Mathematics

=== Pure and applied mathematics ===
{{main|Applied mathematics|Pure mathematics}}
{{multiple image
| footer            = Isaac Newton (left) and [[Gottfried Wilhelm Leibniz]] developed infinitesimal calculus.
| total_width       = 330
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| height1           = 559
| image1            = GodfreyKneller-IsaacNewton-1689.jpg
| alt1              = Isaac Newton
| width2            = 320
| height2           = 390
| image2            = Gottfried Wilhelm Leibniz, Bernhard Christoph Francke.jpg
| alt2              = Gottfried Wilhelm von Leibniz
}}

Until the 19th century, the development of mathematics in the West was mainly motivated by the needs of [[technology]] and science, and there was no clear distinction between pure and applied mathematics.<ref name="Ferreirós_2007">{{cite book
 | title=The Shaping of Arithmetic after C.F. Gauss's Disquisitiones Arithmeticae
 | last=Ferreirós | first=J.
 | chapter=Ό Θεὸς Άριθμητίζει: The Rise of Pure Mathematics as Arithmetic with Gauss
 | pages=235–268 | year=2007 | isbn=978-3-540-34720-0
 | editor1-first=Catherine | editor1-last=Goldstein | editor1-link=Catherine Goldstein
 | editor2-first=Norbert | editor2-last=Schappacher
 | editor3-first=Joachim | editor3-last=Schwermer
 | publisher=Springer Science & Business Media
 | chapter-url={{GBurl|id=IUFTcOsMTysC|p=235}}
}}</ref> For example, the natural numbers and arithmetic were introduced for the need of counting, and geometry was motivated by surveying, architecture and astronomy. Later, [[Isaac Newton]] introduced infinitesimal calculus for explaining the movement of the [[planet]]s with his law of gravitation. Moreover, most mathematicians were also scientists, and many scientists were also mathematicians.<ref>{{cite journal
 | title=Mathematical vs. Experimental Traditions in the Development of Physical Science
 | first=Thomas S. | last=Kuhn | author-link=Thomas Kuhn
 | journal=The Journal of Interdisciplinary History
 | year=1976 | volume=7 | issue=1 | pages=1–31 | publisher=The MIT Press
 | jstor=202372 | doi=10.2307/202372
}}</ref> However, a notable exception occurred with the tradition of [[pure mathematics in Ancient Greece]].<ref>{{cite book
 | chapter=The two cultures of mathematics in ancient Greece
 | first=Markus
 | last=Asper
 | year=2009
 | title=The Oxford Handbook of the History of Mathematics
 | editor1-first=Eleanor
 | editor1-last=Robson
 | editor2-first=Jacqueline
 | editor2-last=Stedall
 | pages=107–132
 | isbn=978-0-19-921312-2
 | publisher=OUP Oxford
 | series=Oxford Handbooks in Mathematics
 | chapter-url={{GBurl|id=xZMSDAAAQBAJ|p=107}}
 | access-date=November 18, 2022
}}</ref> The problem of [[integer factorization]], for example, which goes back to [[Euclid]] in 300 BC, had no practical application before its use in the [[RSA cryptosystem]], now widely used for the security of [[computer network]]s.<ref>{{cite book |last1=Gozwami |first1=Pinkimani |last2=Singh |first2=Madan Mohan |editor-last1=Ahmad |editor-first1=Khaleel |editor-last2=Doja |editor-first2=M. N. |editor-last3=Udzir |editor-first3=Nur Izura |editor-last4=Singh |editor-first4=Manu Pratap |year=2019 |pages=59–60 |chapter=Integer Factorization Problem |title=Emerging Security Algorithms and Techniques |publisher=CRC Press |isbn=978-0-8153-6145-9 |lccn=2019010556 |oclc=1082226900}}</ref>

In the 19th century, mathematicians such as [[Karl Weierstrass]] and [[Richard Dedekind]] increasingly focused their research on internal problems, that is, ''pure mathematics''.<ref name="Ferreirós_2007"/><ref>{{cite journal
 | title=How applied mathematics became pure
 | last=Maddy | first=P. | author-link=Penelope Maddy
 | journal=The Review of Symbolic Logic
 | year=2008
 | volume=1
 | issue=1
 | pages=16–41
 | doi=10.1017/S1755020308080027
 | s2cid=18122406
 | url=http://pgrim.org/philosophersannual/pa28articles/maddyhowapplied.pdf
 | access-date=November 19, 2022
 | archive-date=August 12, 2017
| archive-url=https://web.archive.org/web/20170812012210/http://pgrim.org/philosophersannual/pa28articles/maddyhowapplied.pdf
 | url-status=live
 }}</ref> This led to split mathematics into ''pure mathematics'' and ''applied mathematics'', the latter being often considered as having a lower value among mathematical purists. However, the lines between the two are frequently blurred.<ref>{{cite book
 | title=The Best Writing on Mathematics, 2016
 | chapter=In Defense of Pure Mathematics
 | first=Daniel S.
 | last=Silver
 | pages=17–26
 | isbn=978-0-691-17529-4
 | year=2017
 | editor1-first=Mircea
 | editor1-last=Pitici
 | publisher=Princeton University Press
 | chapter-url={{GBurl|id=RXGYDwAAQBAJ|p=17}}
 | access-date=November 19, 2022
}}</ref>

The aftermath of [[World War II]] led to a surge in the development of applied mathematics in the US and elsewhere.<ref>{{cite journal | title=The American Mathematical Society and Applied Mathematics from the 1920s to the 1950s: A Revisionist Account | first=Karen Hunger | last=Parshall | author-link=Karen Hunger Parshall | journal=Bulletin of the American Mathematical Society | volume=59 | year=2022 | issue=3 | pages=405–427 | doi=10.1090/bull/1754 | s2cid=249561106 | url=https://www.ams.org/journals/bull/2022-59-03/S0273-0979-2022-01754-5/home.html | access-date=November 20, 2022 | doi-access=free | archive-date=November 20, 2022 | archive-url=https://web.archive.org/web/20221120151259/https://www.ams.org/journals/bull/2022-59-03/S0273-0979-2022-01754-5/home.html | url-status=live }}</ref><ref>{{cite journal
 | title=The History Of Applied Mathematics And The History Of Society
 | first=Michael | last=Stolz
 | journal=Synthese
 | volume=133 | pages=43–57 | year=2002
 | doi=10.1023/A:1020823608217
 | s2cid=34271623 | url=https://www.researchgate.net/publication/226795930
 | access-date=November 20, 2022
}}</ref> Many of the theories developed for applications were found interesting from the point of view of pure mathematics, and many results of pure mathematics were shown to have applications outside mathematics; in turn, the study of these applications may give new insights on the "pure theory".<ref>{{cite journal
 | title=On the role of applied mathematics
 | journal=[[Advances in Mathematics]] | first=C. C . | last=Lin
 | volume=19 | issue=3 | date=March 1976 | pages=267–288
 | doi=10.1016/0001-8708(76)90024-4 | doi-access=free
}}</ref><ref>{{cite conference
 | title=Applying Pure Mathematics
 | first=Anthony
 | last=Peressini
 | conference=Philosophy of Science. Proceedings of the 1998 Biennial Meetings of the Philosophy of Science Association. Part I: Contributed Papers
 | volume=66
 | date=September 1999
 | pages=S1–S13
 | jstor=188757
 | access-date=November 30, 2022
 | url=https://www.academia.edu/download/32799272/ApplyingMathPSA.pdf
 | archive-url=https://web.archive.org/web/20240102210931/https://d1wqtxts1xzle7.cloudfront.net/32799272/ApplyingMathPSA-libre.pdf?1391205742=&response-content-disposition=inline%3B+filename%3DApplying_Pure_Mathematics.pdf&Expires=1704233371&Signature=BvNJyYufdj9BiKFe94w6gdXLpAfr7T5JIv~RU74R2uT0O9Ngj6i4cdBtYYOSB6D4V-MgButb6lKNhIGGQogw0e0sHVFkJUy5TRsoCiQ-MLabpZOf74E5SGLMFIExhGVAw7SKrSFaQsFGhfbaRMxbMP~u-wRdJAz6ve6kbWr6oq-doQeEOlRfO4EByNCUYx-KAk3~cBsH1Q2WNZ5QiVObMI1ufQ7zkQM1bqzOumLu6g07F~pt~Cds~lftuQufHomoTH-V9H9iKQgUyc3-4bEB1y1Jdngs7WWg76LcSGn65bPK8dxvsZzKaLDGfoK5jamZkA8z3-xxiMIPL8c6YETjZA__&Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA
 | archive-date=January 2, 2024
 | url-status=live
 }}</ref>

An example of the first case is the [[theory of distributions]], introduced by [[Laurent Schwartz]] for validating computations done in [[quantum mechanics]], which became immediately an important tool of (pure) mathematical analysis.<ref>{{cite conference
 | title=Mathematics meets physics: A contribution to their interaction in the 19th and the first half of the 20th century
 | last=Lützen
 | first=J.
 | year=2011
 | editor1-last=Schlote
 | editor1-first=K. H.
 | editor2-last=Schneider
 | editor2-first=M.
 | publisher=Verlag Harri Deutsch
 | publication-place=Frankfurt am Main
 | chapter=Examples and reflections on the interplay between mathematics and physics in the 19th and 20th century
 | chapter-url=https://slub.qucosa.de/api/qucosa%3A16267/zip/
 | access-date=November 19, 2022
 | archive-date=March 23, 2023
| archive-url=https://web.archive.org/web/20230323164143/https://slub.qucosa.de/api/qucosa%3A16267/zip/
 | url-status=live
 }}</ref> An example of the second case is the [[decidability of the first-order theory of the real numbers]], a problem of pure mathematics that was proved true by [[Alfred Tarski]], with an algorithm that is impossible to [[implementation (computer science)|implement]] because of a computational complexity that is much too high.<ref>{{cite journal
 | title=Model theory and exponentiation
 | last=Marker
 | first=Dave
 | journal=Notices of the American Mathematical Society
 | volume=43
 | issue=7
 | date=July 1996
 | pages=753–759
 | url=https://www.ams.org/notices/199607/
 | access-date=November 19, 2022
 | archive-date=March 13, 2014
| archive-url=https://web.archive.org/web/20140313004011/http://www.ams.org/notices/199607/
 | url-status=live
 }}</ref> For getting an algorithm that can be implemented and can solve systems of polynomial equations and inequalities, [[George E. Collins|George Collins]] introduced the [[cylindrical algebraic decomposition]] that became a fundamental tool in [[real algebraic geometry]].<ref>{{cite conference
 | title=Cylindrical Algebraic Decomposition in the RegularChains Library
 | first1=Changbo | last1=Chen | first2=Marc Moreno | last2=Maza
 | date=August 2014 | volume=8592
 | publisher=Springer | publication-place=Berlin
 | conference=International Congress on Mathematical Software 2014
 | series=Lecture Notes in Computer Science
 | url=https://www.researchgate.net/publication/268067322
 | access-date=November 19, 2022 | doi=10.1007/978-3-662-44199-2_65 }}</ref>

In the present day, the distinction between pure and applied mathematics is more a question of personal research aim of mathematicians than a division of mathematics into broad areas.<ref>{{cite journal
 | title=Purifying applied mathematics and applying pure mathematics: how a late Wittgensteinian perspective sheds light onto the dichotomy
 | first1=José Antonio | last1=Pérez-Escobar | first2=Deniz | last2=Sarikaya
 | journal=European Journal for Philosophy of Science
 | volume=12 | issue=1 | pages=1–22 | year=2021
 | doi=10.1007/s13194-021-00435-9 | s2cid=245465895
| doi-access=free }}</ref><ref>{{cite book
 | chapter=Pure Mathematics and Applied Mathematics are Inseparably Intertwined: Observation of the Early Analysis of the Infinity
 | last=Takase
 | first=M.
 | title=A Mathematical Approach to Research Problems of Science and Technology
 | series=Mathematics for Industry
 | volume=5
 | year=2014
 | pages=393–399
 | publisher=Springer
 | publication-place=Tokyo
 | chapter-url={{GBurl|id=UeElBAAAQBAJ|p=393}}
 | doi=10.1007/978-4-431-55060-0_29
 | isbn=978-4-431-55059-4
 | access-date=November 20, 2022
}}</ref> The Mathematics Subject Classification has a section for "general applied mathematics" but does not mention "pure mathematics".<ref name=MSC/> However, these terms are still used in names of some [[university]] departments, such as at the [[Faculty of Mathematics, University of Cambridge|Faculty of Mathematics]] at the [[University of Cambridge]].