Friday, 28 October 2016

Achilles and a Tortoise

Zeno, a Greek philosopher, lived in the Italian town of Elea in the 5th century BC. Unfortunately nothing of what he wrote survives firsthand and we are reliant on philosophers such as Plato and Aristotle for a glimpse into his philosophical synthesis. We know that Zeno thought that it was unwise to rely on the senses and sense experience in order to obtain knowledge about the world. Instead he taught that 'true' knowledge could only be reliably obtained through mathematics and logic. In this regard he is truly right.

He is remembered mainly for a series of paradoxes. A paradox is a statement that appears to be self-contradictory or silly but may include a latent truth. Here is an example: ‘All Cretans are liars. I am a Cretan’. As you can no doubt see, these deceptively simple sentences harbour an inherent contradiction. Are all Cretans liars? Well according to the first sentence, yes. However, the second sentence casts doubt on the validity of the first sentence and we become stuck in a hopeless cycle of absurdity.

Although Zeno came up with many paradoxes, they seem to involve the same basic principle. Here is an example using the Homeric hero, Achilles, and a tortoise: A race is organised between the fleet of foot Achilles and a slow plodding tortoise. In the spirit of sportsmanship, Achilles allows the tortoise, Terry (for it is he) a head start of 10 metres. For the purpose of this example I'm going to assume that Achilles travels at the blistering speed of 10 metres per second. Terry, on the other hand, can only cover one metre in the same time. After the tortoise has been travelling for 10 seconds (10 metres covered), Achilles begins his sprint. By the time Achilles reaches the 10 metre mark the tortoise will have moved on a further 1 metre (11 metres in total). When Achilles reaches 11 metres the tortoise will be still ahead at 11.1 metres, and so on. According to this reasoning the Tortoise will always be ahead of the mighty Achilles by an ever decreasing amount. Therefore Achilles can never beat the tortoise and plodding Terry must win the race. Of course, all this is contrary to expectation and reality. Indeed, we expect Achilles to pass the Tortoise after an interval of just 1.11 seconds. So why does Zeno's argument suggest that Achilles never catches the wily reptile? Was Zeno an intellectual scamp and merely embroiling his audience in a clever piece of sophistry? In refutation, this conundrum was considered a serious problem by the ancient Greeks and occupied the minds of very smart men for over a millennia. The more we ponder the conundrum the more we come to appreciate its subtle, profound nature and its implications concerning notions of space and time.

We can represent the problem in mathematical format as an ever decreasing geometric series, thusly: 1 + 1/2 + 1/4 + 1/8 + 1/16 + 1/32............

This of course represents an infinite series and no matter how far you run your calculation the sum will never quite add up to 2, although at each stage it becomes just a little bit closer.

The ancient Greeks struggled with the concept of infinity and converging infinite series. Today we can do much better. While it is true that we can divide distance into an infinite set of intervals, we don't have to. Furthermore, the time involved in covering the distance is finite and it is this fact that explains the apparent contradiction. Is that it Flaxen? Is this your answer to the Great Mystery? Gentle readers you have a right to feel cheated and consequently I feel suitably chastised. I built up the paradox to an incandescence frenzy and then dismissed it with a single, glib, sentence. Mayhap you feel cheated. Unfortunately to express the solution in cogent technical terms I need to invoke some gentle mathematics. I'm not going to introduce calculus to this blog, otherwise you may get the impression that I'm weird.

A mind game is just that. It confronts our intellect and deflects it from our comfortable fluffy surroundings and makes us think, anew- if we are lucky. Otherwise what is the point?

The unexamined life is not worth living. Discuss.

1. I once got a fierce belt over the head in maths class when I was asked to define a circle. I said it was a regular polygon with an infinite number of sides. Sadly, the teacher lacked a sense of humour.

1. Sad. You should have lauded as you gave the correct answer.

2. I think Zeno's Achilles/tortoise paradox suggests that time and space are granular. Isn't this what the Planck Length asserts?

1. I'm going to be a pedant and suggest that any length is finite as the smallest possible unit is the Planck length. Thus we have a true dichotomy between mathematics, which permits infinity, and base reality. String theory rules ultimately, or does it? You raise much in your comment Sir, which is best addressed in another article....Watch this space.

3. The sum of an infinite number of non - zero quantities may yet be finite. That is the nub of the paradox.

1. Yes Mark that is another way of looking at the problem. Not all are satisfied that the problem has been resolved.

4. The length of piece of string is twice what it would be if you cut in half. Or is it?

Often I look at the night sky, examine the myriad of twinkling stars and ask myself what it all means. Then I realise it's fuck all to do with me and I go back to sleep...

5. Daft bugger. Expect mail. I'm composing a missive which will be winging/whinging your way later today. Arse.

6. Tortoise Terry? If his surname's "Pin" then he's no tortoise!

1. Ted, yahm a poet an ya don't know it.