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Mathematical structure
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<h2 id="history">History</h2> <p>In 1939, the French group with the pseudonym "<a href="/facts/Nicolas_Bourbaki/qRJhs9nr">Nicolas Bourbaki</a>" saw structures as the root of mathematics. They first mentioned them in their "Fascicule" of <i>Theory of Sets</i> and expanded it into Chapter IV of the 1957 edition.<a class="footnote-ref" id="fnref:2" href="#fn:2"><sup>2</sup></a> They identified three <i>mother structures</i>: algebraic, topological, and order.<a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a><a class="footnote-ref" id="fnref:4" href="#fn:4"><sup>4</sup></a> </p> <h2 id="example-the-real-numbers">Example: the real numbers</h2> <p>The set of <a href="/facts/Real_number/R02gw5Pb">real numbers</a> has several standard structures: </p> <ul><li>An order: each number is either less than or greater than any other number.</li> <li>Algebraic structure: there are operations of addition and multiplication, the first of which makes it into a <a href="/facts/Group_theory/NfowcI5H">group</a> and the pair of which together make it into a <a href="/facts/Field_(mathematics)/xAjAS4ko">field</a>.</li> <li>A measure: <a href="/facts/Interval_(mathematics)/e9se3vTe">intervals</a> of the real line have a specific <a href="/facts/Length/EwFyAqGH">length</a>, which can be extended to the <a href="/facts/Lebesgue_measure/8us9cGoW">Lebesgue measure</a> on many of its <a href="/facts/Subset/SJGERmbA">subsets</a>.</li> <li>A metric: there is a notion of <a href="/facts/Metric_(mathematics)/RkUptSo8">distance</a> between points.</li> <li>A geometry: it is equipped with a <a href="/facts/Metric_(mathematics)/RkUptSo8">metric</a> and is <a href="/facts/Flat_space/vDkfkQJn">flat</a>.</li> <li>A topology: there is a notion of <a href="/facts/Open_set/QfTCIqMu">open sets</a>.</li></ul> <p>There are interfaces among these: </p> <ul><li>Its order and, independently, its metric structure induce its topology.</li> <li>Its order and algebraic structure make it into an <a href="/facts/Ordered_field/an0prSgG">ordered field</a>.</li> <li>Its algebraic structure and topology make it into a <a href="/facts/Lie_group/a9jCQyjF">Lie group</a>, a type of <a href="/facts/Topological_group/cAgNMMRU">topological group</a>.</li></ul> <h2 id="see-also">See also</h2> <ul><li><a href="/facts/Abstract_structure/EkSkcwdQ">Abstract structure</a></li> <li><a href="/facts/Isomorphism/pj8KgkxU">Isomorphism</a></li> <li><a href="/facts/Equivalent_definitions_of_mathematical_structures/g62kyEjQ">Equivalent definitions of mathematical structures</a></li> <li><a href="/facts/Forgetful_functor/onEwu6HA">Forgetful functor</a></li> <li><a href="/facts/Intuitionistic_type_theory/qwkrpKlP">Intuitionistic type theory</a></li> <li><a href="/facts/Mathematical_object/Juhyow9M">Mathematical object</a></li> <li><a href="/facts/Algebraic_structure/TkuXca24">Algebraic structure</a></li> <li><a href="/facts/Space_(mathematics)/G6rt4bQ8">Space (mathematics)</a></li> <li><a href="/facts/Category_(mathematics)/7WzxUThf">Category (mathematics)</a></li></ul> <h2 id="further-reading">Further reading</h2> <ul><li>Bourbaki, Nikolas (1968). "Elements of Mathematics: Theory of Sets". Hermann, Addison-Wesley. pp. 259–346, 383–385.</li> <li>Foldes, Stephan (1994). <a href="https://archive.org/details/fundamentalstruc0000fold"><i>Fundamental Structures of Algebra and Discrete Mathematics</i></a>. Hoboken: John Wiley & Sons. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 9781118031438.</li> <li>Hegedus, Stephen John; Moreno-Armella, Luis (2011). "The emergence of mathematical structures". <i>Educational Studies in Mathematics</i>. 77 (2): 369–388. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1007%2Fs10649-010-9297-7">10.1007/s10649-010-9297-7</a>. <a href="/facts/S2CID_(identifier)/ldJsHa2Y">S2CID</a> <a href="https://api.semanticscholar.org/CorpusID:119981368">119981368</a>.</li> <li>Kolman, Bernard; Busby, Robert C.; Ross, Sharon Cutler (2000). <i>Discrete mathematical structures</i> (4th ed.). Upper Saddle River, NJ: Prentice Hall. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-13-083143-9.</li> <li>Malik, D.S.; Sen, M.K. (2004). <i>Discrete mathematical structures : theory and applications</i>. Australia: Thomson/Course Technology. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-619-21558-3.</li> <li>Pudlák, Pavel (2013). "Mathematical structures". <i>Logical foundations of mathematics and computational complexity a gentle introduction</i>. Cham: Springer. pp. 2–24. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 9783319001197.</li> <li><a href="/facts/Marjorie_Senechal/oYH1VtET">Senechal, M.</a> (21 May 1993). "Mathematical Structures". <i>Science</i>. 260 (5111): 1170–1173. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1126%2Fscience.260.5111.1170">10.1126/science.260.5111.1170</a>. <a href="/facts/PMID_(identifier)/JlHAvMHt">PMID</a> <a href="https://pubmed.ncbi.nlm.nih.gov/17806355">17806355</a>.</li></ul> <h2 id="external-links">External links</h2> <ul><li><a href="https://planetmath.org/Structure">"Structure"</a>. <i><a href="/facts/PlanetMath/wphPCtLf">PlanetMath</a></i>. <i>(provides a model theoretic definition.)</i></li> <li><a href="http://journals.cambridge.org/action/displayJournal?jid=MSC">Mathematical structures in computer science</a> (journal)</li></ul> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>Saunders, Mac Lane (1996). "Structure in Mathematics" (PDF). Philosoph1A Mathemat1Ca. 4 (3): 176. <a href="http://www2.mat.ulaval.ca/fileadmin/Pages_personnelles_des_profs/hm/H14_Mac_Lane_Phil_Math_1996.pdf" target="_blank">http://www2.mat.ulaval.ca/fileadmin/Pages_personnelles_des_profs/hm/H14_Mac_Lane_Phil_Math_1996.pdf</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> <li id="fn:2"><p>Corry, Leo (September 1992). "Nicolas Bourbaki and the concept of mathematical structure". Synthese. 92 (3): 315–348. doi:10.1007/bf00414286. JSTOR 20117057. S2CID 16981077. <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li> <li id="fn:3"><p>Corry, Leo (September 1992). "Nicolas Bourbaki and the concept of mathematical structure". Synthese. 92 (3): 315–348. doi:10.1007/bf00414286. JSTOR 20117057. S2CID 16981077. <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li> <li id="fn:4"><p>Wells, Richard B. (2010). Biological signal processing and computational neuroscience (PDF). pp. 296–335. Retrieved 7 April 2016. <a href="http://www.mrc.uidaho.edu/~rwells/techdocs/Biological%20Signal%20Processing/Chapter%2010%20Mathematical%20Structures.pdf" target="_blank">http://www.mrc.uidaho.edu/~rwells/techdocs/Biological%20Signal%20Processing/Chapter%2010%20Mathematical%20Structures.pdf</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li> </ol>