Mathematics

=== 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}}