Elementarmathematik vom, höheren [Standpunkte] aus (Leipzig, 1909), Bd. 2, p. 382.
[2010]. The characteristic features of our space are not necessities of thought, and the truth of Euclid’s axioms, in so far as they specially differentiate our space from other conceivable spaces, must be established by experience and by experience only.—Ball, R. S.
Encyclopedia Britannica, 9th Edition; Article “Measurement”
[2011]. Mathematical and physiological researches have shown that the space of experience is simply an actual case of many conceivable cases, about whose peculiar properties experience alone can instruct us.—Mach, Ernst.
Popular Scientific Lectures (Chicago, 1910), p. 205.
[2012]. The familiar definition: An axiom is a self-evident truth, means if it means anything, that the proposition which we call an axiom has been approved by us in the light of our experience and intuition. In this sense mathematics has no axioms, for mathematics is a formal subject over which formal and not material implication reigns.—Wilson, E. B.
Bulletin American Mathematical Society, Vol. 2 (1904-1905), p. 81.
[2013]. The proof of self-evident propositions may seem, to the uninitiated, a somewhat frivolous occupation. To this we might reply that it is often by no means self-evident that one obvious proposition follows from another obvious proposition; so that we are really discovering new truths when we prove what is evident by a method which is not evident. But a more interesting retort is, that since people have tried to prove obvious propositions, they have found that many of them are false. Self-evidence is often a mere will-o’-the-wisp, which is sure to lead us astray if we take it as our guide.—Russell, Bertrand.
Recent Work on the Principles of Mathematics; International Monthly, Vol. 4 (1901), p. 86.