Editing Foundations of mathematics (section)

=== Real analysis === 
{{See also|Mathematical analysis#History}}

[[Cauchy]] (1789–1857) started the project of giving rigorous bases to [[infinitesimal calculus]]. In particular, he rejected the heuristic principle that he called the [[generality of algebra]], which consisted to apply properties of [[algebraic operation]]s to [[infinite sequences]] without proper proofs. In his ''[[Cours d'Analyse]]'' (1821), he considered ''very small quantities'', which could presently be called "sufficiently small quantities"; that is, a sentence such that "if {{mvar|x}} is very small {{nowrap|then ..."}} must be understood as "there is a (sufficiently large) [[natural number]] {{mvar|n}} such that {{math|{{abs|''x''}} < 1/''n''}}". In the proofs he used this in a way that predated the modern [[(ε, δ)-definition of limit]].<ref name="Grabiner1983">{{Citation
| last = Grabiner | first = Judith V.
| title = Who Gave You the Epsilon? Cauchy and the Origins of Rigorous Calculus
| journal = [[American Mathematical Monthly]]
| volume = 90
| issue = 3
| pages = 185–194
| doi = 10.2307/2975545
| year = 1983
| jstor = 2975545
}}, collected in [http://www.maa.org/ebooks/spectrum/WGE.html Who Gave You the Epsilon?], {{isbn|978-0-88385-569-0}} pp. 5–13. Also available at: http://www.maa.org/pubs/Calc_articles/ma002.pdf</ref>

The modern [[(ε, δ)-definition of limit]]s and [[continuous functions]] was first developed by [[Bernard Bolzano|Bolzano]] in 1817, but remained relatively unknown, and Cauchy probably did know Bolzano's work.

[[Karl Weierstrass]] (1815–1897) formalized and popularized the (ε, δ)-definition of limits, and discovered some pathological functions that seemed paradoxical at this time, such as [[Weierstrass function|continuous, nowhere-differentiable functions]]. Indeed, such functions contradict previous conceptions of a function as a rule for computation or a smooth graph. 

At this point, the program of [[arithmetization of analysis]] (reduction of [[mathematical analysis]] to arithmetic and algebraic operations) advocated by Weierstrass was essentially completed, except for two points.

Firstly, a formal definition of real numbers was still lacking. Indeed, beginning with [[Richard Dedekind]] in 1858, several mathematicians worked on the definition of the real numbers, including [[Hermann Hankel]], [[Charles Méray]], and [[Eduard Heine]], but this is only in 1872 that two independent complete definitions of real numbers were published: one by Dedekind, by means of [[Dedekind cut]]s; the other one by [[Georg Cantor]] as equivalence classes of [[Cauchy sequences]].<ref>{{MacTutor| class=HistTopics| id=Real_numbers_2| title=The real numbers: Stevin to Hilbert|date=October 2005}}</ref> 

Several problems were left open by these definitions, which contributed to the [[foundational crisis of mathematics]]. Firstly both definitions suppose that [[rational number]]s and thus [[natural number]]s are rigorously defined; this was done a few years later with [[Peano axioms]]. Secondly, both definitions involve [[infinite set]]s (Dedekind cuts and sets of the elements of a Cauchy sequence), and Cantor's [[set theory]] was published several years later.

The third problem is more subtle: and is related to the foundations of logic: classical logic is a [[first-order logic]]; that is, [[quantifier (logic)|quantifier]]s apply to variables representing individual elements, not to variables representing (infinite) sets of elements. The basic property of the [[completeness of the real numbers]] that is required for defining and using real numbers involves a quantification on infinite sets. Indeed, this property may be expressed either as ''for every infinite sequence of real numbers, if it is a [[Cauchy sequence]], it has a limit that is a real number'', or as ''every subset of the real numbers that is [[bounded set|bounded]] has a [[least upper bound]] that is a real number''. This need of quantification over infinite sets is one of the motivation of the development of [[higher-order logic]]s during the first half of the 20th century.