Editing History of mathematics (section)

=== 20th century === <!-- Hibert's problems, foundational crisis, Bourbaki -->
The 20th century saw mathematics become a major profession. By the end of the century, thousands of new Ph.D.s in mathematics were being awarded every year, and jobs were available in both teaching and industry.<ref>{{cite web|url=https://dpcpsi.nih.gov/sites/default/files/opep/document/Final_Report_(03-517-OD-OER)%202006.pdf|title=U.S. Doctorates in the 20th Century|access-date=5 April 2023|website=nih.gov|date=June 2006|author1=Lori Thurgood|author2=Mary J. Golladay|author3=Susan T. Hill}}</ref> An effort to catalogue the areas and applications of mathematics was undertaken in [[Klein's encyclopedia]].<ref>{{Cite journal |last=Pitcher |first=A. D. |date=1922 |title=Encyklopâdie der Mathematischen Wissenschaften. |url=https://www.ams.org/journals/bull/1922-28-09/S0002-9904-1922-03635-X/S0002-9904-1922-03635-X.pdf |journal=[[Bulletin of the American Mathematical Society]] |volume=28 |issue=9 |pages=474 |doi=10.1090/s0002-9904-1922-03635-x}}</ref>

In a 1900 speech to the [[International Congress of Mathematicians]], [[David Hilbert]] set out a list of [[Hilbert's problems|23 unsolved problems in mathematics]].<ref>{{Cite journal |last=Hilbert |first=David |date=1902 |title=Mathematical problems |url=https://www.ams.org/bull/1902-08-10/S0002-9904-1902-00923-3/ |journal=Bulletin of the American Mathematical Society |language=en |volume=8 |issue=10 |pages=437–479 |doi=10.1090/S0002-9904-1902-00923-3 |issn=0002-9904|doi-access=free }}</ref> These problems, spanning many areas of mathematics, formed a central focus for much of 20th-century mathematics. Today, 10 have been solved, 7 are partially solved, and 2 are still open. The remaining 4 are too loosely formulated to be stated as solved or not.<ref>{{Cite web |title=Hilbert's 23 problems {{!}} mathematics {{!}} Britannica |url=https://www.britannica.com/science/Hilberts-23-problems |access-date=2025-04-19 |website=www.britannica.com |language=en}}</ref>

[[Image:Four Colour Map Example.svg|thumb|left|upright|A map illustrating the [[Four Color Theorem]]]]
Notable historical conjectures were finally proven. In 1976, [[Wolfgang Haken]] and [[Kenneth Appel]] proved the [[four color theorem]], controversial at the time for the use of a computer to do so.<ref>{{Cite journal |last=Gonthier |first=Georges |date=December 2008 |title=Formal Proof—The Four-Color Theorem |url=https://www.ams.org/notices/200811/tx081101382p.pdf |journal=[[Notices of the AMS]] |volume=55 |issue=11 |pages=1382}}</ref> [[Andrew Wiles]], building on the work of others, proved [[Fermat's Last Theorem]] in 1995.<ref>{{Cite journal |last=Castelvecchi |first=Davide |date=2016-03-01 |title=Fermat's last theorem earns Andrew Wiles the Abel Prize |url=https://www.nature.com/articles/nature.2016.19552 |journal=Nature |language=en |volume=531 |issue=7594 |pages=287 |doi=10.1038/nature.2016.19552 |pmid=26983518 |bibcode=2016Natur.531..287C |issn=1476-4687}}</ref> [[Paul Cohen (mathematician)|Paul Cohen]] and [[Kurt Gödel]] proved that the [[continuum hypothesis]] is [[logical independence|independent]] of (could neither be proved nor disproved from) the [[ZFC|standard axioms of set theory]].<ref>{{Cite journal |last=Cohen |first=Paul |date=2002-12-01 |title=The Discovery of Forcing |url=https://projecteuclid.org/journals/rocky-mountain-journal-of-mathematics/volume-32/issue-4/The-Discovery-of-Forcing/10.1216/rmjm/1181070010.full |journal=Rocky Mountain Journal of Mathematics |volume=32 |issue=4 |doi=10.1216/rmjm/1181070010 |issn=0035-7596}}</ref> In 1998, [[Thomas Callister Hales]] proved the [[Kepler conjecture]], also using a computer.<ref>{{Cite news |last=Wolchover |first=Natalie |date=22 February 2013 |title=In Computers We Trust? |url=https://www.quantamagazine.org/in-computers-we-trust-20130222/ |access-date=28 January 2024 |work=[[Quanta Magazine]]}}</ref>

Mathematical collaborations of unprecedented size and scope took place. An example is the [[classification of finite simple groups]] (also called the "enormous theorem"), whose proof between 1955 and 2004 required 500-odd journal articles by about 100 authors, and filling tens of thousands of pages.<ref>{{Cite web |title=An enormous theorem: the classification of finite simple groups |url=https://plus.maths.org/content/enormous-theorem-classification-finite-simple-groups |access-date=2024-01-28 |website=Plus Maths |language=en}}</ref> A group of French mathematicians, including [[Jean Dieudonné]] and [[André Weil]], publishing under the [[pseudonym]] "[[Nicolas Bourbaki]]", attempted to exposit all of known mathematics as a coherent rigorous whole. The resulting several dozen volumes has had a controversial influence on mathematical education.<ref>Maurice Mashaal, 2006. ''Bourbaki: A Secret Society of Mathematicians''. [[American Mathematical Society]]. {{ISBN|0-8218-3967-5|978-0-8218-3967-6}}.</ref>

[[File:Relativistic precession.svg|thumb|Newtonian (red) vs. Einsteinian orbit (blue) of a lone planet orbiting a star, with [[General relativity#Orbital effects and the relativity of direction|relativistic precession of apsides]]]]
[[Differential geometry]] came into its own when [[Albert Einstein]] used it in [[general relativity]].{{Citation needed|date=January 2024}} Entirely new areas of mathematics such as [[mathematical logic]], [[topology]], and [[John von Neumann]]'s [[game theory]] changed the kinds of questions that could be answered by mathematical methods.{{Citation needed|date=January 2024}} All kinds of [[Mathematical structure|structures]] were abstracted using axioms and given names like [[metric space]]s, [[topological space]]s etc.{{Citation needed|date=January 2024}} As mathematicians do, the concept of an abstract structure was itself abstracted and led to [[category theory]].{{Citation needed|date=January 2024}} [[Grothendieck]] and [[Jean-Pierre Serre|Serre]] recast [[algebraic geometry]] using [[Sheaf (mathematics)|sheaf theory]].{{Citation needed|date=January 2024}} Large advances were made in the qualitative study of [[dynamical systems theory|dynamical systems]] that [[Henri Poincaré|Poincaré]] had begun in the 1890s.{{Citation needed|date=January 2024}}
[[Measure theory]] was developed in the late 19th and early 20th centuries. Applications of measures include the [[Lebesgue integral]], [[Kolmogorov]]'s axiomatisation of [[probability theory]], and [[ergodic theory]].{{Citation needed|date=January 2024}} [[Knot theory]] greatly expanded.{{Citation needed|date=January 2024}} [[Quantum mechanics]] led to the development of [[functional analysis]],{{Citation needed|date=January 2024}} a branch of mathematics that was greatly developed by [[Stefan Banach]] and his collaborators who formed the [[Lwów School of Mathematics]].<ref>{{cite web|url=https://www.britannica.com/biography/Stefan-Banach|title=Stefan Banach - Polish Mathematician|website=britannica.com|date=27 August 2023 }}</ref> Other new areas include [[Laurent Schwartz]]'s [[Distribution (mathematics)|distribution theory]], [[Fixed-point theorem|fixed point theory]], [[singularity theory]] and [[René Thom]]'s [[catastrophe theory]], [[model theory]], and [[Benoit Mandelbrot|Mandelbrot]]'s [[fractals]].{{Citation needed|date=January 2024}} [[Lie theory]] with its [[Lie group]]s and [[Lie algebra]]s became one of the major areas of study.<ref>*{{cite book |first=Thomas |last=Hawkins |authorlink=Thomas W. Hawkins Jr. |year=2000 |title=Emergence of the Theory of Lie Groups: an essay in the history of mathematics, 1869–1926 |url=https://archive.org/details/emergenceoftheor0000hawk |url-access=registration |publisher=Springer |isbn=0-387-98963-3 }}</ref>

[[Non-standard analysis]], introduced by [[Abraham Robinson]], rehabilitated the [[infinitesimal]] approach to calculus, which had fallen into disrepute in favour of the theory of [[Limit of a function|limits]], by extending the field of real numbers to the [[Hyperreal number]]s which include infinitesimal and infinite quantities.{{Citation needed|date=January 2024}} An even larger number system, the [[surreal number]]s were discovered by [[John Horton Conway]] in connection with [[combinatorial game]]s.{{Citation needed|date=January 2024}}

The development and continual improvement of [[computer]]s, at first mechanical analog machines and then digital electronic machines, allowed [[Private industry|industry]] to deal with larger and larger amounts of data to facilitate mass production and distribution and communication, and new areas of mathematics were developed to deal with this: [[Alan Turing]]'s [[computability theory]]; [[Computational complexity theory|complexity theory]]; [[Derrick Henry Lehmer]]'s use of [[ENIAC]] to further number theory and the [[Lucas–Lehmer primality test]]; [[Rózsa Péter]]'s [[recursive function theory]]; [[Claude Shannon]]'s [[information theory]]; [[signal processing]]; [[data analysis]]; [[Mathematical optimization|optimization]] and other areas of [[operations research]].{{Citation needed|date=January 2024}} In the preceding centuries much mathematical focus was on calculus and continuous functions, but the rise of computing and communication networks led to an increasing importance of [[discrete mathematics|discrete]] concepts and the expansion of [[combinatorics]] including [[graph theory]]. The speed and data processing abilities of computers also enabled the handling of mathematical problems that were too time-consuming to deal with by pencil and paper calculations, leading to areas such as [[numerical analysis]] and [[symbolic computation]].{{Citation needed|date=January 2024}} Some of the most important methods and [[algorithm]]s of the 20th century are: the [[simplex algorithm]], the [[fast Fourier transform]], [[error-correcting code]]s, the [[Kalman filter]] from [[control theory]] and the [[RSA algorithm]] of [[public-key cryptography]].{{Citation needed|date=January 2024}}

At the same time, deep insights were made about the limitations to mathematics. In 1929 and 1930, it was proved{{By whom|date=January 2024}} the truth or falsity of all statements formulated about the [[natural number]]s plus either addition or multiplication (but not both), was [[Decidability (logic)|decidable]], i.e. could be determined by some algorithm.{{Citation needed|date=January 2024}} In 1931, [[Kurt Gödel]] found that this was not the case for the natural numbers plus both addition and multiplication; this system, known as [[Peano arithmetic]], was in fact [[incompleteness theorem|incomplete]]. (Peano arithmetic is adequate for a good deal of [[number theory]], including the notion of [[prime number]].) A consequence of Gödel's two [[incompleteness theorem]]s is that in any mathematical system that includes Peano arithmetic (including all of [[mathematical analysis|analysis]] and geometry), truth necessarily outruns proof, i.e. there are true statements that [[Incompleteness theorem|cannot be proved]] within the system. Hence mathematics cannot be reduced to mathematical logic, and [[David Hilbert]]'s dream of making all of mathematics complete and consistent needed to be reformulated.{{Citation needed|date=January 2024}}

[[Image:GammaAbsSmallPlot.png|thumb|right|The [[absolute value]] of the Gamma function on the complex plane]]
One of the more colorful figures in 20th-century mathematics was [[Srinivasa Aiyangar Ramanujan]] (1887–1920), an Indian [[autodidact]]<ref name=":3">{{Cite journal |last=Ono |first=Ken |date=2006 |title=Honoring a Gift from Kumbakonam |url=https://www.ams.org/notices/200606/fea-ono.pdf |journal=[[Notices of the AMS]] |volume=53 |issue=6 |pages=640–651}}</ref> {{Citation needed span|text=who conjectured or proved over 3000 theorems|date=January 2024|reason=theorem count not mentioned in the source}}, including properties of [[highly composite number]]s,<ref>{{Cite journal |last1=Alaoglu |first1=L. |author-link=Leonidas Alaoglu |last2=Erdős |first2=Paul |author-link2=Paul Erdős |date=14 February 1944 |title=On highly composite and similar numbers |url=https://community.ams.org/journals/tran/1944-056-00/S0002-9947-1944-0011087-2/S0002-9947-1944-0011087-2.pdf |journal=[[Transactions of the American Mathematical Society]] |volume=56 |pages=448–469|doi=10.1090/S0002-9947-1944-0011087-2 }}</ref> the [[partition function (number theory)|partition function]]<ref name=":3" /> and its [[asymptotics]],<ref>{{Cite journal |last=Murty |first=M. Ram |date=2013 |title=The Partition Function Revisited |url=https://mast.queensu.ca/~murty/partition.pd |journal=The Legacy of Srinivasa Ramanujan, RMS-Lecture Notes Series |volume=20 |pages=261–279}}</ref> and [[Ramanujan theta function|mock theta functions]].<ref name=":3" /> He also made major investigations in the areas of [[gamma function]]s,<ref>{{Citation |last=Bradley |first=David M. |title=Ramanujan's formula for the logarithmic derivative of the gamma function |date=2005-05-07 |arxiv=math/0505125 |bibcode=2005math......5125B }}</ref><ref>{{Cite journal |last=Askey |first=Richard |date=1980 |title=Ramanujan's Extensions of the Gamma and Beta Functions |url=https://www.jstor.org/stable/2321202 |journal=The American Mathematical Monthly |volume=87 |issue=5 |pages=346–359 |doi=10.2307/2321202 |jstor=2321202 |issn=0002-9890}}</ref> [[modular form]]s,<ref name=":3" /> [[divergent series]],<ref name=":3" /> [[General hypergeometric function|hypergeometric series]]<ref name=":3" /> and prime number theory.<ref name=":3" />

[[Paul Erdős]] published more papers than any other mathematician in history,<ref>{{cite web | url=http://oakland.edu/enp/trivia/ | title=Grossman – the Erdös Number Project }}</ref> working with hundreds of collaborators. Mathematicians have a game equivalent to the [[Kevin Bacon Game]], which leads to the [[Erdős number]] of a mathematician. This describes the "collaborative distance" between a person and Erdős, as measured by joint authorship of mathematical papers.<ref>{{Cite journal |last=Goffman |first=Casper |date=1969 |title=And What Is Your Erdos Number? |url=https://www.jstor.org/stable/2317868 |journal=The American Mathematical Monthly |volume=76 |issue=7 |pages=791 |doi=10.2307/2317868 |jstor=2317868 |issn=0002-9890}}</ref><ref>{{Cite web |title=grossman - The Erdös Number Project |url=https://sites.google.com/oakland.edu/grossman/home/the-erdoes-number-project |access-date=2024-01-28 |website=sites.google.com |language=en-US}}</ref>

[[Emmy Noether]] has been described by many as the most important woman in the history of mathematics.<ref>{{citation|author-link=Pavel Alexandrov|last=Alexandrov|first=Pavel S.|chapter=In Memory of Emmy Noether | title = Emmy Noether: A Tribute to Her Life and Work|editor1-first =James W | editor1-last = Brewer | editor2-first = Martha K | editor2-last = Smith | place = New York | publisher= Marcel Dekker | year= 1981 | isbn = 978-0-8247-1550-2 |pages= 99–111}}.</ref> She studied the theories of [[ring (mathematics)|rings]], [[field (mathematics)|fields]], and [[algebra over a field|algebras]].<ref>{{Cite news |last=Angier |first=Natalie |date=2012-03-26 |title=The Mighty Mathematician You've Never Heard Of |url=https://www.nytimes.com/2012/03/27/science/emmy-noether-the-most-significant-mathematician-youve-never-heard-of.html |access-date=2024-04-20 |work=The New York Times |language=en-US |issn=0362-4331}}</ref>

As in most areas of study, the explosion of knowledge in the scientific age has led to specialization: by the end of the century, there were hundreds of specialized areas in mathematics, and the [[Mathematics Subject Classification]] was dozens of pages long.<ref>{{Cite web|url=https://www.ams.org/mathscinet/msc/pdfs/classifications2000.pdf|title=Mathematics Subject Classification 2000|accessdate=5 April 2023}}</ref> More and more [[mathematical journal]]s were published and, by the end of the century, the development of the [[World Wide Web]] led to online publishing.{{Citation needed|date=January 2024}}