Mathematics

== History ==
{{Main|History of mathematics}}

=== Ancient ===
The history of mathematics is an ever-growing series of abstractions. Evolutionarily speaking, the first abstraction to ever be discovered, one shared by many animals,<ref>{{cite journal |title=Abstract representations of numbers in the animal and human brain |journal=Trends in Neurosciences |volume=21 |issue=8 |date=Aug 1998 |pages=355–361 |doi=10.1016/S0166-2236(98)01263-6 |pmid=9720604 |last1=Dehaene |first1=Stanislas | author1-link=Stanislas Dehaene |last2=Dehaene-Lambertz |first2=Ghislaine |author2-link=Ghislaine Dehaene-Lambertz | last3=Cohen |first3=Laurent|s2cid=17414557 }}</ref> was probably that of numbers: the realization that, for example, a collection of two apples and a collection of two oranges (say) have something in common, namely that there are {{em|two}} of them. As evidenced by [[tally sticks|tallies]] found on bone, in addition to recognizing how to [[counting|count]] physical objects, [[prehistoric]] peoples may have also known how to count abstract quantities, like time{{emdash}}days, seasons, or years.<ref>See, for example, {{cite book | first=Raymond L. | last=Wilder|author-link=Raymond L. Wilder|title=Evolution of Mathematical Concepts; an Elementary Study|at=passim}}</ref><ref>{{Cite book|last=Zaslavsky|first=Claudia|author-link=Claudia Zaslavsky|title=Africa Counts: Number and Pattern in African Culture.|date=1999|publisher=Chicago Review Press|isbn=978-1-61374-115-3|oclc=843204342}}</ref>
[[File:Plimpton 322.jpg|thumb|The Babylonian mathematical tablet ''[[Plimpton 322]]'', dated to 1800&nbsp;BC]]
Evidence for more complex mathematics does not appear until around 3000&nbsp;{{Abbr|BC|Before Christ}}, when the [[Babylonia]]ns and Egyptians began using arithmetic, algebra, and geometry for taxation and other financial calculations, for building and construction, and for astronomy.{{sfn|Kline|1990|loc=Chapter 1}} The oldest mathematical texts from [[Mesopotamia]] and [[Ancient Egypt|Egypt]] are from 2000 to 1800&nbsp;BC. Many early texts mention [[Pythagorean triple]]s and so, by inference, the [[Pythagorean theorem]] seems to be the most ancient and widespread mathematical concept after basic arithmetic and geometry. It is in Babylonian mathematics that [[elementary arithmetic]] ([[addition]], [[subtraction]], [[multiplication]], and [[division (mathematics)|division]]) first appear in the archaeological record. The Babylonians also possessed a place-value system and used a [[sexagesimal]] numeral system which is still in use today for measuring angles and time.{{sfn|Boyer|1991|loc="Mesopotamia" pp. 24–27}}

In the 6th century BC, Greek mathematics began to emerge as a distinct discipline and some [[Ancient Greece|Ancient Greeks]] such as the [[Pythagoreans]] appeared to have considered it a subject in its own right.<ref>{{cite book | last=Heath | first=Thomas Little | author-link=Thomas Heath (classicist) |url=https://archive.org/details/historyofgreekma0002heat/page/n14 |url-access=registration |page=1 |title=A History of Greek Mathematics: From Thales to Euclid |location=New York |publisher=Dover Publications |date=1981 |orig-date=1921 |isbn=978-0-486-24073-2}}</ref> Around 300 BC, Euclid organized mathematical knowledge by way of postulates and first principles, which evolved into the axiomatic method that is used in mathematics today, consisting of definition, axiom, theorem, and proof.<ref>{{Cite journal |last=Mueller |first=I. |date=1969 |title=Euclid's Elements and the Axiomatic Method |journal=The British Journal for the Philosophy of Science |volume=20 |issue=4 |pages=289–309 |doi=10.1093/bjps/20.4.289 |jstor=686258 |issn=0007-0882}}</ref> His book, ''[[Euclid's Elements|Elements]]'', is widely considered the most successful and influential textbook of all time.{{sfn|Boyer|1991|loc="Euclid of Alexandria" p. 119}} The greatest mathematician of antiquity is often held to be [[Archimedes]] ({{Circa|287|212 BC}}) of [[Syracuse, Italy|Syracuse]].{{sfn|Boyer|1991|loc="Archimedes of Syracuse" p. 120}} He developed formulas for calculating the surface area and volume of [[solids of revolution]] and used the [[method of exhaustion]] to calculate the [[area]] under the arc of a [[parabola]] with the [[Series (mathematics)|summation of an infinite series]], in a manner not too dissimilar from modern calculus.{{sfn|Boyer|1991|loc="Archimedes of Syracuse" p. 130}} Other notable achievements of Greek mathematics are [[conic sections]] ([[Apollonius of Perga]], 3rd century BC),{{sfn|Boyer|1991|loc="Apollonius of Perga" p. 145}} [[trigonometry]] ([[Hipparchus of Nicaea]], 2nd century BC),{{sfn|Boyer|1991|loc="Greek Trigonometry and Mensuration" p. 162}} and the beginnings of algebra (Diophantus, 3rd century AD).{{sfn|Boyer|1991|loc="Revival and Decline of Greek Mathematics" p. 180}}
[[File:Bakhshali numerals 2.jpg|thumb|right|upright=1.5|The numerals used in the [[Bakhshali manuscript]], dated between the 2nd century BC and the 2nd century AD]]
The [[Hindu–Arabic numeral system]] and the rules for the use of its operations, in use throughout the world today, evolved over the course of the first millennium AD in [[Indian mathematics|India]] and were transmitted to the [[Western world]] via [[Islamic mathematics]].<ref>{{cite book
 | title=Number Theory and Its History
 | first=Øystein
 | last=Ore
 | author-link=Øystein Ore
 | publisher=Courier Corporation
 | pages=19–24
 | year=1988
 | isbn=978-0-486-65620-5
 | url={{GBurl|id=Sl_6BPp7S0AC|pg=IA19}}
 | access-date=November 14, 2022
}}</ref> Other notable developments of Indian mathematics include the modern definition and approximation of [[sine]] and [[cosine]], and an early form of [[infinite series]].<ref>{{cite journal
 | title=On the Use of Series in Hindu Mathematics
 | first=A. N. | last=Singh | journal=Osiris
 | volume=1 | date=January 1936 | pages=606–628
 | doi=10.1086/368443 | jstor=301627
| s2cid=144760421 }}</ref><ref>{{cite book
 | chapter=Use of series in India
 | last1=Kolachana | first1=A. | last2=Mahesh | first2=K.
 | last3=Ramasubramanian | first3=K.
 | title=Studies in Indian Mathematics and Astronomy
 | series=Sources and Studies in the History of Mathematics and Physical Sciences
 | pages=438–461 | publisher=Springer | publication-place=Singapore
 | isbn=978-981-13-7325-1  | year=2019
 | doi=10.1007/978-981-13-7326-8_20 | s2cid=190176726 }}</ref>

=== Medieval and later ===
[[File:Image-Al-Kitāb al-muḫtaṣar fī ḥisāb al-ğabr wa-l-muqābala.jpg|thumb|A page from al-Khwārizmī's ''Algebra'']]
During the [[Islamic Golden Age|Golden Age of Islam]], especially during the 9th and 10th&nbsp;centuries, mathematics saw many important innovations building on Greek mathematics. The most notable achievement of Islamic mathematics was the development of algebra. Other achievements of the Islamic period include advances in [[spherical trigonometry]] and the addition of the [[decimal point]] to the Arabic numeral system.<ref>{{Cite book | last=Saliba | first=George | author-link=George Saliba | title=A history of Arabic astronomy: planetary theories during the golden age of Islam | date=1994 | publisher=New York University Press | isbn=978-0-8147-7962-0 | oclc=28723059 }}</ref> Many notable mathematicians from this period were Persian, such as Al-Khwarismi, [[Omar Khayyam]] and [[Sharaf al-Dīn al-Ṭūsī]].<ref>{{cite journal
 | title=Contributions of Islamic scholars to the scientific enterprise
 | first=Yasmeen M.
 | last=Faruqi
 | journal=International Education Journal
 | year=2006
 | volume=7
 | issue=4
 | pages=391–399
 | publisher=Shannon Research Press
 | url=https://eric.ed.gov/?id=EJ854295
 | access-date=November 14, 2022
 | archive-date=November 14, 2022
| archive-url=https://web.archive.org/web/20221114165547/https://eric.ed.gov/?id=EJ854295
 | url-status=live
 }}</ref> The Greek and Arabic mathematical texts were in turn translated to Latin during the Middle Ages and made available in Europe.<ref>{{cite journal | title=Greek-Arabic-Latin: The Transmission of Mathematical Texts in the Middle Ages | first=Richard | last=Lorch | journal=Science in Context | volume=14 | issue=1–2 | date=June 2001 | pages=313–331 | publisher=Cambridge University Press | doi=10.1017/S0269889701000114 | s2cid=146539132 | url=https://epub.ub.uni-muenchen.de/15929/1/greek-arabic-latin.pdf | access-date=December 5, 2022 | archive-date=December 17, 2022 | archive-url=https://web.archive.org/web/20221217160922/https://epub.ub.uni-muenchen.de/15929/1/greek-arabic-latin.pdf | url-status=live }}</ref>

During the [[early modern period]], mathematics began to develop at an accelerating pace in [[Western Europe]], with innovations that revolutionized mathematics, such as the introduction of variables and [[#Symbolic notation and terminology|symbolic notation]] by François Viète (1540–1603), the introduction of [[History of logarithms|logarithms]] by [[John Napier]] in 1614, which greatly simplified numerical calculations, especially for [[astronomy]] and [[marine navigation]], the introduction of coordinates by René Descartes (1596–1650) for reducing geometry to algebra, and the development of calculus by Isaac Newton (1642–1726/27) and [[Gottfried Wilhelm Leibniz|Gottfried Leibniz]] (1646–1716). Leonhard Euler (1707–1783), the most notable mathematician of the 18th century, unified these innovations into a single corpus with a standardized terminology, and completed them with the discovery and the proof of numerous theorems.

[[File:Carl Friedrich Gauss 1840 by Jensen.jpg|thumb|left|[[Carl Friedrich Gauss]]]]
Perhaps the foremost mathematician of the 19th century was the German mathematician Carl Gauss, who made numerous contributions to fields such as algebra, analysis, [[differential geometry and topology|differential geometry]], [[matrix theory]], number theory, and [[statistics]].<ref>{{cite journal
 | title=History of Mathematics After the Sixteenth Century
 | first=Raymond Clare | last=Archibald | author-link=Raymond Clare Archibald
 | journal=The American Mathematical Monthly
 | series=Part 2: Outline of the History of Mathematics
 | volume=56 | issue=1 | date=January 1949 | pages=35–56
 | doi=10.2307/2304570 | jstor=2304570
}}</ref> In the early 20th century, [[Kurt Gödel]] transformed mathematics by publishing his incompleteness theorems, which show in part that any consistent axiomatic system{{emdash}}if powerful enough to describe arithmetic{{emdash}}will contain true propositions that cannot be proved.<ref name=Raatikainen_2005/>

Mathematics has since been greatly extended, and there has been a fruitful interaction between mathematics and [[science]], to the benefit of both. Mathematical discoveries continue to be made to this very day. According to Mikhail B. Sevryuk, in the January&nbsp;2006 issue of the ''[[Bulletin of the American Mathematical Society]]'', "The number of papers and books included in the ''[[Mathematical Reviews]]'' database since 1940 (the first year of operation of MR) is now more than 1.9&nbsp;million, and more than 75&nbsp;thousand items are added to the database each year. The overwhelming majority of works in this ocean contain new mathematical theorems and their proofs."{{sfn|Sevryuk|2006|pp=101–109}}