Actual and Potential Infinities

Abstract

Historically, the most popular theory of the impossible infinite is the one handed down by Aristotle: that infinity can only be potential, never actual.¹ What does this mean? Here are two helpful remarks from Aristotle’s discussion of the infinite:

A quantity is infinite if it is such that we can always take a part outside what has already been taken.

Our account does not rob the mathematicians of their science […]. In point of fact they do not need the infinite and do not use it. They postulate only that the finite straight line may be produced as far as they wish.²

The latter passage is particularly instructive. The idea appears to be this: there is an infinite potentiality, because there is no limit to how long a line can be. A line cannot be infinitely long — this would be an ‘actual infinity’ — but a line can be of any finite length. The possible lengths of lines go on up forever — for any line, there could be one twice as long.