Leibniz’ Epistemology of Geometry and the Parallel Postulate

Abstract

The seventeenth-century mathematicians whom we have mentioned so far did not share a common epistemology, and each of them had his own opinions about the nature, object and aims of geometry, as well as on the meaning of geometrical principles or the standards of rigor needed in a proof. This notwithstanding, there was some kind of consensus at least on this point: that the axioms of geometry were provable. Contrary to our modern understanding of the role of an axiom as an indemonstrable statement, they claimed that geometrical principles were self-evident truths that required no proof to be accepted, but nevertheless were not incapable of proof. The dispute was more about the opportunity to prove them than the possibility of doing this.