A Brief History of Infinity

Salviati is rueful. That's the way it is with infinity. It is a problem, he reckons, of dealing with infinite quantities using our finite minds. And he goes on to show how this is perfectly normal behaviour once you are dealing with infinity.

The simple mathematical tool he uses to demonstrate this is the square - that's the square of a number, not the shape. Salviati makes sure Simplicio knows what a square is - any number multiplied by itself. So, he imagines going through the integers, multiplying each one by itself. It's not rocket science. For every single integer there is a square. We have an infinite number of integers, and there's an infinite number squares in a one-to-one correspondence.

But here's the rub. There are lots of numbers that aren't squares of anything. So though there's a square for every single integer - an infinite set of them - there are even more individual numbers than there are squares. Arggh. Simplicio's brain hurts, and it doesn't surprise us. Galileo has spotted something very special about infinity. The normal rules of arithmetic don't really apply to it. You can effectively have 'smaller' and 'bigger' infinities, one a subset of the other, that are nonetheless the same size.

The true implications of Galileo's ponderings would take more than 300 years to come out, but he had sowed the seeds of all that was to come - and clearly enjoyed a delight and fascination with the paradoxical nature of infinity. A fascination we can all share.