Editing Pure mathematics (section)

{{Short description|Mathematics independent of applications}}
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{{Math topics TOC}}
[[File:E8Petrie.svg|thumb|251x251px|Pure mathematics studies the properties and structure of abstract objects,<ref>{{cite web|url=https://www.liverpool.ac.uk/mathematical-sciences/research/pure-mathematics/|title=Pure Mathematics|publisher=[[University of Liverpool]]|access-date=2022-03-24}}</ref> such as the [[E8 (mathematics)|E8 group]], in [[group theory]]. This may be done without focusing on concrete applications of concepts in the physical world.]]
'''Pure mathematics''' is the study of mathematical concepts independently of any application outside [[mathematics]]. These concepts may originate in real-world concerns, and the results obtained may later turn out to be useful for practical applications, but pure mathematicians are not primarily motivated by such applications. Instead, the appeal is attributed to the intellectual challenge and [[Mathematical beauty|aesthetic beauty]] of working out the logical consequences of basic principles.

While pure mathematics has existed as an activity since at least [[ancient Greece]], the concept was elaborated upon around the year 1900,<ref>{{MacTutor|id=Sadleirian_Professors|title=Sadleirian Professors|last=Piaggio|first=H. T. H.|class=Extras}}</ref> after the introduction of theories with counter-intuitive properties (such as [[non-Euclidean geometries]] and [[Georg Cantor|Cantor's]] theory of infinite sets), and the discovery of apparent paradoxes (such as [[continuous function]]s that are nowhere [[differentiable function|differentiable]], and [[Russell's paradox]]). This introduced the need to renew the concept of [[mathematical rigor]] and rewrite all mathematics accordingly, with a systematic use of [[axiomatic method]]s. This led many mathematicians to focus on mathematics for its own sake, that is, pure mathematics.

Nevertheless, almost all mathematical theories remained motivated by problems coming from the real world or from less abstract mathematical theories. Also, many mathematical theories, which had seemed to be totally pure mathematics, were eventually used in applied areas, mainly [[physics]] and [[computer science]]. A famous early example is [[Isaac Newton]]'s demonstration that his [[law of universal gravitation]] implied that [[planet]]s move in orbits that are [[conic section]]s, geometrical curves that had been studied in antiquity by [[Apollonius of Perga|Apollonius]]. Another example is the problem of [[factorization|factoring]] large [[integer]]s, which is the basis of the [[RSA cryptosystem]], widely used to secure [[internet]] communications.<ref>{{cite journal |url=https://www.msri.org/people/members/sara/articles/rsa.pdf |journal=SIAM News |volume=36 |issue=5 |date=June 2003 |title=Still Guarding Secrets after Years of Attacks, RSA Earns Accolades for its Founders |first=Sara |last=Robinson }}</ref>

It follows that, currently, the distinction between pure and [[applied mathematics]] is more a philosophical point of view or a mathematician's preference rather than a rigid subdivision of mathematics.<ref>{{Cite journal |last1=Koperski |first1=Jeffrey |title=Mathematics |journal=European Journal for Philosophy of Science |volume=12 |issue=1 |pages=Article 12 |year=2022 |doi=10.1007/s13194-021-00435-9 |url=https://link.springer.com/content/pdf/10.1007/s13194-021-00435-9.pdf |access-date=October 16, 2024}}</ref>