Karl Aubert and Zeno's arrow
In Karl Egil Aubert, 'Spurious Mathematical Modelling', Mathematical Intelligencer 6 (3) (1984), 54-60, Aubert discusses Zeno's "paradox of the arrow". We present this short extract about Zeno's arrow from Aubert's fascinating paper:
[One] of the paradoxes of Zeno is the "paradox of the arrow". Consider an arrow in its flight. At each point of its trajectory it is at rest because nothing can move during an "instant" - a time interval of length zero. Since the entire trajectory of the arrow is made up of points, or if you will, the time interval is a "sum" or a "union" of instants; it is claimed by this reasoning that it is logically impossible for the arrow to move at all. It is calculus which came to the rescue in the case of the arrow. The language of calculus is able to handle the notion of instant velocity at a given point as a derivative and the length of the travelled trajectory as an integral, the two fitting together in a beautiful way corresponding to the fundamental theorem of calculus. What is at the heart of the "resolution" of the paradox of the arrow is comparable to what is at the heart of integral calculus, namely to give a precise meaning to certain (uncountable) infinite sums of "infinitely small" addends. Again, we should certainly not view calculus as the "logical proof of" or the "reason for" the fact that it is possible for an arrow to move. It should rather be viewed as a consistent language representing a vast extension of every-day language which is able to cope with the notion of infinity at the same time it admirably fits the observed facts.