Editing Mathematics (section)

=== Specific sciences ===
==== 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 considered to be the motivation of major mathematical developments.<ref>{{Cite conference |last=Atiyah |first=Michael |author-link=Michael Atiyah |year=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}}
Computing is closely related to mathematics in several ways.<ref>{{Cite web |title=Course 18C Mathematics with Computer Science |url=https://math.mit.edu/academics/undergrad/major/course18c.html |access-date=June 1, 2024 |website=math.mit.edu}}</ref> [[Theoretical computer science]] is considered to be mathematical in nature.<ref>{{Cite web |title=Theoretical Computer Science |url=https://math.mit.edu/research/applied/comp-science-theory.html |access-date=June 1, 2024 |website=math.mit.edu}}</ref> 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]], and [[graph theory]].<ref>{{Cite web |date=April 8, 2024 |title=Real-Life Applications of Discrete Mathematics |url=https://www.geeksforgeeks.org/real-life-applications-of-discrete-mathematics/ |access-date=May 19, 2024 |website=GeeksforGeeks |language=en-US |archive-date=May 19, 2024 |archive-url=https://web.archive.org/web/20240519225711/https://www.geeksforgeeks.org/real-life-applications-of-discrete-mathematics/ |url-status=live }}</ref> In 1998, the [[Kepler conjecture]] on [[sphere packing]] seemed to also be partially proven by computer.<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 |year=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>

==== 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 in fields such as 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 discussion of probability 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>

[[Statistical hypothesis testing]], is run on data from [[clinical trial]]s to determine whether a new treatment works.<ref>{{Cite journal |last=Salsburg |first=David |date=August 17, 1992 |title=Commentary |url=https://www.dfcm.utoronto.ca/sites/default/files/inline-files/salsburg_1.pdf |journal=The Use of Statistical Methods in the Analysis of Clinical Studies |volume=46 |pages=17 |archive-date=June 1, 2024 |access-date=June 1, 2024 |archive-url=https://web.archive.org/web/20240601211523/https://www.dfcm.utoronto.ca/sites/default/files/inline-files/salsburg_1.pdf |url-status=live }}</ref> Since the start of the 20th century, chemistry has used computing to model molecules in three dimensions.<ref>{{Cite book |url=https://nap.nationalacademies.org/read/10633/chapter/8 |title=Beyond the Molecular Frontier: Challenges for Chemistry and Chemical Engineering |publisher=NAP.edu |year=2003 |isbn=978-0-309-16839-7 |pages=71–73 |language=en |chapter=8 |doi=10.17226/10633|pmid=25032300 |author=National Research Council |author-link=National Research Council (United States) }}</ref>

==== Earth sciences ====
{{Main|Geomathematics}}
[[Structural geology]] and climatology use probabilistic models to predict the risk of natural catastrophes.<ref>{{Cite web |title=Catastrophe Models (Property) |url=https://content.naic.org/cipr-topics/catastrophe-models-property |access-date=May 19, 2024 |website=content.naic.org |language=en |archive-date=May 19, 2024 |archive-url=https://web.archive.org/web/20240519225714/https://content.naic.org/cipr-topics/catastrophe-models-property |url-status=live }}</ref> Similarly, [[meteorology]], [[oceanography]], and [[planetology]] also use mathematics due to their heavy use of models.<ref>{{Cite web |title=MAM2001 Essay |url=https://ww2.amstat.org/mam/01/essay.html |access-date=May 19, 2024 |website=ww2.amstat.org |archive-date=May 19, 2024 |archive-url=https://web.archive.org/web/20240519225711/https://ww2.amstat.org/mam/01/essay.html |url-status=live }}</ref><ref>{{Cite web |last=Hill |first=Mullica |date=September 7, 2022 |title=HOW MATH IS USED IN WEATHER FORECASTING |url=https://www.mathnasium.com/math-centers/mullicahill/news/how-math-used-weather-forecasting |access-date=May 19, 2024 |website=mathnasium.com |archive-date=May 19, 2024 |archive-url=https://web.archive.org/web/20240519225711/https://www.mathnasium.com/math-centers/mullicahill/news/how-math-used-weather-forecasting |url-status=live }}</ref><ref>{{Cite web |title=Using Mathematical Models to Investigate Planetary Habitability |url=https://icp.giss.nasa.gov/education/modules/eccm/eccm_student_3.pdf |access-date=May 19, 2024 |website=[[NASA]] |archive-date=May 19, 2024 |archive-url=https://web.archive.org/web/20240519225714/https://icp.giss.nasa.gov/education/modules/eccm/eccm_student_3.pdf |url-status=live }}</ref>

==== 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. |year=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 |archive-date=November 15, 2021 |access-date=September 30, 2023 |archive-url=https://web.archive.org/web/20211115130943/https://www.annualreviews.org/doi/10.1146/annurev.soc.28.110601.140942 |url-status=live }}</ref> and [[psychology]].<ref>{{Citation |last=Batchelder |first=William H. |title=Mathematical Psychology: History |date=January 1, 2015 |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=September 30, 2023 |place=Oxford |publisher=Elsevier |isbn=978-0-08-097087-5 |archive-date=February 17, 2023 |archive-url=https://web.archive.org/web/20230217140558/https://www.sciencedirect.com/science/article/pii/B978008097086843059X |url-status=live }}</ref>
[[File:Supply-demand-equilibrium.svg|thumb|[[Supply and demand]] curves, like this one, are a staple of mathematical economics.|class=skin-invert-image]]
Often 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>{{cite book |url=https://web.stanford.edu/~jdlevin/Econ%20202/Choice%20Theory.pdf |title=Introduction to Choice Theory |first1=Jonathan |last1=Levin |first2=Paul |last2=Milgrom |date=September 2004}}</ref> 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. Some reject or criticise the concept of ''Homo economicus''. Economists note that real people have limited information, make poor choices and care about fairness, altruism, not just personal gain.<ref>{{cite book |author1=Kremer, Michael |author2=Rao, Gautam |author3=Schilbach, Frank |url=https://economics.mit.edu/sites/default/files/2022-09/behavioral-development-economics.pdf |title=Handbook of Behavioral Economics: Applications and Foundations |chapter=Chapter 5 Behavioral development economics |year=2019 |volume=2 |archive-date=June 2, 2024 |access-date=June 2, 2024 |archive-url=https://web.archive.org/web/20240602215911/https://economics.mit.edu/sites/default/files/2022-09/behavioral-development-economics.pdf |url-status=live }}</ref>

Without mathematical modeling, it is hard to go beyond statistical observations or untestable speculation. Mathematical modeling allows economists to create structured frameworks to test hypotheses and analyze complex interactions. Models provide clarity and precision, enabling the translation of theoretical concepts into quantifiable predictions that can be tested against real-world data.<ref>{{Cite web|url=https://www.mdpi.com/journal/mathematics/special_issues/Mathematical_Modeling_Economics_Ecology_Environment|title=Mathematics|website=mdpi.com}}</ref>

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, mathematicians extended their analysis into [[geopolitics]].<ref>{{Cite web|url=https://onlinebooks.library.upenn.edu/webbin/book/lookupid?key=ha010090244#:~:text=|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|access-date=January 3, 2024|archive-date=January 3, 2024|archive-url=https://web.archive.org/web/20240103142807/https://onlinebooks.library.upenn.edu/webbin/book/lookupid?key=ha010090244#:~:text=|url-status=live}}</ref> [[Peter Turchin]] developed [[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 |publisher=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>

Mathematization of the social sciences is not without risk. 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.<ref>{{Cite web|url=https://www.factcheck.org/2023/01/bidens-misleading-unemployment-statistic/|title=Biden's Misleading Unemployment Statistic – FactCheck.org|date=January 27, 2023 |access-date=June 2, 2024|archive-date=June 2, 2024|archive-url=https://web.archive.org/web/20240602215913/https://www.factcheck.org/2023/01/bidens-misleading-unemployment-statistic/|url-status=live}}</ref><ref>{{Cite web|url=https://www.minneapolisfed.org/article/2010/modern-macroeconomic-models-as-tools-for-economic-policy|title=Modern Macroeconomic Models as Tools for Economic Policy &#124; Federal Reserve Bank of Minneapolis|website=minneapolisfed.org|access-date=June 2, 2024|archive-date=August 3, 2024|archive-url=https://web.archive.org/web/20240803124409/https://www.minneapolisfed.org/article/2010/modern-macroeconomic-models-as-tools-for-economic-policy|url-status=live}}</ref>