Lectures on the Philosophy of Mathematics

by Joel David Hamkins

The early work tended to proceed from a perspective of Euclidean geometry as the one true geometry, with the goal of proving the parallel postulate from the other axioms. Eventually, this perspective began to shift. János Bolyai had aimed to prove the parallel postulate by contradiction. He accordingly assumed the negation of the parallel postulate and proceeded to make geometric deductions. But rather than finding the sought-after contradiction, what he found developing instead was a beautiful new geometric theory—a theory that led him to strange new geometric conclusions, many of which stood in conflict with classical geometry, but a theory displaying such an internal coherence and geometric sense that it seemed consistent and without contradiction. He wrote in 1823, “out of nothing I have created a strange new universe” (see Greenberg 1992, p. 163).