Introducing Infinity

IMAGINE DIVIDING THE CIRCLE INTO LOTS OF THIN SEGMENTS, LIKE PIECES OF AN ORANGE.
This reflected an idea that went back as far as the ancient Greek philosopher Antiphon.
Antiphon was a contemporary of Socrates, born in the 5th century BC. He suggested that you could work out the area of a circle by drawing a regular polygon inside it and gradually increasing the number of sides until it was closer and closer to the circle itself. But 15th-century philosopher Nicholas of Cusa went further. He imagined stacking segments of a circle on top of each other, alternating direction. This made something very close to a rectangle that would be πr in height and r in width, making its area πr2. Of course the edges of those segments would never be quite straight unless the indivisibles were infinitely narrow.

UNFORTUNATELY, IF YOU TOOK GALILEO’S ARGUMENT LITERALLY YOU WOULD THINK THAT EVERYTHING WITH WHITE CURLY FUR AND FOUR LEGS WAS A SHEEP – EVEN IF WHAT YOU WERE LOOKING AT WAS A WHITE SOFA WITH A FURRY COVER.
The indivisibles

Around this time, the idea of indivisibles became popular. This was a similar approach to the ancient Greek idea of atoms. Ancient Greek atoms were the result of cutting something up so small that it was no longer possible to cut any further (a-tomos means “uncuttable”). The use of indivisibles involved dividing an object into smaller and smaller pieces, but not necessarily in all three dimensions. Take the example of the area of a circle.

MEAN THAT UNITY CONTAINS IN ITSELF AS MANY SQUARES AS THERE ARE CUBES AND NATURAL NUMBERS.
A common error

The trap Galileo seems to have fallen into is a common one – the assumption that when two things have similar properties they can be equated. This is part of the basis of homeopathy, where the idea of the “law of similars” is that a poison that causes similar symptoms to a disease will cure that disease. Galileo equated infinity and unity because they had similar properties.

ARE LED TO CONCLUDE THAT THE ATTRIBUTES “LARGER”, “SMALLER”, AND “EQUAL” HAVE NO PLACE IN COMPARING INFINITE QUANTITIES …
The infinity of 1?

This odd arithmetic of infinity led Galileo to a strange (and erroneous) conclusion. His argument went something like this. There have to be as many squares as there are natural numbers*. But the bigger the number, the more sparsely distributed are the squares (there are many more numbers that aren’t squares). So the bigger the number gets, the further you get from infinity. Since the further you get down the number line* the further you are from infinity, it follows that by turning back we discover that if any number is truly infinite it is 1. And just like infinity, 1 x 1 = 1.

THIS PRESENTATION STRIKES ME AS SO CLEVER AND NOVEL THAT, EVEN IF I WERE ABLE, I WOULD NOT BE WILLING TO OPPOSE IT.
The normal rules don’t apply

Simplicio doesn’t feel this helps, so Galileo tries again. He makes sure Simplicio knows what a square is – any number multiplied by itself. Then he imagines going through the real numbers, multiplying each by itself. For every positive integer there is a square. We’ve an infinite set of integers, and there’s a square for each integer. But here’s the rub. There are far more integers than there are squares.

infinite number of points on each of the wheels – but somehow, one infinity is bigger than the other. The response is rueful. That’s just the way it is with infinity – a problem, Galileo reckons, of dealing with infinite quantities using finite minds. And he goes on to show how this is perfectly normal behaviour for the infinite.

REMEMBER THAT WE ARE DEALING WITH INFINITIES AND INDIVISIBLES, BOTH OF WHICH TRANSCEND OUR FINITE UNDERSTANDING … IN SPITE OF THIS, MEN CANNOT REFRAIN FROM DISCUSSING THEM.
Back to geometry

One way Galileo demonstrates the odd mathematics of infinity is to use geometry, the favourite tool of the ancient Greeks. He gives a geometrical proof that you can design a cone and a bowl (the latter carved from a solid slice of a cylinder) in such a way that when you make a straight line cut through them, each has the same area and volume at the point of the cut, at whatever level you make that cut. Yet if we move the cut to the top, we appear to have a point and a circle, both proved to be the same “size”.

smaller wheel has to move this distance too, even though its sides are shorter, because the wheels are fixed together. It manages to do this by lifting off its rail for just far enough to keep up with the larger wheel.
Now here’s the clever bit. Galileo imagines similar wheels but with more and more sides. The more sides, the smaller the jump the small wheel has to make to catch up. At the extreme we get to two circular wheels. Say we give them a quarter-turn. They both move forward a quarter of the circumference of the big wheel.

The small wheel has never left its rail – yet it has moved much further than a quarter of its circumference. Galileo argued that it manages this by making an infinite set of infinitely small jumps to bridge the gap.
One of the characters in the book, Simplicio, is a little slow and is there to say “Duh, I don’t understand”, so the others can explain. After letting the circular wheels percolate through his brain, Simplicio has a complaint. Galileo seems to be saying that there are an infi

I HAD REAL TROUBLE GETTING THIS PUBLISHED - THE INQUISITION WERE NOT TOO KEEN AFTER MY PREVIOUS BOOK. BUT TO MY GREAT SURPRISE, IT WAS EVENTUALLY TAKEN UP BY THE DUTCH PUBLISHER ELSEVIER.
The book (like his near-fatal work on the motion of the Earth) took the form of a conversation between characters. After wondering about what holds matter together, they have a diversion, just for the fun of it, into the nature of infinity.
Infinity on wheels

The most dramatic demonstration of Galileo’s ideas on infinity involved wheels. He imagined a pair of multi-sided wheels, one stuck to the face of the other, running on rails. Say they’re hexagons. We give them a turn until they move from one face on the rails to the next face. The bigger wheel will have moved forward by the length of one of its sides.

WHEN A DUTCH SPECTACLE-MAKER HEADED FOR VENICE TO SHOW OFF THIS NEW DEVICE, I HAD A FRIEND DELAY MY RIVAL SO I COULD QUICKLY CONSTRUCT A TELESCOPE.
He was rewarded with a well-paid job and would have been a great success if he hadn’t written his book supporting Copernicus’ Sun-centred model of the universe. The way he wrote this offended the religious hierarchy; he was put on trial and sentenced to life imprisonment.

Galileo tends to be remembered for dropping balls off the Tower of Pisa, something he probably never did (he was a great self-publicist, but he never mentioned it – it was only recorded many years later by an assistant) and for being locked up for daring to suggest that the Earth rotates around the Sun. But he also undertook some remarkable thinking on the subject of infinity.
Galileo’s infinite pondering took place after his trial. Until then, his career had been going brilliantly. Although he didn’t, as is often suggested, invent the telescope, he heard about its development in Holland.