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.

by Brian Clegg

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.

After Galileo's house arrest began in 1634, following the trial over his heretical work on the motion of the Earth around the Sun, Galileo was anything but inactive. It was then that he produced the book that was arguably his greatest work of science, his equivalent of Newton's Principia, named Discorsi e dimostrationi matematiche, intorno a due nuove scienze, or Discourses and Mathematical Demonstrations Concerning Two New Sciences. Galileo had considerable trouble getting this published - the Inquisition made it clear that no work by this heretic would be published in any country where it held sway. When the book was eventually taken up by the great Dutch publisher Elsevier, Galileo expressed his great surprise that it had been published at all, which he claimed had never been his intention. He commented in his introduction:

I was notified by the Elzevirs that they had these works of mine in press and that I ought to decide upon a dedication and send them a reply at once. This sudden and unexpected news led me to think that the eagerness of your Lordship to revive and spread my name by passing these works on to various friends was the real cause their falling into the hands of printers who, because they had already published other works of mine, now wished to honour me with a beautiful and ornate edition of this work.

Hardly "sudden and unexpected news" - Galileo was closely involved in the publication, but at least he was protecting his back.

The book (as did his near-fatal work Dialogue Concerning the Two Chief World Systems on the motion of the Earth) took the form of a conversation between a number of characters, largely over serious matters. But after wondering about what holds matter together (Galileo thought it was tiny pockets of vacuum between the particles of matter), they have a diversion, just for the fun of it, into the nature of infinity.

Galileo brings out a number of points, but two are particularly worthy of note. The first involves the rotation of a pair of wheels.

He starts with wheels with a few sides - for example, they could be hexagons. These are three dimensional shapes - imagine the hexagons are cut out of sheets of marble. The smaller hexagon is stuck to the larger one, and each of them rests on its own horizontal rail.

Now we roll the combined wheel along so that it moves onto its next side. As the big wheel turns it pivots on the corner and moves along the track by the length of one side. But what has happened to the smaller wheel? Not only has the big wheel moved on by that distance, so has the small one. It has to: they're fixed together. Yet in turning 1/6 of a rotation, the small one should only have rolled along the track by the length of its own side - a much smaller distance, marked in red on the diagram. To achieve the extra movement, the smaller wheel was lifted entirely off the track.

Now here's the clever bit. Galileo imagined increasing the number of sides. The more sides, the more sets of small movements along the rail and small jumps you get as the wheels rotate. Finally, let's imagine, were it possible, that we take that number of sides to infinity. We end up with circular wheels.

Again we roll the two wheels, joined together, along their respective rails. Again they both travel the same distance - in this case a quarter of the circumference of the big wheel. But now something strange has happened. The rim of the big wheel has rolled out a quarter of its circumference on its track. The rim of the smaller wheel has only rolled out its smaller quarter circumference, but the small wheel still has to travel the same distance as the bigger one, without ever leaving the track. There were no jumps, or at least so it seems.

What Galileo imagined had happened was that as the smaller wheel turns there are an infinite number of infinitesimally small gaps, which add up to make the difference between the circumference of the wheel and the distance it moves. Infinity has come into play in a physical device to make the seemingly impossible happen.

After letting this percolate through his brain in the background, Galileo's traditionalist and frankly rather dim character, Simplicio, has a complaint. What Galileo seems to be saying (or technically Salviati, the character that represents Galileo's voice in the book) is that there are an infinite number of points in one circular wheel and an infinite number of points in the other. But somehow, though each had the same infinity of points, one added up to a greater distance than the other. One infinity was both the same as the other and larger.

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.