I begin by exploring the old adage that ‘the Greeks abhorred the infinite’

and show why and how they did so. I focus in particular on Pythagoras and his followers

who divided the world into two fundamental cosmic principles: what they called

Peras

(the Limited

or the Finite) and

Apeiron

(the Unlimited

or the Infinite). I also consider Zeno of Elea

who

after wrestling with the notion of infinity and discovering various associated paradoxes involving motion

came to the conclusion that motion itself was impossible. But the primary focus of this lecture is Aristotle

who sought a reconciliation between the idea that things can go on for ever and the Greeks’ abhorrence of any such idea. Aristotle famously distinguished between the potential infinite and the actual infinite

and argued that the former

and the former alone

was the acceptable face of infinity. Aristotle’s views held sway for thousands of years. In particular

they had a very significant impact on medieval thought. This was largely through the intermediary St Thomas Aquinas

who attempted to reconcile Aristotle’s teachings on the infinite with the doctrines of the Catholic church. He enjoyed some success in this endeavour

and

once the church had embraced Aristotle’s teachings as the

new orthodoxy

philosophers stepped out of line at their peril. I shall signal in particular Galilei Galileo

who did just that. He dared to add some new paradoxes of his own to discussion of the infinite: these foreshadowed later thinking on the infinite (as I shall discuss in Lecture Three) and showed that Aristotle’s views were not as straightforward as they had come to seem.