We're going to take a look at Galileo's surprisingly insightful
contribution to our understanding of infinity. Galileo Galilei was born in 1564, the son of a musician and scientific dabbler. He tends to be remembered for dropping balls of the Tower of Pisa, something which be probably didn't do (it’s only recorded many years later by his assistant) and for being locked up for daring to suggest that the Earth rotates around the Sun rather than the other way round (not exactly accurate historically). Also inventing the telescope (he didn't) and featuring in that Queen song. But he undertook some remarkable thinking on the subject of infinity.
It took place after his house arrest. Galileo had put together his masterpiece, Discourses and Mathematical Demonstrations Concerning Two New Sciences. He had real trouble getting this published – the Inquisition were not too keen after his previous book – and when it 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. The book took the form of a conversation between characters, and here, having wondered about what holds matter together, they have a diversion, just for the fun of it, into the nature of infinity.
Galileo brings out a number of points, but I'll concentrate on two. The first involves the rotation of a pair of wheels. He starts with wheels with a few sides - in the example below I’ve made them 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 a rail. I've made a poor quality video below to illustrate this.
We roll the wheel along a little. I'm going to do it twice. First watch what the big wheel does, then the small wheel. The big wheel moves along the track by the length of one of its sides. But not only has the big wheel moved on by that distance, so has the small one. It has to. They’re fixed together. Yet the small one should only have rolled along the track by the distance of its smaller side. How did it achieve it? It lifts off the rail entirely. There’s a gap of just the right distance.
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 gaps you get. Now let’s imagine 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 look what’s happening. The rim of the big wheel has rolled out a quarter of its circumference on its track. The rim of the smaller wheel has rolled out a smaller quarter circumference, but the wheel has travelled the same distance, without ever leaving the track. There were no jumps, or at least so it seems.
Galileo imagined 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. After letting this percolate through his brain in the background, Galileo’s thoughtfully challenged character, Simplicio, has a complaint. What Galileo seems to be saying (or technically Salviati, the character that is Galileo’s voice) is that there are an infinite number of points in one circular wheel and an infinite number of points in the other – but somehow, one infinity is bigger than the other.
Salviati is rueful. That’s the way it is with infinity: 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’ve an infinite set of integers, and there’s an infinite set of 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 number – 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. We’ll come back to this in a big way in the final post of the series.
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