Zeno's Paradox: Is Motion Impossible?

Zeno was a Greek philosopher who was born in 490 BC. He is famous for introducing several very simple but profound paradoxes - which are basically described in form of thought experiments. In this post we are going to discuss the most famous paradox of Zeno. This paradox is known as the Dichotomy paradox.

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.    

A Criticism of the “Intelligent Design Argument”

The term “Intelligent Design” (also known as Teleological argument) refers to a view or argument often put forward to prove that there is a creator of the Universe. In this post we are not trying to refute its final claim, but we shall criticize the argument itself by showing that it (the argument) is not logically adequate to prove the existence of any intelligent designer.

The Intelligent Design argument says - the universe is so precisely designed for us to exist that this is impossible to be just an accident. Hence it is claimed that there must be an intelligent designer who has created the universe in this way.

The first part of this argument is of course correct. Our universe is very precisely configured. The laws of Physics are just the way they should be. The physical constants just have the perfect values. Even our earth is at a very suitable distance from the Sun (i.e. otherwise it would be too hot or too cold for life to exist). The configuration of the universe as a whole rests on such a knife edge that any slightest change in the way it was created would make our existence impossible.

Once again, saying that our universe seems to be very precisely designed for our existence is a claim that cannot be denied. There can be countless examples given to its support. So we all agree on this. But there remains an important question about the second part of the argument.

Although the universe is precisely designed for us to exist, was this designing done intentionally? Or was it just an accident? The supporters of Intelligent Design argument will say that this cannot be an accident, because an accident cannot result that much precision. And here is where the weakness of Intelligent Design theory lies.

Suppose someone buys a glass cup form the market. After reaching home he / she stumbles at the doorstep. The cup falls down and breaks into pieces. Now let us consider a certain piece of the broken glass. If its particular shape is considered we can say it is very precisely designed. We can say the person who bought the cup has very cleverly bought the right cup, with right ingredients, stumbled just at the right time, in a right way, so that the cup has fallen at the right place of the ground, in a perfect angle and with perfect speed, at a specific orientation (and we can go on…) so that that perfect piece of broken glass was created in a precise way. So following the logic of Intelligent Design argument we can say that the owner of the cup deliberately designed the broken piece of glass in a particular fashion. While actually it is clear that the creation of that particular broken piece of glass was just an accident.

From the example above, it is quite clear that any accident like this can later be described as a planned or designed incident if it is described as such. Therefore, our perfectly designed universe might as well be an accident.

Actually there are an infinite number of ways the universe could have been configured. It is not strange that one of these configurations will support life. Why we live in a particular configuration which is perfect for us to live? Because that is the only place we can live. As simple as that. We don’t need any intelligent designer to justify the perfect design of our universe.

In the conclusion, the Intelligent Design theory is based on a wrong premise. Precision of the universe doesn’t guarantee that it was not just an accident. Hence the theory fails to prove what it intends to. And the argument is futile. 

Java Programming (Lesson 0): Setting up the JDK and Atom text editor

This is going to be an ongoing series of blog posts on Java programming, starting form the very beginning. Some elementary ideas about programming is assumed (for example: what is programming, how a computer works etc). A certain degree of mathematical maturity and interest in scientific topics will be helpful.

In this first lesson, we are going to set up the Java programming environment for running our first program. To run Java programs in your PC, you need to install the Java Development Kit (JDK) and a text editor. Any simple text editor will do. Here, we will be using the Atom text editor - which is specially designed for programmers.

You can find the JDK from the Oracle site (or via Google search). Note that you must install the JDK, not JRE. The installation site will probably have both of them listed. Download the appropriate version of JDK for your system. If you are using a 32 bit machine download the x86 version, if your machine is 64 bit download the x64 version,

(The instructions in this post are going to be for a Windows PC. If you are using Linux or MAC, you will just need to do a little web search on setting up the JDK and Atom in your particular OS. It is not that hard.)

After downloading install the JDK. Next, download and install the Atom text editor from https://atom.io

To compile and run Java programs, we need to configure one more thing. Go to the Control Panel and search for environmental variables. Open Edit the system environmental variables and you will see this:

  
Now click on the Environment Variables button as shown above. There you will find a list of System variables. Select the variable name Path and click Edit...

In the box Variable value, type the location of your JDK bin folder (separated by a ";"). An example is shown in the image below: (note that your JDK version may be different)

Save the changes. Now, open Atom. Go to File > New File. You will see a space to write your code.

Our set up is done. You are ready to write your first Java program.