Mathematics
<p ><b>Mathematics</b> (from <a href="http://en.wikipedia.org/wiki/Greek_language" title="Greek language" >Greek</a> μάθημα <i>máthēma</i>, “knowledge, study, learning”) is the study of <a href="http://en.wikipedia.org/wiki/Quantity" title="Quantity" >quantity</a>, <a href="http://en.wikipedia.org/wiki/Structure" title="Structure" >structure</a>, <a href="http://en.wikipedia.org/wiki/Space" title="Space" >space</a>, and <a href="http://en.wikipedia.org/wiki/Calculus" title="Calculus" >change</a>.<sup id="cite_ref-artsci_1-0" class="reference" ><a href="http://en.wikipedia.org/wiki/Mathematics#cite_note-artsci-1" >[2]</a></sup></p>
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<a href="http://en.wikipedia.org/wiki/Mathematician" title="Mathematician" >Mathematicians</a> seek out <a href="http://en.wikipedia.org/wiki/Patterns" title="Patterns" class="mw-redirect" >patterns</a><sup id="cite_ref-future_2-0" class="reference" ><a href="http://en.wikipedia.org/wiki/Mathematics#cite_note-future-2" >[3]</a></sup><sup id="cite_ref-devlin_3-0" class="reference" ><a href="http://en.wikipedia.org/wiki/Mathematics#cite_note-devlin-3" >[4]</a></sup> and formulate new <a href="http://en.wikipedia.org/wiki/Conjecture" title="Conjecture" >conjectures</a>. Mathematicians resolve the truth or falsity of conjectures by<a href="http://en.wikipedia.org/wiki/Mathematical_proof" title="Mathematical proof" >mathematical proof</a>. The research required to solve mathematical problems can take years or even centuries of sustained inquiry. </p>
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Since the pioneering work of <a href="http://en.wikipedia.org/wiki/Giuseppe_Peano" title="Giuseppe Peano" >Giuseppe Peano</a> (1858–1932), <a href="http://en.wikipedia.org/wiki/David_Hilbert" title="David Hilbert" >David Hilbert</a> (1862–1943), and others on axiomatic systems in the late 19th century, it has become customary to view mathematical research as establishing <a href="http://en.wikipedia.org/wiki/Truth" title="Truth" >truth</a> by <a href="http://en.wikipedia.org/wiki/Mathematical_rigour" title="Mathematical rigour" class="mw-redirect" >rigorous</a> <a href="http://en.wikipedia.org/wiki/Deductive_reasoning" title="Deductive reasoning" >deduction</a> from appropriately chosen<a href="http://en.wikipedia.org/wiki/Axiom" title="Axiom" >axioms</a> and <a href="http://en.wikipedia.org/wiki/Definition" title="Definition" >definitions</a>. When those mathematical structures are good models of real phenomena, then mathematical reasoning often provides insight or predictions.</p>
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Through the use of <a href="http://en.wikipedia.org/wiki/Abstraction(mathematics)" title="Abstraction (mathematics)" >abstraction</a> and <a href="http://en.wikipedia.org/wiki/Logic" title="Logic" >logical</a> <a href="http://en.wikipedia.org/wiki/Reasoning" title="Reasoning" class="mw-redirect" >reasoning</a>, mathematics developed from <a href="http://en.wikipedia.org/wiki/Counting" title="Counting" >counting</a>, <a href="http://en.wikipedia.org/wiki/Calculation" title="Calculation" >calculation</a>, <a href="http://en.wikipedia.org/wiki/Measurement" title="Measurement" >measurement</a>, and the systematic study of the <a href="http://en.wikipedia.org/wiki/Shape" title="Shape" >shapes</a> and <a href="http://en.wikipedia.org/wiki/Motion(physics)" title="Motion (physics)" >motions</a> of physical objects. Practical mathematics has been a human activity for as far back as<a href="http://en.wikipedia.org/wiki/History_of_Mathematics" title="History of Mathematics" class="mw-redirect" >written records</a> exist. <a href="http://en.wikipedia.org/wiki/Logic" title="Logic" >Rigorous arguments</a> first appeared in <a href="http://en.wikipedia.org/wiki/Greek_mathematics" title="Greek mathematics" >Greek mathematics</a>, most notably in <a href="http://en.wikipedia.org/wiki/Euclid" title="Euclid" >Euclid's</a> <i><a href="http://en.wikipedia.org/wiki/Euclid%27s_Elements" title="Euclid's Elements" >Elements</a></i>. Mathematics developed at a relatively slow pace until the <a href="http://en.wikipedia.org/wiki/Renaissance" title="Renaissance" >Renaissance</a>, when mathematical innovations interacting with new <a href="http://en.wikipedia.org/wiki/Timeline_of_scientific_discoveries" title="Timeline of scientific discoveries" >scientific discoveries</a> led to a rapid increase in the rate of mathematical discovery that has continued to the present day.<sup id="cite_ref-4" class="reference" ><a href="http://en.wikipedia.org/wiki/Mathematics#cite_note-4" >[5]</a></sup>
<a href="http://en.wikipedia.org/wiki/Galileo_Galilei" title="Galileo Galilei" >Galileo Galilei</a> (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'.<sup id="cite_ref-5" class="reference" ><a href="http://en.wikipedia.org/wiki/Mathematics#cite_note-5" >[6]</a></sup> <a href="http://en.wikipedia.org/wiki/Carl_Friedrich_Gauss" title="Carl Friedrich Gauss" >Carl Friedrich Gauss</a> (1777–1855) referred to mathematics as "the Queen of the Sciences".<sup id="cite_ref-Waltershausen_6-0" class="reference" ><a href="http://en.wikipedia.org/wiki/Mathematics#cite_note-Waltershausen-6" >[7]</a></sup> <a href="http://en.wikipedia.org/wiki/Benjamin_Peirce" title="Benjamin Peirce" >Benjamin Peirce</a> (1809–1880) called mathematics "the science that draws necessary conclusions".<sup id="cite_ref-7" class="reference" ><a href="http://en.wikipedia.org/wiki/Mathematics#cite_note-7" >[8]</a></sup> 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."<sup id="cite_ref-8" class="reference" ><a href="http://en.wikipedia.org/wiki/Mathematics#cite_note-8" >[9]</a></sup> <a href="http://en.wikipedia.org/wiki/Albert_Einstein" title="Albert Einstein" >Albert Einstein</a> (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".<sup id="cite_ref-certain_9-0" class="reference" ><a href="http://en.wikipedia.org/wiki/Mathematics#cite_note-certain-9" >[10]</a></sup></p>
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Mathematics is used throughout the world as an essential tool in many fields, including <a href="http://en.wikipedia.org/wiki/Natural_science" title="Natural science" >natural science</a>, <a href="http://en.wikipedia.org/wiki/Engineering" title="Engineering" >engineering</a>, <a href="http://en.wikipedia.org/wiki/Medicine" title="Medicine" >medicine</a>, and the<a href="http://en.wikipedia.org/wiki/Social_sciences" title="Social sciences" class="mw-redirect" >social sciences</a>. <a href="http://en.wikipedia.org/wiki/Applied_mathematics" title="Applied mathematics" >Applied mathematics</a>, the branch of mathematics concerned with application of mathematical knowledge to other fields, inspires and makes use of new mathematical discoveries, which has led to the development of entirely new mathematical disciplines, such as <a href="http://en.wikipedia.org/wiki/Statistics" title="Statistics" >statistics</a> and <a href="http://en.wikipedia.org/wiki/Game_theory" title="Game theory" >game theory</a>. Mathematicians also engage in <a href="http://en.wikipedia.org/wiki/Pure_mathematics" title="Pure mathematics" >pure mathematics</a>, 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.<sup id="cite_ref-10" class="reference" ><a href="http://en.wikipedia.org/wiki/Mathematics#cite_note-10" >[11]</a></sup></p>