Foundations of Mathematics Vol . 1 ( 1934 )

2. Elementary number theory.—Finite inferring and its limits. The question raised at the end of the previous paragraph, whether we couldn’t found arithmetic directly by a method independent of axiomatics and make a special proof of consistency superfluous, gives us reason to recall that the method of rigorous axiomatics, especially existential inference, presupposing a fixed domain of individuals, is by no means the original procedure of mathematics. Geometry was indeed built up axiomatically from the beginning. But Euclid’s axiomatic system is intended to be contentual and intuitive. There is no abstraction from the intuitive meaning of figures in it. Moreover, the axioms are not in existential form. Euclid does not presuppose that points and lines constitute any fixed domain of individuals whatsoever. And that is why he does not formulate any existence-axioms but only constructionpostulates.