"It was not until the nineteenth century, chiefly through... Gauss, Bolyai, Lobachevsky, and Riemann, that the impossibility of deducing the parallel axiom from the others was demonstrated. This outcome was of the greatest intellectual importance. ...[I]t called attention... to the fact that a proof can be given of the impossibility of proving certain propositions within a given system. ...Gödel's paper is a proof of the impossibilty of formally demonstrating certain important propositions in number theory. ...[T]he resolution of that parallel axiom question forced the realization that Euclid was not the last word on the subject of geometry, since new systems of geometry can be constructed... incompatible with those adopted by Euclid. ...[I]t gradually became clear that the proper business of pure mathematicians is to derive theorems from postulated assumptions, and that it is not their concern whether the axioms are actually true."