Let’s discuss Paradoxes
What is a paradox? There can be two ways people define it.
One, it is a seemingly absurd or contradictory statement which when investigated may prove to be well founded or true, or,
Two, a statement which, despite sound (or apparently sound) reasoning from acceptable premises, leads to a conclusion that seems logically unacceptable or self-contradictory.
People can have two types of feelings when they come across a paradox. Either they think it’s useless cause it just does not make any sense, or, they get positively fascinated by it.
So, in this article I am mainly going to talk about a few paradoxes that really bent my brain.
First, The Zeno’s Paradox.
As the name suggests, these paradoxes was thought about by, Greek philosopher, Zeno of Elea. There are about 9 paradoxes he talks about but, I would be only discussing one of them.
Clapping. A process we are all aware of when we want to appreciate someone. While clapping, say, you keep the left hand still and bring the right hand towards it. So, the right first moves half the distance, then the next quarter, then the next one eighth, then the next one sixteenth, then one by 32 and this goes on as the gap between the hands getting infinitely small but never reaching 0. Then, when do the hands finally meet and clap? We have all clapped, so we know that it happens eventually, but again, in this case, it is not seeming to be true. How can an infinite process end? We do not know it’s end point, so how can we know the end answer of the infinite process? This is the paradox. It’s not only you and me, but, many great philosophers, mathematicians and physicists struggled with this paradox. Surprisingly, mathematicians did find a solution to this. They said, let your hands be about 2 metres apart. So you first move 1 m, then 1/2m, then 1/4m and so on.
S = 1 + 1/2 + 1/4 + 1/8 + 1/16 + 1/32 + …… — (i)
Here’s the trick. You multiply both sides by 1/2.
1/2(S) = 1/2 + 1/4 + 1/8 + 1/16 + 1/32 + ……
Then if you subtract this equation from equation (i), you get,
1/2(S) = 1
Thus, S = 2.
This shows that the hand did move a total of 2 metres even though it’s an infinite process. An infinite sum IS measurable. For example, if you take a right angled triangle with it’s legs 1 unit each, it’s diagonal, according to Pythagoras, will be sqrt{2}. This is an infinitely going on decimal value (1.4142135623731…..) . But, we can still draw it.
Next, Russel’s paradox.
Russel stated this paradox, and also gave the barber’s paradox as an example to explain this better. So The barber’s paradox goes somewhat like this,
Suppose every man in a city shaves and that city has only one barber. The barber shaves only those who do not shave themselves. He does not shave those who shave themselves. But, what about the barber himself ? He is the barber, so, he should shave himself. But he is not supposed to shave himself because the barber does not shave those who can shave themselves. Boom, paradox.
Russell’s paradox was based on this, but a bit more deeper.
We know that a power set is a set that contains all the sets. So, does this set also contain itself? If it contains itself then that should again contain itself. But having a power set and then that again containing a power set, that again containing a power set, that again containing a power set, and so on, infinitely. But this is impossible to prove. On the other hand, If we say that the power set does not contain itself, then, it will contradict its own definition of ‘a set containing all the sets’ and it will be incomplete. This is the Paradox.
Next, let’s talk about Hilbert’s paradox. This paradox seriously seems impossible. This is the paradox of the infinite hotel.
This infinite hotel is about a hotel having infinite number of rooms. Say the entire hotel is full with one guest in each room. Then one guest comes there. Where will he go? Well since it’s an infinite hotel, there are never ending number of rooms. So every guest moves from, say, room number n to room number n+1. And the new guest moves into room 1. There are many situations that have been tested using this hotel. One of them being, what if infinite number of people come to live in the hotel? Well, then every guest in the hotel should move from room number n to room number 2n. On doing so, the new infinite number of guests can go to all odd numbered rooms cause it is believed that the number of even numbers is equal to the number of odd numbers. So, there is always room for more, no matter how many guests come in. The paradox is that even though the infinite hotel is full, it can still take in more guests. But a hotel that’s full cannot take in more guests. Thus, it’s a paradox.
I mentioned the Grandi series in my previous article, which is also a paradox. I’ve explained it’s contradictions in the article below.
But, paradoxes don’t only occur in math. They are present in many areas of study and fields. For example in Physics, we have many Temporal or time travel paradoxes. One very famous one being the Grandfather paradox.
This paradox says that if time travelling was possible and you travel back in time and accidentally kill one of your grandparents or, any ancestors who are directly linked to your birth, then your birth will no longer happen. You will cease to exist. But, if you aren’t born and don’t exist, then you would not be able to kill that ancestor. But then if the ancestor is not dead, that would mean that your birth did in fact happen. And then it loops back to saying that you could kill the ancestor — this is the paradox. It is known to be self-contradictory. The below image will hopefully clear the loop forming the paradox.
So, are there any interesting paradoxes you know that amazed you and completely captivated your mind?
