Editing Foundations of mathematics (section)

== Before infinitesimal calculus ==
During [[Middle Ages]], Euclid's ''Elements'' stood as a perfectly solid foundation for mathematics, and [[philosophy of mathematics]] concentrated on the [[ontology|ontological status]] of mathematical concepts; the question was whether they exist independently of perception ([[philosophical realism|realism]]) or within the mind only ([[conceptualism]]); or even whether they are simply names of collection of individual objects ([[nominalism]]). 

In ''Elements'', the only numbers that are considered are [[natural number]]s and ratios of lengths. This geometrical view of non-integer numbers remained dominant until the end of Middle Ages, although the rise of [[algebra]] led to consider them independently from geometry, which implies implicitly that there are foundational primitives of mathematics. For example, the transformations of equations introduced by [[Al-Khwarizmi]] and the [[cubic formula|cubic]] and [[quartic formula|quartic]] formulas discovered in the 16th century result from algebraic manipulations that have no geometric counterpart.

Nevertheless, this did not challenge the classical foundations of mathematics since all properties of numbers that were used can be deduced from their geometrical definition.

In 1637, [[René Descartes]] published ''[[La Géométrie]]'', in which he showed that geometry can be reduced to algebra by means of [[Cartesian coordinate system|coordinates]], which are numbers determining the position of a point. This gives to the numbers that he called [[real number]]s a more foundational role (before him, numbers were defined as the ratio of two lengths). Descartes' book became famous after 1649 and paved the way to [[infinitesimal calculus]].