Editing Foundations of mathematics (section)

=== Non-Euclidean geometries ===
{{See also|Non-Euclidean geometry#History}}

Before the 19th century, there were many failed attempts to derive the [[parallel postulate]] from other axioms of geometry. In an attempt to prove that its negation leads to a contradiction, [[Johann Heinrich Lambert]] (1728–1777) started to build [[hyperbolic geometry]] and introduced the [[hyperbolic functions]] and computed the area of a [[hyperbolic triangle]] (where the sum of angles is less than 180°). 

Continuing the construction of this new geometry, several mathematicians proved independently that if it is [[inconsistent]], then [[Euclidean geometry]] is also inconsistent and thus that the parallel postulate cannot be proved. This was proved by [[Nikolai Lobachevsky]] in 1826, [[János Bolyai]] (1802–1860) in 1832 and [[Carl Friedrich Gauss]] (unpublished).

Later in the 19th century, the German mathematician [[Bernhard Riemann]] developed [[Elliptic geometry]], another [[non-Euclidean geometry]] where no parallel can be found and the sum of angles in a triangle is more than 180°. It was proved consistent by defining points as pairs of [[antipodal point]]s on a sphere (or [[hypersphere]]), and lines as [[great circle]]s on the sphere. 

These proofs of unprovability of the parallel postulate lead to several philosophical problems, the main one being that before this discovery, the parallel postulate and all its consequences were considered as ''true''. So, the non-Euclidean geometries challenged the concept of [[mathematical truth]].