Achilles and the Arrow

Abstract

When we look at the world one of the first questions to strike us is to do with motion: why do things move? Do they really move? Such questions, asked by the Greeks, created kinematics, the science of motion. Zeno of Elea, with his antinomies of motion (Can the flying arrow reach its target? Can Achilles overcome the tortoise? and others), went even further. In his persistent questioning we can identify the seeds of serious problems: the problem of continuity and infinite divisibility, the nature of infinite sets, the problem of limit and of instantaneous velocity. Twenty-five centuries and new branches of mathematics (set theory, topology and calculus) were necessary to resolve Zeno’s paradoxes. Are they indeed resolved? Mathematics is a purely formal science and as such it does not refer to reality, but motion belongs to the real world, and consequently no problem of motion can be solved by mathematics. Correct. But mathematics is used to model the world, and if this is the case, it becomes physics. First, we must correctly guess a mathematical structure, and then interpret it as representing the structure of a certain aspect of the world. The history of science teaches us that more often than not our interpretations are correct. And when this is the case, the mathematical model of a physical phenomenon that we have created rewards us with empirical predictions. In this sense, calculus, together with other co-working mathematical disciplines and suitable physical interpretations, solve the problem of kinematics.