Infinity in Greek Mathematics

Abstract

Reasoning about infinity is one of the characteristic features of mathematics as well as its main source of conflict. We saw, in Chapter 1, the conflict that arose from the discovery of irrationals, and in this chapter we shall see that the rejection of irrational numbers by the Greeks was just part of a general rejection of infinite processes. In fact, until the late nineteenth century most mathematicians were reluctant to accept infinity as more than “potential.” The infinitude of a process, collection, or magnitude was understood as the possibility of its indefinite continuation, and no more—certainly not the possibility of eventual completion. For example, the natural numbers 1, 2, 3,..., can be accepted as a potential infinity—generated from 1 by the process of adding 1—without accepting that there is a completed totality (1, 2, 3,...). The same applies to any sequence x ₁ x ₂, x ₃,... (of rational numbers, say), where x _n+1 is obtained from x _n by a definite rule.