Editing History of mathematics (section)

=== 19th century ===
<!-- Modern period stars here:
 * Mathematical analysis: Bolzano, Cauchy, Riemann, Weierstrass
 * "Purely existential" proofs by Dedekind and Hilbert
 * Dirichlet's "arbitrary function"
 * Cantor's different kinds of infinity
 * Concentration on structures instead of calculation (abstract algebra, non-Euclidean geometry)
 * Institutionalization
-->
[[Image:Carl Friedrich Gauss.jpg|thumb|right|upright|[[Carl Friedrich Gauss]]]]
Throughout the 19th century mathematics became increasingly abstract.<ref>Howard Eves, An Introduction to the History of Mathematics, 6th edition, 1990, "In the nineteenth century, mathematics underwent a great forward surge ... . The new mathematics began to free itself from its ties to mechanics and astronomy, and a purer outlook evolved." p. 493</ref> [[Carl Friedrich Gauss]] (1777–1855) epitomizes this trend.{{Citation needed|date=April 2023}} He did revolutionary work on [[function (mathematics)|functions]] of [[complex variable]]s, in [[geometry]], and on the convergence of [[series (mathematics)|series]], leaving aside his many contributions to science. He also gave the first satisfactory proofs of the [[fundamental theorem of algebra]] and of the [[quadratic reciprocity law]].{{Citation needed|date=January 2024}}

[[Image:noneuclid.svg|thumb|left|upright=1.5|Behavior of lines with a common perpendicular in each of the three types of geometry]]
This century saw the development of the two forms of [[non-Euclidean geometry]], where the [[parallel postulate]] of Euclidean geometry no longer holds.
The Russian mathematician [[Nikolai Ivanovich Lobachevsky]] and his rival, the Hungarian mathematician [[János Bolyai]], independently defined and studied [[hyperbolic geometry]], where uniqueness of parallels no longer holds. In this geometry the sum of angles in a triangle add up to less than 180°. [[Elliptic geometry]] was developed later in the 19th century by the German mathematician [[Bernhard Riemann]]; here no parallel can be found and the angles in a triangle add up to more than 180°. Riemann also developed [[Riemannian geometry]], which unifies and vastly generalizes the three types of geometry, and he defined the concept of a [[manifold]], which generalizes the ideas of [[curve]]s and [[Surface (topology)|surfaces]], and set the mathematical foundations for the [[General relativity|theory of general relativity]].<ref>{{Cite web |last=Wendorf |first=Marcia |date=2020-09-23 |title=Bernhard Riemann Laid the Foundations for Einstein's Theory of Relativity |url=https://interestingengineering.com/science/bernhard-riemann-the-mind-who-laid-the-foundations-for-einsteins-theory-of-relativity |access-date=2023-10-14 |website=interestingengineering.com |language=en-US}}</ref>

The 19th century saw the beginning of a great deal of [[abstract algebra]]. [[Hermann Grassmann]] in Germany gave a first version of [[vector space]]s, [[William Rowan Hamilton]] in Ireland developed [[noncommutative algebra]].{{Citation needed|date=January 2024}} The British mathematician [[George Boole]] devised an algebra that soon evolved into what is now called [[Boolean algebra]], in which the only numbers were 0 and 1. Boolean algebra is the starting point of [[mathematical logic]] and has important applications in [[electrical engineering]] and [[computer science]].{{Citation needed|date=January 2024}}<ref>Mari, C. (2012). George Boole. ''Great Lives from History: Scientists & Science'', N.PAG. Salem Press. <nowiki>https://search.ebscohost.com/login.aspx?AN=</nowiki>

176953509</ref> 
[[Augustin-Louis Cauchy]], [[Bernhard Riemann]], and [[Karl Weierstrass]] reformulated the calculus in a more rigorous fashion.{{Citation needed|date=January 2024}}

Also, for the first time, the limits of mathematics were explored. [[Niels Henrik Abel]], a Norwegian, and [[Évariste Galois]], a Frenchman, proved that there is no general algebraic method for solving polynomial equations of degree greater than four ([[Abel–Ruffini theorem]]).<ref>{{Cite journal |last=Ayoub |first=Raymond G. |date=1980-09-01 |title=Paolo Ruffini's contributions to the quintic |url=https://doi.org/10.1007/BF00357046 |journal=Archive for History of Exact Sciences |language=en |volume=23 |issue=3 |pages=253–277 |doi=10.1007/BF00357046 |s2cid=123447349 |issn=1432-0657}}</ref> Other 19th-century mathematicians used this in their proofs that straight edge and compass alone are not sufficient to [[trisect an arbitrary angle]], to construct the side of a cube twice the volume of a given cube, [[Squaring the circle|nor to construct a square equal in area to a given circle]].{{Citation needed|date=January 2024}} Mathematicians had vainly attempted to solve all of these problems since the time of the ancient Greeks.{{Citation needed|date=January 2024}} On the other hand, the limitation of three [[dimension]]s in geometry was surpassed in the 19th century through considerations of [[parameter space]] and [[hypercomplex number]]s.{{Citation needed|date=January 2024}}

Abel and Galois's investigations into the solutions of various polynomial equations laid the groundwork for further developments of [[group theory]], and the associated fields of [[abstract algebra]]. In the 20th century physicists and other scientists have seen group theory as the ideal way to study [[symmetry]].{{Citation needed|date=January 2024}}

[[Image:Georg Cantor (Porträt).jpg|thumb|right|upright|[[Georg Cantor]]]]
In the later 19th century, [[Georg Cantor]] established the first foundations of [[set theory]], which enabled the rigorous treatment of the notion of infinity and has become the common language of nearly all mathematics. Cantor's set theory, and the rise of [[mathematical logic]] in the hands of [[Peano]], [[L.E.J. Brouwer]], [[David Hilbert]], [[Bertrand Russell]], and [[A.N. Whitehead]], initiated a long running debate on the [[foundations of mathematics]].{{Citation needed|date=January 2024}}

The 19th century saw the founding of a number of national mathematical societies: the [[London Mathematical Society]] in 1865,<ref>{{Cite journal |last=Collingwood |first=E. F. |date=1966 |title=A Century of the London Mathematical Society |url=http://doi.wiley.com/10.1112/jlms/s1-41.1.577 |journal=Journal of the London Mathematical Society |language=en |volume=s1-41 |issue=1 |pages=577–594 |doi=10.1112/jlms/s1-41.1.577}}</ref> the [[Société Mathématique de France]] in 1872,<ref>{{Cite web |title=Nous connaître {{!}} Société Mathématique de France |url=https://smf.emath.fr/la-smf/connaitre-la-smf |access-date=2024-01-28 |website=smf.emath.fr}}</ref> the [[Circolo Matematico di Palermo]] in 1884,<ref>{{Cite web |title=Mathematical Circle of Palermo |url=https://mathshistory.st-andrews.ac.uk/Societies/Palermo/ |access-date=2024-01-28 |website=Maths History |language=en}}</ref><ref>{{Cite book |last1=Grattan-Guinness |first1=Ivor |url=https://books.google.com/books?id=mC9GcTdHqpcC&pg=PA656 |title=The Rainbow of Mathematics: A History of the Mathematical Sciences |last2=Grattan-Guinness |first2=I. |date=2000 |publisher=W. W. Norton & Company |isbn=978-0-393-32030-5 |language=en}}</ref> the [[Edinburgh Mathematical Society]] in 1883,<ref>{{Cite journal |last=Rankin |first=R. A. |date=June 1986 |title=The first hundred years (1883–1983) |url=https://www.cambridge.org/core/services/aop-cambridge-core/content/view/23AAB4A7D96568FC8E9003DE64AA8EF3/S0013091500016849a.pdf/div-class-title-the-first-hundred-years-1883-1983-div.pdf |journal=Proceedings of the Edinburgh Mathematical Society |language=en |volume=26 |issue=2 |pages=135–150 |doi=10.1017/S0013091500016849 |issn=1464-3839}}</ref> and the [[American Mathematical Society]] in 1888.<ref>{{Cite journal |last=Archibald |first=Raymond Clare |date=January 1939 |title=History of the American Mathematical Society, 1888–1938 |url=https://projecteuclid.org/journals/bulletin-of-the-american-mathematical-society/volume-45/issue-1/History-of-the-American-Mathematical-Society-18881938/bams/1183501056.full |journal=Bulletin of the American Mathematical Society |volume=45 |issue=1 |pages=31–46 |doi=10.1090/S0002-9904-1939-06908-5 |issn=0002-9904|doi-access=free }}</ref> The first international, special-interest society, the [[Quaternion Society]], was formed in 1899, in the context of a [[hyperbolic quaternion#Historical review|vector controversy]].<ref>{{Cite journal |last1=Molenbroek |first1=P. |last2=Kimura |first2=Shunkichi |date=3 October 1895 |title=To Friends and Fellow Workers in Quaternions |url=https://www.nature.com/articles/052545a0.pdf |journal=Nature |language=en |volume=52 |issue=1353 |pages=545–546 |doi=10.1038/052545a0 |bibcode=1895Natur..52..545M |s2cid=4008586 |issn=1476-4687}}</ref>

In 1897, [[Kurt Hensel]] introduced [[p-adic number]]s.<ref>{{Cite book |last=Murty |first=M. Ram |url=https://books.google.com/books?id=SseFAwAAQBAJ&dq=p-adic+numbers+hensel+1897&pg=PR9 |title=Introduction to $p$-adic Analytic Number Theory |date=2009-02-09 |publisher=American Mathematical Soc. |isbn=978-0-8218-4774-9 |language=en}}</ref>