Euclid's Fifth Postulate and the nature of geometrical truth

Michael Anthony Augros, Boston College

Abstract

Euclid's Fifth Postulate states that two converging straight lines must eventually intersect in the direction in which they converge. This apparently simple statement has been the subject of intense controversy from Euclid's time to the present day; almost all mathematicians after Euclid admitted his postulate to be true, but for various reasons they almost all denied it to be primary. Many serious mathematicians for two thousand years after Euclid attempted to prove his Fifth Postulate, which he seemed to think unsusceptible of strict proof. Surprisingly, all these attempts at proof failed, usually by tacitly assuming the Postulate itself in trying to prove it. Consequently, mathematicians became suspicious about the possibility of proving the Postulate at all, and when at last in the nineteenth and twentieth centuries it was proven indemonstrable, mathematicians reasoned thus: the Fifth Postulate is not self-evidently true or false, but neither is it demonstrably true or false, and therefore it is neither true nor false but is only arbitrary. The non-Euclidean geometries and the collapse of mathematics into a hypothetico-deductive logical system are two direct consequences of this reasoning. In this thesis, the author attempts to manifest grave oversights in this reasoning by showing that Euclid's Fifth Postulate is indeed self-evidently true. This is accomplished through an investigation into the nature of science and geometry according to the philosophical principles of Aristotle and Thomas Aquinas.

Recommended Citation

Michael Anthony Augros, "Euclid's Fifth Postulate and the nature of geometrical truth" (January 1, 1995). Boston College Dissertations and Theses. Paper AAI9605385.
http://escholarship.bc.edu/dissertations/AAI9605385