How Long Does It Take to Do an Infinite Number of Things?

Infinite processes shred our intuition to Hell like nothing other than probabilities and relativity. This was the infernal currency of Zeno of Elea, who proposed devilish paradoxes in order to prove the conjecture that movement is impossible. Zeno would construct a simple scenario that would show that everyday aspects of life, like chasing a tortoise or shooting an arrow. We really only have accounts from other authors, basically Aristotle, who disagree with him. Some of these arguments intertwine ideas about space and time, the arrow argues that an object is motionless at any instant, and both the dichotomy and Achilles and the Tortoise make arguments about the impossibility of doing an infinite number of things. I will eventually move on to an example to calculate the total time that it takes for a ball to bounce an infinite amount of time, but I will first describe the dichotomy using Philocetes' Arrow as a story (rather than Aristotle's bare-boned description from the Physics [1]).

Philocetes looses an arrow from the Bow of Heracles at the Trojan prince, Paris. In order for the arrow to strike the Son of Priam, first, it must fly half the distance between the heroes. But, it is clear that in order for an arrow to travel half way to the midpoint before it can get to the midpoint. And it has to travel half way to the point before that, ad infinitum. Therefore, in order to move from one spot to another, no matter how close, you have to move an infinite number of times in a finite amount of time to get anywhere, so the arrow never flies and Trojan War never ends.

This is paradox because we do move, but at least the first time your hear the argument, you don't have a good reason why it is wrong. The arrow must move, but logically it cannot. And this is applicable to every kind of motion. Before you can eat your Wheaties in the morning, you have to get out of bed and get into the kitchen. But, you have to move through an infinite number of small separations to get there.

And you can't do an infinite number of things before breakfast.

Or can you?

Most people think that the invention of Calculus resolved Zeno's paradoxes. This is certainly true in the case of the dichotomy: Leibniz used and even dirtier trick with infinity than Zeno did. Leibniz built calculus out of the idea of an infinitesimal to align with his cosmological ideas. An infinitesimal is a chunk of the universe that is smaller than the smallest division, basically the reciprocal of infinity. The integral calculus would define the distance that the arrow must travel as a sum of all of the infinitesimal chunks of space between Philocretes and Paris. The nature of infinitesimals is that they are smaller than the smallest fraction, there are an infinite number of them between each rational number. The infinity of the infinitesimals is that of the real numbers (the continuum) and the infinity of the dichotomy is that of the rational numbers (countable). So, if you can construct a theory of motion that adds up all the infinitesimal points, it will automatically encompass the infinity of halves used by Zeno.

This solve the dichotomy, but it does so indirectly with an end around.

By subsuming the motion of the dichotomy into a single, continuous process that can be analyzed separately, we show that motion can exist. We solve the riddle by changing the problem, though. However, this leaves Zeno's premise unchallenged: an infinite number of processes take an infinite amount of time. What I'd like to do here is take on the premise that an infinite number of discrete, sequential processes needs to take an infinite amount of time. You could do the same thing with the dichotomy, as well, but since that is an arbitrary partition of a single, continuous process, which I feel is a little different.

The specific question I ask is: how long does it take a rubber ball to stop bouncing? The physics here is quite simple. It can be done with kinematics using the simplest of deflection theories: the coefficient of restitution. The model uses the simple rule that the velocity of the rebound of an object is proportional to its original speed, and that proportionality (the coefficient of restitution) remains the same after each bounce. The duration of the air time of the ball is given by uniformly accelerated motion. The sum of a sequence of such bounces will lead to an infinite series with a known sum (thanks again, calculus), and this sum will be finite.

Where can this go wrong? Well, it's not in the assumption of uniform acceleration. Yes, it's not quite true, but it's pretty accurate at low velocities for short times, which a rubber ball acts in. If we complicate the problem by adding in air resistance, that will give us a slightly more accurate estimate at the cost of an annoying integral (no thanks, calculus). This accuracy will give us a time that is strictly smaller than the uniform acceleration version by giving us a factor similar to the coefficient of restitution itself. The significant assumption that would break this analysis, if it were relevant, would be that the time of the bounce itself will be the same each time the ball hits the ground if the bounce is modeled on an elastic restoring force, which is probably the best model available. Even though this will be small, at some point it will be larger than the air time per bounce, and since it remains the same, adding an infinite number of them would create an infinite time for the bounce.

But, to answer the basic question, can an infinite number of processes be completed in a finite amount of time, eliminating the time of the bounce is justifiable.

So what happens in this case? Well, from basic kinematics, we find that the time of an individual flight is proportional to the initial speed of that bounce.* Since the initial speed of each process is the coefficient of restitution is just the initial speed of the previous process, the duration of the subsequent process is scaled down by the sane proportionality,

t_n = r t_n-1 = rⁿ t₀

which means that flight is scaled down by a power of the coefficient of restitution.
When these are summed, we find an infinite series in powers of the coefficient of restitution that has a known sum: the inverse of one less the coefficient [ 1/(1-r) ]. So, the total time the ball bounces is finite if r < 1 (which is must be unless it is gaining energy from the environment somehow).

So, an infinite number of bounces takes a finite amount of time

t = 1/(1-r) t₀.

This is a reasonable answer because if r = 1 the bouncing goes on forever and if r = 0 it stops after the first flight. This should be the same result you'd find if you were to sum the time to travel each segment of the arrow's path, but here we have distinctive processes represented by the flights between bounces. Our hero Leibniz has defeated Zeno of Elea's Satanic dichotomy.

So the Trojan War terminates, and you can do an infinite number of things before breakfast.



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* The proportionality constant is 2/g.

[1] All of my Aristotle is missing. Most of what I know about this comes from Sainsbury's Paradoxes, >although I was using the Stanford Encyclopedia of Philosophy.