Editing Gottfried Wilhelm Leibniz (section)

===Calculus===
Leibniz is credited, along with [[Isaac Newton]], with the invention of [[calculus]] (differential and integral calculus). According to Leibniz's notebooks, a critical breakthrough occurred on 11 November 1675, when he employed integral calculus for the first time to find the area under the graph of a function {{math|1={{var|y}} = {{var|f}}({{var|x}})}}.<ref name="Leibniz1920">{{cite book|last1=Leibniz|first1=Gottfried Wilhelm Freiherr von|last2=Gerhardt|first2=Carl Immanuel (trans.)|title=The Early Mathematical Manuscripts of Leibniz|url=https://archive.org/details/earlymathematic01gerhgoog|access-date=10 November 2013|year=1920|publisher=Open Court Publishing|page=[https://archive.org/details/earlymathematic01gerhgoog/page/n100 93]}}</ref> He introduced several notations used to this day, for instance the [[integral sign]] {{math|∫}} (<math>\displaystyle\int f(x)\,dx</math>), representing an elongated S, from the Latin word ''summa'', and the {{math|d}} used for [[Differential (infinitesimal)|differentials]] (<math>\frac{dy}{dx}</math>), from the Latin word ''differentia''. Leibniz did not publish anything about his calculus until 1684.<ref>For an English translation of this paper, see Struik (1969: 271–284), who also translates parts of two other key papers by Leibniz on calculus.</ref> Leibniz expressed the inverse relation of integration and differentiation, later called the [[fundamental theorem of calculus]], by means of a figure<ref>[[Dirk Jan Struik]], ''A Source Book in Mathematics'' (1969) pp.&nbsp;282–284</ref> in his 1693 paper ''Supplementum geometriae dimensoriae...''.<ref>''Supplementum geometriae dimensoriae, seu generalissima omnium tetragonismorum effectio per motum: similiterque multiplex constructio lineae ex data tangentium conditione'', ''Acta Euriditorum'' (Sep. 1693) pp. 385–392</ref> However, [[James Gregory (mathematician)|James Gregory]] is credited for the theorem's discovery in geometric form, [[Isaac Barrow]] proved a more generalized geometric version, and [[Isaac Newton|Newton]] developed supporting theory. The concept became more transparent as developed through Leibniz's formalism and new notation.<ref>[[John Stillwell]], ''Mathematics and its History'' (1989, 2002) p.159</ref> The [[product rule]] of [[differential calculus]] is still called "Leibniz's law". In addition, the theorem that tells how and when to differentiate under the integral sign is called the [[Leibniz integral rule]].

Leibniz exploited [[infinitesimal]]s in developing calculus, manipulating them in ways suggesting that they had [[paradox]]ical [[algebra]]ic properties. [[George Berkeley]], in a tract called ''[[The Analyst]]'' and also in ''De Motu'', criticized these. A recent study argues that Leibnizian calculus was free of contradictions, and was better grounded than Berkeley's empiricist criticisms.<ref>{{citation
 | last1 = Katz | first1 = Mikhail
 | author1-link = Mikhail Katz
 | last2 = Sherry | first2 = David
 | arxiv = 1205.0174
 | doi = 10.1007/s10670-012-9370-y
 | journal = [[Erkenntnis]]
 | volume = 78
 | issue = 3
 | pages = 571–625
 | title = Leibniz's Infinitesimals: Their Fictionality, Their Modern Implementations, and Their Foes from Berkeley to Russell and Beyond
 | year = 2012| s2cid = 119329569
 }}</ref>

Leibniz introduced [[fractional calculus]] in a letter written to [[Guillaume de l'Hôpital]] in 1695.<ref name="Derivative">{{cite journal |last=Katugampola |first=Udita N. |date=15 October 2014 |title=A New Approach To Generalized Fractional Derivatives |url=https://www.emis.de/journals/BMAA/repository/docs/BMAA6-4-1.pdf |journal=Bulletin of Mathematical Analysis and Applications |volume=6 |issue=4 |pages=1–15 |arxiv=1106.0965}}</ref> At the same time, Leibniz wrote to [[Johann Bernoulli]] about derivatives of "general order".<ref name=":12">{{Cite book |last=Miller |first=Kenneth S. |title=An Introduction to the Fractional Calculus and Fractional Differential Equations |last2=Ross |first2=Bertram |date=1993 |publisher=Wiley |isbn=978-0-471-58884-9 |location=New York |pages=1–2}}</ref> In the correspondence between Leibniz and [[John Wallis]] in 1697, Wallis's infinite product for <math>\frac{1}{2}</math>π is discussed. Leibniz suggested using differential calculus to achieve this result. Leibniz further used the notation <math>{d}^{1/2}{y}</math> to denote the derivative of order <math>\frac{1}{2}</math>.<ref name=":12"/>

From 1711 until his death, Leibniz was engaged in a dispute with [[John Keill]], Newton and others, over [[Leibniz–Newton calculus controversy|whether Leibniz had invented calculus independently of Newton]].

The use of infinitesimals in mathematics was frowned upon by followers of [[Karl Weierstrass]],<ref>{{Cite journal|last=Dauben|first=Joseph W|date=December 2003|title=Mathematics, ideology, and the politics of infinitesimals: mathematical logic and nonstandard analysis in modern China|journal=History and Philosophy of Logic|volume=24|issue=4|pages=327–363|doi=10.1080/01445340310001599560|s2cid=120089173|issn=0144-5340}}</ref> but survived in science and engineering, and even in rigorous mathematics, via the fundamental computational device known as the [[Differential (infinitesimal)|differential]]. Beginning in 1960, [[Abraham Robinson]] worked out a rigorous foundation for Leibniz's infinitesimals, using [[model theory]], in the context of a field of [[hyperreal number]]s. The resulting [[non-standard analysis]] can be seen as a belated vindication of Leibniz's mathematical reasoning. Robinson's [[transfer principle]] is a mathematical implementation of Leibniz's heuristic [[law of continuity]], while the [[standard part function]] implements the Leibnizian [[transcendental law of homogeneity]].