A Brief History of Infinity

The paradoxical twists and turns of infinity have baffled many great thinkers. The first person to truly come to grips with the concept was the remarkable Galileo Galilei.

Infinity has long been treated with a mixture of fascination and awe. Some have equated it with godhead - others see it as a concept with no practical value in the real world, arguing that even mathematics apparently dependent on infinity such as calculus could be made to work by resorting to inexhaustible but finite quantities. The ancient Greeks were uncomfortable with the concept, giving their word for it, apeiron, the same sort of negative connotations we now apply to the word chaos. Apeiron was out of control, wild and dangerous.

It took Aristotle to put infinity in its place so firmly that hardly anyone would give it direct consideration again until the nineteenth century. The approach he took was surprisingly pragmatic. Infinity, Aristotle decided had to exist, because time appeared to have no beginning and no end. Nor was it possible to say that the counting numbers ever finished. If there were a biggest number - call it 'max', then what was wrong with max+1 or max+2? But on the other hand infinity could not exist in the real world. If there were, for example, a physical body that was infinite, he argued, it would be boundless - yet to be a body, by definition an object has to have bounds.

The compromise Aristotle developed - and it's a clever one - was to say that infinity both existed and didn't exist. Instead of being a true property of anything real, he argued, there was just potential infinity. Infinity that could in principle be, but in practice never was. Aristotle gives us an excellent example to illustrate this. The Olympic Games exist - it is impossible to deny this. Yet were an alien to beam in (Aristotle didn't actually include an alien in his example) and ask us 'show me this Olympic Games of which you speak', we couldn't do it. At the moment they don't exist in reality but they do exist as a potential. And infinity, Aristotle argued, was in exactly the same potential state.

Galileo Galilei

It was this form of infinity that would spawn calculus and would be included in practically all mathematical considerations of the infinite until Georg Cantor blew the whole topic apart with the revelations that would eventually contribute to his decline into madness. But one man bucked the trend early, considering the implications of the real thing - and that was a man who was no stranger to standing out from the crowd, Galileo Galilei.