Mathematics

== Relationship with sciences ==
Mathematics is used in most [[science]]s for [[Mathematical model|modeling]] phenomena, which then allows predictions to be made from experimental laws.<ref>{{cite book | title=Modelling Mathematical Methods and Scientific Computation | first1=Nicola | last1=Bellomo | first2=Luigi | last2=Preziosi | publisher=CRC Press | date=December 22, 1994 | page=1 | isbn=978-0-8493-8331-1 | series=Mathematical Modeling | volume=1 | url={{GBurl|id=pJAvWaRYo3UC}} | access-date=November 16, 2022 }}</ref> The independence of mathematical truth from any experimentation implies that the accuracy of such predictions depends only on the adequacy of the model.<ref>{{cite journal
 | title=Mathematical Models and Reality: A Constructivist Perspective
 | first=Christian | last=Hennig
 | journal=Foundations of Science
 | volume=15 | pages=29–48 | year=2010
 | doi=10.1007/s10699-009-9167-x
 | s2cid=6229200 | url=https://www.researchgate.net/publication/225691477
 | access-date=November 17, 2022
}}</ref> Inaccurate predictions, rather than being caused by invalid mathematical concepts, imply the need to change the mathematical model used.<ref>{{cite journal | title=Models in Science | date=February 4, 2020 | first1=Roman | last1=Frigg | author-link=Roman Frigg | first2=Stephan | last2=Hartmann | author2-link=Stephan Hartmann | website=Stanford Encyclopedia of Philosophy | url=https://seop.illc.uva.nl/entries/models-science/ | access-date=November 17, 2022 | archive-date=November 17, 2022 | archive-url=https://web.archive.org/web/20221117162412/https://seop.illc.uva.nl/entries/models-science/ | url-status=live }}</ref> For example, the [[perihelion precession of Mercury]] could only be explained after the emergence of [[Einstein]]'s [[general relativity]], which replaced [[Newton's law of gravitation]] as a better mathematical model.<ref>{{cite book | last=Stewart | first=Ian | author-link=Ian Stewart (mathematician) | chapter=Mathematics, Maps, and Models | title=The Map and the Territory: Exploring the Foundations of Science, Thought and Reality | pages=345–356 | publisher=Springer | year=2018 | editor1-first=Shyam | editor1-last=Wuppuluri | editor2-first=Francisco Antonio | editor2-last=Doria | isbn=978-3-319-72478-2 | series=The Frontiers Collection | chapter-url={{GBurl|id=mRBMDwAAQBAJ|p=345}} | doi=10.1007/978-3-319-72478-2_18 | access-date=November 17, 2022 }}</ref>

There is still a [[philosophy of mathematics|philosophical]] debate whether mathematics is a science. However, in practice, mathematicians are typically grouped with scientists, and mathematics shares much in common with the physical sciences. Like them, it is [[falsifiable]], which means in mathematics that, if a result or a theory is wrong, this can be proved by providing a [[counterexample]]. Similarly as in science, [[mathematical theory|theories]] and results (theorems) are often obtained from [[experimentation]].<ref>{{Cite web|url=https://undsci.berkeley.edu/article/mathematics|title=The science checklist applied: Mathematics|website=Understanding Science |publisher=University of California, Berkeley |access-date=October 27, 2019|archive-url=https://web.archive.org/web/20191027021023/https://undsci.berkeley.edu/article/mathematics|archive-date=October 27, 2019|url-status=live}}</ref> In mathematics, the experimentation may consist of computation on selected examples or of the study of figures or other representations of mathematical objects (often mind representations without physical support). For example, when asked how he came about his theorems, Gauss once replied "durch planmässiges Tattonieren" (through systematic experimentation).<ref>{{cite book | last=Mackay | first=A. L. | year=1991 | title=Dictionary of Scientific Quotations | location=London | page=100 | isbn=978-0-7503-0106-0 | publisher=Taylor & Francis | url={{GBurl|id=KwESE88CGa8C|q=durch planmässiges Tattonieren}} | access-date=March 19, 2023 }}</ref> However, some authors emphasize that mathematics differs from the modern notion of science by not {{em|relying}} on empirical evidence.<ref name="Bishop1991">{{cite book | last1 = Bishop | first1 = Alan | year = 1991 | chapter = Environmental activities and mathematical culture | title = Mathematical Enculturation: A Cultural Perspective on Mathematics Education | chapter-url = {{GBurl|id=9AgrBgAAQBAJ|p=54}} | pages = 20–59 | location = Norwell, Massachusetts | publisher = Kluwer Academic Publishers | isbn = 978-0-7923-1270-3 | access-date = April 5, 2020 }}</ref><ref>{{cite book | title=Out of Their Minds: The Lives and Discoveries of 15 Great Computer Scientists | last1=Shasha | first1=Dennis Elliot | author1-link=Dennis Elliot Shasha | last2=Lazere | first2=Cathy A. | publisher=Springer | year=1998 | page=228 | isbn=978-0-387-98269-4 }}</ref><ref name="Nickles2013">{{cite book | last=Nickles | first=Thomas | year=2013 | chapter=The Problem of Demarcation | title=Philosophy of Pseudoscience: Reconsidering the Demarcation Problem | page=104 | location=Chicago | publisher=The University of Chicago Press | isbn=978-0-226-05182-6 }}</ref><ref name="Pigliucci2014">{{Cite magazine | year=2014| last=Pigliucci| first=Massimo | author-link=Massimo Pigliucci | title=Are There 'Other' Ways of Knowing? | magazine=[[Philosophy Now]]| url=https://philosophynow.org/issues/102/Are_There_Other_Ways_of_Knowing | access-date=April 6, 2020| archive-date=May 13, 2020 | archive-url=https://web.archive.org/web/20200513190522/https://philosophynow.org/issues/102/Are_There_Other_Ways_of_Knowing | url-status=live}}</ref>
<!-- What precedes is only one aspect of the relationship between mathematics and other sciences. Other aspects are considered in the next subsections. -->

=== 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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| image1            = GodfreyKneller-IsaacNewton-1689.jpg
| alt1              = Isaac Newton
| width2            = 320
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| 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]].

=== Unreasonable effectiveness ===

The [[The Unreasonable Effectiveness of Mathematics in the Natural Sciences|unreasonable effectiveness of mathematics]] is a phenomenon that was named and first made explicit by physicist [[Eugene Wigner]].<ref name=wigner1960>{{cite journal
 | title=The Unreasonable Effectiveness of Mathematics in the Natural Sciences
 | last=Wigner | first=Eugene | author-link=Eugene Wigner
 | journal=[[Communications on Pure and Applied Mathematics]]
 | volume=13 | issue=1 | pages=1–14 | year=1960
 | doi=10.1002/cpa.3160130102 | bibcode=1960CPAM...13....1W
 | s2cid=6112252 | url=https://math.dartmouth.edu/~matc/MathDrama/reading/Wigner.html
 | url-status=live | archive-url=https://web.archive.org/web/20110228152633/http://www.dartmouth.edu/~matc/MathDrama/reading/Wigner.html
 | archive-date=February 28, 2011 | df=mdy-all
}}</ref> It is the fact that many mathematical theories (even the "purest") have applications outside their initial object. These applications may be completely outside their initial area of mathematics, and may concern physical phenomena that were completely unknown when the mathematical theory was introduced.<ref>{{cite journal
 | title=Revisiting the 'unreasonable effectiveness' of mathematics
 | first=Sundar | last=Sarukkai
 | journal=Current Science
 | volume=88 | issue=3 | date=February 10, 2005 | pages=415–423
 | jstor=24110208
}}</ref> Examples of unexpected applications of mathematical theories can be found in many areas of mathematics.

A notable example is the [[prime factorization]] of natural numbers that was discovered more than 2,000 years before its common use for secure [[internet]] communications through the [[RSA cryptosystem]].<ref>{{cite book
 | chapter=History of Integer Factoring
 | pages=41–77
 | first=Samuel S. Jr.
 | last=Wagstaff
 | title=Computational Cryptography, Algorithmic Aspects of Cryptography, A Tribute to AKL
 | editor1-first=Joppe W.
 | editor1-last=Bos
 | editor2-first=Martijn
 | editor2-last=Stam
 | series=London Mathematical Society Lecture Notes Series 469
 | publisher=Cambridge University Press
 | year=2021
 | chapter-url=https://www.cs.purdue.edu/homes/ssw/chapter3.pdf
 | access-date=November 20, 2022
 | archive-date=November 20, 2022
| archive-url=https://web.archive.org/web/20221120155733/https://www.cs.purdue.edu/homes/ssw/chapter3.pdf
 | url-status=live
 }}</ref> A second historical example is the theory of [[ellipse]]s. They were studied by the [[Greek mathematics|ancient Greek mathematicians]] as [[conic section]]s (that is, intersections of [[cone]]s with planes). It is almost 2,000 years later that [[Johannes Kepler]] discovered that the [[trajectories]] of the planets are ellipses.<ref>{{cite web
 | title=Curves: Ellipse
 | website=MacTutor
 | publisher=School of Mathematics and Statistics, University of St Andrews, Scotland
 | url=https://mathshistory.st-andrews.ac.uk/Curves/Ellipse/
 | access-date=November 20, 2022
 | archive-date=October 14, 2022
| archive-url=https://web.archive.org/web/20221014051943/https://mathshistory.st-andrews.ac.uk/Curves/Ellipse/
 | url-status=live
 }}</ref>

In the 19th century, the internal development of geometry (pure mathematics) led to definition and study of non-Euclidean geometries, spaces of dimension higher than three and [[manifold]]s. At this time, these concepts seemed totally disconnected from the physical reality, but at the beginning of the 20th century, [[Albert Einstein]] developed the [[theory of relativity]] that uses fundamentally these concepts. In particular, [[spacetime]] of [[special relativity]] is a non-Euclidean space of dimension four, and spacetime of [[general relativity]] is a (curved) manifold of dimension four.<ref>{{cite web
 | title=Beyond the Surface of Einstein's Relativity Lay a Chimerical Geometry
 | first=Vasudevan
 | last=Mukunth
 | website=The Wire
 | date=September 10, 2015
| url=https://thewire.in/science/beyond-the-surface-of-einsteins-relativity-lay-a-chimerical-geometry
 | access-date=November 20, 2022
 | archive-date=November 20, 2022
| archive-url=https://web.archive.org/web/20221120191206/https://thewire.in/science/beyond-the-surface-of-einsteins-relativity-lay-a-chimerical-geometry
 | url-status=live
 }}</ref><ref>{{cite journal
 | title=The Space-Time Manifold of Relativity. The Non-Euclidean Geometry of Mechanics and Electromagnetics
 | first1=Edwin B. | last1=Wilson | first2=Gilbert N. | last2=Lewis
 | journal=Proceedings of the American Academy of Arts and Sciences
 | volume=48 | issue=11 | date=November 1912 | pages=389–507
 | doi=10.2307/20022840 | jstor=20022840 }}</ref>

A striking aspect of the interaction between mathematics and physics is when mathematics drives research in physics. This is illustrated by the discoveries of the [[positron]] and the [[omega baryon|baryon]] <math>\Omega^{-}.</math> In both cases, the equations of the theories had unexplained solutions, which led to conjecture of the existence of an unknown [[particle]], and the search for these particles. In both cases, these particles were discovered a few years later by specific experiments.<ref name=borel /><ref>{{cite journal
 | title=Discovering the Positron (I)
 | first=Norwood Russell | last=Hanson | author-link=Norwood Russell Hanson
 | journal=The British Journal for the Philosophy of Science
 | volume=12 | issue=47 | date=November 1961 | pages=194–214
 | publisher=The University of Chicago Press
 | jstor=685207 | doi=10.1093/bjps/xiii.49.54
}}</ref><ref>{{cite journal
 | title=Avoiding reification: Heuristic effectiveness of mathematics and the prediction of the Ω<sup>–</sup> particle
 | first=Michele | last=Ginammi
 | journal=Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics
 | volume=53 | date=February 2016 | pages=20–27
 | doi=10.1016/j.shpsb.2015.12.001
| bibcode=2016SHPMP..53...20G }}</ref>

=== Specific sciences ===
{{Essay-like|date=December 2022|subsection}}

==== Physics ====
{{Main|Relationship between mathematics and physics}}
[[File:Pendule schema.gif|thumb|Diagram of a pendulum]]
Mathematics and physics have influenced each other over their modern history. Modern physics uses mathematics abundantly,<ref>{{Cite book |last1=Wagh |first1=Sanjay Moreshwar |url={{GBurl|id=-DmfVjBUPksC|p=3}} |title=Essentials of Physics  |last2=Deshpande |first2=Dilip Abasaheb |date=September 27, 2012 |publisher=PHI Learning Pvt. Ltd. |isbn=978-81-203-4642-0 |page=3 |language=en |access-date=January 3, 2023 }}</ref> and is also the motivation of major mathematical developments.<ref>{{Cite conference |last=Atiyah |first=Michael |author-link=Michael Atiyah |date=1990 |title=On the Work of Edward Witten |url=http://www.mathunion.org/ICM/ICM1990.1/Main/icm1990.1.0031.0036.ocr.pdf |conference=Proceedings of the International Congress of Mathematicians |page=31 |archive-url=https://web.archive.org/web/20130928095313/http://www.mathunion.org/ICM/ICM1990.1/Main/icm1990.1.0031.0036.ocr.pdf |archive-date=September 28, 2013 |access-date=December 29, 2022}}</ref>

==== Computing ====
{{Further|Theoretical computer science|Computational mathematics}}
The rise of technology in the 20th century opened the way to a new science: [[computing]].{{Efn|[[Ada Lovelace]], in the 1840s, is known for having written the first computer program ever in collaboration with [[Charles Babbage]]}} This field is closely related to mathematics in several ways. [[Theoretical computer science]] is essentially mathematical in nature. Communication technologies apply branches of mathematics that may be very old (e.g., arithmetic), especially with respect to transmission security, in [[cryptography]] and [[coding theory]]. [[Discrete mathematics]] is useful in many areas of computer science, such as [[Computational complexity theory|complexity theory]], [[information theory]], [[graph theory]], and so on.{{Citation needed|date=December 2022}}

In return, computing has also become essential for obtaining new results. This is a group of techniques known as [[experimental mathematics]], which is the use of ''experimentation'' to discover mathematical insights.<ref>{{Cite web |last1=Borwein |first1=J. |last2=Borwein |first2=P. |last3=Girgensohn |first3=R. |last4=Parnes |first4=S. |date=1996 |title=Conclusion |url=http://oldweb.cecm.sfu.ca/organics/vault/expmath/expmath/html/node16.html |url-status=dead |archive-url=https://web.archive.org/web/20080121081424/http://oldweb.cecm.sfu.ca/organics/vault/expmath/expmath/html/node16.html |archive-date=January 21, 2008 |website=oldweb.cecm.sfu.ca}}</ref> The most well-known example is the [[Four color theorem|four-color theorem]], which was proven in 1976 with the help of a computer. This revolutionized traditional mathematics, where the rule was that the mathematician should verify each part of the proof. In 1998, the [[Kepler conjecture]] on [[sphere packing]] seemed to also be partially proven by computer. An international team had since worked on writing a formal proof; it was finished (and verified) in 2015.<ref>{{cite journal |last1=Hales |first1=Thomas |last2=Adams |first2=Mark |last3=Bauer |first3=Gertrud |last4=Dang |first4=Tat Dat |last5=Harrison |first5=John |last6=Hoang |first6=Le Truong |last7=Kaliszyk |first7=Cezary |last8=Magron |first8=Victor |last9=Mclaughlin |first9=Sean |last10=Nguyen |first10=Tat Thang |last11=Nguyen |first11=Quang Truong |last12=Nipkow |first12=Tobias |last13=Obua |first13=Steven |last14=Pleso |first14=Joseph |last15=Rute |first15=Jason |last16=Solovyev |first16=Alexey |last17=Ta |first17=Thi Hoai An |last18=Tran |first18=Nam Trung |last19=Trieu |first19=Thi Diep |last20=Urban |first20=Josef |last21=Vu |first21=Ky |last22=Zumkeller |first22=Roland |title=A Formal Proof of the Kepler Conjecture |journal=Forum of Mathematics, Pi |date=2017 |volume=5 |page=e2 |doi=10.1017/fmp.2017.1 |s2cid=216912822 |url=https://www.cambridge.org/core/journals/forum-of-mathematics-pi/article/formal-proof-of-the-kepler-conjecture/78FBD5E1A3D1BCCB8E0D5B0C463C9FBC |language=en |issn=2050-5086 |access-date=February 25, 2023 |archive-date=December 4, 2020 |archive-url=https://web.archive.org/web/20201204053232/https://www.cambridge.org/core/journals/forum-of-mathematics-pi/article/formal-proof-of-the-kepler-conjecture/78FBD5E1A3D1BCCB8E0D5B0C463C9FBC |url-status=live |hdl=2066/176365 |hdl-access=free }}</ref>

Once written formally, a proof can be verified using a program called a [[proof assistant]].<ref name=":1">{{Cite journal |last=Geuvers |first=H. |date=February 2009 |title=Proof assistants: History, ideas and future |url=https://www.ias.ac.in/article/fulltext/sadh/034/01/0003-0025 |journal=Sādhanā |volume=34 |pages=3–4 |doi=10.1007/s12046-009-0001-5 |s2cid=14827467 |doi-access=free |access-date=December 29, 2022 |archive-date=December 29, 2022 |archive-url=https://web.archive.org/web/20221229204107/https://www.ias.ac.in/article/fulltext/sadh/034/01/0003-0025 |url-status=live |hdl=2066/75958 |hdl-access=free }}</ref> These programs are useful in situations where one is uncertain about a proof's correctness.<ref name=":1" />

A major open problem in theoretical computer science is [[P versus NP problem|P versus NP]]. It is one of the seven [[Millennium Prize Problems]].<ref>{{Cite web |title=P versus NP problem {{!}} mathematics |url=https://www.britannica.com/science/P-versus-NP-problem |access-date=December 29, 2022 |website=Britannica |language=en |archive-date=December 6, 2022 |archive-url=https://web.archive.org/web/20221206044556/https://www.britannica.com/science/P-versus-NP-problem |url-status=live }}</ref>

==== Biology and chemistry ====
{{Main|Mathematical and theoretical biology|Mathematical chemistry}}
[[File:Giant Pufferfish skin pattern detail.jpg|thumb|The skin of this [[giant pufferfish]] exhibits a [[Turing pattern]], which can be modeled by [[reaction–diffusion system]]s.]]
[[Biology]] uses probability extensively – for example, in ecology or [[neurobiology]].<ref name=":2">{{Cite book |last=Millstein |first=Roberta |author-link=Roberta Millstein |title=The Oxford Handbook of Probability and Philosophy |date=September 8, 2016 |editor-last=Hájek |editor-first=Alan |pages=601–622 |chapter=Probability in Biology: The Case of Fitness |doi=10.1093/oxfordhb/9780199607617.013.27 |editor-last2=Hitchcock |editor-first2=Christopher |chapter-url=http://philsci-archive.pitt.edu/10901/1/Millstein-fitness-v2.pdf |access-date=December 29, 2022 |archive-date=March 7, 2023 |archive-url=https://web.archive.org/web/20230307054456/http://philsci-archive.pitt.edu/10901/1/Millstein-fitness-v2.pdf |url-status=live }}</ref> Most of the discussion of probability in biology, however, centers on the concept of [[evolutionary fitness]].<ref name=":2" />

Ecology heavily uses modeling to simulate [[population dynamics]],<ref name=":2" /><ref>See for example Anne Laurent, Roland Gamet, Jérôme Pantel, ''Tendances nouvelles en modélisation pour l'environnement, actes du congrès «Programme environnement, vie et sociétés»'' 15-17 janvier 1996, CNRS</ref> study ecosystems such as the predator-prey model, measure pollution diffusion,{{Sfn|Bouleau|1999|pp=282–283}} or to assess climate change.{{Sfn|Bouleau|1999|p=285}} The dynamics of a population can be modeled by coupled differential equations, such as the [[Lotka–Volterra equations]].<ref>{{Cite web |date=January 5, 2022 |title=1.4: The Lotka-Volterra Predator-Prey Model |url=https://math.libretexts.org/Bookshelves/Applied_Mathematics/Mathematical_Biology_(Chasnov)/01%3A_Population_Dynamics/1.04%3A_The_Lotka-Volterra_Predator-Prey_Model |access-date=December 29, 2022 |website=Mathematics LibreTexts |language=en |archive-date=December 29, 2022 |archive-url=https://web.archive.org/web/20221229204111/https://math.libretexts.org/Bookshelves/Applied_Mathematics/Mathematical_Biology_(Chasnov)/01:_Population_Dynamics/1.04:_The_Lotka-Volterra_Predator-Prey_Model |url-status=live }}</ref> However, there is the problem of [[model validation]]. This is particularly acute when the results of modeling influence political decisions; the existence of contradictory models could allow nations to choose the most favorable model.{{Sfn|Bouleau|1999|p=287}}

Genotype evolution can be modeled with the [[Hardy-Weinberg principle]].{{Citation needed|date=December 2022}}

[[Phylogeography]] uses probabilistic models.{{Citation needed|date=December 2022}}

Medicine uses [[statistical hypothesis testing]], run on data from [[clinical trial]]s, to determine whether a new treatment works.{{Citation needed|date=December 2022}}

Since the start of the 20th century, chemistry has used computing to model molecules in three dimensions. It turns out that the form of [[macromolecules]] in biology is variable and determines the action. Such modeling uses Euclidean geometry; neighboring atoms form a [[polyhedron]] whose distances and angles are fixed by the laws of interaction.{{Citation needed|date=December 2022}}

==== Earth sciences ====
{{Main|Geomathematics}}
[[Structural geology]] and climatology use probabilistic models to predict the risk of natural catastrophes.{{Citation needed|date=December 2022}} Similarly, [[meteorology]], [[oceanography]], and [[planetology]] also use mathematics due to their heavy use of models.{{Citation needed|date=December 2022}}

==== Social sciences ====
{{Further|Mathematical economics|Historical dynamics}}
Areas of mathematics used in the social sciences include probability/statistics and differential equations. These are used in linguistics, [[economics]], [[sociology]],<ref>{{Cite journal |last=Edling |first=Christofer R. |date=2002 |title=Mathematics in Sociology |url=https://www.annualreviews.org/doi/10.1146/annurev.soc.28.110601.140942 |journal=Annual Review of Sociology |language=en |volume=28 |issue=1 |pages=197–220 |doi=10.1146/annurev.soc.28.110601.140942 |issn=0360-0572}}</ref> and [[psychology]].<ref>{{Citation |last=Batchelder |first=William H. |title=Mathematical Psychology: History |date=2015-01-01 |url=https://www.sciencedirect.com/science/article/pii/B978008097086843059X |encyclopedia=International Encyclopedia of the Social & Behavioral Sciences (Second Edition) |pages=808–815 |editor-last=Wright |editor-first=James D. |access-date=2023-09-30 |place=Oxford |publisher=Elsevier |isbn=978-0-08-097087-5}}</ref>
[[File:Supply-demand-equilibrium.svg|thumb|[[Supply and demand|Supply and demand curves]], like this one, are a staple of mathematical economics.]]
The fundamental postulate of mathematical economics is that of the rational individual actor – ''[[Homo economicus]]'' ({{Literal translation|economic man}}).<ref name=":3">{{Cite book |last=Zak |first=Paul J. |url={{GBurl|id=6QrvmNo2qD4C|p=158}} |title=Moral Markets: The Critical Role of Values in the Economy |date=2010 |page=158 |publisher=Princeton University Press |isbn=978-1-4008-3736-6 |language=en |access-date=January 3, 2023 }}</ref> In this model, the individual seeks to maximize their [[rational choice theory|self-interest]],<ref name=":3" /> and always makes optimal choices using [[perfect information]].<ref name=":4">{{Cite web |last=Kim |first=Oliver W. |date=May 29, 2014 |title=Meet Homo Economicus |url=https://www.thecrimson.com/column/homo-economicus/article/2014/9/19/Harvard-homo-economicus-fiction/ |access-date=December 29, 2022 |website=The Harvard Crimson |archive-date=December 29, 2022 |archive-url=https://web.archive.org/web/20221229204106/https://www.thecrimson.com/column/homo-economicus/article/2014/9/19/Harvard-homo-economicus-fiction/ |url-status=live }}</ref>{{Better source needed|reason=this is an opinion essay, not a scholarly work|date=December 2022}} This atomistic view of economics allows it to relatively easily mathematize its thinking, because individual [[calculations]] are transposed into mathematical calculations. Such mathematical modeling allows one to probe economic mechanisms which would be difficult to discover by a "literary" analysis.{{Citation needed|date=December 2022}} For example, explanations of [[economic cycles]] are not trivial. Without mathematical modeling, it is hard to go beyond statistical observations or unproven speculation.{{Citation needed|date=December 2022}}

However, many people have rejected or criticized the concept of ''Homo economicus''.<ref name=":4" />{{Better source needed|reason=this is an opinion essay, not a scholarly work|date=December 2022}} Economists note that real people have limited information, make poor choices and care about fairness, altruism, not just personal gain.<ref name=":4" />{{Better source needed|reason=this is an opinion essay, not a scholarly work|date=December 2022}}

At the start of the 20th century, there was a development to express historical movements in formulas. In 1922, [[Nikolai Kondratiev]] discerned the ~50-year-long [[Kondratiev cycle]], which explains phases of economic growth or crisis.<ref>{{Cite web |title=Kondratiev, Nikolai Dmitrievich {{!}} Encyclopedia.com |url=https://www.encyclopedia.com/history/encyclopedias-almanacs-transcripts-and-maps/kondratiev-nikolai-dmitrievich |access-date=December 29, 2022 |website=www.encyclopedia.com |archive-date=July 1, 2016 |archive-url=https://web.archive.org/web/20160701224009/http://www.encyclopedia.com/doc/1G2-3404100667.html |url-status=live }}</ref> Towards the end of the 19th century, {{Ill|Nicolas-Remi Brück|fr}} and {{Ill|Charles Henri Lagrange|fr}} extended their analysis into [[geopolitics]].<ref>{{Cite web|url=https://onlinebooks.library.upenn.edu/webbin/book/lookupid?key=ha010090244#:~:text=##+Math%C3%A9matique+de+l'histoire,org%E3%80%91|title=Mathématique de l'histoire-géometrie et cinématique. Lois de Brück. Chronologie géodésique de la Bible., by Charles LAGRANGE et al. &#124; The Online Books Page|website=onlinebooks.library.upenn.edu}}</ref> [[Peter Turchin]] has worked on developing [[cliodynamics]] since the 1990s.<ref>{{Cite web |title=Cliodynamics: a science for predicting the future |url=https://www.zdnet.com/article/cliodynamics-a-science-for-predicting-the-future/ |access-date=December 29, 2022 |website=ZDNET |language=en |archive-date=December 29, 2022 |archive-url=https://web.archive.org/web/20221229204104/https://www.zdnet.com/article/cliodynamics-a-science-for-predicting-the-future/ |url-status=live }}</ref>

Even so, mathematization of the social sciences is not without danger. In the controversial book ''[[Fashionable Nonsense]]'' (1997), [[Alan Sokal|Sokal]] and [[Jean Bricmont|Bricmont]] denounced the unfounded or abusive use of scientific terminology, particularly from mathematics or physics, in the social sciences.<ref>{{cite book|last=Sokal|first=Alan|url=https://archive.org/details/fashionablenonse00soka|title=Fashionable Nonsense|author2=Jean Bricmont|publisher=Picador|year=1998|isbn=978-0-312-19545-8|location=New York|oclc=39605994|author-link=Alan Sokal|author2-link=Jean Bricmont}}</ref> The study of [[complex systems]] (evolution of unemployment, business capital, demographic evolution of a population, etc.) uses mathematical knowledge. However, the choice of counting criteria, particularly for unemployment, or of models, can be subject to controversy.{{Citation needed|date=December 2022}}