Zeno’s Paradoxes

Zeno of Elea devised several paradoxical conundra to demonstrate the problematic nature of our perception of space and time. His paradoxes suggest that motion and change are illusory phenomena.

In his famous paradox, Achilles and the Tortoise the two eponymous characters race one another. The tortoise is given a start of a specified distance – say 50 metres. Achilles is faster than the tortoise but for every stretch of distance covered by the former, the latter also covers a stretch of ground. For example, when Achilles has run 10 metres, the tortoise will have covered one metre. Every time Achilles advances the tortoise will also have proceeded; this means that Achilles has continually more ground to make up in his attempt to win the race. This ongoing process means that Achilles has to cover an infinite range of points which the tortoise has already covered, which means that Achilles can never catch up.

BBC pundit Alan Hansen once said something along the lines of “pace is no longer a luxury in football, it’s a necessity”. But Achilles and the Tortoise demonstrates that pace is actually redundant. The real necessity is a keen footballing brain which enables appropriate on-pitch positioning. Intelligent players will ensure they have a headstart over their opponents when running for the ball, it is then impossible for their rivals to catch them! The paradox therefore explains why a great “reader” of the game like former England forward Teddy Sheringham was able to continue playing top-flight club football up until the age of 67. So we don’t need to concern ourselves with the pace of opposition forwards in South Africa; as long as John Terry maintains a small headstart then he must always make it to the ball first.

However, a more extreme problem suggested by Zeno is represented in his dichotomy paradox. When moving towards a designated point we must first of all get half-way there, and before that a quarter of the way there, and before that an eighth of the way there, and so on. Each distance can be halved and this abridged distance can also be halved. This means that an infinite number of stages must be traversed, which means we can never reach our destination, even when this spatial target is itself motionless. A match played with strict reference to the dichotomy principle would not be much of a spectacle: as kick-off would take an infinite amount of time, there’s little chance that many goals would be scored within 90 minutes.

As motion is an illusion there is really no need for teams preparing for World Cup 2010 to worry about tactics, technique or fitness. Football is not about movement of either the players or the ball, as motion is impossible. Empirical evidence does call these views into question, however, which is why these problems are paradoxical. It will thus be a brave coach who decides to embrace a stationary team policy based on insight from Zeno’s paradoxes. England didn’t move much at the last World Cup, and this did appear to reduce their effectiveness quite considerably!