Editing Infinitesimal (section)

== History of the infinitesimal ==
The notion of infinitely small quantities was discussed by the [[Eleatic School]]. The [[Greek mathematics|Greek]] mathematician [[Archimedes]] (c.&nbsp;287&nbsp;BC – c.&nbsp;212&nbsp;BC), in ''[[The Method of Mechanical Theorems]]'', was the first to propose a logically rigorous definition of infinitesimals.<ref>Archimedes, ''The Method of Mechanical Theorems''; see [[Archimedes Palimpsest]]</ref> His [[Archimedean property]] defines a number ''x'' as infinite if it satisfies the conditions {{nowrap|{{abs|''x''}} > 1,}} {{nowrap|{{abs|''x''}} > 1 + 1,}} {{nowrap|{{abs|''x''}} > 1 + 1 + 1,}} ..., and infinitesimal if {{nowrap|''x'' ≠ 0}} and a similar set of conditions holds for ''x'' and the reciprocals of the positive integers. A number system is said to be Archimedean if it contains no infinite or infinitesimal members.

The English mathematician [[John Wallis]] introduced the expression 1/∞ in his 1655 book ''Treatise on the Conic Sections''. The symbol, which denotes the reciprocal, or inverse, of&nbsp;[[∞]], is the symbolic representation of the mathematical concept of an infinitesimal. In his ''Treatise on the Conic Sections'', Wallis also discusses the concept of a relationship between the symbolic representation of infinitesimal 1/∞ that he introduced and the concept of infinity for which he introduced the symbol&nbsp;∞. The concept suggests a [[thought experiment]] of adding an infinite number of [[parallelogram]]s of infinitesimal width to form a finite area. This concept was the predecessor to the modern method of integration used in [[integral calculus]]. The conceptual origins of the concept of the infinitesimal 1/∞ can be traced as far back as the Greek philosopher [[Zeno of Elea]], whose [[Zeno's dichotomy paradox]] was the first mathematical concept to consider the relationship between a finite interval and an interval approaching that of an infinitesimal-sized interval.

Infinitesimals were the subject of political and religious controversies in 17th century Europe, including a ban on infinitesimals issued by clerics in Rome in 1632.<ref>{{cite book|title=Infinitesimal: How a Dangerous Mathematical Theory Shaped the Modern World|last=Alexander|first=Amir|publisher=Scientific American / Farrar, Straus and Giroux|year=2014|isbn=978-0-374-17681-5|author-link=Amir Alexander}}</ref>

Prior to the invention of calculus mathematicians were able to calculate tangent lines using [[Pierre de Fermat]]'s method of [[adequality]] and [[René Descartes]]' [[method of normals]]. There is debate among scholars as to whether the method was infinitesimal or algebraic in nature. When [[Isaac Newton|Newton]] and [[Gottfried Leibniz|Leibniz]] invented the [[Infinitesimal calculus|calculus]], they made use of infinitesimals, Newton's ''[[fluxion (mathematics)|fluxions]]'' and Leibniz' ''[[differential (infinitesimal)|differential]]''. The use of infinitesimals was attacked as incorrect by [[George Berkeley|Bishop Berkeley]] in his work ''[[The Analyst]]''.<ref>{{Cite book|url=https://archive.org/details/theanalystoradis00berkuoft/page/n4|title=The Analyst: A Discourse Addressed to an Infidel Mathematician.|last=Berkeley|first=George|year=1734|location=London|author-link=George Berkeley}}</ref> Mathematicians, scientists, and engineers continued to use infinitesimals to produce correct results. In the second half of the nineteenth century, the calculus was reformulated by [[Augustin-Louis Cauchy]], [[Bernard Bolzano]], [[Karl Weierstrass]], [[Georg Cantor|Cantor]], [[Richard Dedekind|Dedekind]], and others using the [[(ε, δ)-definition of limit]] and [[set theory]].
While the followers of Cantor, Dedekind, and Weierstrass sought to rid analysis of infinitesimals, and their philosophical allies like [[Bertrand Russell]] and [[Rudolf Carnap]] declared that infinitesimals are ''pseudoconcepts'', [[Hermann Cohen]] and his [[Marburg school]] of [[neo-Kantianism]] sought to develop a working logic of infinitesimals.<ref>{{Cite journal|last1=Mormann|first1=Thomas|author-link=Thomas Mormann|last2=Katz|first2=Mikhail|author-link2=Mikhail Katz|date=Fall 2013|title=Infinitesimals as an Issue of Neo-Kantian Philosophy of Science|journal=[[HOPOS|HOPOS: The Journal of the International Society for the History of Philosophy of Science]]|volume=3|issue=2|pages=236–280|arxiv=1304.1027|doi=10.1086/671348|jstor=10.1086/671348|s2cid=119128707}}</ref> The mathematical study of systems containing infinitesimals continued through the work of [[Tullio Levi-Civita|Levi-Civita]], [[Giuseppe Veronese]], [[Paul du Bois-Reymond]], and others, throughout the late nineteenth and the twentieth centuries, as documented by Philip Ehrlich (2006). In the 20th century, it was found that infinitesimals could serve as a basis for calculus and analysis (see [[hyperreal number]]s).