Zeno of Elea was a Greek philosopher of the 5th century BC, best known for his paradoxes of motion, described by Aristotle in his ‘Physics’. Of these perhaps the best known is his paradox of the tortoise and Achilles, in its various forms. In a modern version, the antelope starts 100 metres ahead of the cheetah and moves at half the speed of the cheetah. Will the cheetah ever catch the antelope, assuming they don’t slow down?

Zeno’s paradox relies on the fact that when the cheetah reaches the starting position of the antelope, the antelope will have travelled 50 metres further. When the cheetah arrives at that point, the antelope will have travelled a further 25 metres, and so on. Zeno argued that this was an infinite process, and so does not have a final, finite step. So how can the cheetah ever catch the antelope?

There is a mathematical solution to the paradox, which goes like this:

Let S be the distance the cheetah runs and let 1 = 100 metres.

So S = 1 + ½ + ¼ + 1/8 + 1/16 + 1/32 …..

½ S = ½ + ¼ + 1/8 + 1/16 + 1/32 …..

Therefore, S – ½ S = 1

Therefore, S = 2

So the cheetah catches the antelope in 200 metres.

So an infinite process, with no final step, has a finite conclusion.

That’s the mathematical solution, but does that solve the intuitive paradox?   How can an infinite process, with no final step, come to an end? I understand the mathematical solution, but somehow it is as unsatisfying as the wrapper of a chocolate bar. To me, the real chocolate remains untouched. Such paradoxes I refer to as ‘chocolate paradoxes.’ What they have in common is that they can be solved mathematically without really being solved at all.

For those who might differ with me, the Thomson’s Lamp thought experiment offers a related challenge. Devised by philosopher James F. Thomson in 1954, it goes like this. Think of a lamp with a switch. You flick the switch to turn the light on. At the end of one minute exactly you flick it off. At the end of a further half minute, you turn it on again. At the end of a further quarter minute you turn it off. And so on. The time between each turning on and off the lamp is always half the duration of the time before. Assume you have the superpower to do each turning on and turning off instantaneously.

Adding these up gives: 1 minute plus half a minute plus a quarter of a minute ….

1 + ½ + ¼ + 1/8 + 1/16 + 1/32 + … = 2.

In other words, all of these infinitely many time intervals add up to exactly two minutes.

So here’s the question. At the end of two minutes, is the lamp on or off?

And here’s a second question. Say the lamp starts out being off and you turn it on after one minute, then off after a further half minute and so on. Does this make any difference to your answer?

Thomson claimed there was no solution, and that the problem led to a contradiction.

“It seems impossible to answer this question. It cannot be on, because I did not ever turn it on without at once turning it off. It cannot be off, because I did in the first place turn it on, and thereafter I never turned it off without at once turning it on. But the lamp must be either on or off. This is a contradiction.”

While considering the relationship between the infinite and the finite, consider in conclusion the following.

Can a number of infinite length be represented by a line of finite length? Solution below.

The square root of 2 is an irrational number, with no finite solution. In other words, it goes on for ever. ‎1.4142135623730950488……………………….. for ever…..

So can a line with a finite length exactly equal to this infinitely long number be drawn?

Draw a right-angled triangle, of vertical length (a) and horizontal length (b) equal to 1.

Then, the length of the hypoteneuse of the triangle, c, can be derived from the length of the adjacent (a) and opposite (b) sides, using Pythagoras’ Theorem.

a2 + b2 = c2

So, 12 + 12 = c2

So c2 = 2

c = √2

This is a line of finite length, representing a number of infinite length. So the answer to the question is yes. Strange? Indeed. Another of those tantalising ‘chocolate paradoxes.’