The question is nonsense. One might as well ask whether the metric system is true and the old measures false; whether Cartesian co-ordinates are true and polar co-ordinates false.—Poincaré, H.
Non-Euclidean Geometry; Nature, Vol 45 (1891-1892), p. 407.
[2006]. I do in no wise share this view [that the axioms are arbitrary propositions which we assume wholly at will, and that in like manner the fundamental conceptions are in the end only arbitrary symbols with which we operate] but consider it the death of all science: in my judgment the axioms of geometry are not arbitrary, but reasonable propositions which generally have the origin in space intuition and whose separate content and sequence is controlled by reasons of expediency.—Klein, F.
Elementarmathematik vom höheren Standpunkte aus (Leipzig, 1909), Bd. 2, p. 384.
[2007]. Euclid’s Postulate 5 [The Parallel Axiom].
That, if a straight line falling on two straight lines make the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles.—Euclid.
The Thirteen Books of Euclid’s Elements [T. L. Heath] Vol. 1 (Cambridge, 1908), p. 202.
[2008]. It must be admitted that Euclid’s [Parallel] Axiom is unsatisfactory as the basis of a theory of parallel straight lines. It cannot be regarded as either simple or self-evident, and it therefore falls short of the essential characteristics of an axiom....—Hall, H. S. and Stevens, F. H.
Euclid’s Elements (London, 1892), p. 55.
[2009]. We may still well declare the parallel axiom the simplest assumption which permits us to represent spatial relations, and so it will be true generally, that concepts and axioms are not immediate facts of intuition, but rather the idealizations of these facts chosen for reasons of expediency.—Klein, F.