At first sight, chains of numbers that go on forever seem harmless and without consequence, but it doesn’t take long to find some that will cause difficulties. Let’s say that rather than just list the numbers we add them all up as we go along to produce a sum. And let’s take a very simple series – just alternating 1 and -1: something like 1-1+1-1+1-1+1-1...
It’s hardly rocket science. We can see how it will total by adding some brackets:
(1-1)+(1-1)+(1-1)...
Each 1 is cancelled out by a -1, so the total of the series is 0. Or is it? Just shift the brackets and we still have a series that cancels out, but now we’ve got a 1 left over:
1+(-1+1)+(-1+1)...
So the same series has a value of 0 and 1. Scary. This has been rephrased as 'If you turn a light bulb off and on for infinity, does it end up on or off?' It could be either. That is clearly an example told by a mathematician – a physicist would tell you that it will obviously be off, because the bulb will have blown.
Or take the simple series at the top of this post where each item is half the value of the last:
1 + ½ + ¼ + ⅛...
It seems, as we add in element after element, that it’s going to eventually reach two, though in practice with any particular number there’s always a little gap left. In fact you could say it added up to two if you had an infinite set of numbers – but what does that mean? And how can an infinite list of things (whatever it is) add up to a finite quantity?
This was the basis of one Zeno’s famous paradoxes. Zeno, who was born as far back as around 539 BC, belonged to the school of Parmenides, which considered reality to be unchanging and movement to be an illusion. Zeno knocked up a number of rather entertaining examples to demonstrate the faulty nature of our attitude to change and motion.
Probably the best known is the arrow, which encourages us to imagine two arrows. One floats stationary in space. The other is flying at full speed. Now catch them at a snapshot in time. How do we tell the difference? For that matter, how does one arrow know to move in the next fraction of time while the other doesn’t?
But the paradox that reflects our sequence concerns Achilles and the Tortoise. This unlikely pair are setting out on a race. Achilles, being after all a hero, gives the slower tortoise a lead. And they’re off.
In a very small amount of time, Achilles has reached the Tortoise’s position. But by then, the tortoise has moved on. In an even shorter amount of time, Achilles has reached the Tortoise’s new spot. And again it has moved on. And it doesn’t matter how many times you go through this procedure – an infinite set of times if you like – Achilles will never catch the Tortoise up.
It’s easy to see the relationship of this paradox and the series if we pretend that Achilles only runs twice as fast as the tortoise (perhaps he’s pulled a hamstring or his Achilles tendon). In the time Achilles covers the first distance, the Tortoise moves half that distance. While Achilles is catching up, the tortoise moves ¼ the original distance. In an infinite number of moves they will only get to twice the original distance (which is where, of course the paradox falls down as Achilles powers through that mark).
There certainly is something unsettling, both about the idea of infinity itself and some infinite series. The Greeks weren’t really sure what to do with it. They called infinity 'apeiron', a word that had the same sort of negative connotations as chaos does today. It meant unbounded, uncontrolled, dangerous.
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Image from Unsplash by Joel Mathey (Achilles is just out of shot)
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