ABSTRACT
The turn of the century witnessed a philosophic crisis in the foundation of mathematics. Logicians, philosophers of mathematics and formal semantics, such as Frege and Russell, investigated the axiomatic fabric of mathematical reasoning and proof. Ancient logical and metaphysical disputes as to the true nature of mathematics-is it arbitrarily conventional? Is it ‘a natural’ construct corresponding to realities in the empirical order of the world?—were revived and given rigorous philosophical and technical expression. Gödel’s celebrated proof of the necessity for an ‘outside’ addition to all self-consistent mathematical systems and operational rules, took on formal and applied significance far beyond the strictly mathematical domain. It is, at the same time, fair to say that certain of the questions raised in the late nineteenth and early twentieth centuries as to the logical foundations, internal coherence and psychological or existential sources of mathematical reasoning and proof, remain open.
