The notion of infinity, and the problems that arise from it, both philosophical and mathematical, have been a central concern for millennia.
The 5th-century-BCE Greek philosopher Zeno presented four paradoxes that, through clever uses of infinity, demonstrated (so it was claimed) that motion was impossible. Here we consider the paradox of Achilles and the tortoise. We, however, will be able to go further than Zeno with the use of mathematical tools that were unknown to the Ancient Greeks. Rather than demonstrating motion is impossible, Zeno’s paradox can be used to analyse the notion of infinity, and show that not all infinities are created equal.
But first, the paradox:
Achilles, known for his speed, is to run a race over 100 metres against a tortoise. To make the race fairer, Achillies decides to offer the tortoise a head start:
Achilles: “Now tortoise, we want this to be a close race, so how large a headstart would you like?”
Tortoise: “A one meter headstart will be more than adequate, thank you Achilles”
Achilles: “Now don’t be ludicrous, the race is a full 100 metres long! With a mere one metre head start I shall overtake you in an instant and win the race comfortably”
Tortoise: “Not so my dear friend. For you see if I start a meter ahead of you and we both start the race at the same time, then by the time you have travelled a meter, I will have moved on and will still be ahead of you”
Achilles: “Well, yes, but…”
Tortoise: “And then by the time you have traversed that distance I would have moved forward again so I will still be ahead of you. And so on and so on. And so you see my dear Achilles that whenever you get to where I have been I will have moved on, and you will in fact never be able to overtake me!”
Thus, according to Zeno, Achilles can never pass the tortoise and win the race, because, no matter how fast he runs, each time he reaches a point where the tortoise was, the tortoise will have moved a bit farther on.
With the problem so stated, we can begin to analyse it mathematically to see what we can learn. For simplicity sake we will assume the speed of the tortoise is half the speed of Achilles. As above we will give the tortoise a one metre head start.
By the time Achilles has travelled one metre (to where the tortoise started), the tortoise will have travelled half a metre. Then, by the time Achilles travels this half metre, the tortoise would have travelled a further quarter of a metre. When Achilles has travelled this quarter metre the tortoise will have moved an eight of a metre and so on and so on. This will happen an infinite number of times as the tortoise suggests.
We can now write the distance travelled (D) by the tortoise as an infinite sum:
D = 1 +½ + ¼ + ⅛ + . . . .
First we must ask if this is an equation that can be solved, and if so is it necessarily infinite? We start by multiplying both sides of the equation by a factor ½:
½ * D = ½ + ¼ + ⅛ + 1/16 + . . . .
Next we add 1 to both sides:
1 + ½ * D = 1 + ½ + ¼ + ⅛ + 1/16 + . . . .
But at this stage we notice that the sum on the right is exactly the original sum, D, that we started with, so we have:
1 + ½ * D = D
Which, rearranged gives D = 2. So we have found the total distance travelled by the tortoise until Achilles overtakes him is 2 metres, and in doing so we have stumbled upon a (perhaps surprising) mathematical fact: Some infinitely long lists of numbers (or infinite series) have a finite sum.
Written by William Brooke, Director of Witherow Brooke