Mathematics

Mathematics (from <a href="http://en.wikipedia.org/wiki/Ancient_Greek" title="Ancient Greek">Greek</a> μάθημα máthēma, “knowledge, study, learning”) is the study of topics such as <a href="http://en.wikipedia.org/wiki/Quantity" title="Quantity">quantity</a> (<a href="http://en.wikipedia.org/wiki/Number" title="Number">numbers</a>),<a href="http://en.wikipedia.org/wiki/Mathematics#cite_note-OED-2">[2]</a> <a href="http://en.wikipedia.org/wiki/Structure" title="Structure">structure</a>,<a href="http://en.wikipedia.org/wiki/Mathematics#cite_note-Kneebone-3">[3]</a> <a href="http://en.wikipedia.org/wiki/Space" title="Space">space</a>,<a href="http://en.wikipedia.org/wiki/Mathematics#cite_note-OED-2">[2]</a> and <a href="http://en.wikipedia.org/wiki/Calculus" title="Calculus">change</a>.<a href="http://en.wikipedia.org/wiki/Mathematics#cite_note-LaTorre-4">[4]</a><a href="http://en.wikipedia.org/wiki/Mathematics#cite_note-Ramana-5">[5]</a><a href="http://en.wikipedia.org/wiki/Mathematics#cite_note-Ziegler-6">[6]</a>

<a href="http://en.wikipedia.org/wiki/Aristotle" title="Aristotle">Aristotle</a> defined mathematics as:

The science of <a href="http://en.wikipedia.org/wiki/Quantity" title="Quantity">quantity</a>.

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In Aristotle's classification of the <a href="http://en.wikipedia.org/wiki/Sciences" title="Sciences" class="mw-redirect">sciences</a>, discrete quantities were studied by <a href="http://en.wikipedia.org/wiki/Arithmetic" title="Arithmetic">arithmetic</a>, continuous quantities by <a href="http://en.wikipedia.org/wiki/Geometry" title="Geometry">geometry</a>.<a href="http://en.wikipedia.org/wiki/Definitions_of_mathematics#cite_note-Franklin-3">[3]</a>

<a href="http://en.wikipedia.org/wiki/Auguste_Comte" title="Auguste Comte">Auguste Comte</a>'s definition tried to explain the role of mathematics in coordinating phenomena in all other <a href="http://en.wikipedia.org/wiki/Fields_of_study" title="Fields of study" class="mw-redirect">fields</a>:<a href="http://en.wikipedia.org/wiki/Definitions_of_mathematics#cite_note-4">[4]</a>

The science of indirect measurement.<a href="http://en.wikipedia.org/wiki/Definitions_of_mathematics#cite_note-Cajori-5">[5]</a> <a href="http://en.wikipedia.org/wiki/Auguste_Comte" title="Auguste Comte">Auguste Comte</a> 1851

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The "indirectness" in Comte's definition refers to determining quantities that cannot be measured directly, such as the distance to planets or the size of atoms, by means of their relations to quantities that can be measured directly.<a href="http://en.wikipedia.org/wiki/Definitions_of_mathematics#cite_note-6">[6]</a>

All Mathematics is Symbolic Logic.<a href="http://en.wikipedia.org/wiki/Definitions_of_mathematics#cite_note-Russell-7">[7]</a> <a href="http://en.wikipedia.org/wiki/Bertrand_Russell" title="Bertrand Russell">Bertrand Russell</a> 1903

<h2>Playful, metaphorical, and poetic definitions</h2>

Bertrand Russell wrote this famous tongue-in-cheek definition, describing the way all terms in mathematics are ultimately defined by reference to undefined terms:

The subject in which we never know what we are talking about, nor whether what we are saying is true.<a href="http://en.wikipedia.org/wiki/Definitions_of_mathematics#cite_note-12">[12]</a> <a href="http://en.wikipedia.org/wiki/Bertrand_Russell" title="Bertrand Russell">Bertrand Russell</a> 1901

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Many other attempts to characterize mathematics have led to humor or poetic prose:

A mathematician is a blind man in a dark room looking for a black cat which isn't there.<a href="http://en.wikipedia.org/wiki/Definitions_of_mathematics#cite_note-13">[13]</a> <a href="http://en.wikipedia.org/wiki/Charles_Darwin" title="Charles Darwin">Charles Darwin</a>

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A mathematician, like a painter or poet, is a maker of patterns. If his patterns are more permanent than theirs, it is because they are made with ideas. <a href="http://en.wikipedia.org/wiki/G.H.Hardy" title="G. H. Hardy">G. H. Hardy</a>, 1940

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Mathematics is the art of giving the same name to different things.<a href="http://en.wikipedia.org/wiki/Definitions_of_mathematics#cite_note-eves-8">[8]</a> <a href="http://en.wikipedia.org/wiki/Henri_Poincar%C3%A9" title="Henri Poincaré">Henri Poincaré</a>

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Mathematics is the science of skilful operations with concepts and rules invented just for this purpose. [this purpose being the skilful operation ....]<a href="http://en.wikipedia.org/wiki/Definitions_of_mathematics#cite_note-14">[14]</a> <a href="http://en.wikipedia.org/wiki/Eugene_Wigner" title="Eugene Wigner">Eugene Wigner</a>

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Mathematics is the study of topics such as <a href="http://en.wikipedia.org/wiki/Quantity" title="Quantity">quantity</a> (<a href="http://en.wikipedia.org/wiki/Number" title="Number">numbers</a>),<a href="http://en.wikipedia.org/wiki/Mathematics#cite_note-OED-2">[2]</a> <a href="http://en.wikipedia.org/wiki/Structure" title="Structure">structure</a>,<a href="http://en.wikipedia.org/wiki/Mathematics#cite_note-Kneebone-3">[3]</a> <a href="http://en.wikipedia.org/wiki/Space" title="Space">space</a>,<a href="http://en.wikipedia.org/wiki/Mathematics#cite_note-OED-2">[2]</a> and <a href="http://en.wikipedia.org/wiki/Calculus" title="Calculus">change</a>.<a href="http://en.wikipedia.org/wiki/Mathematics#cite_note-LaTorre-4">[4]</a><a href="http://en.wikipedia.org/wiki/Mathematics#cite_note-Ramana-5">[5]</a><a href="http://en.wikipedia.org/wiki/Mathematics#cite_note-Ziegler-6">[6]</a> There is a range of views among mathematicians and philosophers as to the exact scope and <a href="http://en.wikipedia.org/wiki/Definition_of_mathematics" title="Definition of mathematics" class="mw-redirect">definition of mathematics</a>.<a href="http://en.wikipedia.org/wiki/Mathematics#cite_note-Mura-7">[7]</a><a href="http://en.wikipedia.org/wiki/Mathematics#cite_note-Runge-8">[8]</a>

<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><a href="http://en.wikipedia.org/wiki/Mathematics#cite_note-future-9">[9]</a><a href="http://en.wikipedia.org/wiki/Mathematics#cite_note-devlin-10">[10]</a> and use them to 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>. When mathematical structures are good models of real phenomena, then mathematical reasoning can provide insight or predictions about nature. Through the use of <a href="http://en.wikipedia.org/wiki/Abstraction%28mathematics%29" title="Abstraction (mathematics)">abstraction</a> and <a href="http://en.wikipedia.org/wiki/Logic" title="Logic">logic</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%28physics%29" 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. The research required to solve mathematical problems can take years or even centuries of sustained inquiry.

<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</a>'s <a href="http://en.wikipedia.org/wiki/Euclid%27s_Elements" title="Euclid's Elements">Elements</a>. 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 <a href="http://en.wikipedia.org/wiki/Foundations_of_mathematics" title="Foundations of mathematics">on axiomatic systems in the late 19th century</a>, 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>. Mathematics developed at a relatively slow pace until the <a href="http://en.wikipedia.org/wiki/Renaissance" title="Renaissance" class="mw-redirect">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.<a href="http://en.wikipedia.org/wiki/Mathematics#cite_note-11">[11]</a>

<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."<a href="http://en.wikipedia.org/wiki/Mathematics#cite_note-12">[12]</a> <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".<a href="http://en.wikipedia.org/wiki/Mathematics#cite_note-Waltershausen-13">[13]</a> <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".<a href="http://en.wikipedia.org/wiki/Mathematics#cite_note-14">[14]</a> 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."<a href="http://en.wikipedia.org/wiki/Mathematics#cite_note-15">[15]</a> <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."<a href="http://en.wikipedia.org/wiki/Mathematics#cite_note-certain-16">[16]</a> French mathematician <a href="http://en.wikipedia.org/wiki/Claire_Voisin" title="Claire Voisin">Claire Voisin</a> states "There is creative drive in mathematics, it's all about movement trying to express itself." <a href="http://en.wikipedia.org/wiki/Mathematics#cite_note-17">[17]</a>

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>, <a href="http://en.wikipedia.org/wiki/Finance" title="Finance">finance</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.<a href="http://en.wikipedia.org/wiki/Mathematics#cite_note-18">[18]</a>

Mathematics

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