Editing Calculus (section)

=== Foundations ===
In calculus, ''foundations'' refers to the [[Rigorous#Mathematical rigor |rigorous]] development of the subject from [[axiom]]s and definitions.  In early calculus, the use of [[infinitesimal]] quantities was thought unrigorous and was fiercely criticized by several authors, most notably [[Michel Rolle]] and [[George Berkeley|Bishop Berkeley]]. Berkeley famously described infinitesimals as the [[ghosts of departed quantities]] in his book ''[[The Analyst]]'' in 1734.  Working out a rigorous foundation for calculus occupied mathematicians for much of the century following Newton and Leibniz, and is still to some extent an active area of research today.<ref name="Bell-SEP">{{cite web |url=https://plato.stanford.edu/entries/continuity/ |title=Continuity and Infinitesimals |date=6 September 2013 |website=[[Stanford Encyclopedia of Philosophy]] |first=John L. |last=Bell |access-date=20 February 2022 |author-link=John Lane Bell |archive-date=16 March 2022 |archive-url=https://web.archive.org/web/20220316170134/https://plato.stanford.edu/entries/continuity/ |url-status=live }}</ref>

Several mathematicians, including [[Colin Maclaurin|Maclaurin]], tried to prove the soundness of using infinitesimals, but it would not be until 150 years later when, due to the work of [[Augustin-Louis Cauchy|Cauchy]] and [[Karl Weierstrass|Weierstrass]], a way was finally found to avoid mere "notions" of infinitely small quantities.<ref>{{Cite book |last=Russell |first=Bertrand |author-link=Bertrand Russell |year=1946 |title=History of Western Philosophy |location=London |publisher=[[George Allen & Unwin Ltd]] |page=[https://archive.org/stream/westernphilosoph035502mbp#page/n857/mode/2up 857] |quote=The great mathematicians of the seventeenth century were optimistic and anxious for quick results; consequently they left the foundations of analytical geometry and the infinitesimal calculus insecure. Leibniz believed in actual infinitesimals, but although this belief suited his metaphysics it had no sound basis in mathematics. Weierstrass, soon after the middle of the nineteenth century, showed how to establish calculus without infinitesimals, and thus, at last, made it logically secure. Next came Georg Cantor, who developed the theory of continuity and infinite number. "Continuity" had been, until he defined it, a vague word, convenient for philosophers like Hegel, who wished to introduce metaphysical muddles into mathematics. Cantor gave a precise significance to the word and showed that continuity, as he defined it, was the concept needed by mathematicians and physicists. By this means a great deal of mysticism, such as that of Bergson, was rendered antiquated. |title-link= A History of Western Philosophy }}</ref> The foundations of differential and integral calculus had been laid. In Cauchy's ''[[Cours d'Analyse]]'', we find a broad range of foundational approaches, including a definition of [[continuous function|continuity]] in terms of infinitesimals, and a (somewhat imprecise) prototype of an [[(ε, δ)-definition of limit]] in the definition of differentiation.<ref>{{cite book |first=Judith V. |last=Grabiner |author-link=Judith Grabiner |title=The Origins of Cauchy's Rigorous Calculus |url=https://archive.org/details/originsofcauchys00judi |url-access=registration |location=Cambridge |publisher=MIT Press |year=1981 |isbn=978-0-387-90527-3 }}</ref> In his work, Weierstrass formalized the concept of [[Limit of a function|limit]] and eliminated infinitesimals (although his definition can validate [[nilsquare]] infinitesimals). Following the work of Weierstrass, it eventually became common to base calculus on limits instead of infinitesimal quantities, though the subject is still occasionally called "infinitesimal calculus". [[Bernhard Riemann]] used these ideas to give a precise definition of the integral.<ref>{{cite book|first=Tom |last=Archibald |chapter=The Development of Rigor in Mathematical Analysis |pages=117–129 |title=The Princeton Companion to Mathematics |title-link=The Princeton Companion to Mathematics |editor-first1=Timothy |editor-last1=Gowers |editor-link1=Timothy Gowers |editor-first2=June |editor-last2=Barrow-Green |editor-link2=June Barrow-Green |editor-first3=Imre |editor-last3=Leader |editor-link3=Imre Leader |publisher=Princeton University Press |year=2008 |isbn=978-0-691-11880-2 |oclc=682200048}}</ref> It was also during this period that the ideas of calculus were generalized to the [[complex plane]] with the development of [[complex analysis]].<ref>{{cite book|first=Adrian |last=Rice |chapter=A Chronology of Mathematical Events |pages=1010–1014 |title=The Princeton Companion to Mathematics |title-link=The Princeton Companion to Mathematics |editor-first1=Timothy |editor-last1=Gowers |editor-link1=Timothy Gowers |editor-first2=June |editor-last2=Barrow-Green |editor-link2=June Barrow-Green |editor-first3=Imre |editor-last3=Leader |editor-link3=Imre Leader |publisher=Princeton University Press |year=2008 |isbn=978-0-691-11880-2 |oclc=682200048}}</ref>

In modern mathematics, the foundations of calculus are included in the field of [[real analysis]], which contains full definitions and [[mathematical proof|proofs]] of the theorems of calculus. The reach of calculus has also been greatly extended. [[Henri Lebesgue]] invented [[measure theory]], based on earlier developments by [[Émile Borel]], and used it to define integrals of all but the most [[Pathological (mathematics)|pathological]] functions.<ref>{{cite book|first=Reinhard |last=Siegmund-Schultze |chapter=Henri Lebesgue |pages=796–797 |title=The Princeton Companion to Mathematics |title-link=The Princeton Companion to Mathematics |editor-first1=Timothy |editor-last1=Gowers |editor-link1=Timothy Gowers |editor-first2=June |editor-last2=Barrow-Green |editor-link2=June Barrow-Green |editor-first3=Imre |editor-last3=Leader |editor-link3=Imre Leader |publisher=Princeton University Press |year=2008 |isbn=978-0-691-11880-2 |oclc=682200048}}</ref> [[Laurent Schwartz]] introduced [[Distribution (mathematics)|distributions]], which can be used to take the derivative of any function whatsoever.<ref>{{Cite journal |last1=Barany |first1=Michael J. |last2=Paumier |first2=Anne-Sandrine |last3=Lützen |first3=Jesper |date=November 2017 |title=From Nancy to Copenhagen to the World: The internationalization of Laurent Schwartz and his theory of distributions |journal=[[Historia Mathematica]] |volume=44 |issue=4 |pages=367–394 |doi=10.1016/j.hm.2017.04.002|doi-access=free }}</ref>

Limits are not the only rigorous approach to the foundation of calculus. Another way is to use [[Abraham Robinson]]'s [[non-standard analysis]]. Robinson's approach, developed in the 1960s, uses technical machinery from [[mathematical logic]] to augment the real number system with [[infinitesimal]] and [[Infinity|infinite]] numbers, as in the original Newton-Leibniz conception. The resulting numbers are called [[hyperreal number]]s, and they can be used to give a Leibniz-like development of the usual rules of calculus.<ref>{{cite book|first=Joseph W. |last=Daubin |chapter=Abraham Robinson |pages=822–823 |title=The Princeton Companion to Mathematics |title-link=The Princeton Companion to Mathematics |editor-first1=Timothy |editor-last1=Gowers |editor-link1=Timothy Gowers |editor-first2=June |editor-last2=Barrow-Green |editor-link2=June Barrow-Green |editor-first3=Imre |editor-last3=Leader |editor-link3=Imre Leader |publisher=Princeton University Press |year=2008 |isbn=978-0-691-11880-2 |oclc=682200048}}</ref> There is also [[smooth infinitesimal analysis]], which differs from non-standard analysis in that it mandates neglecting higher-power infinitesimals during derivations.<ref name="Bell-SEP"/> Based on the ideas of [[F. W. Lawvere]] and employing the methods of [[category theory]], smooth infinitesimal analysis views all functions as being [[continuous function|continuous]] and incapable of being expressed in terms of [[Discrete mathematics|discrete]] entities. One aspect of this formulation is that the [[law of excluded middle]] does not hold.<ref name="Bell-SEP" /> The law of excluded middle is also rejected in [[constructive mathematics]], a branch of mathematics that insists that proofs of the existence of a number, function, or other mathematical object should give a construction of the object. Reformulations of calculus in a constructive framework are generally part of the subject of [[constructive analysis]].<ref name="Bell-SEP"/>