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Euclid (holding calipers), Greek mathematician, 3rd century BC, as imagined by Raphael in this detail from The School of Athens.<ref>No likeness or description of Euclid's physical appearance made during his lifetime survived antiquity. Therefore, Euclid's depiction in works of art depends on the artist's imagination (see Euclid).</ref>

Mathematics (from Greek μάθημα máthēma, “knowledge, study, learning”) is the study of topics such as quantity (numbers),<ref name="OED">{{#invoke:citation/CS1|citation |CitationClass=web }}</ref> structure,<ref name="Kneebone">{{#invoke:citation/CS1|citation |CitationClass=book }}</ref> space,<ref name=OED/> and change.<ref name="LaTorre">{{#invoke:citation/CS1|citation |CitationClass=book }}</ref><ref name="Ramana">{{#invoke:citation/CS1|citation |CitationClass=book }}</ref><ref name="Ziegler">{{#invoke:citation/CS1|citation |CitationClass=book }}</ref> There is a range of views among mathematicians and philosophers as to the exact scope and definition of mathematics.<ref name=Mura/><ref name=Runge/>

Mathematicians seek out patterns<ref name=future/><ref name=devlin/> 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, and the systematic study of the shapes and motions of physical objects. Practical mathematics has been a human activity for 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 Euclid's Elements. Since the pioneering work of Giuseppe Peano (1858–1932), David Hilbert (1862–1943), and others on axiomatic systems in the late 19th century, it has become customary to view mathematical research as establishing truth by rigorous deduction from appropriately chosen axioms and definitions. Mathematics developed at a relatively slow pace until the Renaissance, when mathematical innovations interacting with new scientific discoveries led to a rapid increase in the rate of mathematical discovery that has continued to the present day.<ref>Eves</ref>

Galileo Galilei (1564–1642) said, "The universe cannot be read until we have learned the language and become familiar with the characters in which it is written. 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."<ref>Marcus du Sautoy, A Brief History of Mathematics: 1. Newton and Leibniz, BBC Radio 4, September 27, 2010.</ref> Carl Friedrich Gauss (1777–1855) referred to mathematics as "the Queen of the Sciences".<ref name="Waltershausen">Waltershausen</ref> Benjamin Peirce (1809–1880) called mathematics "the science that draws necessary conclusions".<ref>Peirce, p. 97.</ref> 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."<ref>Hilbert, D. (1919–20), Natur und Mathematisches Erkennen: Vorlesungen, gehalten 1919–1920 in Göttingen. Nach der Ausarbeitung von Paul Bernays (Edited and with an English introduction by David E. Rowe), Basel, Birkhäuser (1992).</ref> Albert Einstein (1879–1955) stated that "as far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality."<ref name=certain/> French mathematician Claire Voisin states "There is creative drive in mathematics, it's all about movement trying to express itself." <ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref>

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 and game theory. 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, and practical applications for what began as pure mathematics are often discovered.<ref>Peterson</ref>

Mathematics sections
Intro  History  Definitions of mathematics  Inspiration, pure and applied mathematics, and aesthetics  Notation, language, and rigor  Fields of mathematics  Mathematical awards  Common misconceptions  See also  Notes  References  Further reading  External links  

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