Kea recently brought up the subject of Zeno of Elea and his now long lost book of 40 paradoxes dealing with the continuum. His nominal 2500th birthday should be celebrated relatively soon. Let me paraphrase an example paradox is the following:
If one assumes that space and time are continuous, then an arrow shot from a bow, before reaching its target, must first travel half the distance. And then travel half the remaining distance. And so on. And therefore, there are an infinite number of distances to be travelled and the arrow could never reach the target. But arrows do reach targets. Therefore, space and time are not continuous.
Surprisingly, there is an echo of this thought in quantum mecahnics. The echo is so close to the original paradox that it is known as the Quantum Zeno’s Effect or sometimes “Paradox” depending on the writer. The subject is discussed in many arXiv articles.
In quantum mechanics, when one measures a system, the formalism requires that the system collapse to the result of the measurement. If one examines this carefully, one finds that if one measures a system at a sufficiently high rate, the effect of the repeated measurements is to prevent the quantum system from changing. In effect, if one examines the position of the arrow too frequently, the arrow cannot move. It’s worthwhile looking at the simple mathematics that causes this effect.