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			<h1 id="firstHeading" class="firstHeading" lang="en">Mathematics</h1>
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				<div id="mw-content-text" lang="en" dir="ltr" class="mw-content-ltr"><div role="note" class="hatnote">This article is about the study of topics such as quantity and structure.  For other uses, see <a href="/wiki/Mathematics_(disambiguation)" title="Mathematics (disambiguation)" class="mw-disambig">Mathematics (disambiguation)</a>.</div>
<div role="note" class="hatnote">"Math" redirects here. For other uses, see <a href="/wiki/Math_(disambiguation)" title="Math (disambiguation)" class="mw-disambig">Math (disambiguation)</a>.</div>
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<div class="thumbinner" style="width:222px;"><a href="/wiki/File:Euclid.jpg" class="image"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/21/Euclid.jpg/220px-Euclid.jpg" width="220" height="184" class="thumbimage" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/21/Euclid.jpg/330px-Euclid.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/21/Euclid.jpg/440px-Euclid.jpg 2x" data-file-width="806" data-file-height="675" /></a>
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<a href="/wiki/Euclid" title="Euclid">Euclid</a> (holding <a href="/wiki/Calipers" title="Calipers">calipers</a>), Greek mathematician, 3rd century BC, as imagined by <a href="/wiki/Raphael" title="Raphael">Raphael</a> in this detail from <i><a href="/wiki/The_School_of_Athens" title="The School of Athens">The School of Athens</a></i>.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span>[</span>1<span>]</span></a></sup></div>
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<p><b>Mathematics</b> (from <a href="/wiki/Ancient_Greek" title="Ancient Greek">Greek</a> μάθημα <i>máthēma</i>, “knowledge, study, learning”) is the study of topics such as <a href="/wiki/Quantity" title="Quantity">quantity</a> (<a href="/wiki/Number" title="Number">numbers</a>),<sup id="cite_ref-OED_2-0" class="reference"><a href="#cite_note-OED-2"><span>[</span>2<span>]</span></a></sup> <a href="/wiki/Structure" title="Structure">structure</a>,<sup id="cite_ref-Kneebone_3-0" class="reference"><a href="#cite_note-Kneebone-3"><span>[</span>3<span>]</span></a></sup> <a href="/wiki/Space" title="Space">space</a>,<sup id="cite_ref-OED_2-1" class="reference"><a href="#cite_note-OED-2"><span>[</span>2<span>]</span></a></sup> and <a href="/wiki/Calculus" title="Calculus">change</a>.<sup id="cite_ref-LaTorre_4-0" class="reference"><a href="#cite_note-LaTorre-4"><span>[</span>4<span>]</span></a></sup><sup id="cite_ref-Ramana_5-0" class="reference"><a href="#cite_note-Ramana-5"><span>[</span>5<span>]</span></a></sup><sup id="cite_ref-Ziegler_6-0" class="reference"><a href="#cite_note-Ziegler-6"><span>[</span>6<span>]</span></a></sup> There is a range of views among mathematicians and philosophers as to the exact scope and <a href="/wiki/Definition_of_mathematics" title="Definition of mathematics" class="mw-redirect">definition of mathematics</a>.<sup id="cite_ref-Mura_7-0" class="reference"><a href="#cite_note-Mura-7"><span>[</span>7<span>]</span></a></sup><sup id="cite_ref-Runge_8-0" class="reference"><a href="#cite_note-Runge-8"><span>[</span>8<span>]</span></a></sup></p>
<p><a href="/wiki/Mathematician" title="Mathematician">Mathematicians</a> seek out <a href="/wiki/Patterns" title="Patterns" class="mw-redirect">patterns</a><sup id="cite_ref-future_9-0" class="reference"><a href="#cite_note-future-9"><span>[</span>9<span>]</span></a></sup><sup id="cite_ref-devlin_10-0" class="reference"><a href="#cite_note-devlin-10"><span>[</span>10<span>]</span></a></sup> and use them to formulate new <a href="/wiki/Conjecture" title="Conjecture">conjectures</a>. Mathematicians resolve the truth or falsity of conjectures by <a href="/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="/wiki/Abstraction_(mathematics)" title="Abstraction (mathematics)">abstraction</a> and <a href="/wiki/Logic" title="Logic">logic</a>, mathematics developed from <a href="/wiki/Counting" title="Counting">counting</a>, <a href="/wiki/Calculation" title="Calculation">calculation</a>, <a href="/wiki/Measurement" title="Measurement">measurement</a>, and the systematic study of the <a href="/wiki/Shape" title="Shape">shapes</a> and <a href="/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="/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.</p>
<p><a href="/wiki/Logic" title="Logic">Rigorous arguments</a> first appeared in <a href="/wiki/Greek_mathematics" title="Greek mathematics">Greek mathematics</a>, most notably in <a href="/wiki/Euclid" title="Euclid">Euclid</a>'s <i><a href="/wiki/Euclid%27s_Elements" title="Euclid's Elements">Elements</a></i>. Since the pioneering work of <a href="/wiki/Giuseppe_Peano" title="Giuseppe Peano">Giuseppe Peano</a> (1858–1932), <a href="/wiki/David_Hilbert" title="David Hilbert">David Hilbert</a> (1862–1943), and others <a href="/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="/wiki/Truth" title="Truth">truth</a> by <a href="/wiki/Mathematical_rigour" title="Mathematical rigour" class="mw-redirect">rigorous</a> <a href="/wiki/Deductive_reasoning" title="Deductive reasoning">deduction</a> from appropriately chosen <a href="/wiki/Axiom" title="Axiom">axioms</a> and <a href="/wiki/Definition" title="Definition">definitions</a>. Mathematics developed at a relatively slow pace until the <a href="/wiki/Renaissance" title="Renaissance">Renaissance</a>, when mathematical innovations interacting with new <a href="/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-11" class="reference"><a href="#cite_note-11"><span>[</span>11<span>]</span></a></sup></p>
<p><a href="/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-12" class="reference"><a href="#cite_note-12"><span>[</span>12<span>]</span></a></sup> <a href="/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_13-0" class="reference"><a href="#cite_note-Waltershausen-13"><span>[</span>13<span>]</span></a></sup> <a href="/wiki/Benjamin_Peirce" title="Benjamin Peirce">Benjamin Peirce</a> (1809–1880) called mathematics "the science that draws necessary conclusions".<sup id="cite_ref-14" class="reference"><a href="#cite_note-14"><span>[</span>14<span>]</span></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-15" class="reference"><a href="#cite_note-15"><span>[</span>15<span>]</span></a></sup> <a href="/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_16-0" class="reference"><a href="#cite_note-certain-16"><span>[</span>16<span>]</span></a></sup></p>
<p>Mathematics is essential in many fields, including <a href="/wiki/Natural_science" title="Natural science">natural science</a>, <a href="/wiki/Engineering" title="Engineering">engineering</a>, <a href="/wiki/Medicine" title="Medicine">medicine</a>, <a href="/wiki/Finance" title="Finance">finance</a> and the <a href="/wiki/Social_sciences" title="Social sciences" class="mw-redirect">social sciences</a>. <a href="/wiki/Applied_mathematics" title="Applied mathematics">Applied mathematics</a> has led to entirely new mathematical disciplines, such as <a href="/wiki/Statistics" title="Statistics">statistics</a> and <a href="/wiki/Game_theory" title="Game theory">game theory</a>. Mathematicians also engage in <a href="/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-17" class="reference"><a href="#cite_note-17"><span>[</span>17<span>]</span></a></sup></p>
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<h2>Contents</h2>
</div>
<ul>
<li class="toclevel-1 tocsection-1"><a href="#History"><span class="tocnumber">1</span> <span class="toctext">History</span></a>
<ul>
<li class="toclevel-2 tocsection-2"><a href="#Etymology"><span class="tocnumber">1.1</span> <span class="toctext">Etymology</span></a></li>
</ul>
</li>
<li class="toclevel-1 tocsection-3"><a href="#Definitions_of_mathematics"><span class="tocnumber">2</span> <span class="toctext">Definitions of mathematics</span></a>
<ul>
<li class="toclevel-2 tocsection-4"><a href="#Mathematics_as_science"><span class="tocnumber">2.1</span> <span class="toctext">Mathematics as science</span></a></li>
</ul>
</li>
<li class="toclevel-1 tocsection-5"><a href="#Inspiration.2C_pure_and_applied_mathematics.2C_and_aesthetics"><span class="tocnumber">3</span> <span class="toctext">Inspiration, pure and applied mathematics, and aesthetics</span></a></li>
<li class="toclevel-1 tocsection-6"><a href="#Notation.2C_language.2C_and_rigor"><span class="tocnumber">4</span> <span class="toctext">Notation, language, and rigor</span></a></li>
<li class="toclevel-1 tocsection-7"><a href="#Fields_of_mathematics"><span class="tocnumber">5</span> <span class="toctext">Fields of mathematics</span></a>
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<li class="toclevel-2 tocsection-8"><a href="#Foundations_and_philosophy"><span class="tocnumber">5.1</span> <span class="toctext">Foundations and philosophy</span></a></li>
<li class="toclevel-2 tocsection-9"><a href="#Pure_mathematics"><span class="tocnumber">5.2</span> <span class="toctext">Pure mathematics</span></a>
<ul>
<li class="toclevel-3 tocsection-10"><a href="#Quantity"><span class="tocnumber">5.2.1</span> <span class="toctext">Quantity</span></a></li>
<li class="toclevel-3 tocsection-11"><a href="#Structure"><span class="tocnumber">5.2.2</span> <span class="toctext">Structure</span></a></li>
<li class="toclevel-3 tocsection-12"><a href="#Space"><span class="tocnumber">5.2.3</span> <span class="toctext">Space</span></a></li>
<li class="toclevel-3 tocsection-13"><a href="#Change"><span class="tocnumber">5.2.4</span> <span class="toctext">Change</span></a></li>
</ul>
</li>
<li class="toclevel-2 tocsection-14"><a href="#Applied_mathematics"><span class="tocnumber">5.3</span> <span class="toctext">Applied mathematics</span></a>
<ul>
<li class="toclevel-3 tocsection-15"><a href="#Statistics_and_other_decision_sciences"><span class="tocnumber">5.3.1</span> <span class="toctext">Statistics and other decision sciences</span></a></li>
<li class="toclevel-3 tocsection-16"><a href="#Computational_mathematics"><span class="tocnumber">5.3.2</span> <span class="toctext">Computational mathematics</span></a></li>
</ul>
</li>
</ul>
</li>
<li class="toclevel-1 tocsection-17"><a href="#Mathematical_awards"><span class="tocnumber">6</span> <span class="toctext">Mathematical awards</span></a></li>
<li class="toclevel-1 tocsection-18"><a href="#See_also"><span class="tocnumber">7</span> <span class="toctext">See also</span></a></li>
<li class="toclevel-1 tocsection-19"><a href="#Notes"><span class="tocnumber">8</span> <span class="toctext">Notes</span></a></li>
<li class="toclevel-1 tocsection-20"><a href="#References"><span class="tocnumber">9</span> <span class="toctext">References</span></a></li>
<li class="toclevel-1 tocsection-21"><a href="#Further_reading"><span class="tocnumber">10</span> <span class="toctext">Further reading</span></a></li>
<li class="toclevel-1 tocsection-22"><a href="#External_links"><span class="tocnumber">11</span> <span class="toctext">External links</span></a></li>
</ul>
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</div>
<h2><span class="mw-headline" id="History">History</span></h2>
<div role="note" class="hatnote relarticle mainarticle">Main article: <a href="/wiki/History_of_mathematics" title="History of mathematics">History of mathematics</a></div>
<p>The history of mathematics can be seen as an ever-increasing series of <a href="/wiki/Abstraction_(mathematics)" title="Abstraction (mathematics)">abstractions</a>. The first abstraction, which is shared by many animals,<sup id="cite_ref-18" class="reference"><a href="#cite_note-18"><span>[</span>18<span>]</span></a></sup> was probably that of <a href="/wiki/Number" title="Number">numbers</a>: the realization that a collection of two apples and a collection of two oranges (for example) have something in common, namely quantity of their members.</p>
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<div class="thumbinner" style="width:172px;"><a href="/wiki/File:Kapitolinischer_Pythagoras_adjusted.jpg" class="image"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/1/1a/Kapitolinischer_Pythagoras_adjusted.jpg/170px-Kapitolinischer_Pythagoras_adjusted.jpg" width="170" height="227" class="thumbimage" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/1a/Kapitolinischer_Pythagoras_adjusted.jpg/255px-Kapitolinischer_Pythagoras_adjusted.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/1a/Kapitolinischer_Pythagoras_adjusted.jpg/340px-Kapitolinischer_Pythagoras_adjusted.jpg 2x" data-file-width="449" data-file-height="599" /></a>
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Greek mathematician <a href="/wiki/Pythagoras" title="Pythagoras">Pythagoras</a> (<span class="nowrap">c. 570 – c. 495 BC</span>), commonly credited with discovering the <a href="/wiki/Pythagorean_theorem" title="Pythagorean theorem">Pythagorean theorem</a></div>
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<div class="thumbinner" style="width:222px;"><a href="/wiki/File:Maya.svg" class="image"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/1/1b/Maya.svg/220px-Maya.svg.png" width="220" height="254" class="thumbimage" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/1b/Maya.svg/330px-Maya.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/1b/Maya.svg/440px-Maya.svg.png 2x" data-file-width="248" data-file-height="286" /></a>
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<a href="/wiki/Mayan_numerals" title="Mayan numerals" class="mw-redirect">Mayan numerals</a></div>
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<p>As evidenced by <a href="/wiki/Tally_sticks" title="Tally sticks" class="mw-redirect">tallies</a> found on bone, in addition to recognizing how to <a href="/wiki/Counting" title="Counting">count</a> physical objects, <a href="/wiki/Prehistoric" title="Prehistoric" class="mw-redirect">prehistoric</a> peoples may have also recognized how to count abstract quantities, like time&#160;– days, <a href="/wiki/Season" title="Season">seasons</a>, years.<sup id="cite_ref-19" class="reference"><a href="#cite_note-19"><span>[</span>19<span>]</span></a></sup></p>
<p>Evidence for more complex mathematics does not appear until around 3000&#160;BC, when the <a href="/wiki/Babylonia" title="Babylonia">Babylonians</a> and Egyptians began using arithmetic, algebra and geometry for <a href="/wiki/Taxation" title="Taxation" class="mw-redirect">taxation</a> and other financial calculations, for building and construction, and for <a href="/wiki/Astronomy" title="Astronomy">astronomy</a>.<sup id="cite_ref-20" class="reference"><a href="#cite_note-20"><span>[</span>20<span>]</span></a></sup> The earliest uses of mathematics were in <a href="/wiki/Trade" title="Trade">trading</a>, <a href="/wiki/Land_measurement" title="Land measurement" class="mw-redirect">land measurement</a>, <a href="/wiki/Painting" title="Painting">painting</a> and <a href="/wiki/Weaving" title="Weaving">weaving</a> patterns and the recording of time.</p>
<p>In <a href="/wiki/Babylonian_mathematics" title="Babylonian mathematics">Babylonian mathematics</a> <a href="/wiki/Elementary_arithmetic" title="Elementary arithmetic">elementary arithmetic</a> (<a href="/wiki/Addition" title="Addition">addition</a>, <a href="/wiki/Subtraction" title="Subtraction">subtraction</a>, <a href="/wiki/Multiplication" title="Multiplication">multiplication</a> and <a href="/wiki/Division_(mathematics)" title="Division (mathematics)">division</a>) first appears in the archaeological record. <a href="/wiki/Numeracy" title="Numeracy">Numeracy</a> pre-dated <a href="/wiki/Writing" title="Writing">writing</a> and <a href="/wiki/Numeral_system" title="Numeral system">numeral systems</a> have been many and diverse, with the first known written numerals created by <a href="/wiki/Ancient_Egypt" title="Ancient Egypt">Egyptians</a> in <a href="/wiki/Middle_Kingdom_of_Egypt" title="Middle Kingdom of Egypt">Middle Kingdom</a> texts such as the <a href="/wiki/Rhind_Mathematical_Papyrus" title="Rhind Mathematical Papyrus">Rhind Mathematical Papyrus</a>.<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources. (August 2009)">citation needed</span></a></i>]</sup></p>
<p>Between 600 and 300&#160;BC the <a href="/wiki/Ancient_Greeks" title="Ancient Greeks" class="mw-redirect">Ancient Greeks</a> began a systematic study of mathematics in its own right with <a href="/wiki/Greek_mathematics" title="Greek mathematics">Greek mathematics</a>.<sup id="cite_ref-21" class="reference"><a href="#cite_note-21"><span>[</span>21<span>]</span></a></sup></p>
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<div class="thumbinner" style="width:222px;"><a href="/wiki/File:Persian_Khwarazmi.jpg" class="image"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b7/Persian_Khwarazmi.jpg/220px-Persian_Khwarazmi.jpg" width="220" height="280" class="thumbimage" srcset="//upload.wikimedia.org/wikipedia/commons/b/b7/Persian_Khwarazmi.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/b/b7/Persian_Khwarazmi.jpg 2x" data-file-width="299" data-file-height="380" /></a>
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Persian mathematician <a href="/wiki/Muhammad_ibn_Musa_al-Khwarizmi" title="Muhammad ibn Musa al-Khwarizmi">Al-Khwarizmi</a> ( c. 780 - c. 850 ), the inventor of the <a href="/wiki/Algebra" title="Algebra">Algebra</a>.</div>
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<p>During the <a href="/wiki/Islamic_Golden_Age" title="Islamic Golden Age">Golden Age of Islam</a>, especially during the 9th and 10th centuries, mathematics saw many important innovations building on Greek mathematics: most of them include the contributions from Persian mathematicians such as <a href="/wiki/Muhammad_ibn_Musa_al-Khwarizmi" title="Muhammad ibn Musa al-Khwarizmi">Al-Khwarismi</a>, <a href="/wiki/Omar_Khayyam" title="Omar Khayyam">Omar Khayyam</a> and <a href="/wiki/Sharaf_al-D%C4%ABn_al-%E1%B9%AC%C5%ABs%C4%AB" title="Sharaf al-Dīn al-Ṭūsī">Sharaf al-Dīn al-Ṭūsī</a>.</p>
<p>Mathematics has since been greatly extended, and there has been a fruitful interaction between mathematics and <a href="/wiki/Science" title="Science">science</a>, to the benefit of both. Mathematical discoveries continue to be made today. According to Mikhail B. Sevryuk, in the January 2006 issue of the <i><a href="/wiki/Bulletin_of_the_American_Mathematical_Society" title="Bulletin of the American Mathematical Society">Bulletin of the American Mathematical Society</a></i>, "The number of papers and books included in the <i><a href="/wiki/Mathematical_Reviews" title="Mathematical Reviews">Mathematical Reviews</a></i> database since 1940 (the first year of operation of MR) is now more than 1.9&#160;million, and more than 75&#160;thousand items are added to the database each year. The overwhelming majority of works in this ocean contain new mathematical <a href="/wiki/Theorem" title="Theorem">theorems</a> and their <a href="/wiki/Mathematical_proof" title="Mathematical proof">proofs</a>."<sup id="cite_ref-FOOTNOTESevryuk2006101.E2.80.93109_22-0" class="reference"><a href="#cite_note-FOOTNOTESevryuk2006101.E2.80.93109-22"><span>[</span>22<span>]</span></a></sup></p>
<h3><span class="mw-headline" id="Etymology">Etymology</span></h3>
<p>The word <i>mathematics</i> comes from the <a href="/wiki/Ancient_Greek" title="Ancient Greek">Greek</a> μάθημα (<i>máthēma</i>), which, in the ancient Greek language, means "that which is learnt",<sup id="cite_ref-23" class="reference"><a href="#cite_note-23"><span>[</span>23<span>]</span></a></sup> "what one gets to know", hence also "study" and "science", and in modern Greek just "lesson". The word <i>máthēma</i> is derived from μανθάνω (<i>manthano</i>), while the modern Greek equivalent is μαθαίνω (<i>mathaino</i>), both of which mean "to learn". In Greece, the word for "mathematics" came to have the narrower and more technical meaning "mathematical study" even in Classical times.<sup id="cite_ref-24" class="reference"><a href="#cite_note-24"><span>[</span>24<span>]</span></a></sup> Its adjective is <span lang="grc">μαθηματικός</span> (<i>mathēmatikós</i>), meaning "related to learning" or "studious", which likewise further came to mean "mathematical". In particular, <span lang="grc">μαθηματικὴ τέχνη</span> (<i>mathēmatikḗ tékhnē</i>), <a href="/wiki/Latin_language" title="Latin language" class="mw-redirect">Latin</a>: <span lang="la"><i>ars mathematica</i></span>, meant "the mathematical art".</p>
<p>In Latin, and in English until around 1700, the term <i>mathematics</i> more commonly meant "astrology" (or sometimes "astronomy") rather than "mathematics"; the meaning gradually changed to its present one from about 1500 to 1800. This has resulted in several mistranslations: a particularly notorious one is <a href="/wiki/Saint_Augustine" title="Saint Augustine" class="mw-redirect">Saint Augustine</a>'s warning that Christians should beware of <i>mathematici</i> meaning astrologers, which is sometimes mistranslated as a condemnation of mathematicians.<sup id="cite_ref-ohiostateuniversity_25-0" class="reference"><a href="#cite_note-ohiostateuniversity-25"><span>[</span>25<span>]</span></a></sup></p>
<p>The apparent plural form in English, like the French plural form <span lang="fr"><i>les mathématiques</i></span> (and the less commonly used singular derivative <span lang="fr"><i>la mathématique</i></span>), goes back to the Latin neuter plural <span lang="la"><i>mathematica</i></span> (<a href="/wiki/Cicero" title="Cicero">Cicero</a>), based on the Greek plural <span lang="el">τα μαθηματικά</span> (<i>ta mathēmatiká</i>), used by <a href="/wiki/Aristotle" title="Aristotle">Aristotle</a> (384–322&#160;BC), and meaning roughly "all things mathematical"; although it is plausible that English borrowed only the adjective <i>mathematic(al)</i> and formed the noun <i>mathematics</i> anew, after the pattern of <a href="/wiki/Physics" title="Physics">physics</a> and <a href="/wiki/Metaphysics" title="Metaphysics">metaphysics</a>, which were inherited from the Greek.<sup id="cite_ref-26" class="reference"><a href="#cite_note-26"><span>[</span>26<span>]</span></a></sup> In English, the noun <i>mathematics</i> takes singular verb forms. It is often shortened to <i>maths</i> or, in English-speaking North America, <i>math</i>.<sup id="cite_ref-maths_27-0" class="reference"><a href="#cite_note-maths-27"><span>[</span>27<span>]</span></a></sup></p>
<h2><span class="mw-headline" id="Definitions_of_mathematics">Definitions of mathematics</span></h2>
<div role="note" class="hatnote relarticle mainarticle">Main article: <a href="/wiki/Definitions_of_mathematics" title="Definitions of mathematics">Definitions of mathematics</a></div>
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<div class="thumbinner" style="width:222px;"><a href="/wiki/File:Fibonacci.jpg" class="image"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a2/Fibonacci.jpg/220px-Fibonacci.jpg" width="220" height="297" class="thumbimage" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a2/Fibonacci.jpg/330px-Fibonacci.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/a/a2/Fibonacci.jpg 2x" data-file-width="356" data-file-height="480" /></a>
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<a href="/wiki/Leonardo_Fibonacci" title="Leonardo Fibonacci" class="mw-redirect">Leonardo Fibonacci</a>, the <a href="/wiki/Italians" title="Italians">Italian</a> mathematician who established the Hindu–Arabic numeral system to the Western World</div>
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<p><a href="/wiki/Aristotle" title="Aristotle">Aristotle</a> defined mathematics as "the science of quantity", and this definition prevailed until the 18th century.<sup id="cite_ref-Franklin_28-0" class="reference"><a href="#cite_note-Franklin-28"><span>[</span>28<span>]</span></a></sup> Starting in the 19th&#160;century, when the study of mathematics increased in rigor and began to address abstract topics such as <a href="/wiki/Group_theory" title="Group theory">group theory</a> and <a href="/wiki/Projective_geometry" title="Projective geometry">projective geometry</a>, which have no clear-cut relation to quantity and measurement, mathematicians and philosophers began to propose a variety of new definitions.<sup id="cite_ref-Cajori_29-0" class="reference"><a href="#cite_note-Cajori-29"><span>[</span>29<span>]</span></a></sup> Some of these definitions emphasize the deductive character of much of mathematics, some emphasize its abstractness, some emphasize certain topics within mathematics. Today, no consensus on the definition of mathematics prevails, even among professionals.<sup id="cite_ref-Mura_7-1" class="reference"><a href="#cite_note-Mura-7"><span>[</span>7<span>]</span></a></sup> There is not even consensus on whether mathematics is an art or a science.<sup id="cite_ref-Runge_8-1" class="reference"><a href="#cite_note-Runge-8"><span>[</span>8<span>]</span></a></sup> A great many professional mathematicians take no interest in a definition of mathematics, or consider it undefinable.<sup id="cite_ref-Mura_7-2" class="reference"><a href="#cite_note-Mura-7"><span>[</span>7<span>]</span></a></sup> Some just say, "Mathematics is what mathematicians do."<sup id="cite_ref-Mura_7-3" class="reference"><a href="#cite_note-Mura-7"><span>[</span>7<span>]</span></a></sup></p>
<p>Three leading types of definition of mathematics are called <a href="/wiki/Logicist" title="Logicist" class="mw-redirect">logicist</a>, <a href="/wiki/Intuitionist" title="Intuitionist" class="mw-redirect">intuitionist</a>, and <a href="/wiki/Formalism_(mathematics)" title="Formalism (mathematics)">formalist</a>, each reflecting a different philosophical school of thought.<sup id="cite_ref-Snapper_30-0" class="reference"><a href="#cite_note-Snapper-30"><span>[</span>30<span>]</span></a></sup> All have severe problems, none has widespread acceptance, and no reconciliation seems possible.<sup id="cite_ref-Snapper_30-1" class="reference"><a href="#cite_note-Snapper-30"><span>[</span>30<span>]</span></a></sup></p>
<p>An early definition of mathematics in terms of logic was <a href="/wiki/Benjamin_Peirce" title="Benjamin Peirce">Benjamin Peirce</a>'s "the science that draws necessary conclusions" (1870).<sup id="cite_ref-Peirce_31-0" class="reference"><a href="#cite_note-Peirce-31"><span>[</span>31<span>]</span></a></sup> In the <i><a href="/wiki/Principia_Mathematica" title="Principia Mathematica">Principia Mathematica</a></i>, <a href="/wiki/Bertrand_Russell" title="Bertrand Russell">Bertrand Russell</a> and <a href="/wiki/Alfred_North_Whitehead" title="Alfred North Whitehead">Alfred North Whitehead</a> advanced the philosophical program known as <a href="/wiki/Logicism" title="Logicism">logicism</a>, and attempted to prove that all mathematical concepts, statements, and principles can be defined and proven entirely in terms of <a href="/wiki/Symbolic_logic" title="Symbolic logic" class="mw-redirect">symbolic logic</a>. A logicist definition of mathematics is Russell's "All Mathematics is Symbolic Logic" (1903).<sup id="cite_ref-Russell_32-0" class="reference"><a href="#cite_note-Russell-32"><span>[</span>32<span>]</span></a></sup></p>
<p><a href="/wiki/Intuitionist" title="Intuitionist" class="mw-redirect">Intuitionist</a> definitions, developing from the philosophy of mathematician <a href="/wiki/L.E.J._Brouwer" title="L.E.J. Brouwer" class="mw-redirect">L.E.J. Brouwer</a>, identify mathematics with certain mental phenomena. An example of an intuitionist definition is "Mathematics is the mental activity which consists in carrying out constructs one after the other."<sup id="cite_ref-Snapper_30-2" class="reference"><a href="#cite_note-Snapper-30"><span>[</span>30<span>]</span></a></sup> A peculiarity of intuitionism is that it rejects some mathematical ideas considered valid according to other definitions. In particular, while other philosophies of mathematics allow objects that can be proven to exist even though they cannot be constructed, intuitionism allows only mathematical objects that one can actually construct.</p>
<p><a href="/wiki/Formalism_(mathematics)" title="Formalism (mathematics)">Formalist</a> definitions identify mathematics with its symbols and the rules for operating on them. <a href="/wiki/Haskell_Curry" title="Haskell Curry">Haskell Curry</a> defined mathematics simply as "the science of formal systems".<sup id="cite_ref-Curry_33-0" class="reference"><a href="#cite_note-Curry-33"><span>[</span>33<span>]</span></a></sup> A <a href="/wiki/Formal_system" title="Formal system">formal system</a> is a set of symbols, or <i>tokens</i>, and some <i>rules</i> telling how the tokens may be combined into <i>formulas</i>. In formal systems, the word <i>axiom</i> has a special meaning, different from the ordinary meaning of "a self-evident truth". In formal systems, an axiom is a combination of tokens that is included in a given formal system without needing to be derived using the rules of the system.</p>
<h3><span class="mw-headline" id="Mathematics_as_science">Mathematics as science</span></h3>
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<div class="thumbinner" style="width:172px;"><a href="/wiki/File:Carl_Friedrich_Gauss.jpg" class="image"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/9/9b/Carl_Friedrich_Gauss.jpg/170px-Carl_Friedrich_Gauss.jpg" width="170" height="218" class="thumbimage" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/9b/Carl_Friedrich_Gauss.jpg/255px-Carl_Friedrich_Gauss.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/9b/Carl_Friedrich_Gauss.jpg/340px-Carl_Friedrich_Gauss.jpg 2x" data-file-width="576" data-file-height="738" /></a>
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<a href="/wiki/Carl_Friedrich_Gauss" title="Carl Friedrich Gauss">Carl Friedrich Gauss</a>, known as the prince of mathematicians</div>
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<p><a href="/wiki/Carl_Friedrich_Gauss" title="Carl Friedrich Gauss">Gauss</a> referred to mathematics as "the Queen of the Sciences".<sup id="cite_ref-Waltershausen_13-1" class="reference"><a href="#cite_note-Waltershausen-13"><span>[</span>13<span>]</span></a></sup> In the original Latin <i>Regina Scientiarum</i>, as well as in <a href="/wiki/German_language" title="German language">German</a> <i>Königin der Wissenschaften</i>, the word corresponding to <i>science</i> means a "field of knowledge", and this was the original meaning of "science" in English, also; mathematics is in this sense a field of knowledge. The specialization restricting the meaning of "science" to <i><a href="/wiki/Natural_science" title="Natural science">natural science</a></i> follows the rise of <a href="/wiki/Baconian_method" title="Baconian method">Baconian science</a>, which contrasted "natural science" to <a href="/wiki/Scholasticism" title="Scholasticism">scholasticism</a>, the <a href="/wiki/Organon" title="Organon">Aristotelean method</a> of inquiring from <a href="/wiki/First_principles" title="First principles" class="mw-redirect">first principles</a>. The role of empirical experimentation and observation is negligible in mathematics, compared to natural sciences such as <a href="/wiki/Biology" title="Biology">biology</a>, <a href="/wiki/Chemistry" title="Chemistry">chemistry</a>, or <a href="/wiki/Physics" title="Physics">physics</a>. <a href="/wiki/Albert_Einstein" title="Albert Einstein">Albert Einstein</a> 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_16-1" class="reference"><a href="#cite_note-certain-16"><span>[</span>16<span>]</span></a></sup> More recently, <a href="/wiki/Marcus_du_Sautoy" title="Marcus du Sautoy">Marcus du Sautoy</a> has called mathematics "the Queen of Science&#160;... the main driving force behind scientific discovery".<sup id="cite_ref-34" class="reference"><a href="#cite_note-34"><span>[</span>34<span>]</span></a></sup></p>
<p>Many philosophers believe that mathematics is not experimentally <a href="/wiki/Falsifiability" title="Falsifiability">falsifiable</a>, and thus not a science according to the definition of <a href="/wiki/Karl_Popper" title="Karl Popper">Karl Popper</a>.<sup id="cite_ref-35" class="reference"><a href="#cite_note-35"><span>[</span>35<span>]</span></a></sup> However, in the 1930s <a href="/wiki/G%C3%B6del%27s_incompleteness_theorems" title="Gödel's incompleteness theorems">Gödel's incompleteness theorems</a> convinced many mathematicians<sup class="noprint Inline-Template" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Manual_of_Style/Words_to_watch#Unsupported_attributions" title="Wikipedia:Manual of Style/Words to watch"><span title="The material near this tag possibly uses too-vague attribution or weasel words. (January 2011)">who?</span></a></i>]</sup> that mathematics cannot be reduced to logic alone, and Karl Popper concluded that "most mathematical theories are, like those of <a href="/wiki/Physics" title="Physics">physics</a> and <a href="/wiki/Biology" title="Biology">biology</a>, <a href="/wiki/Hypothesis" title="Hypothesis">hypothetico</a>-<a href="/wiki/Deductive" title="Deductive" class="mw-redirect">deductive</a>: pure mathematics therefore turns out to be much closer to the natural sciences whose hypotheses are conjectures, than it seemed even recently."<sup id="cite_ref-36" class="reference"><a href="#cite_note-36"><span>[</span>36<span>]</span></a></sup> Other thinkers, notably <a href="/wiki/Imre_Lakatos" title="Imre Lakatos">Imre Lakatos</a>, have applied a version of <a href="/wiki/Falsificationism" title="Falsificationism" class="mw-redirect">falsificationism</a> to mathematics itself.</p>
<p>An alternative view is that certain scientific fields (such as <a href="/wiki/Theoretical_physics" title="Theoretical physics">theoretical physics</a>) are mathematics with axioms that are intended to correspond to reality. The theoretical physicist <a href="/wiki/J.M._Ziman" title="J.M. Ziman" class="mw-redirect">J.M. Ziman</a> proposed that science is <i>public knowledge</i>, and thus includes mathematics.<sup id="cite_ref-37" class="reference"><a href="#cite_note-37"><span>[</span>37<span>]</span></a></sup> Mathematics shares much in common with many fields in the physical sciences, notably the <a href="/wiki/Deductive_reasoning" title="Deductive reasoning">exploration of the logical consequences</a> of assumptions. <a href="/wiki/Intuition_(knowledge)" title="Intuition (knowledge)" class="mw-redirect">Intuition</a> and <a href="/wiki/Experiment" title="Experiment">experimentation</a> also play a role in the formulation of <a href="/wiki/Conjecture" title="Conjecture">conjectures</a> in both mathematics and the (other) sciences. <a href="/wiki/Experimental_mathematics" title="Experimental mathematics">Experimental mathematics</a> continues to grow in importance within mathematics, and computation and simulation are playing an increasing role in both the sciences and mathematics.</p>
<p>The opinions of mathematicians on this matter are varied. Many mathematicians<sup class="noprint Inline-Template" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Manual_of_Style/Words_to_watch#Unsupported_attributions" title="Wikipedia:Manual of Style/Words to watch"><span title="The material near this tag possibly uses too-vague attribution or weasel words. (August 2009)">who?</span></a></i>]</sup> feel that to call their area a science is to downplay the importance of its aesthetic side, and its history in the traditional seven <a href="/wiki/Liberal_arts" title="Liberal arts" class="mw-redirect">liberal arts</a>; others<sup class="noprint Inline-Template" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Manual_of_Style/Words_to_watch#Unsupported_attributions" title="Wikipedia:Manual of Style/Words to watch"><span title="The material near this tag possibly uses too-vague attribution or weasel words. (August 2009)">who?</span></a></i>]</sup> feel that to ignore its connection to the sciences is to turn a blind eye to the fact that the interface between mathematics and its applications in science and <a href="/wiki/Engineering" title="Engineering">engineering</a> has driven much development in mathematics. One way this difference of viewpoint plays out is in the philosophical debate as to whether mathematics is <i>created</i> (as in art) or <i>discovered</i> (as in science). It is common to see <a href="/wiki/University" title="University">universities</a> divided into sections that include a division of <i>Science and Mathematics</i>, indicating that the fields are seen as being allied but that they do not coincide. In practice, mathematicians are typically grouped with scientists at the gross level but separated at finer levels. This is one of many issues considered in the <a href="/wiki/Philosophy_of_mathematics" title="Philosophy of mathematics">philosophy of mathematics</a>.<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources. (August 2009)">citation needed</span></a></i>]</sup></p>
<h2><span class="mw-headline" id="Inspiration.2C_pure_and_applied_mathematics.2C_and_aesthetics">Inspiration, pure and applied mathematics, and aesthetics</span></h2>
<div role="note" class="hatnote relarticle mainarticle">Main article: <a href="/wiki/Mathematical_beauty" title="Mathematical beauty">Mathematical beauty</a></div>
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<div class="thumbimage"><a href="/wiki/File:Gottfried_Wilhelm_von_Leibniz.jpg" class="image"><img alt="Gottfried Wilhelm von Leibniz" src="//upload.wikimedia.org/wikipedia/commons/thumb/6/6a/Gottfried_Wilhelm_von_Leibniz.jpg/200px-Gottfried_Wilhelm_von_Leibniz.jpg" width="200" height="253" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/6a/Gottfried_Wilhelm_von_Leibniz.jpg/300px-Gottfried_Wilhelm_von_Leibniz.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/6/6a/Gottfried_Wilhelm_von_Leibniz.jpg 2x" data-file-width="316" data-file-height="400" /></a></div>
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<div class="thumbcaption" style="clear:left;text-align:left;background-color:transparent"><a href="/wiki/Isaac_Newton" title="Isaac Newton">Isaac Newton</a> (left) and <a href="/wiki/Gottfried_Wilhelm_Leibniz" title="Gottfried Wilhelm Leibniz">Gottfried Wilhelm Leibniz</a> (right), developers of infinitesimal calculus</div>
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<p>Mathematics arises from many different kinds of problems. At first these were found in <a href="/wiki/Commerce" title="Commerce">commerce</a>, <a href="/wiki/Land_measurement" title="Land measurement" class="mw-redirect">land measurement</a>, <a href="/wiki/Architecture" title="Architecture">architecture</a> and later <a href="/wiki/Astronomy" title="Astronomy">astronomy</a>; today, all sciences suggest problems studied by mathematicians, and many problems arise within mathematics itself. For example, the <a href="/wiki/Physicist" title="Physicist">physicist</a> <a href="/wiki/Richard_Feynman" title="Richard Feynman">Richard Feynman</a> invented the <a href="/wiki/Path_integral_formulation" title="Path integral formulation">path integral formulation</a> of <a href="/wiki/Quantum_mechanics" title="Quantum mechanics">quantum mechanics</a> using a combination of mathematical reasoning and physical insight, and today's <a href="/wiki/String_theory" title="String theory">string theory</a>, a still-developing scientific theory which attempts to unify the four <a href="/wiki/Fundamental_interaction" title="Fundamental interaction">fundamental forces of nature</a>, continues to inspire new mathematics.<sup id="cite_ref-38" class="reference"><a href="#cite_note-38"><span>[</span>38<span>]</span></a></sup></p>
<p>Some mathematics is relevant only in the area that inspired it, and is applied to solve further problems in that area. But often mathematics inspired by one area proves useful in many areas, and joins the general stock of mathematical concepts. A distinction is often made between <a href="/wiki/Pure_mathematics" title="Pure mathematics">pure mathematics</a> and <a href="/wiki/Applied_mathematics" title="Applied mathematics">applied mathematics</a>. However pure mathematics topics often turn out to have applications, e.g. <a href="/wiki/Number_theory" title="Number theory">number theory</a> in <a href="/wiki/Cryptography" title="Cryptography">cryptography</a>. This remarkable fact, that even the "purest" mathematics often turns out to have practical applications, is what <a href="/wiki/Eugene_Wigner" title="Eugene Wigner">Eugene Wigner</a> has called "<a href="/wiki/The_Unreasonable_Effectiveness_of_Mathematics_in_the_Natural_Sciences" title="The Unreasonable Effectiveness of Mathematics in the Natural Sciences">the unreasonable effectiveness of mathematics</a>".<sup id="cite_ref-39" class="reference"><a href="#cite_note-39"><span>[</span>39<span>]</span></a></sup> As in most areas of study, the explosion of knowledge in the scientific age has led to specialization: there are now hundreds of specialized areas in mathematics and the latest <a href="/wiki/Mathematics_Subject_Classification" title="Mathematics Subject Classification">Mathematics Subject Classification</a> runs to 46&#160;pages.<sup id="cite_ref-40" class="reference"><a href="#cite_note-40"><span>[</span>40<span>]</span></a></sup> Several areas of applied mathematics have merged with related traditions outside of mathematics and become disciplines in their own right, including <a href="/wiki/Statistics" title="Statistics">statistics</a>, <a href="/wiki/Operations_research" title="Operations research">operations research</a>, and <a href="/wiki/Computer_science" title="Computer science">computer science</a>.</p>
<p>For those who are mathematically inclined, there is often a definite aesthetic aspect to much of mathematics. Many mathematicians talk about the <i>elegance</i> of mathematics, its intrinsic <a href="/wiki/Aesthetics" title="Aesthetics">aesthetics</a> and inner <a href="/wiki/Beauty" title="Beauty">beauty</a>. <a href="/wiki/Simplicity" title="Simplicity">Simplicity</a> and generality are valued. There is beauty in a simple and elegant <a href="/wiki/Proof_(mathematics)" title="Proof (mathematics)" class="mw-redirect">proof</a>, such as <a href="/wiki/Euclid" title="Euclid">Euclid</a>'s proof that there are infinitely many <a href="/wiki/Prime_number" title="Prime number">prime numbers</a>, and in an elegant <a href="/wiki/Numerical_method" title="Numerical method" class="mw-redirect">numerical method</a> that speeds calculation, such as the <a href="/wiki/Fast_Fourier_transform" title="Fast Fourier transform">fast Fourier transform</a>. <a href="/wiki/G.H._Hardy" title="G.H. Hardy" class="mw-redirect">G.H. Hardy</a> in <i><a href="/wiki/A_Mathematician%27s_Apology" title="A Mathematician's Apology">A Mathematician's Apology</a></i> expressed the belief that these aesthetic considerations are, in themselves, sufficient to justify the study of pure mathematics. He identified criteria such as significance, unexpectedness, inevitability, and economy as factors that contribute to a mathematical aesthetic.<sup id="cite_ref-41" class="reference"><a href="#cite_note-41"><span>[</span>41<span>]</span></a></sup> Mathematicians often strive to find proofs that are particularly elegant, proofs from "The Book" of God according to <a href="/wiki/Paul_Erd%C5%91s" title="Paul Erdős">Paul Erdős</a>.<sup id="cite_ref-42" class="reference"><a href="#cite_note-42"><span>[</span>42<span>]</span></a></sup><sup id="cite_ref-43" class="reference"><a href="#cite_note-43"><span>[</span>43<span>]</span></a></sup> The popularity of <a href="/wiki/Recreational_mathematics" title="Recreational mathematics">recreational mathematics</a> is another sign of the pleasure many find in solving mathematical questions.</p>
<h2><span class="mw-headline" id="Notation.2C_language.2C_and_rigor">Notation, language, and rigor</span></h2>
<div role="note" class="hatnote relarticle mainarticle">Main article: <a href="/wiki/Mathematical_notation" title="Mathematical notation">Mathematical notation</a></div>
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<div class="thumbinner" style="width:172px;"><a href="/wiki/File:Leonhard_Euler_2.jpg" class="image"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/6/60/Leonhard_Euler_2.jpg/170px-Leonhard_Euler_2.jpg" width="170" height="212" class="thumbimage" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/60/Leonhard_Euler_2.jpg/255px-Leonhard_Euler_2.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/60/Leonhard_Euler_2.jpg/340px-Leonhard_Euler_2.jpg 2x" data-file-width="614" data-file-height="767" /></a>
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<a href="/wiki/Leonhard_Euler" title="Leonhard Euler">Leonhard Euler</a>, who created and popularized much of the mathematical notation used today</div>
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<p>Most of the mathematical notation in use today was not invented until the 16th century.<sup id="cite_ref-44" class="reference"><a href="#cite_note-44"><span>[</span>44<span>]</span></a></sup> Before that, mathematics was written out in words, limiting mathematical discovery.<sup id="cite_ref-45" class="reference"><a href="#cite_note-45"><span>[</span>45<span>]</span></a></sup> <a href="/wiki/Leonhard_Euler" title="Leonhard Euler">Euler</a> (1707–1783) was responsible for many of the notations in use today. Modern notation makes mathematics much easier for the professional, but beginners often find it daunting. It is compressed: a few symbols contain a great deal of information. Like <a href="/wiki/Musical_notation" title="Musical notation">musical notation</a>, modern mathematical notation has a strict syntax and encodes information that would be difficult to write in any other way.</p>
<p>Mathematical <a href="/wiki/Language" title="Language">language</a> can be difficult to understand for beginners. Common words such as <i>or</i> and <i>only</i> have more precise meanings than in everyday speech. Moreover, words such as <i><a href="/wiki/Open_set" title="Open set">open</a></i> and <i><a href="/wiki/Field_(mathematics)" title="Field (mathematics)">field</a></i> have specialized mathematical meanings. Technical terms such as <i><a href="/wiki/Homeomorphism" title="Homeomorphism">homeomorphism</a></i> and <i><a href="/wiki/Integral" title="Integral">integrable</a></i> have precise meanings in mathematics. Additionally, shorthand phrases such as <i>iff</i> for "<a href="/wiki/If_and_only_if" title="If and only if">if and only if</a>" belong to <a href="/wiki/Mathematical_jargon" title="Mathematical jargon" class="mw-redirect">mathematical jargon</a>. There is a reason for special notation and technical vocabulary: mathematics requires more precision than everyday speech. Mathematicians refer to this precision of language and logic as "rigor".</p>
<p><a href="/wiki/Mathematical_proof" title="Mathematical proof">Mathematical proof</a> is fundamentally a matter of <a href="/wiki/Rigor" title="Rigor" class="mw-redirect">rigor</a>. Mathematicians want their theorems to follow from axioms by means of systematic reasoning. This is to avoid mistaken "<a href="/wiki/Theorem" title="Theorem">theorems</a>", based on fallible intuitions, of which many instances have occurred in the history of the subject.<sup id="cite_ref-46" class="reference"><a href="#cite_note-46"><span>[</span>46<span>]</span></a></sup> The level of rigor expected in mathematics has varied over time: the Greeks expected detailed arguments, but at the time of <a href="/wiki/Isaac_Newton" title="Isaac Newton">Isaac Newton</a> the methods employed were less rigorous. Problems inherent in the definitions used by Newton would lead to a resurgence of careful analysis and formal proof in the 19th&#160;century. Misunderstanding the rigor is a cause for some of the common misconceptions of mathematics. Today, mathematicians continue to argue among themselves about <a href="/wiki/Computer-assisted_proof" title="Computer-assisted proof">computer-assisted proofs</a>. Since large computations are hard to verify, such proofs may not be sufficiently rigorous.<sup id="cite_ref-47" class="reference"><a href="#cite_note-47"><span>[</span>47<span>]</span></a></sup></p>
<p><a href="/wiki/Axiom" title="Axiom">Axioms</a> in traditional thought were "self-evident truths", but that conception is problematic.<sup id="cite_ref-48" class="reference"><a href="#cite_note-48"><span>[</span>48<span>]</span></a></sup> At a formal level, an axiom is just a string of symbols, which has an intrinsic meaning only in the context of all derivable formulas of an <a href="/wiki/Axiomatic_system" title="Axiomatic system">axiomatic system</a>. It was the goal of <a href="/wiki/Hilbert%27s_program" title="Hilbert's program">Hilbert's program</a> to put all of mathematics on a firm axiomatic basis, but according to <a href="/wiki/G%C3%B6del%27s_incompleteness_theorem" title="Gödel's incompleteness theorem" class="mw-redirect">Gödel's incompleteness theorem</a> every (sufficiently powerful) axiomatic system has <a href="/wiki/Independence_(mathematical_logic)" title="Independence (mathematical logic)">undecidable</a> formulas; and so a final <a href="/wiki/Axiomatization" title="Axiomatization" class="mw-redirect">axiomatization</a> of mathematics is impossible. Nonetheless mathematics is often imagined to be (as far as its formal content) nothing but <a href="/wiki/Set_theory" title="Set theory">set theory</a> in some axiomatization, in the sense that every mathematical statement or proof could be cast into formulas within set theory.<sup id="cite_ref-49" class="reference"><a href="#cite_note-49"><span>[</span>49<span>]</span></a></sup></p>
<h2><span class="mw-headline" id="Fields_of_mathematics">Fields of mathematics</span></h2>
<div role="note" class="hatnote">See also: <a href="/wiki/Areas_of_mathematics" title="Areas of mathematics">Areas of mathematics</a> and <a href="/wiki/Glossary_of_areas_of_mathematics" title="Glossary of areas of mathematics">Glossary of areas of mathematics</a></div>
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<div class="thumbinner" style="width:222px;"><a href="/wiki/File:Abacus_6.png" class="image"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/a/af/Abacus_6.png/220px-Abacus_6.png" width="220" height="129" class="thumbimage" srcset="//upload.wikimedia.org/wikipedia/commons/a/af/Abacus_6.png 1.5x, //upload.wikimedia.org/wikipedia/commons/a/af/Abacus_6.png 2x" data-file-width="247" data-file-height="145" /></a>
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<div class="magnify"><a href="/wiki/File:Abacus_6.png" class="internal" title="Enlarge"></a></div>
An <a href="/wiki/Abacus" title="Abacus">abacus</a>, a simple calculating tool used since ancient times</div>
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<p>Mathematics can, broadly speaking, be subdivided into the study of quantity, structure, space, and change (i.e. <a href="/wiki/Arithmetic" title="Arithmetic">arithmetic</a>, <a href="/wiki/Algebra" title="Algebra">algebra</a>, <a href="/wiki/Geometry" title="Geometry">geometry</a>, and <a href="/wiki/Mathematical_analysis" title="Mathematical analysis">analysis</a>). In addition to these main concerns, there are also subdivisions dedicated to exploring links from the heart of mathematics to other fields: to <a href="/wiki/Mathematical_logic" title="Mathematical logic">logic</a>, to <a href="/wiki/Set_theory" title="Set theory">set theory</a> (<a href="/wiki/Foundations_of_mathematics" title="Foundations of mathematics">foundations</a>), to the empirical mathematics of the various sciences (<a href="/wiki/Applied_mathematics" title="Applied mathematics">applied mathematics</a>), and more recently to the rigorous study of <a href="/wiki/Uncertainty" title="Uncertainty">uncertainty</a>. While some areas might seem unrelated, the <a href="/wiki/Langlands_program" title="Langlands program">Langlands program</a> has found connections between areas previously thought unconnected, such as <a href="/wiki/Galois_groups" title="Galois groups" class="mw-redirect">Galois groups</a>, <a href="/wiki/Riemann_surface" title="Riemann surface">Riemann surfaces</a> and <a href="/wiki/Number_theory" title="Number theory">number theory</a>.</p>
<h3><span class="mw-headline" id="Foundations_and_philosophy">Foundations and philosophy</span></h3>
<p>In order to clarify the <a href="/wiki/Foundations_of_mathematics" title="Foundations of mathematics">foundations of mathematics</a>, the fields of <a href="/wiki/Mathematical_logic" title="Mathematical logic">mathematical logic</a> and <a href="/wiki/Set_theory" title="Set theory">set theory</a> were developed. Mathematical logic includes the mathematical study of <a href="/wiki/Logic" title="Logic">logic</a> and the applications of formal logic to other areas of mathematics; set theory is the branch of mathematics that studies <a href="/wiki/Set_(mathematics)" title="Set (mathematics)">sets</a> or collections of objects. <a href="/wiki/Category_theory" title="Category theory">Category theory</a>, which deals in an abstract way with <a href="/wiki/Mathematical_structure" title="Mathematical structure">mathematical structures</a> and relationships between them, is still in development. The phrase "crisis of foundations" describes the search for a rigorous foundation for mathematics that took place from approximately 1900 to 1930.<sup id="cite_ref-50" class="reference"><a href="#cite_note-50"><span>[</span>50<span>]</span></a></sup> Some disagreement about the foundations of mathematics continues to the present day. The crisis of foundations was stimulated by a number of controversies at the time, including the <a href="/wiki/Controversy_over_Cantor%27s_theory" title="Controversy over Cantor's theory">controversy over Cantor's set theory</a> and the <a href="/wiki/Brouwer%E2%80%93Hilbert_controversy" title="Brouwer–Hilbert controversy">Brouwer–Hilbert controversy</a>.</p>
<p>Mathematical logic is concerned with setting mathematics within a rigorous <a href="/wiki/Axiom" title="Axiom">axiomatic</a> framework, and studying the implications of such a framework. As such, it is home to <a href="/wiki/G%C3%B6del%27s_incompleteness_theorems" title="Gödel's incompleteness theorems">Gödel's incompleteness theorems</a> which (informally) imply that any effective <a href="/wiki/Formal_system" title="Formal system">formal system</a> that contains basic arithmetic, if <i>sound</i> (meaning that all theorems that can be proven are true), is necessarily <i>incomplete</i> (meaning that there are true theorems which cannot be proved <i>in that system</i>). Whatever finite collection of number-theoretical axioms is taken as a foundation, Gödel showed how to construct a formal statement that is a true number-theoretical fact, but which does not follow from those axioms. Therefore, no formal system is a complete axiomatization of full number theory. Modern logic is divided into <a href="/wiki/Recursion_theory" title="Recursion theory" class="mw-redirect">recursion theory</a>, <a href="/wiki/Model_theory" title="Model theory">model theory</a>, and <a href="/wiki/Proof_theory" title="Proof theory">proof theory</a>, and is closely linked to <a href="/wiki/Theoretical_computer_science" title="Theoretical computer science">theoretical computer science</a>,<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources. (March 2011)">citation needed</span></a></i>]</sup> as well as to <a href="/wiki/Category_theory" title="Category theory">category theory</a>.</p>
<p><a href="/wiki/Theoretical_computer_science" title="Theoretical computer science">Theoretical computer science</a> includes <a href="/wiki/Computability_theory_(computation)" title="Computability theory (computation)" class="mw-redirect">computability theory</a>, <a href="/wiki/Computational_complexity_theory" title="Computational complexity theory">computational complexity theory</a>, and <a href="/wiki/Information_theory" title="Information theory">information theory</a>. Computability theory examines the limitations of various theoretical models of the computer, including the most well-known model&#160;– the <a href="/wiki/Turing_machine" title="Turing machine">Turing machine</a>. Complexity theory is the study of tractability by computer; some problems, although theoretically solvable by computer, are so expensive in terms of time or space that solving them is likely to remain practically unfeasible, even with the rapid advancement of computer hardware. A famous problem is the "<a href="/wiki/P_%3D_NP_problem" title="P = NP problem" class="mw-redirect"><span class="nowrap"><b>P</b> = <b>NP</b>?</span></a>" problem, one of the <a href="/wiki/Millennium_Prize_Problems" title="Millennium Prize Problems">Millennium Prize Problems</a>.<sup id="cite_ref-51" class="reference"><a href="#cite_note-51"><span>[</span>51<span>]</span></a></sup> Finally, information theory is concerned with the amount of data that can be stored on a given medium, and hence deals with concepts such as <a href="/wiki/Data_compression" title="Data compression">compression</a> and <a href="/wiki/Entropy_(information_theory)" title="Entropy (information theory)">entropy</a>.</p>
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<dd>
<table style="border:1px solid #ddd; text-align:center; margin:auto" cellspacing="15">
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<td><img class="mwe-math-fallback-image-inline tex" alt="p \Rightarrow q \," src="//upload.wikimedia.org/math/a/6/4/a644166cefb23015623cb1670becf7b2.png" /></td>
<td><a href="/wiki/File:Venn_A_intersect_B.svg" class="image"><img alt="Venn A intersect B.svg" src="//upload.wikimedia.org/wikipedia/commons/thumb/6/6d/Venn_A_intersect_B.svg/128px-Venn_A_intersect_B.svg.png" width="128" height="91" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/6d/Venn_A_intersect_B.svg/192px-Venn_A_intersect_B.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/6d/Venn_A_intersect_B.svg/256px-Venn_A_intersect_B.svg.png 2x" data-file-width="350" data-file-height="250" /></a></td>
<td><a href="/wiki/File:Commutative_diagram_for_morphism.svg" class="image"><img alt="Commutative diagram for morphism.svg" src="//upload.wikimedia.org/wikipedia/commons/thumb/e/ef/Commutative_diagram_for_morphism.svg/96px-Commutative_diagram_for_morphism.svg.png" width="96" height="96" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/ef/Commutative_diagram_for_morphism.svg/144px-Commutative_diagram_for_morphism.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/ef/Commutative_diagram_for_morphism.svg/192px-Commutative_diagram_for_morphism.svg.png 2x" data-file-width="100" data-file-height="100" /></a></td>
<td><a href="/wiki/File:DFAexample.svg" class="image"><img alt="DFAexample.svg" src="//upload.wikimedia.org/wikipedia/commons/thumb/9/9d/DFAexample.svg/96px-DFAexample.svg.png" width="96" height="57" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/9d/DFAexample.svg/144px-DFAexample.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/9d/DFAexample.svg/192px-DFAexample.svg.png 2x" data-file-width="500" data-file-height="299" /></a></td>
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<td><a href="/wiki/Mathematical_logic" title="Mathematical logic">Mathematical logic</a></td>
<td><a href="/wiki/Set_theory" title="Set theory">Set theory</a></td>
<td><a href="/wiki/Category_theory" title="Category theory">Category theory</a></td>
<td><a href="/wiki/Theory_of_computation" title="Theory of computation">Theory of computation</a></td>
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<h3><span class="mw-headline" id="Pure_mathematics">Pure mathematics</span></h3>
<h4><span class="mw-headline" id="Quantity">Quantity</span></h4>
<p>The study of quantity starts with <a href="/wiki/Number" title="Number">numbers</a>, first the familiar <a href="/wiki/Natural_number" title="Natural number">natural numbers</a> and <a href="/wiki/Integer" title="Integer">integers</a> ("whole numbers") and arithmetical operations on them, which are characterized in <a href="/wiki/Arithmetic" title="Arithmetic">arithmetic</a>. The deeper properties of integers are studied in <a href="/wiki/Number_theory" title="Number theory">number theory</a>, from which come such popular results as <a href="/wiki/Fermat%27s_Last_Theorem" title="Fermat's Last Theorem">Fermat's Last Theorem</a>. The <a href="/wiki/Twin_prime" title="Twin prime">twin prime</a> conjecture and <a href="/wiki/Goldbach%27s_conjecture" title="Goldbach's conjecture">Goldbach's conjecture</a> are two unsolved problems in number theory.</p>
<p>As the number system is further developed, the integers are recognized as a <a href="/wiki/Subset" title="Subset">subset</a> of the <a href="/wiki/Rational_number" title="Rational number">rational numbers</a> ("<a href="/wiki/Fraction_(mathematics)" title="Fraction (mathematics)">fractions</a>"). These, in turn, are contained within the <a href="/wiki/Real_number" title="Real number">real numbers</a>, which are used to represent <a href="/wiki/Continuous_function" title="Continuous function">continuous</a> quantities. Real numbers are generalized to <a href="/wiki/Complex_number" title="Complex number">complex numbers</a>. These are the first steps of a hierarchy of numbers that goes on to include <a href="/wiki/Quaternion" title="Quaternion">quaternions</a> and <a href="/wiki/Octonion" title="Octonion">octonions</a>. Consideration of the natural numbers also leads to the <a href="/wiki/Transfinite_number" title="Transfinite number">transfinite numbers</a>, which formalize the concept of "<a href="/wiki/Infinity" title="Infinity">infinity</a>". Another area of study is size, which leads to the <a href="/wiki/Cardinal_number" title="Cardinal number">cardinal numbers</a> and then to another conception of infinity: the <a href="/wiki/Aleph_number" title="Aleph number">aleph numbers</a>, which allow meaningful comparison of the size of infinitely large sets.</p>
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<table style="border:1px solid #ddd; text-align:center; margin:auto" cellspacing="20">
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<td><img class="mwe-math-fallback-image-inline tex" alt="1, 2, 3,\ldots\!" src="//upload.wikimedia.org/math/b/c/f/bcfbd40d90bda583181a7100023303f4.png" /></td>
<td><img class="mwe-math-fallback-image-inline tex" alt="\ldots,-2, -1, 0, 1, 2\,\ldots\!" src="//upload.wikimedia.org/math/d/8/1/d812341015aca2fc966e102be586bc58.png" /></td>
<td><img class="mwe-math-fallback-image-inline tex" alt=" -2, \frac{2}{3}, 1.21\,\!" src="//upload.wikimedia.org/math/9/b/6/9b6892bffb24f4e8eb088036e5f7efff.png" /></td>
<td><img class="mwe-math-fallback-image-inline tex" alt="-e, \sqrt{2}, 3, \pi\,\!" src="//upload.wikimedia.org/math/9/d/6/9d6f418bda5bf70193627a3ee78805f4.png" /></td>
<td><img class="mwe-math-fallback-image-inline tex" alt="2, i, -2+3i, 2e^{i\frac{4\pi}{3}}\,\!" src="//upload.wikimedia.org/math/7/5/9/759cf14c729639e5c1152dad2c4843e7.png" /></td>
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<td><a href="/wiki/Natural_number" title="Natural number">Natural numbers</a></td>
<td><a href="/wiki/Integer" title="Integer">Integers</a></td>
<td><a href="/wiki/Rational_number" title="Rational number">Rational numbers</a></td>
<td><a href="/wiki/Real_number" title="Real number">Real numbers</a></td>
<td><a href="/wiki/Complex_number" title="Complex number">Complex numbers</a></td>
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<h4><span class="mw-headline" id="Structure">Structure</span></h4>
<p>Many mathematical objects, such as <a href="/wiki/Set_(mathematics)" title="Set (mathematics)">sets</a> of numbers and <a href="/wiki/Function_(mathematics)" title="Function (mathematics)">functions</a>, exhibit internal structure as a consequence of <a href="/wiki/Operation_(mathematics)" title="Operation (mathematics)">operations</a> or <a href="/wiki/Relation_(mathematics)" title="Relation (mathematics)" class="mw-redirect">relations</a> that are defined on the set. Mathematics then studies properties of those sets that can be expressed in terms of that structure; for instance <a href="/wiki/Number_theory" title="Number theory">number theory</a> studies properties of the set of <a href="/wiki/Integer" title="Integer">integers</a> that can be expressed in terms of <a href="/wiki/Arithmetic" title="Arithmetic">arithmetic</a> operations. Moreover, it frequently happens that different such structured sets (or <a href="/wiki/Mathematical_structure" title="Mathematical structure">structures</a>) exhibit similar properties, which makes it possible, by a further step of <a href="/wiki/Abstraction" title="Abstraction">abstraction</a>, to state <a href="/wiki/Axiom" title="Axiom">axioms</a> for a class of structures, and then study at once the whole class of structures satisfying these axioms. Thus one can study <a href="/wiki/Group_(mathematics)" title="Group (mathematics)">groups</a>, <a href="/wiki/Ring_(mathematics)" title="Ring (mathematics)">rings</a>, <a href="/wiki/Field_(mathematics)" title="Field (mathematics)">fields</a> and other abstract systems; together such studies (for structures defined by algebraic operations) constitute the domain of <a href="/wiki/Abstract_algebra" title="Abstract algebra">abstract algebra</a>.</p>
<p>By its great generality, abstract algebra can often be applied to seemingly unrelated problems; for instance a number of ancient problems concerning <a href="/wiki/Compass_and_straightedge_constructions" title="Compass and straightedge constructions" class="mw-redirect">compass and straightedge constructions</a> were finally solved using <a href="/wiki/Galois_theory" title="Galois theory">Galois theory</a>, which involves field theory and group theory. Another example of an algebraic theory is <a href="/wiki/Linear_algebra" title="Linear algebra">linear algebra</a>, which is the general study of <a href="/wiki/Vector_space" title="Vector space">vector spaces</a>, whose elements called <a href="/wiki/Vector_(geometric)" title="Vector (geometric)" class="mw-redirect">vectors</a> have both quantity and direction, and can be used to model (relations between) points in space. This is one example of the phenomenon that the originally unrelated areas of <a href="/wiki/Geometry" title="Geometry">geometry</a> and <a href="/wiki/Algebra" title="Algebra">algebra</a> have very strong interactions in modern mathematics. <a href="/wiki/Combinatorics" title="Combinatorics">Combinatorics</a> studies ways of enumerating the number of objects that fit a given structure.</p>
<dl>
<dd>
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<td><img class="mwe-math-fallback-image-inline tex" alt="\begin{matrix} (1,2,3) &amp; (1,3,2) \\ (2,1,3) &amp; (2,3,1) \\ (3,1,2) &amp; (3,2,1) \end{matrix}" src="//upload.wikimedia.org/math/b/c/a/bca5b51d15b30266dc37decb94175dc2.png" /></td>
<td><a href="/wiki/File:Elliptic_curve_simple.svg" class="image"><img alt="Elliptic curve simple.svg" src="//upload.wikimedia.org/wikipedia/commons/thumb/d/da/Elliptic_curve_simple.svg/96px-Elliptic_curve_simple.svg.png" width="96" height="107" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/da/Elliptic_curve_simple.svg/144px-Elliptic_curve_simple.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/da/Elliptic_curve_simple.svg/192px-Elliptic_curve_simple.svg.png 2x" data-file-width="225" data-file-height="250" /></a></td>
<td><a href="/wiki/File:Rubik%27s_cube.svg" class="image"><img alt="Rubik's cube.svg" src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a6/Rubik%27s_cube.svg/96px-Rubik%27s_cube.svg.png" width="96" height="100" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a6/Rubik%27s_cube.svg/144px-Rubik%27s_cube.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a6/Rubik%27s_cube.svg/192px-Rubik%27s_cube.svg.png 2x" data-file-width="480" data-file-height="500" /></a></td>
<td><a href="/wiki/File:Group_diagdram_D6.svg" class="image"><img alt="Group diagdram D6.svg" src="//upload.wikimedia.org/wikipedia/commons/thumb/0/0e/Group_diagdram_D6.svg/96px-Group_diagdram_D6.svg.png" width="96" height="96" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/0e/Group_diagdram_D6.svg/144px-Group_diagdram_D6.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/0e/Group_diagdram_D6.svg/192px-Group_diagdram_D6.svg.png 2x" data-file-width="180" data-file-height="180" /></a></td>
<td><a href="/wiki/File:Lattice_of_the_divisibility_of_60.svg" class="image"><img alt="Lattice of the divisibility of 60.svg" src="//upload.wikimedia.org/wikipedia/commons/thumb/5/51/Lattice_of_the_divisibility_of_60.svg/96px-Lattice_of_the_divisibility_of_60.svg.png" width="96" height="77" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/51/Lattice_of_the_divisibility_of_60.svg/144px-Lattice_of_the_divisibility_of_60.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/51/Lattice_of_the_divisibility_of_60.svg/192px-Lattice_of_the_divisibility_of_60.svg.png 2x" data-file-width="313" data-file-height="250" /></a></td>
<td><a href="/wiki/File:Braid-modular-group-cover.svg" class="image"><img alt="Braid-modular-group-cover.svg" src="//upload.wikimedia.org/wikipedia/commons/thumb/d/da/Braid-modular-group-cover.svg/96px-Braid-modular-group-cover.svg.png" width="96" height="39" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/da/Braid-modular-group-cover.svg/144px-Braid-modular-group-cover.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/da/Braid-modular-group-cover.svg/192px-Braid-modular-group-cover.svg.png 2x" data-file-width="376" data-file-height="153" /></a></td>
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<td><a href="/wiki/Combinatorics" title="Combinatorics">Combinatorics</a></td>
<td><a href="/wiki/Number_theory" title="Number theory">Number theory</a></td>
<td><a href="/wiki/Group_theory" title="Group theory">Group theory</a></td>
<td><a href="/wiki/Graph_theory" title="Graph theory">Graph theory</a></td>
<td><a href="/wiki/Order_theory" title="Order theory">Order theory</a></td>
<td><a href="/wiki/Algebra" title="Algebra">Algebra</a></td>
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</table>
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<h4><span class="mw-headline" id="Space">Space</span></h4>
<p>The study of space originates with <a href="/wiki/Geometry" title="Geometry">geometry</a>&#160;– in particular, <a href="/wiki/Euclidean_geometry" title="Euclidean geometry">Euclidean geometry</a>. <a href="/wiki/Trigonometry" title="Trigonometry">Trigonometry</a> is the branch of mathematics that deals with relationships between the sides and the angles of triangles and with the trigonometric functions; it combines space and numbers, and encompasses the well-known <a href="/wiki/Pythagorean_theorem" title="Pythagorean theorem">Pythagorean theorem</a>. The modern study of space generalizes these ideas to include higher-dimensional geometry, <a href="/wiki/Non-Euclidean_geometries" title="Non-Euclidean geometries" class="mw-redirect">non-Euclidean geometries</a> (which play a central role in <a href="/wiki/General_relativity" title="General relativity">general relativity</a>) and <a href="/wiki/Topology" title="Topology">topology</a>. Quantity and space both play a role in <a href="/wiki/Analytic_geometry" title="Analytic geometry">analytic geometry</a>, <a href="/wiki/Differential_geometry" title="Differential geometry">differential geometry</a>, and <a href="/wiki/Algebraic_geometry" title="Algebraic geometry">algebraic geometry</a>. <a href="/wiki/Convex_geometry" title="Convex geometry">Convex</a> and <a href="/wiki/Discrete_geometry" title="Discrete geometry">discrete geometry</a> were developed to solve problems in <a href="/wiki/Geometry_of_numbers" title="Geometry of numbers">number theory</a> and <a href="/wiki/Functional_analysis" title="Functional analysis">functional analysis</a> but now are pursued with an eye on applications in <a href="/wiki/Convex_optimization" title="Convex optimization">optimization</a> and <a href="/wiki/Computational_geometry" title="Computational geometry">computer science</a>. Within differential geometry are the concepts of <a href="/wiki/Fiber_bundles" title="Fiber bundles" class="mw-redirect">fiber bundles</a> and calculus on <a href="/wiki/Manifold" title="Manifold">manifolds</a>, in particular, <a href="/wiki/Vector_calculus" title="Vector calculus">vector</a> and <a href="/wiki/Tensor_calculus" title="Tensor calculus">tensor calculus</a>. Within algebraic geometry is the description of geometric objects as solution sets of <a href="/wiki/Polynomial" title="Polynomial">polynomial</a> equations, combining the concepts of quantity and space, and also the study of <a href="/wiki/Topological_groups" title="Topological groups" class="mw-redirect">topological groups</a>, which combine structure and space. <a href="/wiki/Lie_group" title="Lie group">Lie groups</a> are used to study space, structure, and change. <a href="/wiki/Topology" title="Topology">Topology</a> in all its many ramifications may have been the greatest growth area in 20th-century mathematics; it includes <a href="/wiki/Point-set_topology" title="Point-set topology" class="mw-redirect">point-set topology</a>, <a href="/wiki/Set-theoretic_topology" title="Set-theoretic topology">set-theoretic topology</a>, <a href="/wiki/Algebraic_topology" title="Algebraic topology">algebraic topology</a> and <a href="/wiki/Differential_topology" title="Differential topology">differential topology</a>. In particular, instances of modern-day topology are <a href="/wiki/Metrizability_theory" title="Metrizability theory" class="mw-redirect">metrizability theory</a>, <a href="/wiki/Axiomatic_set_theory" title="Axiomatic set theory" class="mw-redirect">axiomatic set theory</a>, <a href="/wiki/Homotopy_theory" title="Homotopy theory" class="mw-redirect">homotopy theory</a>, and <a href="/wiki/Morse_theory" title="Morse theory">Morse theory</a>. Topology also includes the now solved <a href="/wiki/Poincar%C3%A9_conjecture" title="Poincaré conjecture">Poincaré conjecture</a>, and the still unsolved areas of the <a href="/wiki/Hodge_conjecture" title="Hodge conjecture">Hodge conjecture</a>. Other results in geometry and topology, including the <a href="/wiki/Four_color_theorem" title="Four color theorem">four color theorem</a> and <a href="/wiki/Kepler_conjecture" title="Kepler conjecture">Kepler conjecture</a>, have been proved only with the help of computers.</p>
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<dd>
<table style="border:1px solid #ddd; text-align:center; margin:auto" cellspacing="15">
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<td><a href="/wiki/File:Illustration_to_Euclid%27s_proof_of_the_Pythagorean_theorem.svg" class="image"><img alt="Illustration to Euclid's proof of the Pythagorean theorem.svg" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Illustration_to_Euclid%27s_proof_of_the_Pythagorean_theorem.svg/96px-Illustration_to_Euclid%27s_proof_of_the_Pythagorean_theorem.svg.png" width="96" height="104" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Illustration_to_Euclid%27s_proof_of_the_Pythagorean_theorem.svg/144px-Illustration_to_Euclid%27s_proof_of_the_Pythagorean_theorem.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Illustration_to_Euclid%27s_proof_of_the_Pythagorean_theorem.svg/192px-Illustration_to_Euclid%27s_proof_of_the_Pythagorean_theorem.svg.png 2x" data-file-width="500" data-file-height="540" /></a></td>
<td><a href="/wiki/File:Sinusv%C3%A5g_400px.png" class="image"><img alt="Sinusvåg 400px.png" src="//upload.wikimedia.org/wikipedia/commons/thumb/8/8c/Sinusv%C3%A5g_400px.png/96px-Sinusv%C3%A5g_400px.png" width="96" height="96" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/8c/Sinusv%C3%A5g_400px.png/144px-Sinusv%C3%A5g_400px.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/8c/Sinusv%C3%A5g_400px.png/192px-Sinusv%C3%A5g_400px.png 2x" data-file-width="400" data-file-height="400" /></a></td>
<td><a href="/wiki/File:Hyperbolic_triangle.svg" class="image"><img alt="Hyperbolic triangle.svg" src="//upload.wikimedia.org/wikipedia/commons/thumb/8/89/Hyperbolic_triangle.svg/96px-Hyperbolic_triangle.svg.png" width="96" height="64" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/89/Hyperbolic_triangle.svg/144px-Hyperbolic_triangle.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/89/Hyperbolic_triangle.svg/192px-Hyperbolic_triangle.svg.png 2x" data-file-width="600" data-file-height="400" /></a></td>
<td><a href="/wiki/File:Torus.png" class="image"><img alt="Torus.png" src="//upload.wikimedia.org/wikipedia/commons/thumb/1/17/Torus.png/96px-Torus.png" width="96" height="61" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/17/Torus.png/144px-Torus.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/17/Torus.png/192px-Torus.png 2x" data-file-width="784" data-file-height="502" /></a></td>
<td><a href="/wiki/File:Mandel_zoom_07_satellite.jpg" class="image"><img alt="Mandel zoom 07 satellite.jpg" src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b3/Mandel_zoom_07_satellite.jpg/96px-Mandel_zoom_07_satellite.jpg" width="96" height="72" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/b3/Mandel_zoom_07_satellite.jpg/144px-Mandel_zoom_07_satellite.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/b3/Mandel_zoom_07_satellite.jpg/192px-Mandel_zoom_07_satellite.jpg 2x" data-file-width="2560" data-file-height="1920" /></a></td>
<td><a href="/wiki/File:Measure_illustration.png" class="image"><img alt="Measure illustration.png" src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a6/Measure_illustration.png/70px-Measure_illustration.png" width="70" height="115" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a6/Measure_illustration.png/105px-Measure_illustration.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a6/Measure_illustration.png/140px-Measure_illustration.png 2x" data-file-width="364" data-file-height="598" /></a></td>
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<tr>
<td><a href="/wiki/Geometry" title="Geometry">Geometry</a></td>
<td><a href="/wiki/Trigonometry" title="Trigonometry">Trigonometry</a></td>
<td><a href="/wiki/Differential_geometry" title="Differential geometry">Differential geometry</a></td>
<td><a href="/wiki/Topology" title="Topology">Topology</a></td>
<td><a href="/wiki/Fractal" title="Fractal">Fractal geometry</a></td>
<td><a href="/wiki/Measure_theory" title="Measure theory" class="mw-redirect">Measure theory</a></td>
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</dd>
</dl>
<h4><span class="mw-headline" id="Change">Change</span></h4>
<p>Understanding and describing change is a common theme in the <a href="/wiki/Natural_science" title="Natural science">natural sciences</a>, and <a href="/wiki/Calculus" title="Calculus">calculus</a> was developed as a powerful tool to investigate it. <a href="/wiki/Function_(mathematics)" title="Function (mathematics)">Functions</a> arise here, as a central concept describing a changing quantity. The rigorous study of <a href="/wiki/Real_number" title="Real number">real numbers</a> and functions of a real variable is known as <a href="/wiki/Real_analysis" title="Real analysis">real analysis</a>, with <a href="/wiki/Complex_analysis" title="Complex analysis">complex analysis</a> the equivalent field for the <a href="/wiki/Complex_number" title="Complex number">complex numbers</a>. <a href="/wiki/Functional_analysis" title="Functional analysis">Functional analysis</a> focuses attention on (typically infinite-dimensional) <a href="/wiki/Space#Mathematics" title="Space">spaces</a> of functions. One of many applications of functional analysis is <a href="/wiki/Quantum_mechanics" title="Quantum mechanics">quantum mechanics</a>. Many problems lead naturally to relationships between a quantity and its rate of change, and these are studied as <a href="/wiki/Differential_equation" title="Differential equation">differential equations</a>. Many phenomena in nature can be described by <a href="/wiki/Dynamical_system" title="Dynamical system">dynamical systems</a>; <a href="/wiki/Chaos_theory" title="Chaos theory">chaos theory</a> makes precise the ways in which many of these systems exhibit unpredictable yet still <a href="/wiki/Deterministic_system_(mathematics)" title="Deterministic system (mathematics)" class="mw-redirect">deterministic</a> behavior.</p>
<table style="border:1px solid #ddd; text-align:center; margin:auto" cellspacing="20">
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<td><a href="/wiki/File:Integral_as_region_under_curve.svg" class="image"><img alt="Integral as region under curve.svg" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/f2/Integral_as_region_under_curve.svg/96px-Integral_as_region_under_curve.svg.png" width="96" height="84" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/f2/Integral_as_region_under_curve.svg/144px-Integral_as_region_under_curve.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/f2/Integral_as_region_under_curve.svg/192px-Integral_as_region_under_curve.svg.png 2x" data-file-width="744" data-file-height="654" /></a></td>
<td><a href="/wiki/File:Vector_field.svg" class="image"><img alt="Vector field.svg" src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c2/Vector_field.svg/96px-Vector_field.svg.png" width="96" height="96" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/c2/Vector_field.svg/144px-Vector_field.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/c2/Vector_field.svg/192px-Vector_field.svg.png 2x" data-file-width="394" data-file-height="394" /></a></td>
<td><a href="/wiki/File:Navier_Stokes_Laminar.svg" class="image"><img alt="Navier Stokes Laminar.svg" src="//upload.wikimedia.org/wikipedia/commons/thumb/7/73/Navier_Stokes_Laminar.svg/96px-Navier_Stokes_Laminar.svg.png" width="96" height="72" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/73/Navier_Stokes_Laminar.svg/144px-Navier_Stokes_Laminar.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/73/Navier_Stokes_Laminar.svg/192px-Navier_Stokes_Laminar.svg.png 2x" data-file-width="720" data-file-height="540" /></a></td>
<td><a href="/wiki/File:Limitcycle.svg" class="image"><img alt="Limitcycle.svg" src="//upload.wikimedia.org/wikipedia/commons/thumb/9/91/Limitcycle.svg/96px-Limitcycle.svg.png" width="96" height="78" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/91/Limitcycle.svg/144px-Limitcycle.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/91/Limitcycle.svg/192px-Limitcycle.svg.png 2x" data-file-width="390" data-file-height="315" /></a></td>
<td><a href="/wiki/File:Lorenz_attractor.svg" class="image"><img alt="Lorenz attractor.svg" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/f4/Lorenz_attractor.svg/96px-Lorenz_attractor.svg.png" width="96" height="96" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/f4/Lorenz_attractor.svg/144px-Lorenz_attractor.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/f4/Lorenz_attractor.svg/192px-Lorenz_attractor.svg.png 2x" data-file-width="750" data-file-height="750" /></a></td>
<td><a href="/wiki/File:Conformal_grid_after_M%C3%B6bius_transformation.svg" class="image"><img alt="Conformal grid after Möbius transformation.svg" src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3f/Conformal_grid_after_M%C3%B6bius_transformation.svg/96px-Conformal_grid_after_M%C3%B6bius_transformation.svg.png" width="96" height="96" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/3f/Conformal_grid_after_M%C3%B6bius_transformation.svg/144px-Conformal_grid_after_M%C3%B6bius_transformation.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/3f/Conformal_grid_after_M%C3%B6bius_transformation.svg/192px-Conformal_grid_after_M%C3%B6bius_transformation.svg.png 2x" data-file-width="288" data-file-height="288" /></a></td>
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<tr>
<td><a href="/wiki/Calculus" title="Calculus">Calculus</a></td>
<td><a href="/wiki/Vector_calculus" title="Vector calculus">Vector calculus</a></td>
<td><a href="/wiki/Differential_equation" title="Differential equation">Differential equations</a></td>
<td><a href="/wiki/Dynamical_system" title="Dynamical system">Dynamical systems</a></td>
<td><a href="/wiki/Chaos_theory" title="Chaos theory">Chaos theory</a></td>
<td><a href="/wiki/Complex_analysis" title="Complex analysis">Complex analysis</a></td>
</tr>
</table>
<h3><span class="mw-headline" id="Applied_mathematics">Applied mathematics</span></h3>
<p><a href="/wiki/Applied_mathematics" title="Applied mathematics">Applied mathematics</a> concerns itself with mathematical methods that are typically used in science, engineering, business, and industry. Thus, "applied mathematics" is a <a href="/wiki/Mathematical_science" title="Mathematical science" class="mw-redirect">mathematical science</a> with specialized knowledge. The term <i>applied mathematics</i> also describes the <a href="/wiki/Professional" title="Professional">professional</a> specialty in which mathematicians work on practical problems; as a profession focused on practical problems, <i>applied mathematics</i> focuses on the "formulation, study, and use of mathematical models" in <a href="/wiki/Science" title="Science">science</a>, <a href="/wiki/Engineering" title="Engineering">engineering</a>, and other areas of mathematical practice.</p>
<p>In the past, practical applications have motivated the development of mathematical theories, which then became the subject of study in pure mathematics, where mathematics is developed primarily for its own sake. Thus, the activity of applied mathematics is vitally connected with research in <a href="/wiki/Pure_mathematics" title="Pure mathematics">pure mathematics</a>.</p>
<h4><span class="mw-headline" id="Statistics_and_other_decision_sciences">Statistics and other decision sciences</span></h4>
<p>Applied mathematics has significant overlap with the discipline of <a href="/wiki/Statistics" title="Statistics">statistics</a>, whose theory is formulated mathematically, especially with <a href="/wiki/Probability_theory" title="Probability theory">probability theory</a>. Statisticians (working as part of a research project) "create data that makes sense" with <a href="/wiki/Random_sampling" title="Random sampling" class="mw-redirect">random sampling</a> and with randomized <a href="/wiki/Design_of_experiments" title="Design of experiments">experiments</a>;<sup id="cite_ref-52" class="reference"><a href="#cite_note-52"><span>[</span>52<span>]</span></a></sup> the design of a statistical sample or experiment specifies the analysis of the data (before the data be available). When reconsidering data from experiments and samples or when analyzing data from <a href="/wiki/Observational_study" title="Observational study">observational studies</a>, statisticians "make sense of the data" using the art of <a href="/wiki/Statistical_model" title="Statistical model">modelling</a> and the theory of <a href="/wiki/Statistical_inference" title="Statistical inference">inference</a>&#160;– with <a href="/wiki/Model_selection" title="Model selection">model selection</a> and <a href="/wiki/Estimation_theory" title="Estimation theory">estimation</a>; the estimated models and consequential <a href="/wiki/Scientific_method#Predictions_from_the_hypothesis" title="Scientific method">predictions</a> should be <a href="/wiki/Statistical_hypothesis_testing" title="Statistical hypothesis testing">tested</a> on <a href="/wiki/Scientific_method#Evaluation_and_improvement" title="Scientific method">new data</a>.<sup id="cite_ref-53" class="reference"><a href="#cite_note-53"><span>[</span>53<span>]</span></a></sup></p>
<p><a href="/wiki/Statistical_theory" title="Statistical theory">Statistical theory</a> studies <a href="/wiki/Statistical_decision_theory" title="Statistical decision theory" class="mw-redirect">decision problems</a> such as minimizing the <a href="/wiki/Risk" title="Risk">risk</a> (<a href="/wiki/Expected_loss" title="Expected loss">expected loss</a>) of a statistical action, such as using a <a href="/wiki/Statistical_method" title="Statistical method" class="mw-redirect">procedure</a> in, for example, <a href="/wiki/Parameter_estimation" title="Parameter estimation" class="mw-redirect">parameter estimation</a>, <a href="/wiki/Hypothesis_testing" title="Hypothesis testing" class="mw-redirect">hypothesis testing</a>, and <a href="/wiki/Selection_algorithm" title="Selection algorithm">selecting the best</a>. In these traditional areas of <a href="/wiki/Mathematical_statistics" title="Mathematical statistics">mathematical statistics</a>, a statistical-decision problem is formulated by minimizing an <a href="/wiki/Objective_function" title="Objective function" class="mw-redirect">objective function</a>, like expected loss or <a href="/wiki/Cost" title="Cost">cost</a>, under specific constraints: For example, designing a survey often involves minimizing the cost of estimating a population mean with a given level of confidence.<sup id="cite_ref-RaoOpt_54-0" class="reference"><a href="#cite_note-RaoOpt-54"><span>[</span>54<span>]</span></a></sup> Because of its use of <a href="/wiki/Mathematical_optimization" title="Mathematical optimization">optimization</a>, the mathematical theory of statistics shares concerns with other <a href="/wiki/Decision_science" title="Decision science" class="mw-redirect">decision sciences</a>, such as <a href="/wiki/Operations_research" title="Operations research">operations research</a>, <a href="/wiki/Control_theory" title="Control theory">control theory</a>, and <a href="/wiki/Mathematical_economics" title="Mathematical economics">mathematical economics</a>.<sup id="cite_ref-Whittle_55-0" class="reference"><a href="#cite_note-Whittle-55"><span>[</span>55<span>]</span></a></sup></p>
<h4><span class="mw-headline" id="Computational_mathematics">Computational mathematics</span></h4>
<p><a href="/wiki/Computational_mathematics" title="Computational mathematics">Computational mathematics</a> proposes and studies methods for solving <a href="/wiki/Mathematical_problem" title="Mathematical problem">mathematical problems</a> that are typically too large for human numerical capacity. <a href="/wiki/Numerical_analysis" title="Numerical analysis">Numerical analysis</a> studies methods for problems in <a href="/wiki/Analysis_(mathematics)" title="Analysis (mathematics)" class="mw-redirect">analysis</a> using <a href="/wiki/Functional_analysis" title="Functional analysis">functional analysis</a> and <a href="/wiki/Approximation_theory" title="Approximation theory">approximation theory</a>; numerical analysis includes the study of <a href="/wiki/Approximation" title="Approximation">approximation</a> and <a href="/wiki/Discretization" title="Discretization">discretization</a> broadly with special concern for <a href="/wiki/Rounding_error" title="Rounding error" class="mw-redirect">rounding errors</a>. Numerical analysis and, more broadly, scientific computing also study non-analytic topics of mathematical science, especially <a href="/wiki/Algorithm" title="Algorithm">algorithmic</a> <a href="/wiki/Numerical_linear_algebra" title="Numerical linear algebra">matrix</a> and <a href="/wiki/Graph_theory" title="Graph theory">graph theory</a>. Other areas of computational mathematics include <a href="/wiki/Computer_algebra" title="Computer algebra" class="mw-redirect">computer algebra</a> and <a href="/wiki/Symbolic_computation" title="Symbolic computation">symbolic computation</a>.</p>
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<td><a href="/wiki/File:Gravitation_space_source.png" class="image"><img alt="Gravitation space source.png" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Gravitation_space_source.png/96px-Gravitation_space_source.png" width="96" height="64" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Gravitation_space_source.png/144px-Gravitation_space_source.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Gravitation_space_source.png/192px-Gravitation_space_source.png 2x" data-file-width="300" data-file-height="200" /></a></td>
<td><a href="/wiki/File:BernoullisLawDerivationDiagram.svg" class="image"><img alt="BernoullisLawDerivationDiagram.svg" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/20/BernoullisLawDerivationDiagram.svg/96px-BernoullisLawDerivationDiagram.svg.png" width="96" height="45" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/20/BernoullisLawDerivationDiagram.svg/144px-BernoullisLawDerivationDiagram.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/20/BernoullisLawDerivationDiagram.svg/192px-BernoullisLawDerivationDiagram.svg.png 2x" data-file-width="790" data-file-height="370" /></a></td>
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<td><a href="/wiki/File:Maximum_boxed.png" class="image"><img alt="Maximum boxed.png" src="//upload.wikimedia.org/wikipedia/commons/thumb/1/1a/Maximum_boxed.png/96px-Maximum_boxed.png" width="96" height="91" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/1a/Maximum_boxed.png/144px-Maximum_boxed.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/1a/Maximum_boxed.png/192px-Maximum_boxed.png 2x" data-file-width="634" data-file-height="599" /></a></td>
<td><a href="/wiki/File:Two_red_dice_01.svg" class="image"><img alt="Two red dice 01.svg" src="//upload.wikimedia.org/wikipedia/commons/thumb/3/36/Two_red_dice_01.svg/96px-Two_red_dice_01.svg.png" width="96" height="62" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/36/Two_red_dice_01.svg/144px-Two_red_dice_01.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/36/Two_red_dice_01.svg/192px-Two_red_dice_01.svg.png 2x" data-file-width="671" data-file-height="430" /></a></td>
<td><a href="/wiki/File:Oldfaithful3.png" class="image"><img alt="Oldfaithful3.png" src="//upload.wikimedia.org/wikipedia/commons/thumb/0/0f/Oldfaithful3.png/96px-Oldfaithful3.png" width="96" height="96" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/0f/Oldfaithful3.png/144px-Oldfaithful3.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/0f/Oldfaithful3.png/192px-Oldfaithful3.png 2x" data-file-width="401" data-file-height="400" /></a></td>
<td><a href="/wiki/File:Caesar3.svg" class="image"><img alt="Caesar3.svg" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/2b/Caesar3.svg/96px-Caesar3.svg.png" width="96" height="40" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/2b/Caesar3.svg/144px-Caesar3.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/2b/Caesar3.svg/192px-Caesar3.svg.png 2x" data-file-width="856" data-file-height="361" /></a></td>
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<td><a href="/wiki/Mathematical_physics" title="Mathematical physics">Mathematical physics</a></td>
<td><a href="/wiki/Fluid_dynamics" title="Fluid dynamics">Fluid dynamics</a></td>
<td><a href="/wiki/Numerical_analysis" title="Numerical analysis">Numerical analysis</a></td>
<td><a href="/wiki/Mathematical_optimization" title="Mathematical optimization">Optimization</a></td>
<td><a href="/wiki/Probability_theory" title="Probability theory">Probability theory</a></td>
<td><a href="/wiki/Statistics" title="Statistics">Statistics</a></td>
<td><a href="/wiki/Cryptography" title="Cryptography">Cryptography</a></td>
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<td><a href="/wiki/File:Market_Data_Index_NYA_on_20050726_202628_UTC.png" class="image"><img alt="Market Data Index NYA on 20050726 202628 UTC.png" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/46/Market_Data_Index_NYA_on_20050726_202628_UTC.png/96px-Market_Data_Index_NYA_on_20050726_202628_UTC.png" width="96" height="64" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/46/Market_Data_Index_NYA_on_20050726_202628_UTC.png/144px-Market_Data_Index_NYA_on_20050726_202628_UTC.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/46/Market_Data_Index_NYA_on_20050726_202628_UTC.png/192px-Market_Data_Index_NYA_on_20050726_202628_UTC.png 2x" data-file-width="600" data-file-height="400" /></a></td>
<td><a href="/wiki/File:Arbitrary-gametree-solved.svg" class="image"><img alt="Arbitrary-gametree-solved.svg" src="//upload.wikimedia.org/wikipedia/commons/thumb/d/d7/Arbitrary-gametree-solved.svg/96px-Arbitrary-gametree-solved.svg.png" width="96" height="65" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/d7/Arbitrary-gametree-solved.svg/144px-Arbitrary-gametree-solved.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/d7/Arbitrary-gametree-solved.svg/192px-Arbitrary-gametree-solved.svg.png 2x" data-file-width="420" data-file-height="286" /></a></td>
<td><a href="/wiki/File:Signal_transduction_pathways.svg" class="image"><img alt="Signal transduction pathways.svg" src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b0/Signal_transduction_pathways.svg/96px-Signal_transduction_pathways.svg.png" width="96" height="70" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/b0/Signal_transduction_pathways.svg/144px-Signal_transduction_pathways.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/b0/Signal_transduction_pathways.svg/192px-Signal_transduction_pathways.svg.png 2x" data-file-width="1858" data-file-height="1364" /></a></td>
<td><a href="/wiki/File:CH4-structure.svg" class="image"><img alt="CH4-structure.svg" src="//upload.wikimedia.org/wikipedia/commons/thumb/0/0f/CH4-structure.svg/96px-CH4-structure.svg.png" width="96" height="104" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/0f/CH4-structure.svg/144px-CH4-structure.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/0f/CH4-structure.svg/192px-CH4-structure.svg.png 2x" data-file-width="175" data-file-height="190" /></a></td>
<td><a href="/wiki/File:GDP_PPP_Per_Capita_IMF_2008.svg" class="image"><img alt="GDP PPP Per Capita IMF 2008.svg" src="//upload.wikimedia.org/wikipedia/commons/thumb/d/d4/GDP_PPP_Per_Capita_IMF_2008.svg/96px-GDP_PPP_Per_Capita_IMF_2008.svg.png" width="96" height="44" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/d4/GDP_PPP_Per_Capita_IMF_2008.svg/144px-GDP_PPP_Per_Capita_IMF_2008.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/d4/GDP_PPP_Per_Capita_IMF_2008.svg/192px-GDP_PPP_Per_Capita_IMF_2008.svg.png 2x" data-file-width="1800" data-file-height="820" /></a></td>
<td><a href="/wiki/File:Simple_feedback_control_loop2.svg" class="image"><img alt="Simple feedback control loop2.svg" src="//upload.wikimedia.org/wikipedia/commons/thumb/9/90/Simple_feedback_control_loop2.svg/96px-Simple_feedback_control_loop2.svg.png" width="96" height="34" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/90/Simple_feedback_control_loop2.svg/144px-Simple_feedback_control_loop2.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/90/Simple_feedback_control_loop2.svg/192px-Simple_feedback_control_loop2.svg.png 2x" data-file-width="315" data-file-height="110" /></a></td>
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<td><a href="/wiki/Mathematical_finance" title="Mathematical finance">Mathematical finance</a></td>
<td><a href="/wiki/Game_theory" title="Game theory">Game theory</a></td>
<td><a href="/wiki/Mathematical_biology" title="Mathematical biology" class="mw-redirect">Mathematical biology</a></td>
<td><a href="/wiki/Mathematical_chemistry" title="Mathematical chemistry">Mathematical chemistry</a></td>
<td><a href="/wiki/Mathematical_economics" title="Mathematical economics">Mathematical economics</a></td>
<td><a href="/wiki/Control_theory" title="Control theory">Control theory</a></td>
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<h2><span class="mw-headline" id="Mathematical_awards">Mathematical awards</span></h2>
<p>Arguably the most prestigious award in mathematics is the <a href="/wiki/Fields_Medal" title="Fields Medal">Fields Medal</a>,<sup id="cite_ref-FOOTNOTEMonastyrsky2001_56-0" class="reference"><a href="#cite_note-FOOTNOTEMonastyrsky2001-56"><span>[</span>56<span>]</span></a></sup><sup id="cite_ref-FOOTNOTERiehm2002778.E2.80.93782_57-0" class="reference"><a href="#cite_note-FOOTNOTERiehm2002778.E2.80.93782-57"><span>[</span>57<span>]</span></a></sup> established in 1936 and awarded every four years (except around World War II) to as many as four individuals. The Fields Medal is often considered a mathematical equivalent to the <a href="/wiki/Nobel_Prize" title="Nobel Prize">Nobel Prize</a>.</p>
<p>The <a href="/wiki/Wolf_Prize_in_Mathematics" title="Wolf Prize in Mathematics">Wolf Prize in Mathematics</a>, instituted in 1978, recognizes lifetime achievement, and another major international award, the <a href="/wiki/Abel_Prize" title="Abel Prize">Abel Prize</a>, was introduced in 2003. The <a href="/wiki/Chern_Medal" title="Chern Medal">Chern Medal</a> was introduced in 2010 to recognize lifetime achievement. These accolades are awarded in recognition of a particular body of work, which may be innovational, or provide a solution to an outstanding problem in an established field.</p>
<p>A famous list of 23 <a href="/wiki/Open_problem" title="Open problem">open problems</a>, called "<a href="/wiki/Hilbert%27s_problems" title="Hilbert's problems">Hilbert's problems</a>", was compiled in 1900 by German mathematician <a href="/wiki/David_Hilbert" title="David Hilbert">David Hilbert</a>. This list achieved great celebrity among mathematicians, and at least nine of the problems have now been solved. A new list of seven important problems, titled the "<a href="/wiki/Millennium_Prize_Problems" title="Millennium Prize Problems">Millennium Prize Problems</a>", was published in 2000. A solution to each of these problems carries a $1&#160;million reward, and only one (the <a href="/wiki/Riemann_hypothesis" title="Riemann hypothesis">Riemann hypothesis</a>) is duplicated in Hilbert's problems.</p>
<h2><span class="mw-headline" id="See_also">See also</span></h2>
<div role="note" class="hatnote relarticle mainarticle">Main article: <a href="/wiki/Lists_of_mathematics_topics" title="Lists of mathematics topics">Lists of mathematics topics</a></div>
<ul>
<li><a href="/wiki/Mathematics_and_art" title="Mathematics and art">Mathematics and art</a></li>
<li><a href="/wiki/Mathematics_education" title="Mathematics education">Mathematics education</a></li>
<li><a href="/wiki/Relationship_between_mathematics_and_physics" title="Relationship between mathematics and physics">Relationship between mathematics and physics</a></li>
<li><a href="/wiki/STEM_fields" title="STEM fields" class="mw-redirect">STEM fields</a></li>
</ul>
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<h2><span class="mw-headline" id="Notes">Notes</span></h2>
<div class="reflist columns references-column-width" style="-moz-column-width: 30em; -webkit-column-width: 30em; column-width: 30em; list-style-type: decimal;">
<ol class="references">
<li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text">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 <i><a href="/wiki/Euclid" title="Euclid">Euclid</a></i>).</span></li>
<li id="cite_note-OED-2"><span class="mw-cite-backlink">^ <a href="#cite_ref-OED_2-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-OED_2-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><cite class="citation web"><a rel="nofollow" class="external text" href="http://oed.com/view/Entry/114974">"mathematics, <i>n.</i>"</a>. <i>Oxford English Dictionary</i>. Oxford University Press. 2012<span class="reference-accessdate">. Retrieved <span class="nowrap">June 16,</span> 2012</span>. <q>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.</q></cite><span title="ctx_ver=Z39.88-2004&amp;rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematics&amp;rft.atitle=mathematics%2C+n.&amp;rft.date=2012&amp;rft.genre=unknown&amp;rft_id=http%3A%2F%2Foed.com%2Fview%2FEntry%2F114974&amp;rft.jtitle=Oxford+English+Dictionary&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal" class="Z3988"><span style="display:none;">&#160;</span></span></span></li>
<li id="cite_note-Kneebone-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-Kneebone_3-0">^</a></b></span> <span class="reference-text"><cite class="citation book">Kneebone, G.T. (1963). <i>Mathematical Logic and the Foundations of Mathematics: An Introductory Survey</i>. Dover. pp.&#160;<a rel="nofollow" class="external text" href="https://books.google.com/books?id=tCXxf4vbXCcC&amp;pg=PA4">4</a>. <a href="/wiki/International_Standard_Book_Number" title="International Standard Book Number">ISBN</a>&#160;<a href="/wiki/Special:BookSources/0-486-41712-3" title="Special:BookSources/0-486-41712-3">0-486-41712-3</a>. <q>Mathematics&#160;... is simply the study of abstract structures, or formal patterns of connectedness.</q></cite><span title="ctx_ver=Z39.88-2004&amp;rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematics&amp;rft.au=Kneebone%2C+G.T.&amp;rft.btitle=Mathematical+Logic+and+the+Foundations+of+Mathematics%3A+An+Introductory+Survey&amp;rft.date=1963&amp;rft.genre=book&amp;rft.isbn=0-486-41712-3&amp;rft.pages=4&amp;rft.pub=Dover&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;">&#160;</span></span></span></li>
<li id="cite_note-LaTorre-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-LaTorre_4-0">^</a></b></span> <span class="reference-text"><cite class="citation book">LaTorre, Donald R., John W. Kenelly, Iris B. Reed, Laurel R. Carpenter, and Cynthia R Harris (2011). <i>Calculus Concepts: An Informal Approach to the Mathematics of Change</i>. Cengage Learning. pp.&#160;<a rel="nofollow" class="external text" href="https://books.google.com/books?id=1Ebu2Tij4QsC&amp;pg=PA2">2</a>. <a href="/wiki/International_Standard_Book_Number" title="International Standard Book Number">ISBN</a>&#160;<a href="/wiki/Special:BookSources/1-4390-4957-2" title="Special:BookSources/1-4390-4957-2">1-4390-4957-2</a>. <q>Calculus is the study of change—how things change, and how quickly they change.</q></cite><span title="ctx_ver=Z39.88-2004&amp;rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematics&amp;rft.au=LaTorre%2C+Donald+R.%2C+John+W.+Kenelly%2C+Iris+B.+Reed%2C+Laurel+R.+Carpenter%2C+and+Cynthia+R+Harris&amp;rft.btitle=Calculus+Concepts%3A+An+Informal+Approach+to+the+Mathematics+of+Change&amp;rft.date=2011&amp;rft.genre=book&amp;rft.isbn=1-4390-4957-2&amp;rft.pages=2&amp;rft.pub=Cengage+Learning&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;">&#160;</span></span></span></li>
<li id="cite_note-Ramana-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-Ramana_5-0">^</a></b></span> <span class="reference-text"><cite class="citation book">Ramana (2007). <i>Applied Mathematics</i>. Tata McGraw–Hill Education. p.&#160;<a rel="nofollow" class="external text" href="https://books.google.com/books?id=XCRC6BeKhIIC&amp;pg=SA2–PA10">2.10</a>. <a href="/wiki/International_Standard_Book_Number" title="International Standard Book Number">ISBN</a>&#160;<a href="/wiki/Special:BookSources/0-07-066753-5" title="Special:BookSources/0-07-066753-5">0-07-066753-5</a>. <q>The mathematical study of change, motion, growth or decay is calculus.</q></cite><span title="ctx_ver=Z39.88-2004&amp;rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematics&amp;rft.au=Ramana&amp;rft.btitle=Applied+Mathematics&amp;rft.date=2007&amp;rft.genre=book&amp;rft.isbn=0-07-066753-5&amp;rft.pages=2.10&amp;rft.pub=Tata+McGraw%93Hill+Education&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;">&#160;</span></span></span></li>
<li id="cite_note-Ziegler-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-Ziegler_6-0">^</a></b></span> <span class="reference-text"><cite class="citation book"><a href="/wiki/G%C3%BCnter_M._Ziegler" title="Günter M. Ziegler">Ziegler, Günter M.</a> (2011). "What Is Mathematics?". <i>An Invitation to Mathematics: From Competitions to Research</i>. Springer. pp.&#160;<a rel="nofollow" class="external text" href="https://books.google.com/books?id=9TATfteVeVYC&amp;pg=PR7">7</a>. <a href="/wiki/International_Standard_Book_Number" title="International Standard Book Number">ISBN</a>&#160;<a href="/wiki/Special:BookSources/3-642-19532-6" title="Special:BookSources/3-642-19532-6">3-642-19532-6</a>.</cite><span title="ctx_ver=Z39.88-2004&amp;rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematics&amp;rft.atitle=What+Is+Mathematics%3F&amp;rft.au=Ziegler%2C+G%C3%BCnter+M.&amp;rft.btitle=An+Invitation+to+Mathematics%3A+From+Competitions+to+Research&amp;rft.date=2011&amp;rft.genre=bookitem&amp;rft.isbn=3-642-19532-6&amp;rft.pages=7&amp;rft.pub=Springer&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;">&#160;</span></span></span></li>
<li id="cite_note-Mura-7"><span class="mw-cite-backlink">^ <a href="#cite_ref-Mura_7-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Mura_7-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-Mura_7-2"><sup><i><b>c</b></i></sup></a> <a href="#cite_ref-Mura_7-3"><sup><i><b>d</b></i></sup></a></span> <span class="reference-text"><cite id="CITEREFMura.2C_Roberta1993" class="citation journal">Mura, Roberta (Dec 1993). "Images of Mathematics Held by University Teachers of Mathematical Sciences". <i>Educational Studies in Mathematics</i> <b>25</b> (4): <a rel="nofollow" class="external text" href="http://www.jstor.org/stable/10.2307/3482762">375–385</a>.</cite><span title="ctx_ver=Z39.88-2004&amp;rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematics&amp;rft.atitle=Images+of+Mathematics+Held+by+University+Teachers+of+Mathematical+Sciences&amp;rft.au=Mura%2C+Roberta&amp;rft.date=1993-12&amp;rft.genre=article&amp;rft.issue=4&amp;rft.jtitle=Educational+Studies+in+Mathematics&amp;rft.pages=375-385&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&amp;rft.volume=25" class="Z3988"><span style="display:none;">&#160;</span></span></span></li>
<li id="cite_note-Runge-8"><span class="mw-cite-backlink">^ <a href="#cite_ref-Runge_8-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Runge_8-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><cite class="citation book">Tobies, Renate and Helmut Neunzert (2012). <i><a href="/wiki/Iris_Runge" title="Iris Runge">Iris Runge</a>: A Life at the Crossroads of Mathematics, Science, and Industry</i>. Springer. pp.&#160;<a rel="nofollow" class="external text" href="https://books.google.com/books?id=EDm0eQqFUQ4C&amp;pg=PA9">9</a>. <a href="/wiki/International_Standard_Book_Number" title="International Standard Book Number">ISBN</a>&#160;<a href="/wiki/Special:BookSources/3-0348-0229-3" title="Special:BookSources/3-0348-0229-3">3-0348-0229-3</a>. <q>It is first necessary to ask what is meant by <i>mathematics</i> in general. Illustrious scholars have debated this matter until they were blue in the face, and yet no consensus has been reached about whether mathematics is a natural science, a branch of the humanities, or an art form.</q></cite><span title="ctx_ver=Z39.88-2004&amp;rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematics&amp;rft.au=Tobies%2C+Renate+and+Helmut+Neunzert&amp;rft.btitle=Iris+Runge%3A+A+Life+at+the+Crossroads+of+Mathematics%2C+Science%2C+and+Industry&amp;rft.date=2012&amp;rft.genre=book&amp;rft.isbn=3-0348-0229-3&amp;rft.pages=9&amp;rft.pub=Springer&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;">&#160;</span></span></span></li>
<li id="cite_note-future-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-future_9-0">^</a></b></span> <span class="reference-text"><a href="/wiki/Lynn_Steen" title="Lynn Steen">Steen, L.A.</a> (April 29, 1988). <i>The Science of Patterns</i> <a href="/wiki/Science_(journal)" title="Science (journal)">Science</a>, 240: 611–616. And summarized at <a rel="nofollow" class="external text" href="http://www.ascd.org/publications/curriculum-handbook/409/chapters/The-Future-of-Mathematics-Education.aspx">Association for Supervision and Curriculum Development</a>, www.ascd.org.</span></li>
<li id="cite_note-devlin-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-devlin_10-0">^</a></b></span> <span class="reference-text"><a href="/wiki/Keith_Devlin" title="Keith Devlin">Devlin, Keith</a>, <i>Mathematics: The Science of Patterns: The Search for Order in Life, Mind and the Universe</i> (Scientific American Paperback Library) 1996, <a href="/wiki/Special:BookSources/9780716750475" class="internal mw-magiclink-isbn">ISBN 978-0-7167-5047-5</a></span></li>
<li id="cite_note-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-11">^</a></b></span> <span class="reference-text">Eves</span></li>
<li id="cite_note-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-12">^</a></b></span> <span class="reference-text"><a href="/wiki/Marcus_du_Sautoy" title="Marcus du Sautoy">Marcus du Sautoy</a>, <i><a rel="nofollow" class="external text" href="http://www.bbc.co.uk/programmes/b00sr3fm">A Brief History of Mathematics: 1. Newton and Leibniz</a></i>, <a href="/wiki/BBC_Radio_4" title="BBC Radio 4">BBC Radio 4</a>, September 27, 2010.</span></li>
<li id="cite_note-Waltershausen-13"><span class="mw-cite-backlink">^ <a href="#cite_ref-Waltershausen_13-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Waltershausen_13-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text">Waltershausen</span></li>
<li id="cite_note-14"><span class="mw-cite-backlink"><b><a href="#cite_ref-14">^</a></b></span> <span class="reference-text">Peirce, p. 97.</span></li>
<li id="cite_note-15"><span class="mw-cite-backlink"><b><a href="#cite_ref-15">^</a></b></span> <span class="reference-text">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).</span></li>
<li id="cite_note-certain-16"><span class="mw-cite-backlink">^ <a href="#cite_ref-certain_16-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-certain_16-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text">Einstein, p. 28. The quote is Einstein's answer to the question: "how can it be that mathematics, being after all a product of human thought which is independent of experience, is so admirably appropriate to the objects of reality?" He, too, is concerned with <i><a href="/wiki/The_Unreasonable_Effectiveness_of_Mathematics_in_the_Natural_Sciences" title="The Unreasonable Effectiveness of Mathematics in the Natural Sciences">The Unreasonable Effectiveness of Mathematics in the Natural Sciences</a></i>.</span></li>
<li id="cite_note-17"><span class="mw-cite-backlink"><b><a href="#cite_ref-17">^</a></b></span> <span class="reference-text">Peterson</span></li>
<li id="cite_note-18"><span class="mw-cite-backlink"><b><a href="#cite_ref-18">^</a></b></span> <span class="reference-text"><cite id="CITEREFDehaeneDehaene-LambertzCohen1998" class="citation journal">Dehaene, Stanislas; Dehaene-Lambertz, Ghislaine; Cohen, Laurent (Aug 1998). "Abstract representations of numbers in the animal and human brain". <i>Trends in Neuroscience</i> <b>21</b> (8): 355–361. <a href="/wiki/Digital_object_identifier" title="Digital object identifier">doi</a>:<a rel="nofollow" class="external text" href="//dx.doi.org/10.1016%2FS0166-2236%2898%2901263-6">10.1016/S0166-2236(98)01263-6</a>. <a href="/wiki/PubMed_Identifier" title="PubMed Identifier" class="mw-redirect">PMID</a>&#160;<a rel="nofollow" class="external text" href="//www.ncbi.nlm.nih.gov/pubmed/9720604">9720604</a>.</cite><span title="ctx_ver=Z39.88-2004&amp;rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematics&amp;rft.atitle=Abstract+representations+of+numbers+in+the+animal+and+human+brain&amp;rft.au=Cohen%2C+Laurent&amp;rft.au=Dehaene-Lambertz%2C+Ghislaine&amp;rft.aufirst=Stanislas&amp;rft.aulast=Dehaene&amp;rft.date=1998-08&amp;rft.genre=article&amp;rft_id=info%3Adoi%2F10.1016%2FS0166-2236%2898%2901263-6&amp;rft_id=info%3Apmid%2F9720604&amp;rft.issue=8&amp;rft.jtitle=Trends+in+Neuroscience&amp;rft.pages=355-361&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&amp;rft.volume=21" class="Z3988"><span style="display:none;">&#160;</span></span></span></li>
<li id="cite_note-19"><span class="mw-cite-backlink"><b><a href="#cite_ref-19">^</a></b></span> <span class="reference-text">See, for example, Raymond L. Wilder, <i>Evolution of Mathematical Concepts; an Elementary Study</i>, <i>passim</i></span></li>
<li id="cite_note-20"><span class="mw-cite-backlink"><b><a href="#cite_ref-20">^</a></b></span> <span class="reference-text">Kline 1990, Chapter 1.</span></li>
<li id="cite_note-21"><span class="mw-cite-backlink"><b><a href="#cite_ref-21">^</a></b></span> <span class="reference-text">"<i><a rel="nofollow" class="external text" href="https://books.google.com/books?id=drnY3Vjix3kC&amp;pg=PA1&amp;dq&amp;hl=en#v=onepage&amp;q=&amp;f=false">A History of Greek Mathematics: From Thales to Euclid</a></i>". Thomas Little Heath (1981). <a href="/wiki/Special:BookSources/0486240738" class="internal mw-magiclink-isbn">ISBN 0-486-24073-8</a></span></li>
<li id="cite_note-FOOTNOTESevryuk2006101.E2.80.93109-22"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTESevryuk2006101.E2.80.93109_22-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFSevryuk2006">Sevryuk 2006</a>, pp.&#160;101–109.</span></li>
<li id="cite_note-23"><span class="mw-cite-backlink"><b><a href="#cite_ref-23">^</a></b></span> <span class="reference-text"><cite class="citation web"><a rel="nofollow" class="external text" href="http://www.etymonline.com/index.php?term=mathematic&amp;allowed_in_frame=0">"mathematic"</a>. <a href="/wiki/Online_Etymology_Dictionary" title="Online Etymology Dictionary">Online Etymology Dictionary</a>.</cite><span title="ctx_ver=Z39.88-2004&amp;rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematics&amp;rft.btitle=mathematic&amp;rft.genre=unknown&amp;rft_id=http%3A%2F%2Fwww.etymonline.com%2Findex.php%3Fterm%3Dmathematic%26allowed_in_frame%3D0&amp;rft.pub=Online+Etymology+Dictionary&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;">&#160;</span></span></span></li>
<li id="cite_note-24"><span class="mw-cite-backlink"><b><a href="#cite_ref-24">^</a></b></span> <span class="reference-text">Both senses can be found in Plato. <a rel="nofollow" class="external text" href="http://www.perseus.tufts.edu/hopper/text?doc=Perseus:text:1999.04.0057:entry=maqhmatiko/s"><span lang="grc">μαθηματική</span></a>. <a href="/wiki/Henry_Liddell" title="Henry Liddell">Liddell, Henry George</a>; <a href="/wiki/Robert_Scott_(philologist)" title="Robert Scott (philologist)">Scott, Robert</a>; <i><a href="/wiki/A_Greek%E2%80%93English_Lexicon" title="A Greek–English Lexicon">A Greek–English Lexicon</a></i> at the <a href="/wiki/Perseus_Project" title="Perseus Project">Perseus Project</a></span></li>
<li id="cite_note-ohiostateuniversity-25"><span class="mw-cite-backlink"><b><a href="#cite_ref-ohiostateuniversity_25-0">^</a></b></span> <span class="reference-text"><cite class="citation web">Cipra, Barry (1982). <a rel="nofollow" class="external text" href="https://people.math.osu.edu/easwaran.1/augustine.html">"St. Augustine v. The Mathematicians"</a>. <i>osu.edu</i>. <a href="/wiki/Ohio_State_University" title="Ohio State University">Ohio State University</a> Mathematics department<span class="reference-accessdate">. Retrieved <span class="nowrap">July 14,</span> 2014</span>.</cite><span title="ctx_ver=Z39.88-2004&amp;rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematics&amp;rft.atitle=St.+Augustine+v.+The+Mathematicians&amp;rft.aufirst=Barry&amp;rft.aulast=Cipra&amp;rft.date=1982&amp;rft.genre=unknown&amp;rft_id=https%3A%2F%2Fpeople.math.osu.edu%2Feaswaran.1%2Faugustine.html&amp;rft.jtitle=osu.edu&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal" class="Z3988"><span style="display:none;">&#160;</span></span><sup class="noprint Inline-Template"><span style="white-space: nowrap;">[<i><a href="/wiki/Wikipedia:Link_rot" title="Wikipedia:Link rot"><span title="&#160;since March 2016">dead link</span></a></i>]</span></sup></span></li>
<li id="cite_note-26"><span class="mw-cite-backlink"><b><a href="#cite_ref-26">^</a></b></span> <span class="reference-text"><i><a href="/wiki/The_Oxford_Dictionary_of_English_Etymology" title="The Oxford Dictionary of English Etymology">The Oxford Dictionary of English Etymology</a></i>, <i><a href="/wiki/Oxford_English_Dictionary" title="Oxford English Dictionary">Oxford English Dictionary</a></i>, <i>sub</i> "mathematics", "mathematic", "mathematics"</span></li>
<li id="cite_note-maths-27"><span class="mw-cite-backlink"><b><a href="#cite_ref-maths_27-0">^</a></b></span> <span class="reference-text"><a rel="nofollow" class="external text" href="http://oed.com/view/Entry/114982">"maths, <i>n.</i>"</a> and <a rel="nofollow" class="external text" href="http://oed.com/view/Entry/114962">"math, <i>n.3</i>"</a>. <i>Oxford English Dictionary,</i> on-line version (2012).</span></li>
<li id="cite_note-Franklin-28"><span class="mw-cite-backlink"><b><a href="#cite_ref-Franklin_28-0">^</a></b></span> <span class="reference-text">James Franklin, "Aristotelian Realism" in <i>Philosophy of Mathematics", ed. A.D. Irvine, <a rel="nofollow" class="external text" href="https://books.google.com/books?id=mbn35b2ghgkC&amp;pg=PA104#v=onepage&amp;q&amp;f=false">p. 104</a>. Elsevier (2009).</i></span></li>
<li id="cite_note-Cajori-29"><span class="mw-cite-backlink"><b><a href="#cite_ref-Cajori_29-0">^</a></b></span> <span class="reference-text"><cite class="citation book"><a href="/wiki/Florian_Cajori" title="Florian Cajori">Cajori, Florian</a> (1893). <i>A History of Mathematics</i>. American Mathematical Society (1991 reprint). pp.&#160;<a rel="nofollow" class="external text" href="https://books.google.com/books?id=mGJRjIC9fZgC&amp;pg=PA285">285–6</a>. <a href="/wiki/International_Standard_Book_Number" title="International Standard Book Number">ISBN</a>&#160;<a href="/wiki/Special:BookSources/0-8218-2102-4" title="Special:BookSources/0-8218-2102-4">0-8218-2102-4</a>.</cite><span title="ctx_ver=Z39.88-2004&amp;rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematics&amp;rft.au=Cajori%2C+Florian&amp;rft.btitle=A+History+of+Mathematics&amp;rft.date=1893&amp;rft.genre=book&amp;rft.isbn=0-8218-2102-4&amp;rft.pages=285-6&amp;rft.pub=American+Mathematical+Society+%281991+reprint%29&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;">&#160;</span></span></span></li>
<li id="cite_note-Snapper-30"><span class="mw-cite-backlink">^ <a href="#cite_ref-Snapper_30-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Snapper_30-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-Snapper_30-2"><sup><i><b>c</b></i></sup></a></span> <span class="reference-text"><cite id="CITEREFSnapper1979" class="citation journal">Snapper, Ernst (September 1979). "The Three Crises in Mathematics: Logicism, Intuitionism, and Formalism". <i>Mathematics Magazine</i> <b>52</b> (4): 207–16. <a href="/wiki/Digital_object_identifier" title="Digital object identifier">doi</a>:<a rel="nofollow" class="external text" href="//dx.doi.org/10.2307%2F2689412">10.2307/2689412</a>. <a href="/wiki/JSTOR" title="JSTOR">JSTOR</a>&#160;<a rel="nofollow" class="external text" href="//www.jstor.org/stable/2689412">2689412</a>.</cite><span title="ctx_ver=Z39.88-2004&amp;rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematics&amp;rft.atitle=The+Three+Crises+in+Mathematics%3A+Logicism%2C+Intuitionism%2C+and+Formalism&amp;rft.aufirst=Ernst&amp;rft.aulast=Snapper&amp;rft.date=1979-09&amp;rft.genre=article&amp;rft_id=%2F%2Fwww.jstor.org%2Fstable%2F2689412&amp;rft_id=info%3Adoi%2F10.2307%2F2689412&amp;rft.issue=4&amp;rft.jtitle=Mathematics+Magazine&amp;rft.pages=207-16&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&amp;rft.volume=52" class="Z3988"><span style="display:none;">&#160;</span></span></span></li>
<li id="cite_note-Peirce-31"><span class="mw-cite-backlink"><b><a href="#cite_ref-Peirce_31-0">^</a></b></span> <span class="reference-text"><cite class="citation book"><a href="/wiki/Benjamin_Peirce" title="Benjamin Peirce">Peirce, Benjamin</a> (1882). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=De0GAAAAYAAJ&amp;pg=PA1#v=onepage&amp;q&amp;f=false"><i>Linear Associative Algebra</i></a>. p.&#160;1.</cite><span title="ctx_ver=Z39.88-2004&amp;rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematics&amp;rft.au=Peirce%2C+Benjamin&amp;rft.btitle=Linear+Associative+Algebra&amp;rft.date=1882&amp;rft.genre=book&amp;rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DDe0GAAAAYAAJ%26pg%3DPA1%23v%3Donepage%26q%26f%3Dfalse&amp;rft.pages=1&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;">&#160;</span></span></span></li>
<li id="cite_note-Russell-32"><span class="mw-cite-backlink"><b><a href="#cite_ref-Russell_32-0">^</a></b></span> <span class="reference-text">Bertrand Russell, <i>The Principles of Mathematics,</i> <a rel="nofollow" class="external text" href="https://books.google.com/books?id=kj0a_aV2mxIC&amp;pg=PA5#v=onepage&amp;q&amp;f=false">p. 5</a>. University Press, Cambridge (1903)</span></li>
<li id="cite_note-Curry-33"><span class="mw-cite-backlink"><b><a href="#cite_ref-Curry_33-0">^</a></b></span> <span class="reference-text"><cite class="citation book"><a href="/wiki/Haskell_Curry" title="Haskell Curry">Curry, Haskell</a> (1951). <i>Outlines of a Formalist Philosophy of Mathematics</i>. Elsevier. pp.&#160;<a rel="nofollow" class="external text" href="https://books.google.com/books?id=tZHrBQgp1bkC">56</a>. <a href="/wiki/International_Standard_Book_Number" title="International Standard Book Number">ISBN</a>&#160;<a href="/wiki/Special:BookSources/0-444-53368-0" title="Special:BookSources/0-444-53368-0">0-444-53368-0</a>.</cite><span title="ctx_ver=Z39.88-2004&amp;rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematics&amp;rft.au=Curry%2C+Haskell&amp;rft.btitle=Outlines+of+a+Formalist+Philosophy+of+Mathematics&amp;rft.date=1951&amp;rft.genre=book&amp;rft.isbn=0-444-53368-0&amp;rft.pages=56&amp;rft.pub=Elsevier&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;">&#160;</span></span></span></li>
<li id="cite_note-34"><span class="mw-cite-backlink"><b><a href="#cite_ref-34">^</a></b></span> <span class="reference-text"><a href="/wiki/Marcus_du_Sautoy" title="Marcus du Sautoy">Marcus du Sautoy</a>, <i><a rel="nofollow" class="external text" href="http://www.bbc.co.uk/programmes/b00stcgv">A Brief History of Mathematics: 10. Nicolas Bourbaki</a></i>, <a href="/wiki/BBC_Radio_4" title="BBC Radio 4">BBC Radio 4</a>, October 1, 2010.</span></li>
<li id="cite_note-35"><span class="mw-cite-backlink"><b><a href="#cite_ref-35">^</a></b></span> <span class="reference-text"><cite class="citation book">Shasha, Dennis Elliot; Lazere, Cathy A. (1998). <i>Out of Their Minds: The Lives and Discoveries of 15 Great Computer Scientists</i>. Springer. p.&#160;228.</cite><span title="ctx_ver=Z39.88-2004&amp;rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematics&amp;rft.au=Shasha%2C+Dennis+Elliot%3B+Lazere%2C+Cathy+A.&amp;rft.btitle=Out+of+Their+Minds%3A+The+Lives+and+Discoveries+of+15+Great+Computer+Scientists&amp;rft.date=1998&amp;rft.genre=book&amp;rft.pages=228&amp;rft.pub=Springer&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;">&#160;</span></span></span></li>
<li id="cite_note-36"><span class="mw-cite-backlink"><b><a href="#cite_ref-36">^</a></b></span> <span class="reference-text">Popper 1995, p. 56</span></li>
<li id="cite_note-37"><span class="mw-cite-backlink"><b><a href="#cite_ref-37">^</a></b></span> <span class="reference-text">Ziman</span></li>
<li id="cite_note-38"><span class="mw-cite-backlink"><b><a href="#cite_ref-38">^</a></b></span> <span class="reference-text"><cite class="citation book">Johnson, Gerald W.; Lapidus, Michel L. (2002). <i>The Feynman Integral and Feynman's Operational Calculus</i>. <a href="/wiki/Oxford_University_Press" title="Oxford University Press">Oxford University Press</a>. <a href="/wiki/International_Standard_Book_Number" title="International Standard Book Number">ISBN</a>&#160;<a href="/wiki/Special:BookSources/0-8218-2413-9" title="Special:BookSources/0-8218-2413-9">0-8218-2413-9</a>.</cite><span title="ctx_ver=Z39.88-2004&amp;rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematics&amp;rft.au=Johnson%2C+Gerald+W.%3B+Lapidus%2C+Michel+L.&amp;rft.btitle=The+Feynman+Integral+and+Feynman%27s+Operational+Calculus&amp;rft.date=2002&amp;rft.genre=book&amp;rft.isbn=0-8218-2413-9&amp;rft.pub=Oxford+University+Press&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;">&#160;</span></span></span></li>
<li id="cite_note-39"><span class="mw-cite-backlink"><b><a href="#cite_ref-39">^</a></b></span> <span class="reference-text"><cite id="CITEREFWigner1960" class="citation journal">Wigner, Eugene (1960). <a rel="nofollow" class="external text" href="http://www.dartmouth.edu/~matc/MathDrama/reading/Wigner.html">"The Unreasonable Effectiveness of Mathematics in the Natural Sciences"</a>. <i><a href="/wiki/Communications_on_Pure_and_Applied_Mathematics" title="Communications on Pure and Applied Mathematics">Communications on Pure and Applied Mathematics</a></i> <b>13</b> (1): 1–14. <a href="/wiki/Digital_object_identifier" title="Digital object identifier">doi</a>:<a rel="nofollow" class="external text" href="//dx.doi.org/10.1002%2Fcpa.3160130102">10.1002/cpa.3160130102</a>.</cite><span title="ctx_ver=Z39.88-2004&amp;rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematics&amp;rft.atitle=The+Unreasonable+Effectiveness+of+Mathematics+in+the+Natural+Sciences&amp;rft.aufirst=Eugene&amp;rft.aulast=Wigner&amp;rft.date=1960&amp;rft.genre=article&amp;rft_id=http%3A%2F%2Fwww.dartmouth.edu%2F~matc%2FMathDrama%2Freading%2FWigner.html&amp;rft_id=info%3Adoi%2F10.1002%2Fcpa.3160130102&amp;rft.issue=1&amp;rft.jtitle=Communications+on+Pure+and+Applied+Mathematics&amp;rft.pages=1-14&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&amp;rft.volume=13" class="Z3988"><span style="display:none;">&#160;</span></span></span></li>
<li id="cite_note-40"><span class="mw-cite-backlink"><b><a href="#cite_ref-40">^</a></b></span> <span class="reference-text"><cite class="citation web"><a rel="nofollow" class="external text" href="http://www.ams.org/mathscinet/msc/pdfs/classification2010.pdf">"Mathematics Subject Classification 2010"</a> <span style="font-size:85%;">(PDF)</span><span class="reference-accessdate">. Retrieved <span class="nowrap">November 9,</span> 2010</span>.</cite><span title="ctx_ver=Z39.88-2004&amp;rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematics&amp;rft.btitle=Mathematics+Subject+Classification+2010&amp;rft.genre=unknown&amp;rft_id=http%3A%2F%2Fwww.ams.org%2Fmathscinet%2Fmsc%2Fpdfs%2Fclassification2010.pdf&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;">&#160;</span></span></span></li>
<li id="cite_note-41"><span class="mw-cite-backlink"><b><a href="#cite_ref-41">^</a></b></span> <span class="reference-text"><cite class="citation book">Hardy, G.H. (1940). <i>A Mathematician's Apology</i>. Cambridge University Press. <a href="/wiki/International_Standard_Book_Number" title="International Standard Book Number">ISBN</a>&#160;<a href="/wiki/Special:BookSources/0-521-42706-1" title="Special:BookSources/0-521-42706-1">0-521-42706-1</a>.</cite><span title="ctx_ver=Z39.88-2004&amp;rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematics&amp;rft.au=Hardy%2C+G.H.&amp;rft.btitle=A+Mathematician%27s+Apology&amp;rft.date=1940&amp;rft.genre=book&amp;rft.isbn=0-521-42706-1&amp;rft.pub=Cambridge+University+Press&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;">&#160;</span></span></span></li>
<li id="cite_note-42"><span class="mw-cite-backlink"><b><a href="#cite_ref-42">^</a></b></span> <span class="reference-text"><cite class="citation book">Gold, Bonnie; Simons, Rogers A. (2008). <i>Proof and Other Dilemmas: Mathematics and Philosophy</i>. MAA.</cite><span title="ctx_ver=Z39.88-2004&amp;rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematics&amp;rft.au=Gold%2C+Bonnie%3B+Simons%2C+Rogers+A.&amp;rft.btitle=Proof+and+Other+Dilemmas%3A+Mathematics+and+Philosophy&amp;rft.date=2008&amp;rft.genre=book&amp;rft.pub=MAA&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;">&#160;</span></span></span></li>
<li id="cite_note-43"><span class="mw-cite-backlink"><b><a href="#cite_ref-43">^</a></b></span> <span class="reference-text"><cite class="citation book">Aigner, Martin; <a href="/wiki/G%C3%BCnter_M._Ziegler" title="Günter M. Ziegler">Ziegler, Günter&#160;M.</a> (2001). <i>Proofs from</i> The Book. Springer. <a href="/wiki/International_Standard_Book_Number" title="International Standard Book Number">ISBN</a>&#160;<a href="/wiki/Special:BookSources/3-540-40460-0" title="Special:BookSources/3-540-40460-0">3-540-40460-0</a>.</cite><span title="ctx_ver=Z39.88-2004&amp;rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematics&amp;rft.aufirst=Martin&amp;rft.aulast=Aigner&amp;rft.au=Ziegler%2C+G%C3%BCnter+M.&amp;rft.btitle=Proofs+from+The+Book&amp;rft.date=2001&amp;rft.genre=book&amp;rft.isbn=3-540-40460-0&amp;rft.pub=Springer&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;">&#160;</span></span></span></li>
<li id="cite_note-44"><span class="mw-cite-backlink"><b><a href="#cite_ref-44">^</a></b></span> <span class="reference-text"><cite class="citation web"><a rel="nofollow" class="external text" href="http://jeff560.tripod.com/mathsym.html">"Earliest Uses of Various Mathematical Symbols"</a><span class="reference-accessdate">. Retrieved <span class="nowrap">September 14,</span> 2014</span>.</cite><span title="ctx_ver=Z39.88-2004&amp;rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematics&amp;rft.btitle=Earliest+Uses+of+Various+Mathematical+Symbols&amp;rft.genre=unknown&amp;rft_id=http%3A%2F%2Fjeff560.tripod.com%2Fmathsym.html&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;">&#160;</span></span></span></li>
<li id="cite_note-45"><span class="mw-cite-backlink"><b><a href="#cite_ref-45">^</a></b></span> <span class="reference-text">Kline, p. 140, on <a href="/wiki/Diophantus" title="Diophantus">Diophantus</a>; p. 261, on <a href="/wiki/Franciscus_Vieta" title="Franciscus Vieta" class="mw-redirect">Vieta</a>.</span></li>
<li id="cite_note-46"><span class="mw-cite-backlink"><b><a href="#cite_ref-46">^</a></b></span> <span class="reference-text">See <i><a href="/wiki/False_proof" title="False proof" class="mw-redirect">false proof</a></i> for simple examples of what can go wrong in a formal proof.</span></li>
<li id="cite_note-47"><span class="mw-cite-backlink"><b><a href="#cite_ref-47">^</a></b></span> <span class="reference-text">Ivars Peterson, <i>The Mathematical Tourist</i>, Freeman, 1988, <a href="/wiki/Special:BookSources/0716719533" class="internal mw-magiclink-isbn">ISBN 0-7167-1953-3</a>. p. 4 "A few complain that the computer program can't be verified properly", (in reference to the Haken–Apple proof of the Four Color Theorem).</span></li>
<li id="cite_note-48"><span class="mw-cite-backlink"><b><a href="#cite_ref-48">^</a></b></span> <span class="reference-text">" The method of "postulating" what we want has many advantages; they are the same as the advantages of theft over honest toil." <a href="/wiki/Bertrand_Russell" title="Bertrand Russell">Bertrand Russell</a> (1919), <i>Introduction to Mathematical Philosophy</i>, New York and London, <a rel="nofollow" class="external text" href="http://www-history.mcs.st-and.ac.uk/Quotations/Russell.html">p 71.</a></span></li>
<li id="cite_note-49"><span class="mw-cite-backlink"><b><a href="#cite_ref-49">^</a></b></span> <span class="reference-text">Patrick Suppes, <i>Axiomatic Set Theory</i>, Dover, 1972, <a href="/wiki/Special:BookSources/0486616304" class="internal mw-magiclink-isbn">ISBN 0-486-61630-4</a>. p. 1, "Among the many branches of modern mathematics set theory occupies a unique place: with a few rare exceptions the entities which are studied and analyzed in mathematics may be regarded as certain particular sets or classes of objects."</span></li>
<li id="cite_note-50"><span class="mw-cite-backlink"><b><a href="#cite_ref-50">^</a></b></span> <span class="reference-text">Luke Howard Hodgkin &amp; Luke Hodgkin, <i>A History of Mathematics</i>, Oxford University Press, 2005.</span></li>
<li id="cite_note-51"><span class="mw-cite-backlink"><b><a href="#cite_ref-51">^</a></b></span> <span class="reference-text"><a rel="nofollow" class="external text" href="http://www.webcitation.org/5Qj76uCbF">Clay Mathematics Institute</a>, P=NP, claymath.org</span></li>
<li id="cite_note-52"><span class="mw-cite-backlink"><b><a href="#cite_ref-52">^</a></b></span> <span class="reference-text"><a href="/wiki/C.R._Rao" title="C.R. Rao" class="mw-redirect">Rao, C.R.</a> (1997) <i>Statistics and Truth: Putting Chance to Work</i>, World Scientific. <a href="/wiki/Special:BookSources/9810231113" class="internal mw-magiclink-isbn">ISBN 981-02-3111-3</a></span></li>
<li id="cite_note-53"><span class="mw-cite-backlink"><b><a href="#cite_ref-53">^</a></b></span> <span class="reference-text">Like other mathematical sciences such as <a href="/wiki/Physics" title="Physics">physics</a> and <a href="/wiki/Computer_science" title="Computer science">computer science</a>, 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.</span></li>
<li id="cite_note-RaoOpt-54"><span class="mw-cite-backlink"><b><a href="#cite_ref-RaoOpt_54-0">^</a></b></span> <span class="reference-text"><cite id="CITEREFRao1981" class="citation book"><a href="/wiki/C.R._Rao" title="C.R. Rao" class="mw-redirect">Rao, C.R.</a> (1981). "Foreword". In Arthanari, T.S.; <a href="/wiki/Yadolah_Dodge" title="Yadolah Dodge">Dodge, Yadolah</a>. <i>Mathematical programming in statistics</i>. Wiley Series in Probability and Mathematical Statistics. New York: Wiley. pp.&#160;vii–viii. <a href="/wiki/International_Standard_Book_Number" title="International Standard Book Number">ISBN</a>&#160;<a href="/wiki/Special:BookSources/0-471-08073-X" title="Special:BookSources/0-471-08073-X">0-471-08073-X</a>. <a href="/wiki/Mathematical_Reviews" title="Mathematical Reviews">MR</a>&#160;<a rel="nofollow" class="external text" href="//www.ams.org/mathscinet-getitem?mr=607328">607328</a>.</cite><span title="ctx_ver=Z39.88-2004&amp;rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematics&amp;rft.atitle=Foreword&amp;rft.aufirst=C.R.&amp;rft.aulast=Rao&amp;rft.btitle=Mathematical+programming+in+statistics&amp;rft.date=1981&amp;rft.genre=bookitem&amp;rft_id=%2F%2Fwww.ams.org%2Fmathscinet-getitem%3Fmr%3D607328&amp;rft.isbn=0-471-08073-X&amp;rft.pages=vii-viii&amp;rft.place=New+York&amp;rft.pub=Wiley&amp;rft.series=Wiley+Series+in+Probability+and+Mathematical+Statistics&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;">&#160;</span></span></span></li>
<li id="cite_note-Whittle-55"><span class="mw-cite-backlink"><b><a href="#cite_ref-Whittle_55-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFWhittle1994">Whittle (1994</a>, pp.&#160;10–11 and 14–18): <cite id="CITEREFWhittle1994" class="citation book"><a href="/wiki/Peter_Whittle" title="Peter Whittle">Whittle, Peter</a> (1994). "Almost home". In <a href="/wiki/Frank_Kelly_(mathematician)" title="Frank Kelly (mathematician)">Kelly, F.P.</a> <a rel="nofollow" class="external text" href="http://www.statslab.cam.ac.uk/History/2history.html#6._1966--72:_The_Churchill_Chair"><i>Probability, statistics and optimisation: A Tribute to Peter Whittle</i></a> (previously "A realised path: The Cambridge Statistical Laboratory upto 1993 (revised 2002)" ed.). Chichester: John Wiley. pp.&#160;1–28. <a href="/wiki/International_Standard_Book_Number" title="International Standard Book Number">ISBN</a>&#160;<a href="/wiki/Special:BookSources/0-471-94829-2" title="Special:BookSources/0-471-94829-2">0-471-94829-2</a>.</cite><span title="ctx_ver=Z39.88-2004&amp;rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematics&amp;rft.atitle=Almost+home&amp;rft.aufirst=Peter&amp;rft.aulast=Whittle&amp;rft.btitle=Probability%2C+statistics+and+optimisation%3A+A+Tribute+to+Peter+Whittle&amp;rft.date=1994&amp;rft.edition=previously+%22A+realised+path%3A+The+Cambridge+Statistical+Laboratory+upto+1993+%28revised+2002%29%22&amp;rft.genre=bookitem&amp;rft_id=http%3A%2F%2Fwww.statslab.cam.ac.uk%2FHistory%2F2history.html%236._1966--72%3A_The_Churchill_Chair&amp;rft.isbn=0-471-94829-2&amp;rft.pages=1-28&amp;rft.place=Chichester&amp;rft.pub=John+Wiley&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;">&#160;</span></span></span></li>
<li id="cite_note-FOOTNOTEMonastyrsky2001-56"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEMonastyrsky2001_56-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFMonastyrsky2001">Monastyrsky 2001</a>: "<i>The Fields Medal is now indisputably the best known and most influential award in mathematics.</i>"</span></li>
<li id="cite_note-FOOTNOTERiehm2002778.E2.80.93782-57"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTERiehm2002778.E2.80.93782_57-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFRiehm2002">Riehm 2002</a>, pp.&#160;778–782.</span></li>
</ol>
</div>
<h2><span class="mw-headline" id="References">References</span></h2>
<div class="refbegin columns references-column-width" style="-moz-column-width: 30em; -webkit-column-width: 30em; column-width: 30em;">
<ul>
<li><a href="/wiki/Richard_Courant" title="Richard Courant">Courant, Richard</a> and <a href="/wiki/Herbert_Robbins" title="Herbert Robbins">H. Robbins</a>, <i>What Is Mathematics?&#160;: An Elementary Approach to Ideas and Methods</i>, Oxford University Press, USA; 2 edition (July 18, 1996). <a href="/wiki/Special:BookSources/0195105192" class="internal mw-magiclink-isbn">ISBN 0-19-510519-2</a>.</li>
<li><cite class="citation book"><a href="/wiki/Albert_Einstein" title="Albert Einstein">Einstein, Albert</a> (1923). <a rel="nofollow" class="external text" href="http://searchworks.stanford.edu/view/1216826"><i>Sidelights on Relativity: I. Ether and relativity. II. Geometry and experience (translated by G.B. Jeffery, D.Sc., and W. Perrett, Ph.D).</i></a> E.P. Dutton &amp; Co., New York.</cite><span title="ctx_ver=Z39.88-2004&amp;rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematics&amp;rft.aufirst=Albert&amp;rft.aulast=Einstein&amp;rft.btitle=Sidelights+on+Relativity%3A+I.+Ether+and+relativity.+II.+Geometry+and+experience+%28translated+by+G.B.+Jeffery%2C+D.Sc.%2C+and+W.+Perrett%2C+Ph.D%29.&amp;rft.date=1923&amp;rft.genre=book&amp;rft_id=http%3A%2F%2Fsearchworks.stanford.edu%2Fview%2F1216826&amp;rft.pub=E.P.+Dutton+%26+Co.%2C+New+York&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;">&#160;</span></span></li>
<li><a href="/wiki/Marcus_du_Sautoy" title="Marcus du Sautoy">du Sautoy, Marcus</a>, <i><a rel="nofollow" class="external text" href="http://www.bbc.co.uk/podcasts/series/maths">A Brief History of Mathematics</a></i>, <a href="/wiki/BBC_Radio_4" title="BBC Radio 4">BBC Radio 4</a> (2010).</li>
<li>Eves, Howard, <i>An Introduction to the History of Mathematics</i>, Sixth Edition, Saunders, 1990, <a href="/wiki/Special:BookSources/0030295580" class="internal mw-magiclink-isbn">ISBN 0-03-029558-0</a>.</li>
<li><a href="/wiki/Morris_Kline" title="Morris Kline">Kline, Morris</a>, <i>Mathematical Thought from Ancient to Modern Times</i>, Oxford University Press, USA; Paperback edition (March 1, 1990). <a href="/wiki/Special:BookSources/0195061357" class="internal mw-magiclink-isbn">ISBN 0-19-506135-7</a>.</li>
<li><cite id="CITEREFMonastyrsky2001" class="citation journal">Monastyrsky, Michael (2001). <a rel="nofollow" class="external text" href="http://www.fields.utoronto.ca/aboutus/FieldsMedal_Monastyrsky.pdf">"Some Trends in Modern Mathematics and the Fields Medal"</a> <span style="font-size:85%;">(PDF)</span>. Canadian Mathematical Society<span class="reference-accessdate">. Retrieved <span class="nowrap">July 28,</span> 2006</span>.</cite><span title="ctx_ver=Z39.88-2004&amp;rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematics&amp;rft.atitle=Some+Trends+in+Modern+Mathematics+and+the+Fields+Medal&amp;rft.aufirst=Michael&amp;rft.aulast=Monastyrsky&amp;rft.date=2001&amp;rft.genre=article&amp;rft_id=http%3A%2F%2Fwww.fields.utoronto.ca%2Faboutus%2FFieldsMedal_Monastyrsky.pdf&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal" class="Z3988"><span style="display:none;">&#160;</span></span></li>
<li><a href="/wiki/Oxford_English_Dictionary" title="Oxford English Dictionary">Oxford English Dictionary</a>, second edition, ed. John Simpson and Edmund Weiner, <a href="/wiki/Clarendon_Press" title="Clarendon Press" class="mw-redirect">Clarendon Press</a>, 1989, <a href="/wiki/Special:BookSources/0198611862" class="internal mw-magiclink-isbn">ISBN 0-19-861186-2</a>.</li>
<li><i><a href="/wiki/The_Oxford_Dictionary_of_English_Etymology" title="The Oxford Dictionary of English Etymology">The Oxford Dictionary of English Etymology</a></i>, 1983 reprint. <a href="/wiki/Special:BookSources/0198611129" class="internal mw-magiclink-isbn">ISBN 0-19-861112-9</a>.</li>
<li>Pappas, Theoni, <i>The Joy Of Mathematics</i>, Wide World Publishing; Revised edition (June 1989). <a href="/wiki/Special:BookSources/0933174659" class="internal mw-magiclink-isbn">ISBN 0-933174-65-9</a>.</li>
<li><cite id="CITEREFPeirce1881" class="citation journal"><a href="/wiki/Benjamin_Peirce" title="Benjamin Peirce">Peirce, Benjamin</a> (1881). <a href="/wiki/Charles_Sanders_Peirce" title="Charles Sanders Peirce">Peirce, Charles&#160;Sanders</a>, ed. <a rel="nofollow" class="external text" href="https://books.google.com/?id=De0GAAAAYAAJ&amp;pg=PA1&amp;dq=Peirce+Benjamin+Linear+Associative+Algebra+&amp;q=">"Linear associative algebra"</a>. <i>American Journal of Mathematics</i> (Corrected, expanded, and annotated revision with an 1875 paper by B.&#160;Peirce and annotations by his son, C.S. Peirce, of the 1872 lithograph ed.) (Johns Hopkins University) <b>4</b> (1–4): 97–229. <a href="/wiki/Digital_object_identifier" title="Digital object identifier">doi</a>:<a rel="nofollow" class="external text" href="//dx.doi.org/10.2307%2F2369153">10.2307/2369153</a>. <a href="/wiki/JSTOR" title="JSTOR">JSTOR</a>&#160;<a rel="nofollow" class="external text" href="//www.jstor.org/stable/2369153">2369153</a>. Corrected, expanded, and annotated revision with an 1875 paper by B.&#160;Peirce and annotations by his son, C.&#160;S.&#160;Peirce, of the 1872 lithograph ed. <i>Google</i> <a rel="nofollow" class="external text" href="https://books.google.com/books?id=LQgPAAAAIAAJ&amp;pg=PA221">Eprint</a> and as an extract, D.&#160;Van Nostrand, 1882, <i>Google</i> <a rel="nofollow" class="external text" href="https://books.google.com/books?id=De0GAAAAYAAJ&amp;printsec=frontcover">Eprint</a>.</cite><span title="ctx_ver=Z39.88-2004&amp;rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematics&amp;rft.atitle=Linear+associative+algebra&amp;rft.aufirst=Benjamin&amp;rft.aulast=Peirce&amp;rft.date=1881&amp;rft.genre=article&amp;rft_id=%2F%2Fwww.jstor.org%2Fstable%2F2369153&amp;rft_id=https%3A%2F%2Fbooks.google.com%2F%3Fid%3DDe0GAAAAYAAJ%26pg%3DPA1%26dq%3DPeirce%2BBenjamin%2BLinear%2BAssociative%2BAlgebra%2B%26q%3D&amp;rft_id=info%3Adoi%2F10.2307%2F2369153&amp;rft.issue=1%934&amp;rft.jtitle=American+Journal+of+Mathematics&amp;rft.pages=97-229&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&amp;rft.volume=4" class="Z3988"><span style="display:none;">&#160;</span></span>.</li>
<li>Peterson, Ivars, <i>Mathematical Tourist, New and Updated Snapshots of Modern Mathematics</i>, Owl Books, 2001, <a href="/wiki/Special:BookSources/0805071598" class="internal mw-magiclink-isbn">ISBN 0-8050-7159-8</a>.</li>
<li><cite class="citation book"><a href="/wiki/Karl_Popper" title="Karl Popper">Popper, Karl R.</a> (1995). "On knowledge". <i>In Search of a Better World: Lectures and Essays from Thirty Years</i>. Routledge. <a href="/wiki/International_Standard_Book_Number" title="International Standard Book Number">ISBN</a>&#160;<a href="/wiki/Special:BookSources/0-415-13548-6" title="Special:BookSources/0-415-13548-6">0-415-13548-6</a>.</cite><span title="ctx_ver=Z39.88-2004&amp;rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematics&amp;rft.atitle=On+knowledge&amp;rft.aufirst=Karl+R.&amp;rft.aulast=Popper&amp;rft.btitle=In+Search+of+a+Better+World%3A+Lectures+and+Essays+from+Thirty+Years&amp;rft.date=1995&amp;rft.genre=bookitem&amp;rft.isbn=0-415-13548-6&amp;rft.pub=Routledge&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;">&#160;</span></span></li>
<li><cite id="CITEREFRiehm2002" class="citation journal">Riehm, Carl (August 2002). <a rel="nofollow" class="external text" href="http://www.ams.org/notices/200207/comm-riehm.pdf">"The Early History of the Fields Medal"</a> <span style="font-size:85%;">(PDF)</span>. <i>Notices of the AMS</i> (AMS) <b>49</b> (7): 778–782.</cite><span title="ctx_ver=Z39.88-2004&amp;rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematics&amp;rft.atitle=The+Early+History+of+the+Fields+Medal&amp;rft.aufirst=Carl&amp;rft.aulast=Riehm&amp;rft.date=2002-08&amp;rft.genre=article&amp;rft_id=http%3A%2F%2Fwww.ams.org%2Fnotices%2F200207%2Fcomm-riehm.pdf&amp;rft.issue=7&amp;rft.jtitle=Notices+of+the+AMS&amp;rft.pages=778-782&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&amp;rft.volume=49" class="Z3988"><span style="display:none;">&#160;</span></span></li>
<li><cite id="CITEREFSevryuk2006" class="citation journal">Sevryuk, Mikhail B. (January 2006). <a rel="nofollow" class="external text" href="http://www.ams.org/bull/2006-43-01/S0273-0979-05-01069-4/S0273-0979-05-01069-4.pdf">"Book Reviews"</a> <span style="font-size:85%;">(PDF)</span>. <i><a href="/wiki/Bulletin_of_the_American_Mathematical_Society" title="Bulletin of the American Mathematical Society">Bulletin of the American Mathematical Society</a></i> <b>43</b> (1): 101–109. <a href="/wiki/Digital_object_identifier" title="Digital object identifier">doi</a>:<a rel="nofollow" class="external text" href="//dx.doi.org/10.1090%2FS0273-0979-05-01069-4">10.1090/S0273-0979-05-01069-4</a><span class="reference-accessdate">. Retrieved <span class="nowrap">June 24,</span> 2006</span>.</cite><span title="ctx_ver=Z39.88-2004&amp;rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematics&amp;rft.atitle=Book+Reviews&amp;rft.aufirst=Mikhail+B.&amp;rft.aulast=Sevryuk&amp;rft.date=2006-01&amp;rft.genre=article&amp;rft_id=http%3A%2F%2Fwww.ams.org%2Fbull%2F2006-43-01%2FS0273-0979-05-01069-4%2FS0273-0979-05-01069-4.pdf&amp;rft_id=info%3Adoi%2F10.1090%2FS0273-0979-05-01069-4&amp;rft.issue=1&amp;rft.jtitle=Bulletin+of+the+American+Mathematical+Society&amp;rft.pages=101-109&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&amp;rft.volume=43" class="Z3988"><span style="display:none;">&#160;</span></span></li>
<li><cite class="citation book"><a href="/wiki/Wolfgang_Sartorius_von_Waltershausen" title="Wolfgang Sartorius von Waltershausen">Waltershausen, Wolfgang Sartorius von</a> (1965) [first published 1856]. <i>Gauss zum Gedächtniss</i>. Sändig Reprint Verlag H. R. Wohlwend. <a href="/wiki/Amazon_Standard_Identification_Number" title="Amazon Standard Identification Number">ASIN</a>&#160;<a rel="nofollow" class="external text" href="//www.amazon.com/dp/B0000BN5SQ">B0000BN5SQ</a>. <a href="/wiki/International_Standard_Book_Number" title="International Standard Book Number">ISBN</a>&#160;<a href="/wiki/Special:BookSources/3-253-01702-8" title="Special:BookSources/3-253-01702-8">3-253-01702-8</a>. <a href="/wiki/Amazon_Standard_Identification_Number" title="Amazon Standard Identification Number">ASIN</a>&#160;<a rel="nofollow" class="external text" href="http://www.amazon.de/dp/3253017028">3253017028</a>.</cite><span title="ctx_ver=Z39.88-2004&amp;rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematics&amp;rft.aufirst=Wolfgang+Sartorius+von&amp;rft.aulast=Waltershausen&amp;rft.btitle=Gauss+zum+Ged%C3%A4chtniss&amp;rft.date=1965&amp;rft.genre=book&amp;rft.isbn=3-253-01702-8&amp;rft.pub=S%C3%A4ndig+Reprint+Verlag+H.+R.+Wohlwend&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;">&#160;</span></span></li>
</ul>
</div>
<h2><span class="mw-headline" id="Further_reading">Further reading</span></h2>
<div class="refbegin columns references-column-width" style="-moz-column-width: 30em; -webkit-column-width: 30em; column-width: 30em;">
<ul>
<li>Benson, Donald C., <i>The Moment of Proof: Mathematical Epiphanies</i>, <a href="/wiki/Oxford_University_Press" title="Oxford University Press">Oxford University Press</a>, USA; New Ed edition (December 14, 2000). <a href="/wiki/Special:BookSources/0195139194" class="internal mw-magiclink-isbn">ISBN 0-19-513919-4</a>.</li>
<li><a href="/wiki/Carl_B._Boyer" title="Carl B. Boyer" class="mw-redirect">Boyer, Carl B.</a>, <i>A History of Mathematics</i>, Wiley; 2nd edition, revised by Uta C. Merzbach, (March 6, 1991). <a href="/wiki/Special:BookSources/0471543977" class="internal mw-magiclink-isbn">ISBN 0-471-54397-7</a>.—A concise history of mathematics from the Concept of Number to contemporary Mathematics.</li>
<li><a href="/wiki/Philip_J._Davis" title="Philip J. Davis">Davis, Philip J.</a> and <a href="/wiki/Reuben_Hersh" title="Reuben Hersh">Hersh, Reuben</a>, <i><a href="/wiki/The_Mathematical_Experience" title="The Mathematical Experience">The Mathematical Experience</a></i>. Mariner Books; Reprint edition (January 14, 1999). <a href="/wiki/Special:BookSources/0395929687" class="internal mw-magiclink-isbn">ISBN 0-395-92968-7</a>.</li>
<li><a href="/wiki/Jan_Gullberg" title="Jan Gullberg">Gullberg, Jan</a>, <i>Mathematics&#160;– From the Birth of Numbers</i>. <a href="/wiki/W._W._Norton_%26_Company" title="W. W. Norton &amp; Company">W. W. Norton &amp; Company</a>; 1st edition (October 1997). <a href="/wiki/Special:BookSources/039304002X" class="internal mw-magiclink-isbn">ISBN 0-393-04002-X</a>.</li>
<li><a href="/wiki/Hazewinkel,_Michiel" title="Hazewinkel, Michiel" class="mw-redirect">Hazewinkel, Michiel</a> (ed.), <i><a href="/wiki/Encyclopaedia_of_Mathematics" title="Encyclopaedia of Mathematics" class="mw-redirect">Encyclopaedia of Mathematics</a></i>. <a href="/wiki/Kluwer_Academic_Publishers" title="Kluwer Academic Publishers" class="mw-redirect">Kluwer Academic Publishers</a> 2000.&#160;– A translated and expanded version of a Soviet mathematics encyclopedia, in ten (expensive) volumes, the most complete and authoritative work available. Also in paperback and on CD-ROM, and <a rel="nofollow" class="external text" href="http://www.encyclopediaofmath.org/">online</a>.</li>
<li>Jourdain, Philip E. B., <i>The Nature of Mathematics</i>, in <i>The World of Mathematics</i>, James R. Newman, editor, <a href="/wiki/Dover_Publications" title="Dover Publications">Dover Publications</a>, 2003, <a href="/wiki/Special:BookSources/0486432688" class="internal mw-magiclink-isbn">ISBN 0-486-43268-8</a>.</li>
<li>Maier, Annaliese, <i>At the Threshold of Exact Science: Selected Writings of Annaliese Maier on Late Medieval Natural Philosophy</i>, edited by Steven Sargent, Philadelphia: University of Pennsylvania Press, 1982.</li>
</ul>
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<h2><span class="mw-headline" id="External_links">External links</span></h2>
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<td style="padding-top:0.4em;line-height:1.2em"><a href="/wiki/Wikipedia:LIBRARY" title="Wikipedia:LIBRARY" class="mw-redirect">Library resources</a> about<br />
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<li><a class="external text" href="//tools.wmflabs.org/ftl/cgi-bin/ftl?st=wp&amp;su=Mathematics">Resources in your library</a></li>
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<div style="width:52px"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/f/f2/Edit-clear.svg/40px-Edit-clear.svg.png" width="40" height="40" srcset="//upload.wikimedia.org/wikipedia/en/thumb/f/f2/Edit-clear.svg/60px-Edit-clear.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/f/f2/Edit-clear.svg/80px-Edit-clear.svg.png 2x" data-file-width="48" data-file-height="48" /></div>
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<td class="mbox-text"><span class="mbox-text-span">This article's <b>use of <a href="/wiki/Wikipedia:External_links" title="Wikipedia:External links">external links</a> may not follow Wikipedia's policies or guidelines</b>. <span class="hide-when-compact">Please <a class="external text" href="//en.wikipedia.org/w/index.php?title=Mathematics&amp;action=edit">improve this article</a> by removing <a href="/wiki/Wikipedia:What_Wikipedia_is_not#Wikipedia_is_not_a_mirror_or_a_repository_of_links.2C_images.2C_or_media_files" title="Wikipedia:What Wikipedia is not">excessive</a> or <a href="/wiki/Wikipedia:External_links" title="Wikipedia:External links">inappropriate</a> external links, and converting useful links where appropriate into <a href="/wiki/Wikipedia:Citing_sources" title="Wikipedia:Citing sources">footnote references</a>.</span> <small><i>(October 2015)</i></small></span></td>
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<div class="refbegin" style="">
<ul>
<li><a rel="nofollow" class="external text" href="http://www.britannica.com/EBchecked/topic/369194">Mathematics</a> at <i><a href="/wiki/Encyclop%C3%A6dia_Britannica" title="Encyclopædia Britannica">Encyclopædia Britannica</a></i></li>
<li class="mw-empty-li"></li>
<li><a rel="nofollow" class="external text" href="http://www.bbc.co.uk/programmes/p00545hk">Mathematics</a> on <a href="/wiki/In_Our_Time_(BBC_Radio_4)" title="In Our Time (BBC Radio 4)" class="mw-redirect"><i>In Our Time</i></a> at the <a href="/wiki/BBC" title="BBC">BBC</a>. (<a rel="nofollow" class="external text" href="http://www.bbc.co.uk/iplayer/console/p00545hk/In_Our_Time_Mathematics">listen now</a>)</li>
<li><a rel="nofollow" class="external text" href="http://freebookcentre.net/SpecialCat/Free-Mathematics-Books-Download.html">Free Mathematics books</a> Free Mathematics books collection.</li>
<li><a href="/wiki/Encyclopaedia_of_Mathematics" title="Encyclopaedia of Mathematics" class="mw-redirect">Encyclopaedia of Mathematics</a> online encyclopaedia from <a rel="nofollow" class="external text" href="http://www.encyclopediaofmath.org/">Springer</a>, Graduate-level reference work with over 8,000 entries, illuminating nearly 50,000 notions in mathematics.</li>
<li><a rel="nofollow" class="external text" href="http://hyperphysics.phy-astr.gsu.edu/Hbase/hmat.html">HyperMath site at Georgia State University</a></li>
<li><a rel="nofollow" class="external text" href="http://www.freescience.info/mathematics.php">FreeScience Library</a> The mathematics section of FreeScience library</li>
<li>Rusin, Dave: <i><a rel="nofollow" class="external text" href="http://www.math-atlas.org/">The Mathematical Atlas</a><sup class="noprint Inline-Template"><span style="white-space: nowrap;">[<i><a href="/wiki/Wikipedia:Link_rot" title="Wikipedia:Link rot"><span title="&#160;since March 2016">dead link</span></a></i>]</span></sup></i>. A guided tour through the various branches of modern mathematics. (Can also be found at <a rel="nofollow" class="external text" href="http://www.math.niu.edu/~rusin/known-math/index/index.html">NIU.edu</a><sup class="noprint Inline-Template"><span style="white-space: nowrap;">[<i><a href="/wiki/Wikipedia:Link_rot" title="Wikipedia:Link rot"><span title="&#160;since March 2016">dead link</span></a></i>]</span></sup>.)</li>
<li>Polyanin, Andrei: <i><a rel="nofollow" class="external text" href="http://eqworld.ipmnet.ru/">EqWorld: The World of Mathematical Equations</a></i>. An online resource focusing on algebraic, ordinary differential, partial differential (<a href="/wiki/Mathematical_physics" title="Mathematical physics">mathematical physics</a>), integral, and other mathematical equations.</li>
<li>Cain, George: <a rel="nofollow" class="external text" href="http://www.math.gatech.edu/~cain/textbooks/onlinebooks.html">Online Mathematics Textbooks</a> available free online.</li>
<li><a rel="nofollow" class="external text" href="http://www.tricki.org/">Tricki</a>, Wiki-style site that is intended to develop into a large store of useful mathematical problem-solving techniques.</li>
<li><a rel="nofollow" class="external text" href="http://mathworld.wolfram.com/">Wolfram Mathworld</a></li>
<li><a rel="nofollow" class="external text" href="http://www.math.chapman.edu/~jipsen/structures/doku.php/">Mathematical Structures</a>, list information about classes of mathematical structures.</li>
<li><a rel="nofollow" class="external text" href="http://www-history.mcs.st-and.ac.uk/~history/">Mathematician Biographies</a>. The <a href="/wiki/MacTutor_History_of_Mathematics_archive" title="MacTutor History of Mathematics archive">MacTutor History of Mathematics archive</a> Extensive history and quotes from all famous mathematicians.</li>
<li><i><a rel="nofollow" class="external text" href="http://metamath.org/">Metamath</a></i>. A site and a language, that formalize mathematics from its foundations.</li>
<li><a rel="nofollow" class="external text" href="http://www.nrich.maths.org/public/index.php">Nrich</a>, a prize-winning site for students from age five from <a href="/wiki/University_of_Cambridge" title="University of Cambridge">Cambridge University</a></li>
<li><a rel="nofollow" class="external text" href="http://garden.irmacs.sfu.ca/">Open Problem Garden</a>, a <a href="/wiki/Wiki" title="Wiki">wiki</a> of open problems in mathematics</li>
<li><i><a rel="nofollow" class="external text" href="http://planetmath.org/">Planet Math</a></i>. An online mathematics encyclopedia under construction, focusing on modern mathematics. Uses the <a href="/wiki/CC_BY_SA" title="CC BY SA" class="mw-redirect">Attribution-ShareAlike</a> license, allowing article exchange with Wikipedia. Uses <a href="/wiki/TeX" title="TeX">TeX</a> markup.</li>
<li><a rel="nofollow" class="external text" href="http://www-math.mit.edu/daimp">Some mathematics applets, at MIT</a><sup class="noprint Inline-Template"><span style="white-space: nowrap;">[<i><a href="/wiki/Wikipedia:Link_rot" title="Wikipedia:Link rot"><span title="&#160;since March 2016">dead link</span></a></i>]</span></sup></li>
<li>Weisstein, Eric et al.: <i><a rel="nofollow" class="external text" href="http://www.mathworld.com/">MathWorld: World of Mathematics</a></i>. An online encyclopedia of mathematics.</li>
<li>Patrick Jones' <a rel="nofollow" class="external text" href="https://www.youtube.com/user/patrickJMT">Video Tutorials</a> on Mathematics</li>
<li><a rel="nofollow" class="external text" href="http://en.citizendium.org/wiki/Theory_(mathematics)">Citizendium: Theory (mathematics)</a>.</li>
<li><a href="/wiki/Marcus_du_Sautoy" title="Marcus du Sautoy">du Sautoy, Marcus</a>, <i><a rel="nofollow" class="external text" href="http://www.bbc.co.uk/podcasts/series/maths">A Brief History of Mathematics</a></i>, <a href="/wiki/BBC_Radio_4" title="BBC Radio 4">BBC Radio 4</a> (2010).</li>
<li><a rel="nofollow" class="external text" href="http://mathoverflow.net/">MathOverflow</a> A Q&amp;A site for research-level mathematics</li>
<li><a rel="nofollow" class="external text" href="https://www.khanacademy.org/math">Math</a>&#160;– <a href="/wiki/Khan_Academy" title="Khan Academy">Khan Academy</a></li>
<li><a rel="nofollow" class="external text" href="http://momath.org/">National Museum of Mathematics, located in New York City</a></li>
</ul>
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<li><a href="/wiki/Algebra" title="Algebra">Algebra</a>
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<li><a href="/wiki/Elementary_algebra" title="Elementary algebra">elementary</a></li>
<li><a href="/wiki/Linear_algebra" title="Linear algebra">linear</a></li>
<li><a href="/wiki/Multilinear_algebra" title="Multilinear algebra">multilinear</a></li>
<li><a href="/wiki/Abstract_algebra" title="Abstract algebra">abstract</a></li>
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<li><a href="/wiki/Arithmetic" title="Arithmetic">Arithmetic</a>&#160;/ <a href="/wiki/Number_theory" title="Number theory">Number theory</a></li>
<li><a href="/wiki/Calculus" title="Calculus">Calculus</a>&#160;/ <a href="/wiki/Mathematical_analysis" title="Mathematical analysis">Analysis</a></li>
<li><a href="/wiki/Category_theory" title="Category theory">Category theory</a></li>
<li><a href="/wiki/Combinatorics" title="Combinatorics">Combinatorics</a></li>
<li><a href="/wiki/Theory_of_computation" title="Theory of computation">Computation</a></li>
<li><a href="/wiki/Control_theory" title="Control theory">Control theory</a></li>
<li><a href="/wiki/Differential_equation" title="Differential equation">Differential equations</a>&#160;/ <a href="/wiki/Dynamical_systems_theory" title="Dynamical systems theory">Dynamical systems</a></li>
<li><a href="/wiki/Functional_analysis" title="Functional analysis">Functional analysis</a></li>
<li><a href="/wiki/Game_theory" title="Game theory">Game theory</a></li>
<li><a href="/wiki/Geometry" title="Geometry">Geometry</a>
<ul>
<li><a href="/wiki/Discrete_geometry" title="Discrete geometry">discrete</a></li>
<li><a href="/wiki/Algebraic_geometry" title="Algebraic geometry">algebraic</a></li>
<li><a href="/wiki/Differential_geometry" title="Differential geometry">differential</a></li>
<li><a href="/wiki/Finite_geometry" title="Finite geometry">finite</a></li>
</ul>
</li>
<li><a href="/wiki/Graph_theory" title="Graph theory">Graph theory</a></li>
<li><a href="/wiki/Information_theory" title="Information theory">Information theory</a></li>
<li><a href="/wiki/Lie_theory" title="Lie theory">Lie theory</a></li>
<li><a href="/wiki/Mathematical_logic" title="Mathematical logic">Mathematical logic</a></li>
<li><a href="/wiki/Mathematical_physics" title="Mathematical physics">Mathematical physics</a></li>
<li><a href="/wiki/Mathematical_statistics" title="Mathematical statistics">Mathematical statistics</a></li>
<li><a href="/wiki/Numerical_analysis" title="Numerical analysis">Numerical analysis</a></li>
<li><a href="/wiki/Mathematical_optimization" title="Mathematical optimization">Optimization</a></li>
<li><a href="/wiki/Probability_theory" title="Probability theory">Probability</a></li>
<li><a href="/wiki/Representation_theory" title="Representation theory">Representation theory</a></li>
<li><a href="/wiki/Set_theory" title="Set theory">Set theory</a></li>
<li><a href="/wiki/Statistics" title="Statistics">Statistics</a></li>
<li><a href="/wiki/Topology" title="Topology">Topology</a></li>
<li><a href="/wiki/Trigonometry" title="Trigonometry">Trigonometry</a></li>
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<li><a href="/wiki/Pure_mathematics" title="Pure mathematics">Pure</a></li>
<li><a href="/wiki/Applied_mathematics" title="Applied mathematics">Applied</a></li>
<li><a href="/wiki/Discrete_mathematics" title="Discrete mathematics">Discrete</a></li>
<li><a href="/wiki/Computational_mathematics" title="Computational mathematics">Computational</a></li>
<li><a href="/wiki/Metamathematics" title="Metamathematics">Meta-</a></li>
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<li>&#160;<img alt="Category" src="//upload.wikimedia.org/wikipedia/en/thumb/4/48/Folder_Hexagonal_Icon.svg/16px-Folder_Hexagonal_Icon.svg.png" title="Category" width="16" height="14" srcset="//upload.wikimedia.org/wikipedia/en/thumb/4/48/Folder_Hexagonal_Icon.svg/24px-Folder_Hexagonal_Icon.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/4/48/Folder_Hexagonal_Icon.svg/32px-Folder_Hexagonal_Icon.svg.png 2x" data-file-width="36" data-file-height="31" /> <b><a href="/wiki/Category:Fields_of_mathematics" title="Category:Fields of mathematics">Category</a></b></li>
<li>&#160;<img alt="Portal" src="//upload.wikimedia.org/wikipedia/en/thumb/f/fd/Portal-puzzle.svg/16px-Portal-puzzle.svg.png" title="Portal" width="16" height="14" srcset="//upload.wikimedia.org/wikipedia/en/thumb/f/fd/Portal-puzzle.svg/24px-Portal-puzzle.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/f/fd/Portal-puzzle.svg/32px-Portal-puzzle.svg.png 2x" data-file-width="32" data-file-height="28" /> <b><a href="/wiki/Portal:Mathematics" title="Portal:Mathematics">Portal</a></b></li>
<li>&#160;<img alt="Commons page" src="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/12px-Commons-logo.svg.png" title="Commons page" width="12" height="16" srcset="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/18px-Commons-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/24px-Commons-logo.svg.png 2x" data-file-width="1024" data-file-height="1376" /><b><a href="//commons.wikimedia.org/wiki/Category:Mathematics" class="extiw" title="commons:Category:Mathematics">Commons</a></b></li>
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<div style="font-size:114%"><a href="/wiki/Logic" title="Logic">Logic</a></div>
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<li><a href="/wiki/Outline_of_logic" title="Outline of logic">Outline</a></li>
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<li><a href="/wiki/Argumentation_theory" title="Argumentation theory">Argumentation theory</a></li>
<li><a href="/wiki/Axiology" title="Axiology">Axiology</a></li>
<li><a href="/wiki/Critical_thinking" title="Critical thinking">Critical thinking</a></li>
<li><a href="/wiki/Logic_in_computer_science" title="Logic in computer science">Logic in computer science</a></li>
<li><a href="/wiki/Mathematical_logic" title="Mathematical logic">Mathematical logic</a></li>
<li><a href="/wiki/Metalogic" title="Metalogic">Metalogic</a></li>
<li><a href="/wiki/Metamathematics" title="Metamathematics">Metamathematics</a></li>
<li><a href="/wiki/Non-classical_logic" title="Non-classical logic">Non-classical logic</a></li>
<li><a href="/wiki/Philosophical_logic" title="Philosophical logic">Philosophical logic</a></li>
<li><a href="/wiki/Philosophy_of_logic" title="Philosophy of logic">Philosophy of logic</a></li>
<li><a href="/wiki/Set_theory" title="Set theory">Set theory</a></li>
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<ul>
<li><a href="/wiki/Abductive_reasoning" title="Abductive reasoning">Abduction</a></li>
<li><a href="/wiki/Analytic%E2%80%93synthetic_distinction" title="Analytic–synthetic distinction">Analytic and synthetic propositions</a></li>
<li><a href="/wiki/Antinomy" title="Antinomy">Antinomy</a></li>
<li><a href="/wiki/A_priori_and_a_posteriori" title="A priori and a posteriori"><i>A priori</i> and <i>a posteriori</i></a></li>
<li><a href="/wiki/Deductive_reasoning" title="Deductive reasoning">Deduction</a></li>
<li><a href="/wiki/Definition" title="Definition">Definition</a></li>
<li><a href="/wiki/Description" title="Description">Description</a></li>
<li><a href="/wiki/Inductive_reasoning" title="Inductive reasoning">Induction</a></li>
<li><a href="/wiki/Inference" title="Inference">Inference</a></li>
<li><a href="/wiki/Logical_form" title="Logical form">Logical form</a></li>
<li><a href="/wiki/Logical_consequence" title="Logical consequence">Logical consequence</a></li>
<li><a href="/wiki/Logical_truth" title="Logical truth">Logical truth</a></li>
<li><a href="/wiki/Name" title="Name">Name</a></li>
<li><a href="/wiki/Necessity_and_sufficiency" title="Necessity and sufficiency">Necessity and sufficiency</a></li>
<li><a href="/wiki/Meaning_(linguistics)" title="Meaning (linguistics)">Meaning</a></li>
<li><a href="/wiki/Paradox" title="Paradox">Paradox</a></li>
<li><a href="/wiki/Possible_world" title="Possible world">Possible world</a></li>
<li><a href="/wiki/Presupposition" title="Presupposition">Presupposition</a></li>
<li><a href="/wiki/Probability" title="Probability">Probability</a></li>
<li><a href="/wiki/Reason" title="Reason">Reason</a></li>
<li><a href="/wiki/Reference" title="Reference">Reference</a></li>
<li><a href="/wiki/Semantics" title="Semantics">Semantics</a></li>
<li><a href="/wiki/Statement_(logic)" title="Statement (logic)">Statement</a></li>
<li><a href="/wiki/Strict_implication" title="Strict implication" class="mw-redirect">Strict implication</a></li>
<li><a href="/wiki/Substitution_(logic)" title="Substitution (logic)">Substitution</a></li>
<li><a href="/wiki/Syntax_(logic)" title="Syntax (logic)">Syntax</a></li>
<li><a href="/wiki/Truth" title="Truth">Truth</a></li>
<li><a href="/wiki/Validity" title="Validity">Validity</a></li>
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<th scope="row" class="navbox-group" style="font-weight:normal;"><a href="/wiki/Index_of_logic_articles" title="Index of logic articles">topics</a></th>
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<li><a href="/wiki/List_of_mathematical_logic_topics" title="List of mathematical logic topics">Mathematical logic</a></li>
<li><a href="/wiki/List_of_Boolean_algebra_topics" title="List of Boolean algebra topics">Boolean algebra</a></li>
<li><a href="/wiki/List_of_set_theory_topics" title="List of set theory topics">Set theory</a></li>
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<li><a href="/wiki/List_of_logicians" title="List of logicians">Logicians</a></li>
<li><a href="/wiki/List_of_rules_of_inference" title="List of rules of inference">Rules of inference</a></li>
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href="//ba.wikipedia.org/wiki/%D0%9C%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0" title="Математика – Bashkir" lang="ba" hreflang="ba">Башҡортса</a></li><li class="interlanguage-link interwiki-be"><a href="//be.wikipedia.org/wiki/%D0%9C%D0%B0%D1%82%D1%8D%D0%BC%D0%B0%D1%82%D1%8B%D0%BA%D0%B0" title="Матэматыка – Belarusian" lang="be" hreflang="be">Беларуская</a></li><li class="interlanguage-link interwiki-be-x-old"><a href="//be-x-old.wikipedia.org/wiki/%D0%9C%D0%B0%D1%82%D1%8D%D0%BC%D0%B0%D1%82%D1%8B%D0%BA%D0%B0" title="Матэматыка – беларуская (тарашкевіца)‎" lang="be-x-old" hreflang="be-x-old">Беларуская (тарашкевіца)‎</a></li><li class="interlanguage-link interwiki-bh"><a href="//bh.wikipedia.org/wiki/%E0%A4%97%E0%A4%A3%E0%A4%BF%E0%A4%A4" title="गणित – भोजपुरी" lang="bh" hreflang="bh">भोजपुरी</a></li><li class="interlanguage-link interwiki-bg"><a href="//bg.wikipedia.org/wiki/%D0%9C%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0" title="Математика – Bulgarian" lang="bg" hreflang="bg">Български</a></li><li class="interlanguage-link interwiki-bar"><a href="//bar.wikipedia.org/wiki/Mathematik" title="Mathematik – Bavarian" lang="bar" hreflang="bar">Boarisch</a></li><li class="interlanguage-link interwiki-bo"><a href="//bo.wikipedia.org/wiki/%E0%BD%A2%E0%BE%A9%E0%BD%B2%E0%BD%A6%E0%BC%8B%E0%BD%A2%E0%BD%B2%E0%BD%82" title="རྩིས་རིག – Tibetan" lang="bo" hreflang="bo">བོད་ཡིག</a></li><li class="interlanguage-link interwiki-bs"><a href="//bs.wikipedia.org/wiki/Matematika" title="Matematika – Bosnian" lang="bs" hreflang="bs">Bosanski</a></li><li class="interlanguage-link interwiki-br"><a href="//br.wikipedia.org/wiki/Matematik" title="Matematik – Breton" lang="br" hreflang="br">Brezhoneg</a></li><li class="interlanguage-link interwiki-bxr"><a href="//bxr.wikipedia.org/wiki/%D0%A2%D0%BE%D0%BE%D0%B3%D0%BE%D0%B9_%D1%83%D1%85%D0%B0%D0%B0%D0%BD" title="Тоогой ухаан – буряад" lang="bxr" hreflang="bxr">Буряад</a></li><li class="interlanguage-link interwiki-ca"><a href="//ca.wikipedia.org/wiki/Matem%C3%A0tiques" title="Matemàtiques – Catalan" lang="ca" hreflang="ca">Català</a></li><li class="interlanguage-link interwiki-cv"><a href="//cv.wikipedia.org/wiki/%D0%9C%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0" title="Математика – Chuvash" lang="cv" hreflang="cv">Чӑвашла</a></li><li class="interlanguage-link interwiki-ceb"><a href="//ceb.wikipedia.org/wiki/Matematika" title="Matematika – Cebuano" lang="ceb" hreflang="ceb">Cebuano</a></li><li class="interlanguage-link interwiki-cs"><a href="//cs.wikipedia.org/wiki/Matematika" title="Matematika – Czech" lang="cs" hreflang="cs">Čeština</a></li><li class="interlanguage-link interwiki-ch"><a href="//ch.wikipedia.org/wiki/Matematika" title="Matematika – Chamorro" lang="ch" hreflang="ch">Chamoru</a></li><li class="interlanguage-link interwiki-sn"><a href="//sn.wikipedia.org/wiki/Masvomhu" title="Masvomhu – Shona" lang="sn" hreflang="sn">ChiShona</a></li><li class="interlanguage-link interwiki-co"><a href="//co.wikipedia.org/wiki/Matematica" title="Matematica – Corsican" lang="co" hreflang="co">Corsu</a></li><li class="interlanguage-link interwiki-cy"><a href="//cy.wikipedia.org/wiki/Mathemateg" title="Mathemateg – Welsh" lang="cy" hreflang="cy">Cymraeg</a></li><li class="interlanguage-link interwiki-da"><a href="//da.wikipedia.org/wiki/Matematik" title="Matematik – Danish" lang="da" hreflang="da">Dansk</a></li><li class="interlanguage-link interwiki-de badge-Q17437798 badge-goodarticle" title="good article"><a href="//de.wikipedia.org/wiki/Mathematik" title="Mathematik – German" lang="de" hreflang="de">Deutsch</a></li><li class="interlanguage-link interwiki-dv"><a href="//dv.wikipedia.org/wiki/%DE%83%DE%A8%DE%94%DE%A7%DE%9F%DE%A8%DE%87%DE%B0%DE%94%DE%A7%DE%8C%DE%AA" title="ރިޔާޟިއްޔާތު – Divehi" lang="dv" hreflang="dv">ދިވެހިބަސް</a></li><li class="interlanguage-link interwiki-nv"><a href="//nv.wikipedia.org/wiki/A%C5%82hii%CA%BCn%C3%ADn%C3%A1%CA%BCiidz%C3%B3%C3%B3h" title="Ałhiiʼnínáʼiidzóóh – Navajo" lang="nv" hreflang="nv">Diné bizaad</a></li><li class="interlanguage-link interwiki-dsb"><a href="//dsb.wikipedia.org/wiki/Matematika" title="Matematika – Lower Sorbian" lang="dsb" hreflang="dsb">Dolnoserbski</a></li><li class="interlanguage-link interwiki-et"><a href="//et.wikipedia.org/wiki/Matemaatika" title="Matemaatika – Estonian" lang="et" hreflang="et">Eesti</a></li><li class="interlanguage-link interwiki-el"><a href="//el.wikipedia.org/wiki/%CE%9C%CE%B1%CE%B8%CE%B7%CE%BC%CE%B1%CF%84%CE%B9%CE%BA%CE%AC" title="Μαθηματικά – Greek" lang="el" hreflang="el">Ελληνικά</a></li><li class="interlanguage-link interwiki-eml"><a href="//eml.wikipedia.org/wiki/Matem%C3%A2tica" title="Matemâtica – Emiliano-Romagnolo" lang="eml" hreflang="eml">Emiliàn e rumagnòl</a></li><li class="interlanguage-link interwiki-myv"><a href="//myv.wikipedia.org/wiki/%D0%9C%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0" title="Математика – Erzya" lang="myv" hreflang="myv">Эрзянь</a></li><li class="interlanguage-link interwiki-es"><a href="//es.wikipedia.org/wiki/Matem%C3%A1ticas" title="Matemáticas – Spanish" lang="es" hreflang="es">Español</a></li><li class="interlanguage-link interwiki-eo"><a href="//eo.wikipedia.org/wiki/Matematiko" title="Matematiko – Esperanto" lang="eo" hreflang="eo">Esperanto</a></li><li class="interlanguage-link interwiki-ext"><a href="//ext.wikipedia.org/wiki/Matem%C3%A1ticas" title="Matemáticas – Extremaduran" lang="ext" hreflang="ext">Estremeñu</a></li><li class="interlanguage-link interwiki-eu"><a href="//eu.wikipedia.org/wiki/Matematika" title="Matematika – Basque" lang="eu" hreflang="eu">Euskara</a></li><li class="interlanguage-link interwiki-fa"><a href="//fa.wikipedia.org/wiki/%D8%B1%DB%8C%D8%A7%D8%B6%DB%8C%D8%A7%D8%AA" title="ریاضیات – Persian" lang="fa" hreflang="fa">فارسی</a></li><li class="interlanguage-link interwiki-hif"><a href="//hif.wikipedia.org/wiki/Mathematics" title="Mathematics – Fiji Hindi" lang="hif" hreflang="hif">Fiji Hindi</a></li><li class="interlanguage-link interwiki-fo"><a href="//fo.wikipedia.org/wiki/St%C3%B8ddfr%C3%B8%C3%B0i" title="Støddfrøði – Faroese" lang="fo" hreflang="fo">Føroyskt</a></li><li class="interlanguage-link interwiki-fr"><a href="//fr.wikipedia.org/wiki/Math%C3%A9matiques" title="Mathématiques – French" lang="fr" hreflang="fr">Français</a></li><li class="interlanguage-link interwiki-fy"><a href="//fy.wikipedia.org/wiki/Wiskunde" title="Wiskunde – Western Frisian" lang="fy" hreflang="fy">Frysk</a></li><li class="interlanguage-link interwiki-fur"><a href="//fur.wikipedia.org/wiki/Matematiche" title="Matematiche – Friulian" lang="fur" hreflang="fur">Furlan</a></li><li class="interlanguage-link interwiki-ga"><a href="//ga.wikipedia.org/wiki/Matamaitic" title="Matamaitic – Irish" lang="ga" hreflang="ga">Gaeilge</a></li><li class="interlanguage-link interwiki-gv"><a href="//gv.wikipedia.org/wiki/Maddaght" title="Maddaght – Manx" lang="gv" hreflang="gv">Gaelg</a></li><li class="interlanguage-link interwiki-gd"><a href="//gd.wikipedia.org/wiki/Matamataig" title="Matamataig – Scottish Gaelic" lang="gd" hreflang="gd">Gàidhlig</a></li><li class="interlanguage-link interwiki-gl"><a href="//gl.wikipedia.org/wiki/Matem%C3%A1ticas" title="Matemáticas – Galician" lang="gl" hreflang="gl">Galego</a></li><li class="interlanguage-link interwiki-gan"><a href="//gan.wikipedia.org/wiki/%E6%95%B8%E5%AD%B8" title="數學 – Gan Chinese" lang="gan" hreflang="gan">贛語</a></li><li class="interlanguage-link interwiki-gu"><a href="//gu.wikipedia.org/wiki/%E0%AA%97%E0%AA%A3%E0%AA%BF%E0%AA%A4" title="ગણિત – Gujarati" lang="gu" hreflang="gu">ગુજરાતી</a></li><li class="interlanguage-link interwiki-hak"><a href="//hak.wikipedia.org/wiki/S%E1%B9%B3-ho%CC%8Dk" title="Sṳ-ho̍k – Hakka Chinese" lang="hak" hreflang="hak">客家語/Hak-kâ-ngî</a></li><li class="interlanguage-link interwiki-xal"><a href="//xal.wikipedia.org/wiki/%D0%AD%D1%81%D0%B2" title="Эсв – Kalmyk" lang="xal" hreflang="xal">Хальмг</a></li><li class="interlanguage-link interwiki-ko"><a href="//ko.wikipedia.org/wiki/%EC%88%98%ED%95%99" title="수학 – Korean" lang="ko" hreflang="ko">한국어</a></li><li class="interlanguage-link interwiki-haw"><a href="//haw.wikipedia.org/wiki/Makemakika" title="Makemakika – Hawaiian" lang="haw" hreflang="haw">Hawai`i</a></li><li class="interlanguage-link interwiki-hy"><a href="//hy.wikipedia.org/wiki/%D5%84%D5%A1%D5%A9%D5%A5%D5%B4%D5%A1%D5%BF%D5%AB%D5%AF%D5%A1" title="Մաթեմատիկա – Armenian" lang="hy" hreflang="hy">Հայերեն</a></li><li class="interlanguage-link interwiki-hi"><a href="//hi.wikipedia.org/wiki/%E0%A4%97%E0%A4%A3%E0%A4%BF%E0%A4%A4" title="गणित – Hindi" lang="hi" hreflang="hi">हिन्दी</a></li><li class="interlanguage-link interwiki-hr"><a href="//hr.wikipedia.org/wiki/Matematika" title="Matematika – Croatian" lang="hr" hreflang="hr">Hrvatski</a></li><li class="interlanguage-link interwiki-io"><a href="//io.wikipedia.org/wiki/Matematiko" title="Matematiko – Ido" lang="io" hreflang="io">Ido</a></li><li class="interlanguage-link interwiki-ig"><a href="//ig.wikipedia.org/wiki/%E1%BB%8Cm%C3%BAm%C3%BA-%C3%B3n%C3%BA%E1%BB%8Dg%E1%BB%A5g%E1%BB%A5" title="Ọmúmú-ónúọgụgụ – Igbo" lang="ig" hreflang="ig">Igbo</a></li><li class="interlanguage-link interwiki-ilo"><a href="//ilo.wikipedia.org/wiki/Matematika" title="Matematika – Iloko" lang="ilo" hreflang="ilo">Ilokano</a></li><li class="interlanguage-link interwiki-bpy"><a href="//bpy.wikipedia.org/wiki/%E0%A6%97%E0%A6%A3%E0%A6%BF%E0%A6%A4" title="গণিত – Bishnupriya" lang="bpy" hreflang="bpy">বিষ্ণুপ্রিয়া মণিপুরী</a></li><li class="interlanguage-link interwiki-id"><a href="//id.wikipedia.org/wiki/Matematika" title="Matematika – Indonesian" lang="id" hreflang="id">Bahasa Indonesia</a></li><li class="interlanguage-link interwiki-ia badge-Q17437796 badge-featuredarticle" title="featured article"><a href="//ia.wikipedia.org/wiki/Mathematica" title="Mathematica – Interlingua" lang="ia" hreflang="ia">Interlingua</a></li><li class="interlanguage-link interwiki-ie"><a href="//ie.wikipedia.org/wiki/Matematica" title="Matematica – Interlingue" lang="ie" hreflang="ie">Interlingue</a></li><li class="interlanguage-link interwiki-os"><a href="//os.wikipedia.org/wiki/%D0%9C%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%C3%A6" title="Математикæ – Ossetic" lang="os" hreflang="os">Ирон</a></li><li class="interlanguage-link interwiki-xh"><a href="//xh.wikipedia.org/wiki/I-Mathematics" title="I-Mathematics – Xhosa" lang="xh" hreflang="xh">IsiXhosa</a></li><li class="interlanguage-link interwiki-zu"><a href="//zu.wikipedia.org/wiki/Imathemathiki" title="Imathemathiki – Zulu" lang="zu" hreflang="zu">IsiZulu</a></li><li class="interlanguage-link interwiki-is"><a href="//is.wikipedia.org/wiki/St%C3%A6r%C3%B0fr%C3%A6%C3%B0i" title="Stærðfræði – Icelandic" lang="is" hreflang="is">Íslenska</a></li><li class="interlanguage-link interwiki-it"><a href="//it.wikipedia.org/wiki/Matematica" title="Matematica – Italian" lang="it" hreflang="it">Italiano</a></li><li class="interlanguage-link interwiki-he"><a href="//he.wikipedia.org/wiki/%D7%9E%D7%AA%D7%9E%D7%98%D7%99%D7%A7%D7%94" title="מתמטיקה – Hebrew" lang="he" hreflang="he">עברית</a></li><li class="interlanguage-link interwiki-jv"><a href="//jv.wikipedia.org/wiki/Mat%C3%A9matika" title="Matématika – Javanese" lang="jv" hreflang="jv">Basa Jawa</a></li><li class="interlanguage-link interwiki-kl"><a href="//kl.wikipedia.org/wiki/Matematikki" title="Matematikki – Kalaallisut" lang="kl" hreflang="kl">Kalaallisut</a></li><li class="interlanguage-link interwiki-kn"><a href="//kn.wikipedia.org/wiki/%E0%B2%97%E0%B2%A3%E0%B2%BF%E0%B2%A4" title="ಗಣಿತ – Kannada" lang="kn" hreflang="kn">ಕನ್ನಡ</a></li><li class="interlanguage-link interwiki-krc"><a href="//krc.wikipedia.org/wiki/%D0%9C%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0" title="Математика – Karachay-Balkar" lang="krc" hreflang="krc">Къарачай-малкъар</a></li><li class="interlanguage-link interwiki-ka badge-Q17437796 badge-featuredarticle" title="featured article"><a href="//ka.wikipedia.org/wiki/%E1%83%9B%E1%83%90%E1%83%97%E1%83%94%E1%83%9B%E1%83%90%E1%83%A2%E1%83%98%E1%83%99%E1%83%90" title="მათემატიკა – Georgian" lang="ka" hreflang="ka">ქართული</a></li><li class="interlanguage-link interwiki-csb"><a href="//csb.wikipedia.org/wiki/Matematika" title="Matematika – Kashubian" lang="csb" hreflang="csb">Kaszëbsczi</a></li><li class="interlanguage-link interwiki-kk"><a href="//kk.wikipedia.org/wiki/%D0%9C%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0" title="Математика – Kazakh" lang="kk" hreflang="kk">Қазақша</a></li><li class="interlanguage-link interwiki-sw"><a href="//sw.wikipedia.org/wiki/Hisabati" title="Hisabati – Swahili" lang="sw" hreflang="sw">Kiswahili</a></li><li class="interlanguage-link interwiki-ht"><a href="//ht.wikipedia.org/wiki/Matematik" title="Matematik – Haitian Creole" lang="ht" hreflang="ht">Kreyòl ayisyen</a></li><li class="interlanguage-link interwiki-ku"><a href="//ku.wikipedia.org/wiki/Matemat%C3%AEk" title="Matematîk – Kurdish" lang="ku" hreflang="ku">Kurdî</a></li><li class="interlanguage-link interwiki-ky"><a href="//ky.wikipedia.org/wiki/%D0%9C%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0" title="Математика – Kyrgyz" lang="ky" hreflang="ky">Кыргызча</a></li><li class="interlanguage-link interwiki-lad"><a href="//lad.wikipedia.org/wiki/Matem%C3%A1tika" title="Matemátika – Ladino" lang="lad" hreflang="lad">Ladino</a></li><li class="interlanguage-link interwiki-lez"><a href="//lez.wikipedia.org/wiki/%D0%9C%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0" title="Математика – Lezghian" lang="lez" hreflang="lez">Лезги</a></li><li class="interlanguage-link interwiki-lo"><a href="//lo.wikipedia.org/wiki/%E0%BA%84%E0%BA%B0%E0%BA%99%E0%BA%B4%E0%BA%94%E0%BA%AA%E0%BA%B2%E0%BA%94" title="ຄະນິດສາດ – Lao" lang="lo" hreflang="lo">ລາວ</a></li><li class="interlanguage-link interwiki-la badge-Q17437796 badge-featuredarticle" title="featured article"><a href="//la.wikipedia.org/wiki/Mathematica" title="Mathematica – Latin" lang="la" hreflang="la">Latina</a></li><li class="interlanguage-link interwiki-lv"><a href="//lv.wikipedia.org/wiki/Matem%C4%81tika" title="Matemātika – Latvian" lang="lv" hreflang="lv">Latviešu</a></li><li class="interlanguage-link interwiki-lb"><a href="//lb.wikipedia.org/wiki/Mathematik" title="Mathematik – Luxembourgish" lang="lb" hreflang="lb">Lëtzebuergesch</a></li><li class="interlanguage-link interwiki-lt"><a href="//lt.wikipedia.org/wiki/Matematika" title="Matematika – Lithuanian" lang="lt" hreflang="lt">Lietuvių</a></li><li class="interlanguage-link interwiki-lij"><a href="//lij.wikipedia.org/wiki/Matematica" title="Matematica – Ligurian" lang="lij" hreflang="lij">Ligure</a></li><li class="interlanguage-link interwiki-li"><a href="//li.wikipedia.org/wiki/Mathematiek" title="Mathematiek – Limburgish" lang="li" hreflang="li">Limburgs</a></li><li class="interlanguage-link interwiki-jbo"><a href="//jbo.wikipedia.org/wiki/cmaci" title="cmaci – Lojban" lang="jbo" hreflang="jbo">La .lojban.</a></li><li class="interlanguage-link interwiki-lg"><a href="//lg.wikipedia.org/wiki/Ekibalangulo" title="Ekibalangulo – Ganda" lang="lg" hreflang="lg">Luganda</a></li><li class="interlanguage-link interwiki-lmo"><a href="//lmo.wikipedia.org/wiki/Matem%C3%A0tega" title="Matemàtega – Lombard" lang="lmo" hreflang="lmo">Lumbaart</a></li><li class="interlanguage-link interwiki-hu"><a href="//hu.wikipedia.org/wiki/Matematika" title="Matematika – Hungarian" lang="hu" hreflang="hu">Magyar</a></li><li class="interlanguage-link interwiki-mk"><a href="//mk.wikipedia.org/wiki/%D0%9C%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0" title="Математика – Macedonian" lang="mk" hreflang="mk">Македонски</a></li><li class="interlanguage-link interwiki-mg"><a href="//mg.wikipedia.org/wiki/Fanisana" title="Fanisana – Malagasy" lang="mg" hreflang="mg">Malagasy</a></li><li class="interlanguage-link interwiki-ml"><a href="//ml.wikipedia.org/wiki/%E0%B4%97%E0%B4%A3%E0%B4%BF%E0%B4%A4%E0%B4%82" title="ഗണിതം – Malayalam" lang="ml" hreflang="ml">മലയാളം</a></li><li class="interlanguage-link interwiki-mt"><a href="//mt.wikipedia.org/wiki/Matematika" title="Matematika – Maltese" lang="mt" hreflang="mt">Malti</a></li><li class="interlanguage-link interwiki-mr"><a href="//mr.wikipedia.org/wiki/%E0%A4%97%E0%A4%A3%E0%A4%BF%E0%A4%A4" title="गणित – Marathi" lang="mr" hreflang="mr">मराठी</a></li><li class="interlanguage-link interwiki-arz"><a href="//arz.wikipedia.org/wiki/%D8%B1%D9%8A%D8%A7%D8%B6%D9%8A%D8%A7%D8%AA" title="رياضيات – Egyptian Arabic" lang="arz" hreflang="arz">مصرى</a></li><li class="interlanguage-link interwiki-mzn"><a href="//mzn.wikipedia.org/wiki/%D8%B1%DB%8C%D8%A7%D8%B6%DB%8C" title="ریاضی – Mazanderani" lang="mzn" hreflang="mzn">مازِرونی</a></li><li class="interlanguage-link interwiki-ms"><a href="//ms.wikipedia.org/wiki/Matematik" title="Matematik – Malay" lang="ms" hreflang="ms">Bahasa Melayu</a></li><li class="interlanguage-link interwiki-cdo"><a href="//cdo.wikipedia.org/wiki/S%C3%B3-h%C5%8Fk" title="Só-hŏk – Min Dong Chinese" lang="cdo" hreflang="cdo">Mìng-dĕ̤ng-ngṳ̄</a></li><li class="interlanguage-link interwiki-mwl"><a href="//mwl.wikipedia.org/wiki/Matem%C3%A1tica" title="Matemática – Mirandese" lang="mwl" hreflang="mwl">Mirandés</a></li><li class="interlanguage-link interwiki-mn"><a href="//mn.wikipedia.org/wiki/%D0%9C%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA" title="Математик – Mongolian" lang="mn" hreflang="mn">Монгол</a></li><li class="interlanguage-link interwiki-my"><a href="//my.wikipedia.org/wiki/%E1%80%9E%E1%80%84%E1%80%BA%E1%80%B9%E1%80%81%E1%80%BB%E1%80%AC" title="သင်္ချာ – Burmese" lang="my" hreflang="my">မြန်မာဘာသာ</a></li><li class="interlanguage-link interwiki-nah"><a href="//nah.wikipedia.org/wiki/Tlap%C5%8Dhualmatiliztli" title="Tlapōhualmatiliztli – Nāhuatl" lang="nah" hreflang="nah">Nāhuatl</a></li><li class="interlanguage-link interwiki-nl"><a href="//nl.wikipedia.org/wiki/Wiskunde" title="Wiskunde – Dutch" lang="nl" hreflang="nl">Nederlands</a></li><li class="interlanguage-link interwiki-nds-nl"><a href="//nds-nl.wikipedia.org/wiki/Wiskunde" title="Wiskunde – Low Saxon" lang="nds-NL" hreflang="nds-NL">Nedersaksies</a></li><li class="interlanguage-link interwiki-ne"><a href="//ne.wikipedia.org/wiki/%E0%A4%97%E0%A4%A3%E0%A4%BF%E0%A4%A4" title="गणित – Nepali" lang="ne" hreflang="ne">नेपाली</a></li><li class="interlanguage-link interwiki-new"><a href="//new.wikipedia.org/wiki/%E0%A4%B2%E0%A5%8D%E0%A4%AF%E0%A4%BE%E0%A4%83%E0%A4%9C%E0%A5%8D%E0%A4%AF%E0%A4%BE" title="ल्याःज्या – Newari" lang="new" hreflang="new">नेपाल भाषा</a></li><li class="interlanguage-link interwiki-ja"><a href="//ja.wikipedia.org/wiki/%E6%95%B0%E5%AD%A6" title="数学 – Japanese" lang="ja" hreflang="ja">日本語</a></li><li class="interlanguage-link interwiki-frr"><a href="//frr.wikipedia.org/wiki/Matematiik" title="Matematiik – Northern Frisian" lang="frr" hreflang="frr">Nordfriisk</a></li><li class="interlanguage-link interwiki-no"><a href="//no.wikipedia.org/wiki/Matematikk" title="Matematikk – Norwegian" lang="no" hreflang="no">Norsk bokmål</a></li><li class="interlanguage-link interwiki-nn"><a href="//nn.wikipedia.org/wiki/Matematikk" title="Matematikk – Norwegian Nynorsk" lang="nn" hreflang="nn">Norsk nynorsk</a></li><li class="interlanguage-link interwiki-nrm"><a href="//nrm.wikipedia.org/wiki/Caltchul" title="Caltchul – Nouormand" lang="nrm" hreflang="nrm">Nouormand</a></li><li class="interlanguage-link interwiki-nov"><a href="//nov.wikipedia.org/wiki/Matematike" title="Matematike – Novial" lang="nov" hreflang="nov">Novial</a></li><li class="interlanguage-link interwiki-oc"><a href="//oc.wikipedia.org/wiki/Matematicas" title="Matematicas – Occitan" lang="oc" hreflang="oc">Occitan</a></li><li class="interlanguage-link interwiki-mhr"><a href="//mhr.wikipedia.org/wiki/%D0%9C%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B5" title="Математике – Eastern Mari" lang="mhr" hreflang="mhr">Олык марий</a></li><li class="interlanguage-link interwiki-or"><a href="//or.wikipedia.org/wiki/%E0%AC%97%E0%AC%A3%E0%AC%BF%E0%AC%A4" title="ଗଣିତ – Oriya" lang="or" hreflang="or">ଓଡ଼ିଆ</a></li><li class="interlanguage-link interwiki-om"><a href="//om.wikipedia.org/wiki/Herrega" title="Herrega – Oromo" lang="om" hreflang="om">Oromoo</a></li><li class="interlanguage-link interwiki-uz"><a href="//uz.wikipedia.org/wiki/Matematika" title="Matematika – Uzbek" lang="uz" hreflang="uz">Oʻzbekcha/ўзбекча</a></li><li class="interlanguage-link interwiki-pa"><a href="//pa.wikipedia.org/wiki/%E0%A8%B9%E0%A8%BF%E0%A8%B8%E0%A8%BE%E0%A8%AC" title="ਹਿਸਾਬ – Punjabi" lang="pa" hreflang="pa">ਪੰਜਾਬੀ</a></li><li class="interlanguage-link interwiki-pi"><a href="//pi.wikipedia.org/wiki/%E0%A4%97%E0%A4%A3%E0%A4%BF%E0%A4%A4%E0%A4%82" title="गणितं – Pali" lang="pi" hreflang="pi">पालि</a></li><li class="interlanguage-link interwiki-pag"><a href="//pag.wikipedia.org/wiki/Matematiks" title="Matematiks – Pangasinan" lang="pag" hreflang="pag">Pangasinan</a></li><li class="interlanguage-link interwiki-pnb"><a href="//pnb.wikipedia.org/wiki/%D9%85%DB%8C%D8%AA%DA%BE%D9%85%DB%8C%D9%B9%DA%A9%D8%B3" title="میتھمیٹکس – Western Punjabi" lang="pnb" hreflang="pnb">پنجابی</a></li><li class="interlanguage-link interwiki-ps"><a href="//ps.wikipedia.org/wiki/%D8%B4%D9%85%DB%90%D8%B1%D9%BE%D9%88%D9%87%D9%86%D9%87" title="شمېرپوهنه – Pashto" lang="ps" hreflang="ps">پښتو</a></li><li class="interlanguage-link interwiki-km"><a href="//km.wikipedia.org/wiki/%E1%9E%82%E1%9E%8E%E1%9E%B7%E1%9E%8F%E1%9E%9C%E1%9E%B7%E1%9E%91%E1%9F%92%E1%9E%99%E1%9E%B6" title="គណិតវិទ្យា – Khmer" lang="km" hreflang="km">ភាសាខ្មែរ</a></li><li class="interlanguage-link interwiki-pcd"><a href="//pcd.wikipedia.org/wiki/Mat%C3%A9matikes" title="Matématikes – Picard" lang="pcd" hreflang="pcd">Picard</a></li><li class="interlanguage-link interwiki-pms"><a href="//pms.wikipedia.org/wiki/Matem%C3%A0tica" title="Matemàtica – Piedmontese" lang="pms" hreflang="pms">Piemontèis</a></li><li class="interlanguage-link interwiki-tpi"><a href="//tpi.wikipedia.org/wiki/Ol_matematik" title="Ol matematik – Tok Pisin" lang="tpi" hreflang="tpi">Tok Pisin</a></li><li class="interlanguage-link interwiki-nds"><a href="//nds.wikipedia.org/wiki/Mathematik" title="Mathematik – Low German" lang="nds" hreflang="nds">Plattdüütsch</a></li><li class="interlanguage-link interwiki-pl"><a href="//pl.wikipedia.org/wiki/Matematyka" title="Matematyka – Polish" lang="pl" hreflang="pl">Polski</a></li><li class="interlanguage-link interwiki-pt"><a href="//pt.wikipedia.org/wiki/Matem%C3%A1tica" title="Matemática – Portuguese" lang="pt" hreflang="pt">Português</a></li><li class="interlanguage-link interwiki-kaa"><a href="//kaa.wikipedia.org/wiki/Matematika" title="Matematika – Kara-Kalpak" lang="kaa" hreflang="kaa">Qaraqalpaqsha</a></li><li class="interlanguage-link interwiki-crh"><a href="//crh.wikipedia.org/wiki/Riyaziyat" title="Riyaziyat – Crimean Turkish" lang="crh" hreflang="crh">Qırımtatarca</a></li><li class="interlanguage-link interwiki-ro"><a href="//ro.wikipedia.org/wiki/Matematic%C4%83" title="Matematică – Romanian" lang="ro" hreflang="ro">Română</a></li><li class="interlanguage-link interwiki-qu"><a href="//qu.wikipedia.org/wiki/Yupay_yachay" title="Yupay yachay – Quechua" lang="qu" hreflang="qu">Runa Simi</a></li><li class="interlanguage-link interwiki-rue"><a href="//rue.wikipedia.org/wiki/%D0%9C%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D1%96%D0%BA%D0%B0" title="Математіка – Rusyn" lang="rue" hreflang="rue">Русиньскый</a></li><li class="interlanguage-link interwiki-ru"><a href="//ru.wikipedia.org/wiki/%D0%9C%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0" title="Математика – Russian" lang="ru" hreflang="ru">Русский</a></li><li class="interlanguage-link interwiki-sah"><a href="//sah.wikipedia.org/wiki/%D0%9C%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0" title="Математика – Sakha" lang="sah" hreflang="sah">Саха тыла</a></li><li class="interlanguage-link interwiki-sm"><a href="//sm.wikipedia.org/wiki/Matematika" title="Matematika – Samoan" lang="sm" hreflang="sm">Gagana Samoa</a></li><li class="interlanguage-link interwiki-sa"><a href="//sa.wikipedia.org/wiki/%E0%A4%97%E0%A4%A3%E0%A4%BF%E0%A4%A4%E0%A4%AE%E0%A5%8D" title="गणितम् – Sanskrit" lang="sa" hreflang="sa">संस्कृतम्</a></li><li class="interlanguage-link interwiki-sc"><a href="//sc.wikipedia.org/wiki/Matem%C3%A0tica" title="Matemàtica – Sardinian" lang="sc" hreflang="sc">Sardu</a></li><li class="interlanguage-link interwiki-sco"><a href="//sco.wikipedia.org/wiki/Mathematics" title="Mathematics – Scots" lang="sco" hreflang="sco">Scots</a></li><li class="interlanguage-link interwiki-stq"><a href="//stq.wikipedia.org/wiki/Mathematik" title="Mathematik – Saterland Frisian" lang="stq" hreflang="stq">Seeltersk</a></li><li class="interlanguage-link interwiki-sq"><a href="//sq.wikipedia.org/wiki/Matematika" title="Matematika – Albanian" lang="sq" hreflang="sq">Shqip</a></li><li class="interlanguage-link interwiki-scn"><a href="//scn.wikipedia.org/wiki/Matim%C3%A0tica" title="Matimàtica – Sicilian" lang="scn" hreflang="scn">Sicilianu</a></li><li class="interlanguage-link interwiki-si"><a href="//si.wikipedia.org/wiki/%E0%B6%9C%E0%B6%AB%E0%B7%92%E0%B6%AD%E0%B6%BA" title="ගණිතය – Sinhala" lang="si" hreflang="si">සිංහල</a></li><li class="interlanguage-link interwiki-simple"><a href="//simple.wikipedia.org/wiki/Mathematics" title="Mathematics – Simple English" lang="simple" hreflang="simple">Simple English</a></li><li class="interlanguage-link interwiki-sd"><a href="//sd.wikipedia.org/wiki/%D8%B1%D9%8A%D8%A7%D8%B6%D9%8A" title="رياضي – Sindhi" lang="sd" hreflang="sd">سنڌي</a></li><li class="interlanguage-link interwiki-ss"><a href="//ss.wikipedia.org/wiki/Tekubala" title="Tekubala – Swati" lang="ss" hreflang="ss">SiSwati</a></li><li class="interlanguage-link interwiki-sk"><a href="//sk.wikipedia.org/wiki/Matematika" title="Matematika – Slovak" lang="sk" hreflang="sk">Slovenčina</a></li><li class="interlanguage-link interwiki-sl"><a href="//sl.wikipedia.org/wiki/Matematika" title="Matematika – Slovenian" lang="sl" hreflang="sl">Slovenščina</a></li><li class="interlanguage-link interwiki-szl"><a href="//szl.wikipedia.org/wiki/Matymatyka" title="Matymatyka – Silesian" lang="szl" hreflang="szl">Ślůnski</a></li><li class="interlanguage-link interwiki-so"><a href="//so.wikipedia.org/wiki/Xisaab" title="Xisaab – Somali" lang="so" hreflang="so">Soomaaliga</a></li><li class="interlanguage-link interwiki-ckb"><a href="//ckb.wikipedia.org/wiki/%D9%85%D8%A7%D8%AA%D9%85%D8%A7%D8%AA%DB%8C%DA%A9" title="ماتماتیک – Central Kurdish" lang="ckb" hreflang="ckb">کوردیی ناوەندی</a></li><li class="interlanguage-link interwiki-srn"><a href="//srn.wikipedia.org/wiki/Sabi_fu_Teri" title="Sabi fu Teri – Sranan Tongo" lang="srn" hreflang="srn">Sranantongo</a></li><li class="interlanguage-link interwiki-sr"><a href="//sr.wikipedia.org/wiki/%D0%9C%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0" title="Математика – Serbian" lang="sr" hreflang="sr">Српски / srpski</a></li><li class="interlanguage-link interwiki-sh"><a href="//sh.wikipedia.org/wiki/Matematika" title="Matematika – Serbo-Croatian" lang="sh" hreflang="sh">Srpskohrvatski / српскохрватски</a></li><li class="interlanguage-link interwiki-su"><a href="//su.wikipedia.org/wiki/Matematika" title="Matematika – Sundanese" lang="su" hreflang="su">Basa Sunda</a></li><li class="interlanguage-link interwiki-fi"><a href="//fi.wikipedia.org/wiki/Matematiikka" title="Matematiikka – Finnish" lang="fi" hreflang="fi">Suomi</a></li><li class="interlanguage-link interwiki-sv"><a href="//sv.wikipedia.org/wiki/Matematik" title="Matematik – Swedish" lang="sv" hreflang="sv">Svenska</a></li><li class="interlanguage-link interwiki-tl"><a href="//tl.wikipedia.org/wiki/Matematika" title="Matematika – Tagalog" lang="tl" hreflang="tl">Tagalog</a></li><li class="interlanguage-link interwiki-ta"><a href="//ta.wikipedia.org/wiki/%E0%AE%95%E0%AE%A3%E0%AE%BF%E0%AE%A4%E0%AE%AE%E0%AF%8D" title="கணிதம் – Tamil" lang="ta" hreflang="ta">தமிழ்</a></li><li class="interlanguage-link interwiki-kab"><a href="//kab.wikipedia.org/wiki/Tusnakt" title="Tusnakt – Kabyle" lang="kab" hreflang="kab">Taqbaylit</a></li><li class="interlanguage-link interwiki-tt"><a href="//tt.wikipedia.org/wiki/%D0%9C%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0" title="Математика – Tatar" lang="tt" hreflang="tt">Татарча/tatarça</a></li><li class="interlanguage-link interwiki-te"><a href="//te.wikipedia.org/wiki/%E0%B0%97%E0%B0%A3%E0%B0%BF%E0%B0%A4%E0%B0%AE%E0%B1%81" title="గణితము – Telugu" lang="te" hreflang="te">తెలుగు</a></li><li class="interlanguage-link interwiki-tet"><a href="//tet.wikipedia.org/wiki/Matem%C3%A1tika" title="Matemátika – Tetum" lang="tet" hreflang="tet">Tetun</a></li><li class="interlanguage-link interwiki-th"><a href="//th.wikipedia.org/wiki/%E0%B8%84%E0%B8%93%E0%B8%B4%E0%B8%95%E0%B8%A8%E0%B8%B2%E0%B8%AA%E0%B8%95%E0%B8%A3%E0%B9%8C" title="คณิตศาสตร์ – Thai" lang="th" hreflang="th">ไทย</a></li><li class="interlanguage-link interwiki-tg"><a href="//tg.wikipedia.org/wiki/%D0%A0%D0%B8%D1%91%D0%B7%D0%B8%D1%91%D1%82" title="Риёзиёт – Tajik" lang="tg" hreflang="tg">Тоҷикӣ</a></li><li class="interlanguage-link interwiki-tr"><a href="//tr.wikipedia.org/wiki/Matematik" title="Matematik – Turkish" lang="tr" hreflang="tr">Türkçe</a></li><li class="interlanguage-link interwiki-tk"><a href="//tk.wikipedia.org/wiki/Matematika" title="Matematika – Turkmen" lang="tk" hreflang="tk">Türkmençe</a></li><li class="interlanguage-link interwiki-bug"><a href="//bug.wikipedia.org/wiki/%E1%A8%86%E1%A8%88%E1%A8%9B%E1%A8%86%E1%A8%88%E1%A8%97%E1%A8%80" title="ᨆᨈᨛᨆᨈᨗᨀ – Buginese" lang="bug" hreflang="bug">ᨅᨔ ᨕᨘᨁᨗ</a></li><li class="interlanguage-link interwiki-uk"><a href="//uk.wikipedia.org/wiki/%D0%9C%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0" title="Математика – Ukrainian" lang="uk" hreflang="uk">Українська</a></li><li class="interlanguage-link interwiki-ur"><a href="//ur.wikipedia.org/wiki/%D8%B1%DB%8C%D8%A7%D8%B6%DB%8C" title="ریاضی – Urdu" lang="ur" hreflang="ur">اردو</a></li><li class="interlanguage-link interwiki-za"><a href="//za.wikipedia.org/wiki/Soqyoz" title="Soqyoz – Zhuang" lang="za" hreflang="za">Vahcuengh</a></li><li class="interlanguage-link interwiki-vec"><a href="//vec.wikipedia.org/wiki/Matem%C3%A0tega" title="Matemàtega – Venetian" lang="vec" hreflang="vec">Vèneto</a></li><li class="interlanguage-link interwiki-vi badge-Q17437798 badge-goodarticle" title="good article"><a href="//vi.wikipedia.org/wiki/To%C3%A1n_h%E1%BB%8Dc" title="Toán học – Vietnamese" lang="vi" hreflang="vi">Tiếng Việt</a></li><li class="interlanguage-link interwiki-vo badge-Q17437796 badge-featuredarticle" title="featured article"><a href="//vo.wikipedia.org/wiki/Matemat" title="Matemat – Volapük" lang="vo" hreflang="vo">Volapük</a></li><li class="interlanguage-link interwiki-fiu-vro"><a href="//fiu-vro.wikipedia.org/wiki/Mat%C3%B5maatiga" title="Matõmaatiga – Võro" lang="fiu-vro" hreflang="fiu-vro">Võro</a></li><li class="interlanguage-link interwiki-zh-classical"><a href="//zh-classical.wikipedia.org/wiki/%E6%95%B8%E5%AD%B8" title="數學 – Classical Chinese" lang="zh-classical" hreflang="zh-classical">文言</a></li><li class="interlanguage-link interwiki-vls"><a href="//vls.wikipedia.org/wiki/Wiskunde" title="Wiskunde – West Flemish" lang="vls" hreflang="vls">West-Vlams</a></li><li class="interlanguage-link interwiki-war"><a href="//war.wikipedia.org/wiki/Matematika" title="Matematika – Waray" lang="war" hreflang="war">Winaray</a></li><li class="interlanguage-link interwiki-wo"><a href="//wo.wikipedia.org/wiki/Xayma" title="Xayma – Wolof" lang="wo" hreflang="wo">Wolof</a></li><li class="interlanguage-link interwiki-wuu"><a href="//wuu.wikipedia.org/wiki/%E6%95%B0%E5%AD%A6" title="数学 – Wu Chinese" lang="wuu" hreflang="wuu">吴语</a></li><li class="interlanguage-link interwiki-ts"><a href="//ts.wikipedia.org/wiki/Dyondzo-Tinhlayo" title="Dyondzo-Tinhlayo – Tsonga" lang="ts" hreflang="ts">Xitsonga</a></li><li class="interlanguage-link interwiki-yi"><a href="//yi.wikipedia.org/wiki/%D7%9E%D7%90%D7%98%D7%A2%D7%9E%D7%90%D7%98%D7%99%D7%A7" title="מאטעמאטיק – Yiddish" lang="yi" hreflang="yi">ייִדיש</a></li><li class="interlanguage-link interwiki-yo"><a href="//yo.wikipedia.org/wiki/Mathim%C3%A1t%C3%ADk%C3%AC" title="Mathimátíkì – Yoruba" lang="yo" hreflang="yo">Yorùbá</a></li><li class="interlanguage-link interwiki-zh-yue"><a href="//zh-yue.wikipedia.org/wiki/%E6%95%B8%E5%AD%B8" title="數學 – Cantonese" lang="zh-yue" hreflang="zh-yue">粵語</a></li><li class="interlanguage-link interwiki-diq"><a href="//diq.wikipedia.org/wiki/Matematik" title="Matematik – Zazaki" lang="diq" hreflang="diq">Zazaki</a></li><li class="interlanguage-link interwiki-zea"><a href="//zea.wikipedia.org/wiki/Wiskunde" title="Wiskunde – Zeelandic" lang="zea" hreflang="zea">Zeêuws</a></li><li class="interlanguage-link interwiki-bat-smg"><a href="//bat-smg.wikipedia.org/wiki/Matemat%C4%97ka" title="Matematėka – Samogitian" lang="bat-smg" hreflang="bat-smg">Žemaitėška</a></li><li class="interlanguage-link interwiki-zh badge-Q17437798 badge-goodarticle" title="good article"><a href="//zh.wikipedia.org/wiki/%E6%95%B0%E5%AD%A6" title="数学 – Chinese" lang="zh" hreflang="zh">中文</a></li><li class="interlanguage-link interwiki-mai"><a href="//mai.wikipedia.org/wiki/%E0%A4%97%E0%A4%A3%E0%A4%BF%E0%A4%A4" title="गणित – Maithili" lang="mai" hreflang="mai">मैथिली</a></li><li class="uls-p-lang-dummy"><a href="#"></a></li>					</ul>
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