Dichotomy paradox has been very famous since it was first introduced thousands of years ago. The paradox is notable because it is very simple to state but unbelievably difficult to analyze without the right tools.
Statement of Zeno's Dichotomy paradox
"All motion is impossible."At a first glance the statement of Zeno's paradox may seem utterly rubbish. But wait till you read the thought experiment Zeno devised to craft this statement.
Description of the paradox
Suppose you want to move from point A to point B. The distance between the points 1 km. We all can agree that to cover a certain distance we must first cover half of the distance.
So to cover 1 km, first you must move 1/2 km. Then, another 1/2 km of the total distance will be left.
Next to cover the left 1/2 km, you must first cover half of it, that is 1/4 km. Then 1/4 km of the total distance will be left.
Next to cover the left 1/4 km, you must again cove half of it, that is 1/8 km. Then 1/8 km of the total distance will be left.
Then you will cover half of 1/8 km, that is 1/16 km. But still 1/16 km distance will be left to reach B.
Thinking somewhat abstractly like this, no matter how much time you continue this process there will always be some distance left to cover.
Thus, this logical thinking suggests that you can never actually reach to point B!
But what does that mean?! How is this even possible? If you are searching for any logical fallacy, you will be disappointed. To cover a distance you must first cover half of it - the logic here is watertight.
Now, let's continue. This was only the half of Zeno's dichotomy paradox. You will see the more dangerous conclusion next.
Same example. We want to go from A to B. Distance is 1 km. This time we will think in reverse. To cover 1 km one must first cover 1/2 km. But to cover half km, you must first move 1/4 km. Then again, you must move 1/8 km before that/ 1/16 km before that, 1/32 km before that, 1/64 km before that and do on. It looks like not only you can't cover the distance between A and B, but actually you can't cover any distance at all. Because, thinking abstractly, no matter how small the distance is, you must cover even a smaller distance, i.e. half of it, beforehand. You certainly realize that if we keep dividing like this, we can't even move because the distance we need to cover becomes smaller and smaller, reaching zero.
Hence, Zeno proudly declared, "Motion is impossible!"
Evidently, Zeno's paradox was very disturbing. Unlike Zeno, people wasn't pleased with his conclusion. He was attacked by contemporary Greek philosophers. But they weren't much successful. We shall give one example.
Diogenes's reply to Zeno's paradox
The famous Greek philosopher Diogenes, who lived in a barrel and (perhaps rightfully!) believed dogs to be better than humans, replied to Zeno's paradox just like many of us would. He just stood silently and walked a few steps. This might be a clever answer, but it leaves a lot to be desired.
A more rigorous formulation of Dichotomy paradox
To properly understand Zeno's paradox of Dichotomy, we need to reformulate the paradox in a bit more rigorous language. And there is nothing logically more rigorous than mathematics. In fact, mathematics will make this paradox a lot easier to understand and solve.
Let's return to our beginning example. We need to cover 1 km. We first cover 1/2 km, then 1/4 km, 1/8 km and so on. Zeno says we can't finally cover the whole 1 km. Mathematically speaking, he claims,
1/2 + 1/4 + 1/8 + 1/16 + 1/32 + .... ≠ 1
To resolve this paradox (and we must do that, otherwise we won't be able to move!), we have to prove that,
1/2 + 1/4 + 1/8 + 1/16 + 1/32 + .... = 1
It should be understood that the dots (...) in the above two sums denotes an infinite number of terms in the series. That means the series goes on forever.
But is it even possible to add an infinite number of terms to get a finite sum? As small as 1 only? How can we even add them all?
How to calculate the sum of an infinite series?
Well, honestly speaking we can't always add all the terms of a series which has an infinite member. But in some cases we can. These type of series are said to be Convergent. We can mathematically test if a series in convergent or not. We won't describe how to conduct the test here. However, the test shows that the series related to Zeno's paradox is a convergent series. So it should be possible to find a sum of its infinite members.
Now we are going to find the sum of the convergent infinite series 1/2 + 1/4 + 1/8 + 1/16 + 1/32 + .... The procedure may seem a bit naive, but lets go with it anyway.
Let, S = 1/2 + 1/4 + 1/8 + 1/16 + 1/32 + ....
So, 2S = 1 + 1/2 + 1/4 + 1/8 + 1/16 + ....
Or, 2S = 1 + (1/2 + 1/4 + 1/8 + 1/16 + ....)
Or, 2S = 1 + S
Thus, S = 1
Done! We have just added Zeno's series and found that the sum of its infinite terms is indeed equal to a finite number 1.
Therefore, Zeno's suggestion that if we keep going to cover a distance there will always be some distance left to cover - turns out to be mathematically incorrect. By adding the infinite series what we have actually shown is, if one keeps moving, than the distance left will be smaller and smaller and finally one will be able to cover the distance of 1 km; because all those infinite smaller and smaller steps eventually adds up to 1 km.
Note that we have used 1 km as an example. The same logic and math work for any distance.
A more visual way
Ok, so we have solved the Zeno's paradox but you are not quite happy with the way we summed the infinite series, right? If that is the case, than imagine a square. The geometrical square. Suppose the length of each arm is 1 m, Hence, it has a total area of 1 meter-squared.
Now divide the square in two equal parts as below:
We can see that the gray-ish blue part has an area of 1/2 and the sky blue has an area of 1/2. Next, we will divide the sky blue area in two equal parts.
Now, the gray-ish blue part has an area of 1/2, the sky blue has an area of 1/4 and the purple part has an area of 1/4. Let's divide the purple area in two equal parts.
Here, the gray-ish blue part has an area of 1/2, the sky blue has an area of 1/4, the purple part now has an area of 1/2, same as the pink part. We can keep dividing.
Here, the area of gear-ish blue part is 1/2, area of sky blue is 1/4, area of purple is 1/8, area of pink is 1/16 and the new orange part has an area of 1/16. We can continue dividing the orange area into two 1/36 parts and go for more and more steps (i.e infinite number of areas). But the point is that no matter how much new area divisions we make, they are all within the square of area 1. So their sum must equal to 1.
Thus, even though we can divide the area in the above mentioned way for an infinite amount of time, we will still have the total area of 1/2 + 1/4 + 1/8 + 1/16 + 1/32 + .... = 1
Hopefully you are satisfied with the sum of this infinite series now.
Solution of Zeno's paradox
Showing that the infinite terms we get from Zeno's thought experiment actually makes only a finite value when added together, essentially solves the Zeno's paradox of motion. But there remains a few subtleties which are not discussed here. (Note that we haven't clearly mentioned how our solution works for the reverse description of Zeno's thought experiment.)
There are also many other paradoxes of Zeno. This old man has taken away a lot of time from philosophers and scientists alike! Dichotomy paradox is a fine example of how simple logical thoughts may call for a completely new way of looking at things.