Zeno’s paradoxes are amongst the most venerable and to many impenetrable of all philosophical puzzles. There have been many attempts to explain Achilles and the tortoise, the arrow paradox and the so-called dichotomy paradox, often illustrated by ‘proving’ the impossibility of a frog ever managing to hop its way out of a pond. Apparently there were about 40 in all, and were originally created by Zeno of Elea to illustrate Parmenides’ conception of the world as logically changeless by showing that much of what we believed we witnessed in the world simply could not be happening.
This view has undergone a lot of humorous parody and satire, from Zeno’s contemporaries to Terry Pratchett’s occasional sketches of ‘Ephebian’ (i.e., Greek) philosophers deducing all sorts of wonderful things. All in all it tends to remind one of the wonderful line in Monty Python and the Holy Grail, ‘and that, my lord, is how we know that the earth is banana-shaped’.
There have been plenty of attempts to refute Zeno too, involving ideas like infinitesimals (roughly speaking Aristotle’s answer), which works quite well. For we have known since Archimedes that the sum of an infinite number of progressively smaller amounts will add up to a finite total. Or rather, it will tend towards a finite result, which is a problem, since it is not obvious that this is not actually simply a mathematical way of simply restating the original paradox, because this reasoning does not, in itself, conclude that the frog ever actually reaches the side of the pond.
Another use of the infinitesimals inherent in Zeno’s original paradoxes is to reject them. If it simply isn’t true that there is always a point lying between two other points, no matter how close together, to which the frog/ arrow/ Achilles/ whatever can move next, then it must make definite progress and eventually Achilles will be the winner. You can also reject the idea that things are anywhere definite or finite at any given point in time, but I think it’s a bit much to have to rely on relativity or the uncertainty principle to beat up classical philosophers!
I have never quite understood why these paradoxes seem to attract solutions like this. Perhaps I am missing something, but hasn’t Zeno actually defined the problem in such as way that it cannot be solved? He describes all these events in terms of them never being concluded. For example, if Achilles’ movements are defined in terms of reaching where the tortoise was last – a place from which it must have moved on – then plainly he can never overtake it in this framework. But that is only to day you can define a problem in terms of failure, which is what Zeno has done. If at each step Achilles’ motion is defined as reaching only to where the tortoise last was, he cannot have reached where it is now. So the problem is actually stated in terms that permit only failure. In other words, failure is not a paradoxical result of this overall situation but a simple deduction from its definition.
There is no need to appeal to any fancy mathematics. All you have to do is to realise that this is not a paradox at all, but rather the description of an activity in negative terms that cannot be shaken off without seeming to disregard the original problem. Zeno’s paradoxes strike me as valid but uninteresting – except for what they tell us about how philosophers and mathematicians reason.