Editing Foundations of mathematics (section)

==Infinitesimal calculus==
[[Isaac Newton]] (1642–1727) in England and [[Gottfried Wilhelm Leibniz|Leibniz]] (1646–1716) in Germany independently developed the [[infinitesimal calculus]] for dealing with mobile points (such as planets in the sky) and variable quantities.

This needed the introduction of new concepts such as [[continuous function]]s, [[derivative]]s and [[limit (mathematics)|limit]]s. For dealing with these concepts in a logical way, they were defined in terms of [[infinitesimal]]s that are hypothetical numbers that are infinitely close to zero. The strong implications of infinitesimal calculus on foundations of mathematics is illustrated by a pamphlet of the Protestant philosopher [[George Berkeley]] (1685–1753), who wrote "[Infinitesimals] are neither finite quantities, nor quantities infinitely small, nor yet nothing. May we not call them the ghosts of departed quantities?".<ref name="berkeley">''[[The Analyst]], A Discourse Addressed to an Infidel Mathematician''</ref>

Also, a lack of rigor has been frequently invoked, because infinitesimals and the associated concepts were not formally defined ([[line (geometry)|line]]s and [[plane (geometry)|plane]]s were not formally defined either, but people were more accustomed to them). Real numbers, continuous functions, derivatives were not formally defined before the 19th century, as well as [[Euclidean geometry]]. It is only in the 20th century that a formal definition of infinitesimals has been given, with the proof that the whole infinitesimal can be deduced from them.

Despite its lack of firm logical foundations, infinitesimal calculus was quickly adopted by mathematicians, and validated by its numerous applications; in particular the fact that the planet trajectories can be deduced from the [[Newton's law of gravitation]].