The examination paradox and formal prediction

Abstract

In treating paradoxes which are formulated in common language, one is sometimes tempted to go too far, by making a mathematical model that demonstrates in a correct way the fundamental assertion of the paradox or its negation. But this is not always relevant, because in several cases the point is to find a logical formulation of the non formal reasoning given by the paradox, and to show -as exactly as possible- where it is incorrect or paradoxical. For example, we do not say anything relevant to the paradox of Achilles and the tortoise by defining a certain mathematical series and thus demonstrating that Achilles reaches the tortoise in a finite time: the point here is to follow step by step Zeno's argument and to show where it is logically wrong. Perhaps in doing it we are led to make more rigorous or more explicit Zeno's argument, but in any case the logical devices employed must respect the "spirit" of the original reasoning. I emphasize this by saying that the informal reasoning is a substantial part of the paradox itself.