"For Twenty-one hundred years the geometry Euclid had outlined in his Elements stood as a supreme mathematical and logical system. Based on five postulates and five axioms, Euclidean geometry deduced an elaborate system of propositions that seemed both to accurately describe physical reality and to compose a flawlessly logical system. Though innumerable geometers and mathematicians had tried to disprove or refine it, the system had withstood all challengers. Crucial to the development of modern philosophy was the assertion of Immanuel Kant that the example of Euclidean geometry proved that the human mind possessed synthetic a priori  knowledge, a knowledge intuitively perceived as true concerning both the laws of the mind’s operation and the structure of the external world ... During the nineteenth and early twentieth century logicians, mathematicians, and geometers severely restricted the scope of the deductive method, demonstrating that it was wholly formal and experientially void. Pragmatists and positivists had long scorned concepts of a priori self evidence and had tried to limit or even discredit the method of deduction in favor of inductive techniques. The new logico-mathematical theory, however, inspired in large part by the development of non-Euclidean geometries, seemed to demonstrate conclusively that deduction was by its very nature incapable of producing any authoritative system of ethics, and thus seemed to give final theoretical confirmation to the arguments of a line of philosophical empiricists that extended back through Hume." [Edward A. Purcell Jr., The Crisis of Democratic Theory: Scientific Naturalism & The Problem of Value (1973), 49-50].