Paradoxes can be extremely useful. A paradox is something that seems to be both true and not true, something which can simply never happen . If we find a plausible paradox what we've actually discovered is an error in our reasoning: flaws either in our scientific models, or our informal intuition, or our observations of the world. A paradox in a formal argument (such as a mathematical proof) is called a contradiction, or *reductio ad absurdum* -- "proven to be absurd" -- and is pretty much the end of the line for any thesis unless some error can be found. So paradoxes can teach us important things; I want to look at a few of those cases.

Zeno's most famous paradox tries to show that nothing can ever get anywhere. An object moving from A to B has to first move halfway, then halfway from that point, then halfway again, and so on. Therefore because there are an infinite number of waypoints between A and B it must take forever for anything to traverse them all.

Because we observe things arriving at their destinations all the time we clearly have a paradox. So where is the error? Tuns out this is an error in intuition. Zeno is exploiting the fact that humans take time to consider something. Considering and endless series of things will therefore take forever. But the moving object isn't considering those waypoints, it's just passing through them. Provided that hitting each point takes an amount of time proportional to the distance between the points, the object can hit an infinite number of points in finite time.

There are two ways to think about this that can help bootstrap the intuition so that it makes sense. Both are a little bit mathematical (because the paradox itself is math masked as parable) but I'm going to avoid notation to help make it clear to any reader.

1) Let's say that the object is moving at a speed that will (theoretically) take it from A to B in one minute. Say 60 mph for a distance of one mile, for example (assuming movement is possible at all, of course). The first Zeno step is one half mile and takes 30 seconds. The second Zeno step is 1/4 mile and take 15 seconds, and so on. If we extrapolate we can see that the time it takes to get from A to B will be given by the sum of all the steps so far: 1/2 min + 1/4 min + 1/8 min + 1/16 min, etc., and the time remaining will be the same as the last Zeno step. What we can observe from this is that after a while the remaining time is not actually growing that fast. In fact it's going down by half after each step. By Zeno step 10 there's only 0.06 seconds left in our minute. At Zeno step 39 there's only one nanosecond left. No matter how many steps you take, the total time is less than one minute, and the time remaining gets smaller and smaller. It may not be such a big leap to see that if we keep doing this forever it still ends up taking 1 minute to get from A to B.

2) But can we expect a finite result from what's effectively a sum of an infinite number of terms? If that still seems strange consider the second approach. Let's take that sum from before: 1/2 + 1/4 + 1/8 + 1/16 + 1/32 ..., and let's call it Z. Will it sum to a finite value? There's an easy way to show that it does. The first Zeno step splits the interval in half; we've taken a step of 1/2 and there's 1/2 remaining. But we're also going to split the remaining 1/2 into Zeno steps: 1/4 + 1/8 + 1/16..., etc. So that means the the entire second half can be represented as Z/2. Why's that? It just means that we're taking the terms of Z and dividing by 2. So 1/2 becomes 1/4, 1/4 become 1/8, etc. The total is then 1/2 plus the remainder which is Z/2, but the sum of those two terms is Z itself. By definition, Z equals 1/2 plus Z/2. The only result that fits that relation is that Z is 1. Thus the result of an infinite sum can be finite.

This is actually a very important result. Understanding this paradox gives us a handle on infinities. Infinite mathematical expressions no longer need to frighten us. We know that some of them will evaluate to finite and even well-behaved answers. This key unlocks doors in geometry, calculus, the equations of physics, chemistry, environmental science, even sociology, and political science. Zeno may have been genuinely puzzled, but the solution gives us -- generations later -- a greater insight into how our models actually work when taken to their ultimate limits.

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