Arthur Schopenhauer

Arthur Schopenhauer

Arthur Schopenhauer was a German philosopher known for his pessimism and philosophical clarity. At age 25, he published his doctoral dissertation, On the Fourfold Root of the Principle of Sufficient Reason, which examined the four separate manifestations of reason in the phenomenal...

criticized mathematicians' attempts to prove Euclid

Euclid

Euclid , fl. 300 BC, also known as Euclid of Alexandria, was a Greek mathematician, often referred to as the "Father of Geometry". He was active in Alexandria during the reign of Ptolemy I...

's Parallel Postulate

Parallel postulate

In geometry, the parallel postulate, also called Euclid's fifth postulate because it is the fifth postulate in Euclid's Elements, is a distinctive axiom in Euclidean geometry...

because they try to prove from indirect concept

Concept

The word concept is used in ordinary language as well as in almost all academic disciplines. Particularly in philosophy, psychology and cognitive sciences the term is much used and much discussed. WordNet defines concept: "conception, construct ". However, the meaning of the term concept is much...

s that which is directly evident from perception

Perception

Perception is the process of attaining awareness or understanding of the environment by organizing and interpreting sensory information. All perception involves signals in the nervous system, which in turn result from physical stimulation of the sense organs...

.
Throughout his writings, Schopenhauer criticized the logical derivation of philosophies and mathematics from mere concepts, instead of from intuitive perceptions.
In voicing his criticism, Schopenhauer naturally assumed that Euclid was using plane geometry

Plane geometry

In mathematics, plane geometry may refer to:*Euclidean plane geometry, the geometry of plane figures,*geometry of a plane,or sometimes:*geometry of a projective plane, most commonly the real projective plane but possibly the complex projective plane, Fano plane or others;*geometry of the hyperbolic...

 which operates with lines drawn on a flat surface. Neither Schopenhauer nor Euclid was referring to hyperbolic geometry

Hyperbolic geometry

In mathematics, hyperbolic geometry is a non-Euclidean geometry, meaning that the parallel postulate of Euclidean geometry is replaced...

, with its concave surface, or elliptic geometry

Elliptic geometry

Elliptic geometry is a non-Euclidean geometry, in which, given a line L and a point p outside L, there exists no line parallel to L passing through p. Elliptic geometry, like hyperbolic geometry, violates Euclid's parallel postulate, which can be interpreted as asserting that there is exactly one...

's convex surface.
Although Schopenhauer could see no justification for trying to prove Euclid's Parallel Postulate, he did see a reason for examining another of Euclid's axioms.