Online research in philosophy

It is widely held that the paradox of Achilles and the Tortoise, introduced by Zeno of Elea around 460 B.C., was solved by mathematical advances in the nineteenth century. The techniques of Weierstrass, Dedekind and Cantor made it clear, according to this view, that Achilles’ difficulty in traversing an infinite number of intervals while trying to catch up with the tortoise does not involve a contradiction, let alone a logical absurdity. Yet ever since the nineteenth century there have been dissidents claiming that the apparatus of Weierstrass et al. has not resolved the paradox, and that serious problems remain. It seems that these claims have received unexpected support from recent developments in mathematical physics. This support has however remained largely unnoticed by historians of philosophy, presumably because the relevant debates are cast in mathematical-technical terms that are only accessible to people with the relevant training. That is unfortunate, since the debates in question might well profit from input by philosophers in general and historians of philosophy in particular. Below we will first recall the Achilles paradox, and describe the way in which nineteenth century mathematics supposedly solved it. Then we discuss recent work that contests this solution, reiterating the dissident dogma that no mathematical approach whatsoever can even come close to solving the original Achilles. We shall argue that this dissatisfaction with a mathematical solution is inadequate as it stands, but that it can perhaps be reformulated in the light of new developments in mathematical physics.

An example of the second situation is the most famous of the paradoxes of Zeno, the Greek philosopher who lived during the Golden Age of Greece on the island of Elea. Zeno proposed the following "thought experiment". Achilles, a young athlete, runs a race with a tortoise. Achilles can run exactly twice as fast as the tortoise, so to make it fair he gives the tortoise a head start of exactly half the distance from the starting line to the finish line. The starting signal is given and the race begins. Achilles runs to the starting position of the tortoise. In the time it takes to do that, the tortoise has advanced half the distance from his starting position and the finish line. Achilles then advances to the new position of the tortoise. During that time the tortoise again advances half the distance to the finish line. And so on ... Every time Achilles moves ahead by a given distance, the tortoise moves ahead by half that distance. Zeno concluded that Achilles can never catch the tortoise, because in every time interval in which Achilles moves to the tortoise's former position, the tortoise always moves ahead by half that distance.