Achilles and the Tortoise

Achilles and the Tortoise is a well known example of Zeno's Paradoxes. It essentially goes as follows.

Achilles and the Tortoise have a race, but the Tortoise is given a head start; 100 metres, for example. For Achilles to pass the Tortoise, he must first catch up with the Tortoise, but by the time he catches up with where the Tortoise was, the Tortoise has moved further ahead, so Achilles must then catch up with the Tortoise again.

Since Achilles must reach the point where the tortoise was before he can race on to the point where the Tortoise currently is and, since when he gets there, he must repeat the whole process because the Tortoise has again moved forwards, it follows that Achilles can never logically catch up the Tortoise and can never, therefore, pass the Tortoise and win the race.

The paradox is, of course, that we know Achilles is faster than the Tortoise and, therefore, must catch and pass the Tortoise very easily and finally win the race.

There are a number of flaws in the argument, although it appears quite logical. Firstly, there is the assumption that, as the gap between Achilles and the Tortoise diminishes, that there will always be a gap, no matter how small and, secondly, that the gap left will always be meaningful. In practice, the gap left diminishes rapidly until there is, effectively, no gap at all and at that point Achilles passes the Tortoise.

The second flaw is similar to the first but relating to time. The time left in which Achilles will reach the Tortoise diminishes as rapidly as the physical gap left and described above. It is simply not meaningful to continually subdivide time in this way.

The continual redefinition of how much time and space remains for Achilles to catch the Tortoise is obviously an infinite series, but the time in which this series can play out is, most definitely, a finite amount of time and, in which, the infinite series will resolve.

The key flaw is a misunderstanding of abstract mathematical concepts: infinity and zero are not ordinary numbers and cannot be manipulated or applied in quite the same way as other numbers; you cannot meaningfully divide or multiply either number and, in the case of infinity, neither addition nor subtraction give a meaningful result.

When these are applied to real quantities problems quickly arise. For example, a line of any length contains an infinite number of points. If you reduce the length of the line, it still contains an infinite number of points. Since a point has a length of zero, it will always be possible to place an infinite number of points along any line. It would be easy to argue that any line containing an infinite number of points must be the same as any other line containing an infinite number of points, but this is to completely misunderstand the nature of infinity: it is simply not a proper number, it is a concept and far more philosophical than numerical.

Zero is much the same, but people think they understand it better, simply because it is there between 1 and minus 1 (another concept), but the truth is that it does not exist in the same way: there is no difference between minus zero and zero; they are both the same. Zero is simply an infinitely small number: so small that it has no size at all. The main difference is that, if you add one to it, you get one and, if you add one to infinity, you get infinity.

The main flaw of the paradox, as opposed to the key flaw which underlines it is, that by subdividing Achilles task into an infinite number of tasks, we have apparently created a series of obstacles which cannot be resolved in a finite amount of time. The simple truth is that, in reality, the finite cannot be combined with the infinite in any way that is practically meaningful. Mathematicians and phycisists may disagree, but they understand the concepts in use and the rules which need to be applied when doing so.