The Pythagorean Arithmetic of the Fifth Century

Abstract

The Pythagoreans took number to be their universal principle. By designating the suitable numbers and ratios they sought to comprehend the cosmos in all its varied aspects: for instance, in the harmony of the consonant musical intervals, in the relative motions and configurations of the heavenly bodies, and in the properties of geometric shapes. They took the soul to be a harmony and assigned numbers by way of metaphor to such intangibles as justice, opportunity and marriage.¹ Within such a tradition the commensurability of all geometric magnitudes must at first have been an implicit axiom. As we have seen, this axiom met its direct refutation in the discovery of the incommensurability of the side and diameter of the square, a discovery made and disseminated sometime within the last third of the fifth century B.C. I now propose to examine more closely the character of the Pythagorean mathematics, and in particular of the Pythagorean arithmetic, of that time. I will then show how this body of mathematics could form the basis of the investigations on incommensurability attributed to the early Pythagoreans and to Theodorus.