later find practical applications.{{Sfn|Peterson|1988|page=12}}Historically, the concept of a proof and its associated mathematical rigour first appeared in Greek mathematics, most notably in Euclid's Elements.WEB, Wise, David,weblink Eudoxus' Influence on Euclid's Elements with a close look at The Method of Exhaustion, The University of Georgia, live,weblink" title="">weblink June 1, 2019, January 18, 2024, Since its beginning, mathematics was primarily divided into geometry and arithmetic (the manipulation of natural numbers and fractions), until the 16th and 17th centuries, when algebra{{efn|Here, algebra is taken in its modern sense, which is, roughly speaking, the art of manipulating formulas.}} and infinitesimal calculus were introduced as new fields. Since then, the interaction between mathematical innovations and scientific discoveries has led to a correlated increase in the development of both.JOURNAL, Alexander, Amir, Amir Alexander, September 2011, The Skeleton in the Closet: Should Historians of Science Care about the History of Mathematics?, Isis, 102, 3, 475–480, 10.1086/661620, 0021-1753, 2884913, 22073771, 21629993, At the end of the 19th century, the foundational crisis of mathematics led to the systematization of the axiomatic method,JOURNAL, Kleiner, Israel, Israel Kleiner (mathematician), December 1991, Rigor and Proof in Mathematics: A Historical Perspective, Mathematics Magazine, Taylor & Francis, Ltd., 64, 5, 291–314, 10.1080/0025570X.1991.11977625, 2690647, 0025-570X, 1930-0980, 47003192, 1141557, 1756877, 7787171, which heralded a dramatic increase in the number of mathematical areas and their fields of application. The contemporary Mathematics Subject Classification lists more than sixty first-level areas of mathematics.{{TOC limit|3}}


The word mathematics comes from Ancient Greek máthÄ“ma (|label=none}}), meaning "that which is learnt",ENCYCLOPEDIA, Harper, Douglas, March 28, 2019,weblink Mathematic (n.), Online Etymology Dictionary, live,weblink" title="">weblink March 7, 2013, January 25, 2024, "what one gets to know", hence also "study" and "science". The word came to have the narrower and more technical meaning of "mathematical study" even in Classical times.{{efn|This meaning can be found in Plato's Republic, Book 6 Section 510c.BOOK, Plato,weblink Republic, Book 6, Section 510c, live,weblink" title="">weblink February 24, 2021, February 2, 2024, However, Plato did not use a math- word; Aristotle did, commenting on it.DICTIONARY, Liddell, Henry George, Henry Liddell, Scott, Robert, Robert Scott (philologist), 1940, μαθηματική, A Greek–English Lexicon, A Greek–English Lexicon, Clarendon Press,weblink February 2, 2024, {{better source needed |date=February 2024 |reason=This source doesn't identify when Aristotle comments on a "math-" word.}}WEB, Harper, Douglas, April 20, 2022, Online Etymology Dictionary, Mathematics (n.),weblink February 2, 2024, {{better source needed|date=February 2024|reason=This source doesn't identify when Aristotle comments on a "math-" word.}}}} Its adjective is mathÄ“matikós (), meaning "related to learning" or "studious", which likewise further came to mean "mathematical".ENCYCLOPEDIA, Harper, Douglas, December 22, 2018,weblink Mathematical (adj.), Online Etymology Dictionary, live,weblink November 26, 2022, January 25, 2024, In particular, mathÄ“matikḗ tékhnÄ“ (; ) meant "the mathematical art".Similarly, one of the two main schools of thought in Pythagoreanism was known as the mathÄ“matikoi (μαθηματικοί){{emdash}}which at the time meant "learners" rather than "mathematicians" in the modern sense. The Pythagoreans were likely the first to constrain the use of the word to just the study of arithmetic and geometry. By the time of Aristotle (384–322 BC) this meaning was fully established.JOURNAL, Perisho, Margaret W., Spring 1965, The Etymology of Mathematical Terms, Pi Mu Epsilon Journal, 4, 2, 62–66, 0031-952X, 24338341, 58015848, 1762376, 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.BOOK, Boas, Ralph P., Ralph P. Boas Jr., Alexanderson, Gerald L., Mugler, Dale H., 1995, What Augustine Didn't Say About Mathematicians, 257, Lion Hunting and Other Mathematical Pursuits: A Collection of Mathematics, Verse, and Stories, Mathematical Association of America, 978-0-88385-323-8, 94078313, 633018890, The apparent plural form in English goes back to the Latin neuter plural (Cicero), based on the Greek plural ta mathÄ“matiká () and means roughly "all things mathematical", although it is plausible that English borrowed only the adjective mathematic(al) and formed the noun mathematics anew, after the pattern of physics and metaphysics, inherited from Greek.The Oxford Dictionary of English Etymology, Oxford English Dictionary, sub "mathematics", "mathematic", "mathematics". In English, the noun mathematics takes a singular verb. It is often shortened to mathsWEB,weblink Maths (Noun), Oxford English Dictionary, Oxford University Press, January 25, 2024, or, in North America, math.WEB,weblink Math (Noun³), Oxford English Dictionary, Oxford University Press, live,weblink" title="">weblink April 4, 2020, January 25, 2024,

Areas of mathematics

{{anchor|Branches of mathematics}}Before the Renaissance, mathematics was divided into two main areas: arithmetic, regarding the manipulation of numbers, and geometry, regarding the study of shapes.BOOK, Bell, E. T., Eric Temple Bell, 1945, 1940, General Prospectus, The Development of Mathematics, 2nd, 978-0-486-27239-9, 45010599, 523284, 3, Dover Publications, ... mathematics has come down to the present by the two main streams of number and form. The first carried along arithmetic and algebra, the second, geometry., Some types of pseudoscience, such as numerology and astrology, were not then clearly distinguished from mathematics.BOOK, Tiwari, Sarju, 1992, A Mirror of Civilization, Mathematics in History, Culture, Philosophy, and Science, 1st, 27, Mittal Publications, New Delhi, India, 978-81-7099-404-6, 92909575, 28115124, It is unfortunate that two curses of mathematics--Numerology and Astrology were also born with it and have been more acceptable to the masses than mathematics itself., During the Renaissance, two more areas appeared. Mathematical notation led to algebra which, roughly speaking, consists of the study and the manipulation of formulas. Calculus, consisting of the two subfields differential calculus and integral calculus, is the study of continuous functions, which model the typically nonlinear relationships between varying quantities, as represented by variables. This division into four main areasarithmetic, geometry, algebra, calculusBOOK, Restivo, Sal, Sal Restivo, Bunge, Mario, Mario Bunge, 1992, Mathematics from the Ground Up, Mathematics in Society and History, 14, Episteme, 20, Kluwer Academic Publishers, 0-7923-1765-3, 25709270, 92013695, endured until the end of the 19th century. Areas such as celestial mechanics and solid mechanics were then studied by mathematicians, but now are considered as belonging to physics.BOOK, Musielak, Dora, Dora Musielak, 2022, Leonhard Euler and the Foundations of Celestial Mechanics, History of Physics, Springer International Publishing, 10.1007/978-3-031-12322-1, 978-3-031-12321-4, 253240718, 2730-7549, 2730-7557, 1332780664, The subject of combinatorics has been studied for much of recorded history, yet did not become a separate branch of mathematics until the seventeenth century.JOURNAL, May 1979, Biggs, N. L., The roots of combinatorics, Historia Mathematica, 6, 2, 109–136, 10.1016/0315-0860(79)90074-0, free, 0315-0860, 1090-249X, 75642280, 2240703, At the end of the 19th century, the foundational crisis in mathematics and the resulting systematization of the axiomatic method led to an explosion of new areas of mathematics.WEB, Warner, Evan, Splash Talk: The Foundational Crisis of Mathematics, Columbia University,weblink dead,weblink March 22, 2023, February 3, 2024, The 2020 Mathematics Subject Classification contains no less than {{em|sixty-three}} first-level areas.JOURNAL, Dunne, Edward, Hulek, Klaus, Klaus Hulek, March 2020, Mathematics Subject Classification 2020, Notices of the American Mathematical Society, 67, 3, 410–411, 10.1090/noti2052, free, 0002-9920, 1088-9477, sf77000404, 1480366,weblink live,weblink August 3, 2021, February 3, 2024, The new MSC contains 63 two-digit classifications, 529 three-digit classifications, and 6,006 five-digit classifications., Some of these areas correspond to the older division, as is true regarding number theory (the modern name for higher arithmetic) and geometry. Several other first-level areas have "geometry" in their names or are otherwise commonly considered part of geometry. Algebra and calculus do not appear as first-level areas but are respectively split into several first-level areas. Other first-level areas emerged during the 20th century or had not previously been considered as mathematics, such as mathematical logic and foundations.WEB,weblink MSC2020-Mathematics Subject Classification System, zbMath, Associate Editors of Mathematical Reviews and zbMATH, live,weblink January 2, 2024, February 3, 2024,

Number theory

File:Spirale Ulam 150.jpg|thumb|This is the Ulam spiral, which illustrates the distribution of prime numbers. The dark diagonal lines in the spiral hint at the hypothesized approximate independence between being prime and being a value of a quadratic polynomial, a conjecture now known as Hardy and Littlewood's Conjecture F.]]Number theory began with the manipulation of numbers, that is, natural numbers (mathbb{N}), and later expanded to integers (Z) and rational numbers (Q). Number theory was once called arithmetic, but nowadays this term is mostly used for numerical calculations.BOOK, LeVeque, William J., William J. LeVeque, 1977, Introduction, Fundamentals of Number Theory, 1–30, Addison-Wesley Publishing Company, 0-201-04287-8, 76055645, 3519779, 118560854, Number theory dates back to ancient Babylon and probably China. Two prominent early number theorists were Euclid of ancient Greece and Diophantus of Alexandria.BOOK, Goldman, Jay R., 1998, The Founding Fathers, The Queen of Mathematics: A Historically Motivated Guide to Number Theory, 2–3, A K Peters, Wellesley, MA, 10.1201/9781439864623, 1-56881-006-7, 94020017, 30437959, 118934517, The modern study of number theory in its abstract form is largely attributed to Pierre de Fermat and Leonhard Euler. The field came to full fruition with the contributions of Adrien-Marie Legendre and Carl Friedrich Gauss.BOOK, Weil, André, André Weil, 1983, Number Theory: An Approach Through History From Hammurapi to Legendre, Birkhäuser Boston, 2–3, 10.1007/978-0-8176-4571-7, 0-8176-3141-0, 83011857, 9576587, 117789303, Many easily stated number problems have solutions that require sophisticated methods, often from across mathematics. A prominent example is Fermat's Last Theorem. This conjecture was stated in 1637 by Pierre de Fermat, but it was proved only in 1994 by Andrew Wiles, who used tools including scheme theory from algebraic geometry, category theory, and homological algebra.JOURNAL, Kleiner, Israel, Israel Kleiner (mathematician), March 2000, From Fermat to Wiles: Fermat's Last Theorem Becomes a Theorem, Elemente der Mathematik, 55, 1, 19–37, 10.1007/PL00000079, free, 0013-6018, 1420-8962, 66083524, 1567783, 53319514, Another example is Goldbach's conjecture, which asserts that every even integer greater than 2 is the sum of two prime numbers. Stated in 1742 by Christian Goldbach, it remains unproven despite considerable effort.BOOK, Wang, Yuan, 2002, The Goldbach Conjecture, 1–18, 2nd, Series in Pure Mathematics, 4, World Scientific, 10.1142/5096, 981-238-159-7, 2003268597, 51533750, 14555830, Number theory includes several subareas, including analytic number theory, algebraic number theory, geometry of numbers (method oriented), diophantine equations, and transcendence theory (problem oriented).


(File:Triangles (spherical geometry).jpg|thumb|On the surface of a sphere, Euclidean geometry only applies as a local approximation. For larger scales the sum of the angles of a triangle is not equal to 180°.)Geometry is one of the oldest branches of mathematics. It started with empirical recipes concerning shapes, such as lines, angles and circles, which were developed mainly for the needs of surveying and architecture, but has since blossomed out into many other subfields.ARXIV, Straume, Eldar, September 4, 2014, A Survey of the Development of Geometry up to 1870, math.HO, 1409.1140, A fundamental innovation was the ancient Greeks' introduction of the concept of proofs, which require that every assertion must be proved. For example, it is not sufficient to verify by measurement that, say, two lengths are equal; their equality must be proven via reasoning from previously accepted results (theorems) and a few basic statements. The basic statements are not subject to proof because they are self-evident (postulates), or are part of the definition of the subject of study (axioms). This principle, foundational for all mathematics, was first elaborated for geometry, and was systematized by Euclid around 300 BC in his book Elements.BOOK, Hilbert, David, David Hilbert, 1902, The Foundations of Geometry, 1, Open Court Publishing Company, 10.1126/science.16.399.307, 02019303, 996838, 238499430, {{GBurl, 8ZBsAAAAMAAJ, |access-date=February 6, 2024}} {{free access}}BOOK, Hartshorne, Robin, Robin Hartshorne, 2000, Euclid's Geometry, 9–13, Geometry: Euclid and Beyond, Springer New York, 0-387-98650-2, 99044789, 42290188, {{GBurl, EJCSL9S6la0C, 9, |access-date=February 7, 2024}}The resulting Euclidean geometry is the study of shapes and their arrangements constructed from lines, planes and circles in the Euclidean plane (plane geometry) and the three-dimensional Euclidean space.{{efn|This includes conic sections, which are intersections of circular cylinders and planes.}}Euclidean geometry was developed without change of methods or scope until the 17th century, when René Descartes introduced what is now called Cartesian coordinates. This constituted a major change of paradigm: Instead of defining real numbers as lengths of line segments (see number line), it allowed the representation of points using their coordinates, which are numbers. Algebra (and later, calculus) can thus be used to solve geometrical problems. Geometry was split into two new subfields: synthetic geometry, which uses purely geometrical methods, and analytic geometry, which uses coordinates systemically.BOOK, Boyer, Carl B., Carl B. Boyer, 2004, 1956, Fermat and Descartes, 74–102, History of Analytic Geometry, Dover Publications, 0-486-43832-5, 2004056235, 56317813, Analytic geometry allows the study of curves unrelated to circles and lines. Such curves can be defined as the graph of functions, the study of which led to differential geometry. They can also be defined as implicit equations, often polynomial equations (which spawned algebraic geometry). Analytic geometry also makes it possible to consider Euclidean spaces of higher than three dimensions.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.JOURNAL, Stump, 1997, David J., Reconstructing the Unity of Mathematics circa 1900, Perspectives on Science, 5, 3, 383–417, 10.1162/posc_a_00532, 1530-9274, 1063-6145, 94657506, 26085129, 117709681,weblink February 8, 2024, 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.WEB, O'Connor, J. J., Robertson, E. F., February 1996, Non-Euclidean geometry, MacTuror, University of St. Andrews, Scotland, UK,weblink live,weblink November 6, 2022, February 8, 2024, Today's subareas of geometry include:


File:Quadratic formula.svg|thumb|The quadratic formula, which concisely expresses the solutions of all quadratic equationquadratic equation File:Rubik's cube.svg|thumb|The Rubik's Cube group is a concrete application of group theory.BOOK, Joyner, David, 2008, The (legal) Rubik's Cube group, Adventures in Group Theory: Rubik's Cube, Merlin's Machine, and Other Mathematical Toys, 219–232, 2nd, Johns Hopkins University PressJohns Hopkins University PressAlgebra is the art of manipulating equations and formulas. Diophantus (3rd century) and al-Khwarizmi (9th century) were the two main precursors of algebra.JOURNAL, Christianidis, Jean, Oaks, Jeffrey, May 2013, Practicing algebra in late antiquity: The problem-solving of Diophantus of Alexandria, Historia Mathematica, 40, 2, 127–163, 10.1016/, free, 1090-249X, 0315-0860, 75642280, 2240703, 121346342, {{sfn|Kleiner|2007|loc="History of Classical Algebra" pp. 3–5}} 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'WEB, Lim, Lisa, December 21, 2018, Where the x we use in algebra came from, and the X in Xmas, South China Morning Post,weblink limited, live,weblink December 22, 2018, February 8, 2024, that he used for naming one of these methods in the title of his main treatise.Algebra became an area in its own right only with François Viète (1540–1603), who introduced the use of variables for representing unknown or unspecified numbers.JOURNAL, Oaks, Jeffery A., 2018, François Viète's revolution in algebra, Archive for History of Exact Sciences, 72, 3, 245–302, 10.1007/s00407-018-0208-0, 1432-0657, 0003-9519, 63024699, 1482042, 125704699,weblink live,weblink November 8, 2022, February 8, 2024, Variables allow mathematicians to describe the operations that have to be done on the numbers represented using mathematical formulas.Until the 19th century, algebra consisted mainly of the study of linear equations (presently linear algebra), and polynomial equations in a single unknown, which were called algebraic equations (a term still in use, although it may be ambiguous). During the 19th century, mathematicians began to use variables to represent things other than numbers (such as matrices, modular integers, and geometric transformations), on which generalizations of arithmetic operations are often valid.{{sfn|Kleiner|2007|loc="History of Linear Algebra" pp. 79–101}} The concept of algebraic structure addresses this, consisting of a set whose elements are unspecified, of operations acting on the elements of the set, and rules that these operations must follow. The scope of algebra thus grew to include the study of algebraic structures. This object of algebra was called modern algebra or abstract algebra, as established by the influence and works of Emmy Noether.BOOK, Corry, Leo, Leo Corry, 2004, Emmy Noether: Ideals and Structures, Modern Algebra and the Rise of Mathematical Structures, 247–252, 2nd revised, Birkhäuser Basel, Germany, 3-7643-7002-5, 2004556211, 51234417, {{GBurl, WdGbeyehoCoC, 247, |access-date=February 8, 2024}} (The latter term appears mainly in an educational context, in opposition to elementary algebra, which is concerned with the older way of manipulating formulas.)Some types of algebraic structures have useful and often fundamental properties, in many areas of mathematics. Their study became autonomous parts of algebra, and include: The study of types of algebraic structures as mathematical objects is the purpose of universal algebra and category theory.BOOK, Riche, Jacques, Beziau, J. Y., Costa-Leite, Alexandre, 2007, From Universal Algebra to Universal Logic, 3–39, Perspectives on Universal Logic, Polimetrica International Scientific Publisher, Milano, Italy, 978-88-7699-077-9, 647049731, {{GBurl, ZoRN9T1GUVwC, 3, |access-date=February 8, 2024}} The latter applies to every mathematical structure (not only algebraic ones). At its origin, it was introduced, together with homological algebra for allowing the algebraic study of non-algebraic objects such as topological spaces; this particular area of application is called algebraic topology.BOOK, Krömer, Ralph, 2007, Tool and Object: A History and Philosophy of Category Theory, xxi–xxv, 1–91, Science Networks - Historical Studies, 32, Springer Science & Business Media, Germany, 978-3-7643-7523-2, 2007920230, 85242858, {{GBurl, 41bHxtHxjUAC, PR20, |access-date=February 8, 2024}}

Calculus and analysis

File:Cauchy sequence illustration.svg|thumb|A Cauchy sequenceCauchy sequenceCalculus, formerly called infinitesimal calculus, was introduced independently and simultaneously by 17th-century mathematicians Newton and Leibniz.BOOK, Guicciardini, Niccolo, Niccolò Guicciardini, Schliesser, Eric, Smeenk, Chris, 2017, The Newton–Leibniz Calculus Controversy, 1708–1730, The Oxford Handbook of Newton, Oxford Handbooks, Oxford University Press, 10.1093/oxfordhb/9780199930418.013.9, 978-0-19-993041-8, 975829354,weblink live,weblink November 9, 2022, February 9, 2024, 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.WEB, O'Connor, J. J., Robertson, E. F., September 1998, Leonhard Euler, MacTutor, University of St Andrews, Scotland, UK,weblink live,weblink November 9, 2022, February 9, 2024, Presently, "calculus" refers mainly to the elementary part of this theory, and "analysis" is commonly used for advanced parts.Analysis is further subdivided into real analysis, where variables represent real numbers, and complex analysis, where variables represent complex numbers. Analysis includes many subareas shared by other areas of mathematics which include:

Discrete mathematics

File:Markovkate_01.svg|right|thumb|A diagram representing a two-state Markov chainMarkov chainDiscrete mathematics, broadly speaking, is the study of individual, countable mathematical objects. An example is the set of all integers.JOURNAL, Franklin, James, James Franklin (philosopher), July 2017, Discrete and Continuous: A Fundamental Dichotomy in Mathematics, Journal of Humanistic Mathematics, 7, 2, 355–378,weblink 10.5642/jhummath.201702.18, free, 2159-8118, 2011202231, 700943261, 6945363, February 9, 2024, Because the objects of study here are discrete, the methods of calculus and mathematical analysis do not directly apply.{{efn|However, some advanced methods of analysis are sometimes used; for example, methods of complex analysis applied to generating series.}} Algorithms{{emdash}}especially their implementation and computational complexity{{emdash}}play a major role in discrete mathematics.BOOK, Maurer, Stephen B., Rosenstein, Joseph G., Franzblau, Deborah S., Roberts, Fred S., Fred S. Roberts, 1997, What is Discrete Mathematics? The Many Answers, 121–124, Discrete Mathematics in the Schools, DIMACS: Series in Discrete Mathematics and Theoretical Computer Science, 36, American Mathematical Society, 10.1090/dimacs/036/13, 0-8218-0448-0, 1052-1798, 97023277, 37141146, 67358543, {{GBurl, EvuQdO3h-DQC, 121, |access-date=February 9, 2024}}The four color theorem and optimal sphere packing were two major problems of discrete mathematics solved in the second half of the 20th century.BOOK, Hales, Thomas C., Turing's Legacy, Thomas Callister Hales, Downey, Rod, Rod Downey, 2014, 260–261, Turing's Legacy: Developments from Turing's Ideas in Logic, Cambridge University Press, Lecture Notes in Logic, 42, 10.1017/CBO9781107338579.001, 978-1-107-04348-0, 2014000240, 867717052, 19315498, {{GBurl, fYgaBQAAQBAJ, 260, |access-date=February 9, 2024}} The P versus NP problem, which remains open to this day, is also important for discrete mathematics, since its solution would potentially impact a large number of computationally difficult problems.CONFERENCE, Sipser, Michael, Michael Sipser, July 1992, The History and Status of the P versus NP Question, STOC '92: Proceedings of the twenty-fourth annual ACM symposium on Theory of Computing, 603–618, 10.1145/129712.129771, 11678884, Discrete mathematics includes:

Mathematical logic and set theory

File:Venn A intersect B.svg|thumb|The Venn diagramVenn diagramThe two subjects of mathematical logic and set theory have belonged to mathematics since the end of the 19th century.WEB
, William
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, November 2, 2022
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, June 18, 2020, The Early Development of Set Theory
, Stanford Encyclopedia of Philosophy
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, Before this period, sets were not considered to be mathematical objects, and logic, although used for mathematical proofs, belonged to philosophy and was not specifically studied by mathematicians.JOURNAL
, The Road to Modern Logic—An Interpretation
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Before Cantor's study of infinite sets, mathematicians were reluctant to consider actually infinite collections, and considered infinity to be the result of endless enumeration. Cantor's work offended many mathematicians not only by considering actually infinite setsWEB, Natalie, Wolchover, Natalie Wolchover, December 3, 2013, Dispute over Infinity Divides Mathematicians, Scientific American,weblink November 1, 2022, November 2, 2022,weblink live, but by showing that this implies different sizes of infinity, per Cantor's diagonal argument. This led to the controversy over Cantor's set theory.WEB
, Wittgenstein's analysis on Cantor's diagonal argument
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In the same period, various areas of mathematics concluded the former intuitive definitions of the basic mathematical objects were insufficient for ensuring mathematical rigour. Examples of such intuitive definitions are "a set is a collection of objects", "natural number is what is used for counting", "a point is a shape with a zero length in every direction", "a curve is a trace left by a moving point", etc.This became the foundational crisis of mathematics.WEB
, "Clarifying the nature of the infinite": the development of metamathematics and proof theory
, Jeremy
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, 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. 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.BOOK
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, 1982
, 978-0-521-28761-6
, Cambridge University Press
, {hide}GBurl, OXfmTHXvRXMC, 3,
| access-date=November 12, 2022
{edih} This mathematical abstraction from reality is embodied in the modern philosophy of formalism, as founded by David Hilbert around 1910.JOURNAL, 10.2307/2689412, The Three Crises in Mathematics: Logicism, Intuitionism, and Formalism, Mathematics Magazine, September 1979, Ernst, Snapper, Ernst Snapper, 52, 4, 207–216, 2689412, The "nature" of the objects defined this way is a philosophical problem that mathematicians leave to philosophers, even if many mathematicians have opinions on this nature, and use their opinion{{emdash}}sometimes called "intuition"{{emdash}}to guide their study and proofs. The approach allows considering "logics" (that is, sets of allowed deducing rules), theorems, proofs, etc. as mathematical objects, and to prove theorems about them. For example, Gödel's incompleteness theorems assert, roughly speaking that, in every consistent formal system that contains the natural numbers, there are theorems that are true (that is provable in a stronger system), but not provable inside the system.JOURNAL, On the Philosophical Relevance of Gödel's Incompleteness Theorems, Panu, Raatikainen, Revue Internationale de Philosophie, 59, 4, October 2005, 513–534, 10.3917/rip.234.0513,weblink 23955909, 52083793, November 12, 2022, November 12, 2022,weblink live, This approach to the foundations of mathematics was challenged during the first half of the 20th century by mathematicians led by Brouwer, who promoted intuitionistic logic, which explicitly lacks the law of excluded middle.WEB
, Intuitionistic Logic
, September 4, 2018, Joan
, Moschovakis
, Joan Moschovakis
, Stanford Encyclopedia of Philosophy
, November 12, 2022
, December 16, 2022,weblink
, live
, At the Heart of Analysis: Intuitionism and Philosophy
, Charles, McCarty
, Philosophia Scientiæ, Cahier spécial 6
, 2006, 81–94, 10.4000/philosophiascientiae.411, free,
These problems and debates led to a wide expansion of mathematical logic, with subareas such as model theory (modeling some logical theories inside other theories), proof theory, type theory, computability theory and computational complexity theory. Although these aspects of mathematical logic were introduced before the rise of computers, their use in compiler design, program certification, proof assistants and other aspects of computer science, contributed in turn to the expansion of these logical theories.WEB
, Halpern, Joseph, Joseph Halpern
, Harper, Robert, Robert Harper (computer scientist)
, Immerman, Neil, Neil Immerman
, Kolaitis, Phokion, Phokion Kolaitis
, Vardi, Moshe, Moshe Vardi
, Vianu, Victor, Victor Vianu
, On the Unusual Effectiveness of Logic in Computer Science
, January 15, 2021, 2001, March 3, 2021,weblink
, live,

Statistics and other decision sciences

File:IllustrationCentralTheorem.png|upright=1.5|thumb|right|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 theoremcentral limit theoremThe field of statistics is a mathematical application that is employed for the collection and processing of data samples, using procedures based on mathematical methods especially probability theory. Statisticians generate data with random sampling or randomized experiments.BOOK, Rao, C. Radhakrishna, C. R. Rao, 1997, 1989, Statistics and Truth: Putting Chance to Work, 2nd, 3–17, 63–70, World Scientific, 981-02-3111-3, 97010349, 1474730, 36597731, The design of a statistical sample or experiment determines the analytical methods that will be used. Analysis of data from observational studies is done using statistical models and the theory of inference, using model selection and estimation. The models and consequential predictions should then be tested against new data.{{efn|Like other mathematical sciences such as physics and computer science, statistics is an autonomous discipline rather than a branch of applied mathematics. Like research physicists and computer scientists, research statisticians are mathematical scientists. Many statisticians have a degree in mathematics, and some statisticians are also mathematicians.}}Statistical theory studies decision problems such as minimizing the risk (expected loss) of a statistical action, such as using a procedure in, for example, parameter estimation, hypothesis testing, and selecting the best. In these traditional areas of mathematical statistics, a statistical-decision problem is formulated by minimizing an objective function, like expected loss or cost, under specific constraints. For example, designing a survey often involves minimizing the cost of estimating a population mean with a given level of confidence.BOOK, Rao, C. Radhakrishna, C.R. Rao, Arthanari, T.S., Dodge, Yadolah, Yadolah Dodge, Foreword, Mathematical programming in statistics, Wiley Series in Probability and Mathematical Statistics, Wiley, New York, 1981, vii–viii, 978-0-471-08073-2, 80021637, 607328, 6707805, Because of its use of optimization, the mathematical theory of statistics overlaps with other decision sciences, such as operations research, control theory, and mathematical economics.{{sfn|Whittle|1994|pp=10–11, 14–18}}

Computational mathematics

Computational mathematics is the study of mathematical problems that are typically too large for human, numerical capacity.WEB
, G I Marchuk's plenary: ICM 1970
, Gurii Ivanovich
, Marchuk
, MacTutor
, April 2020
, School of Mathematics and Statistics, University of St Andrews, Scotland
, November 13, 2022
, November 13, 2022,weblink
, live
, CONFERENCE, Grand Challenges, High Performance Computing, and Computational Science, Johnson, Gary M., Cavallini, John S., Singapore Supercomputing Conference'90: Supercomputing For Strategic Advantage, September 1991, 28, 91018998, World Scientific, Kang Hoh, Phua, Kia Fock, Loe, {{GBurl, jYNIDwAAQBAJ, 28, | access-date=November 13, 2022 }} Numerical analysis studies methods for problems in analysis using functional analysis and approximation theory; numerical analysis broadly includes the study of approximation and discretization with special focus on rounding errors.BOOK, Trefethen, Lloyd N., Lloyd N. Trefethen, Gowers, Timothy, Timothy Gowers, Barrow-Green, June, June Barrow-Green, Leader, Imre, Imre Leader, 2008, Numerical Analysis, 604–615, The Princeton Companion to Mathematics, Princeton University Press, 978-0-691-11880-2, 2008020450, 2467561, 227205932,weblink live,weblink" title="">weblink March 7, 2023, February 15, 2024, Numerical analysis and, more broadly, scientific computing also study non-analytic topics of mathematical science, especially algorithmic-matrix-and-graph theory. Other areas of computational mathematics include computer algebra and symbolic computation.



The history of mathematics is an ever-growing series of abstractions. Evolutionarily speaking, the first abstraction to ever be discovered, one shared by many animals,JOURNAL, Abstract representations of numbers in the animal and human brain, Trends in Neurosciences, 21, 8, Aug 1998, 355–361, 10.1016/S0166-2236(98)01263-6, 9720604, Dehaene, Stanislas, Stanislas Dehaene, Dehaene-Lambertz, Ghislaine, Ghislaine Dehaene-Lambertz, Cohen, Laurent, 17414557, was probably that of numbers: the realization that, for example, a collection of two apples and a collection of two oranges (say) have something in common, namely that there are {{em|two}} of them. As evidenced by tallies found on bone, in addition to recognizing how to count physical objects, prehistoric peoples may have also known how to count abstract quantities, like time{{emdash}}days, seasons, or years.See, for example, BOOK, Raymond L., Wilder, Raymond L. Wilder, Evolution of Mathematical Concepts; an Elementary Study, passim, BOOK, Zaslavsky, Claudia, Claudia Zaslavsky, Africa Counts: Number and Pattern in African Culture., 1999, Chicago Review Press, 978-1-61374-115-3, 843204342, File:Plimpton 322.jpg|thumb|The Babylonian mathematical tablet Plimpton 322Plimpton 322Evidence for more complex mathematics does not appear until around 3000 {{Abbr|BC|Before Christ}}, when the Babylonians and Egyptians began using arithmetic, algebra, and geometry for taxation and other financial calculations, for building and construction, and for astronomy.{{sfn|Kline|1990|loc=Chapter 1}} The oldest mathematical texts from Mesopotamia and Egypt are from 2000 to 1800 BC. Many early texts mention Pythagorean triples and so, by inference, the Pythagorean theorem seems to be the most ancient and widespread mathematical concept after basic arithmetic and geometry. It is in Babylonian mathematics that elementary arithmetic (addition, subtraction, multiplication, and division) first appear in the archaeological record. The Babylonians also possessed a place-value system and used a sexagesimal numeral system which is still in use today for measuring angles and time.{{sfn|Boyer|1991|loc="Mesopotamia" pp. 24–27}}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.BOOK, Heath, Thomas Little, Thomas Heath (classicist),weblink registration, 1, A History of Greek Mathematics: From Thales to Euclid, New York, Dover Publications, 1981, 1921, 978-0-486-24073-2, 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.JOURNAL, Mueller, I., 1969, Euclid's Elements and the Axiomatic Method, The British Journal for the Philosophy of Science, 20, 4, 289–309, 10.1093/bjps/20.4.289, 686258, 0007-0882, His book, Elements, is widely considered the most successful and influential textbook of all time.{{sfn|Boyer|1991|loc="Euclid of Alexandria" p. 119}} The greatest mathematician of antiquity is often held to be Archimedes ({{Circa|287|212 BC}}) of Syracuse.{{sfn|Boyer|1991|loc="Archimedes of Syracuse" p. 120}} 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.{{sfn|Boyer|1991|loc="Archimedes of Syracuse" p. 130}} Other notable achievements of Greek mathematics are conic sections (Apollonius of Perga, 3rd century BC),{{sfn|Boyer|1991|loc="Apollonius of Perga" p. 145}} trigonometry (Hipparchus of Nicaea, 2nd century BC),{{sfn|Boyer|1991|loc="Greek Trigonometry and Mensuration" p. 162}} and the beginnings of algebra (Diophantus, 3rd century AD).{{sfn|Boyer|1991|loc="Revival and Decline of Greek Mathematics" p. 180}}File:Bakhshali numerals 2.jpg|thumb|right|upright=1.5|The numerals used in the Bakhshali manuscriptBakhshali manuscriptThe Hindu–Arabic numeral system and the rules for the use of its operations, in use throughout the world today, evolved over the course of the first millennium AD in India and were transmitted to the Western world via Islamic mathematics.BOOK
, Number Theory and Its History
, Øystein
, Ore
, Øystein Ore
, Courier Corporation
, 19–24
, 1988
, 978-0-486-65620-5
, {hide}GBurl, Sl_6BPp7S0AC, IA19,
| access-date=November 14, 2022
{edih} Other notable developments of Indian mathematics include the modern definition and approximation of sine and cosine, and an early form of infinite series.JOURNAL
, On the Use of Series in Hindu Mathematics
, A. N., Singh, Osiris
, 1, January 1936, 606–628
, 10.1086/368443, 301627, 144760421, BOOK
, Use of series in India
, Kolachana, A., Mahesh, K.
, Ramasubramanian, K.
, Studies in Indian Mathematics and Astronomy
, Sources and Studies in the History of Mathematics and Physical Sciences
, 438–461, Springer, Singapore
, 978-981-13-7325-1, 2019
, 10.1007/978-981-13-7326-8_20, 190176726,

Medieval and later

File:Image-Al-Kitāb al-muḫtaá¹£ar fÄ« ḥisāb al-ÄŸabr wa-l-muqābala.jpg|thumb|A page from al-KhwārizmÄ«'s Algebra]]During the Golden Age of Islam, especially during the 9th and 10th centuries, mathematics saw many important innovations building on Greek mathematics. The most notable achievement of Islamic mathematics was the development of Algebra. Other achievements of the Islamic period include advances in spherical trigonometry and the addition of the decimal point to the Arabic numeral system.BOOK, Saliba, George, George Saliba, A history of Arabic astronomy: planetary theories during the golden age of Islam, 1994, New York University Press, 978-0-8147-7962-0, 28723059, Many notable mathematicians from this period were Persian, such as Al-Khwarismi, Omar Khayyam and Sharaf al-DÄ«n al-ṬūsÄ«.JOURNAL
, Contributions of Islamic scholars to the scientific enterprise
, Yasmeen M.
, Faruqi
, International Education Journal
, 2006
, 7
, 4
, 391–399
, Shannon Research Press
, November 14, 2022
, November 14, 2022,weblink
, live
, The Greek and Arabic mathematical texts were in turn translated to Latin during the Middle Ages and made available in Europe.JOURNAL, Greek-Arabic-Latin: The Transmission of Mathematical Texts in the Middle Ages, Richard, Lorch, Science in Context, 14, 1–2, June 2001, 313–331, Cambridge University Press, 10.1017/S0269889701000114, 146539132,weblink December 5, 2022, December 17, 2022,weblink live,
During the early modern period, mathematics began to develop at an accelerating pace in Western Europe, with innovations that revolutionized mathematics, such as the introduction of variables and symbolic notation by François Viète (1540–1603), the introduction of logarithms by John Napier in 1614, which greatly simplified numerical calculations, especially for astronomy and marine navigation, the introduction of coordinates by René Descartes (1596–1650) for reducing geometry to algebra, and the development of calculus by Isaac Newton (1642–1726/27) and Gottfried Leibniz (1646–1716). Leonhard Euler (1707–1783), the most notable mathematician of the 18th century, unified these innovations into a single corpus with a standardized terminology, and completed them with the discovery and the proof of numerous theorems.File:Carl Friedrich Gauss 1840 by Jensen.jpg|thumb|left|Carl Friedrich GaussCarl Friedrich GaussPerhaps the foremost mathematician of the 19th century was the German mathematician Carl Gauss, who made numerous contributions to fields such as algebra, analysis, differential geometry, matrix theory, number theory, and statistics.JOURNAL
, History of Mathematics After the Sixteenth Century
, Raymond Clare, Archibald, Raymond Clare Archibald
, The American Mathematical Monthly
, Part 2: Outline of the History of Mathematics
, 56, 1, January 1949, 35–56
, 10.2307/2304570, 2304570
, In the early 20th century, Kurt Gödel transformed mathematics by publishing his incompleteness theorems, which show in part that any consistent axiomatic system{{emdash}}if powerful enough to describe arithmetic{{emdash}}will contain true propositions that cannot be proved.Mathematics has since been greatly extended, and there has been a fruitful interaction between mathematics and science, to the benefit of both. Mathematical discoveries continue to be made to this very day. According to Mikhail B. Sevryuk, in the January 2006 issue of the Bulletin of the American Mathematical Society, "The number of papers and books included in the Mathematical Reviews database since 1940 (the first year of operation of MR) is now more than 1.9 million, and more than 75 thousand items are added to the database each year. The overwhelming majority of works in this ocean contain new mathematical theorems and their proofs."{{sfn|Sevryuk|2006|pp=101–109}}

Symbolic notation and terminology

File:Sigma summation notation.svg|thumb|An explanation of the sigma (Σ) summationsummationMathematical notation is widely used in science and engineering for representing complex concepts and properties in a concise, unambiguous, and accurate way. This notation consists of symbols used for representing operations, unspecified numbers, relations and any other mathematical objects, and then assembling them into expressions and formulas.CONFERENCE, Wolfram, Stephan, October 2000, Stephen Wolfram, Mathematical Notation: Past and Future, MathML and Math on the Web: MathML International Conference 2000, Urbana Champaign, USA,weblink live,weblink November 16, 2022, February 3, 2024, More precisely, numbers and other mathematical objects are represented by symbols called variables, which are generally Latin or Greek letters, and often include subscripts. Operation and relations are generally represented by specific symbols or glyphs,JOURNAL, Douglas, Heather, Headley, Marcia Gail, Hadden, Stephanie, LeFevre, Jo-Anne, Jo-Anne LeFevre, December 3, 2020, Knowledge of Mathematical Symbols Goes Beyond Numbers, Journal of Numerical Cognition, 6, 3, 322–354, 10.5964/jnc.v6i3.293, free, 2363-8761, 228085700, such as {{math|+}} (plus), {{math|×}} (multiplication), int (integral), {{math|1==}} (equal), and {{math|

Pure and applied mathematics

{{multiple image| footer = Isaac Newton (left) and Gottfried Wilhelm Leibniz developed infinitesimal calculus.| total_width = 330| width1 = 407| height1 = 559| image1 = GodfreyKneller-IsaacNewton-1689.jpg| alt1 = Isaac Newton| width2 = 320| height2 = 390| image2 = Gottfried Wilhelm Leibniz, Bernhard Christoph Francke.jpg| alt2 = Gottfried Wilhelm von Leibniz}}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.

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