Professor Stewart's Hoard of Mathematical Treasures (29 page)

As a follow-up, we might add that the infinite sequence
converges to 11
, meaning that it approaches indefinitely close to that value, and that value alone, if you go far enough along the sequence.

The dichotomy paradox can be approached in a similar way. Suppose that the arrow has to travel 1 metre and moves at 1 metre per second. Zeno tells us where the arrow is after
second,
second,
second, and so on. At none of these times has it reached its target. But that doesn’t imply that there is no time at which it reaches the target - just that it’s not one of those considered by Zeno. It doesn’t get there after
seconds, either, for instance. And it clearly does get there after 1 second.

And here, too, we can point out that the infinite sequence
converges to zero, and the corresponding sequence of times
converges to 1, the instant at which the arrow hits the target.

Many philosophers are less satisfied with these resolutions than mathematicians, physicists and engineers are. They argue that these ‘limit’ calculations do not explain why infinitely many different things can happen in turn. Mathematicians tend to reply that they show how infinitely many different things can happen in turn, so the assumption that they can’t is what’s making everything seem paradoxical. When the arrow travels from the 0-metre mark to the 1-metre mark, it does so in the finite time of 1 second. But although the length of the interval from 0 to 1 is finite, the number of points in it (in the usual ‘real number’ model) is infinite. In such a model, all motion involves passing through infinitely many points
³²
in a finite time.

I don’t claim that my discussion knocks the argument on the
head, or covers all relevant points of view. It’s just a quick and broad summary of a few of the main issues.

The arrow paradox is also often resolved by taking the ‘limit’ point of view, or, more precisely, calculus, which is what limits were invented for. In calculus, a moving object can have an instantaneous speed that is not zero, even though it has a fixed location at that instant. Making logical sense of this took a few centuries, and boils down to taking the limit of the average speed over shorter and shorter intervals of time. Again, some philosophers feel that this is not an acceptable approach.

I think there’s another interesting mathematical point buried in this one. Physically, there is a definite difference between an arrow that is moving and one that is not, even if they are both in the same place at some instant. The difference can’t be seen in an instantaneous ‘snapshot’, but nevertheless it is physically real (whatever that means). Anyone who does classical mechanics knows what the difference is: a moving body has momentum (mass times velocity). A snapshot tells you the position of the body, but not its momentum. These are independent variables: in principle, a body can have any position and any momentum.

While position is directly observable (see where the body is), momentum is not. The only way we know to observe it is to measure the velocity, which involves at least two positions, at closely spaced intervals of time. Momentum is a ‘hidden variable’, whose value must be inferred indirectly. Since 1833, the most popular formulation of mechanics has been the one proposed by Sir William Rowan Hamilton, which explicitly works with these two kinds of variables, position and momentum. So the difference between a moving arrow and a fixed one is that the moving one has momentum, whereas the fixed one does not. How can you tell the difference? Not by taking a snapshot. You have to wait and see what happens next. The main thing missing from this approach, philosophically, is any description of what momentum is, physically. And that’s probably a lot harder than anything that worried Zeno.

What of the stadium? One answer is that Zeno was hopelessly
confused, and that his conclusion ‘half the time is equal to double the time’ does not follow from his set-up. But there is an interpretation that puts all four paradoxes in a more interesting light. The suggestion is that Zeno was trying to understand the nature of space and time.

The most obvious models of space are either that it is discrete, with isolated points placed at (say) integer positions 0, 1, 2, 3, and so on; or it is continuous, and points correspond to real numbers, which can be subdivided as finely as we wish. The same goes for time.

Possible structures for space and time.

Altogether, these choices give four distinct combinations for the structure of space and time. And these relate fairly convincingly to the four paradoxes, like this:
Possibly Zeno was trying to show that each combination suffers from logical problems.