Infinity: Everything and Nothing
A number, a place or a finite within our minds? (Part 1)
“Two things are infinite: the universe and human stupidity; and I’m not sure about the universe,” Albert Einstein
As an infant, what was the most fascinating thing you came upon? When I was a kid, I used to love numbers (well, I still do). Back then, counting was a concept I freshly learnt. I used to count random stuff all the time. 1…2…3…4…5… and then I realize that this goes on forever…
What is this forever? Does this forever exist? Does this forever end? Keep these questions in the back of your mind. It’s time that we look for infinity!
So…where should we go looking?
Is infinity a number, a place, an idea or a concept? Some say it’s all of the above, while others think it doesn’t exist. Let’s start by assuming that infinity is a number.
1…2…3…Infinity
Numbers never end. That’s the basic idea of infinity. if you imagine continuing to count for as long as you can count, infinity will definitely last longer than your lifetime. The funny thing is, any number, no matter how big, is just absolutely insignificant compared to infinity. Counting is the earliest writing that we have. Whoever invented writing already had the idea of counting and the idea that you can always count one more. So does infinity exist in math? In math, we have an interesting relationship to things in existence. If you can conceive of something, if you can create rules for handling it, then it exists. So how does infinity fit into all that? It creates many paradoxes which summed up in a parable that goes by the name ‘The Infinite Hotel’.
A Trip to the Infinite Hotel
Imagine a hotel with infinitely many rooms. It’s a very popular hotel and every room is always occupied. Nevertheless, there’s always room at the infinite hotel. So one night, a guest shows up and needs a room. The manager tells the guest to hold on for a minute and tells everone over the loudspeaker, “Please gather your things. Get ready to move to the next room down the hall.” Person in room one moves to room two; person in room two moves to room three and so on. You might think there’s gonna be trouble because all the rooms are occupied. Nevertheless, as it’s an Infinite hotel, I mean this can happen, right? So everybody moves down to the next room, which means room one is open and the new guest can be accommodated. Let’s make this more challenging. The next night, instead of just one guest, an infinite number of people all pound on the bell at the same time saying, “We all want rooms!’ How is it possible to find room for infinitely many new people, given that infinitely many rooms are occupied? It seems that the manager has encountered this problem before and calls out over the loudspeaker, “Everyone to move to the room that is double your current room.” So person in room one goes to room two; person in room two goes to room four, and so on. What you’ll notice is that all the odd-numbered rooms have now been vacated and all those new guests can start filing in. The next time you don’t find room in a hotel, you know the right place to go to now!
So the moral of this crazy parable is that infinity doesn’t behave like anything we’re used to i.e. collections of a finite number of things. So it’s normal for us to not understand how infinite things work.
Infinity plus Infinity
Ok so infinity is the biggest possible number ever. Well what happens if you add one? It should still equal infinity. You’d hate me for this but now cancel infinity from both sides. You get 1=0. Terrible, right? Now try adding infinity to infinity. You still get infinity. Subtract infinity from both sides and you get infinity=0. That’s even worse! But it’s wonderful, because what’s going on? To try understanding this, we gotta change our perspective towards infinity. Now let’s imagine infinity to be a circle.
3.141592653589793238….
People often think about the most perfect shape as a circle- perfectly round, never ending. If we try to think about pointy shapes, say a triangle, if we try to use it as a wheel, it would kind of painful going up a street. If we had 20 corners, that wouldn’t be so bad as a wheel, and then if you keep going, you can go, “Oh wait! If we then had infinity sides, then we would be completely like a circle, and we would have infinite corners which is like having no corners.” And so, infinity has come round to being just like 0!
And this is why mathematicians needed to invent ways to deal with infinity and the ways that they came up with is the reason why you have to study calculus.
The Infinity Principle
There’s one idea at the heart of calculus, let’s name it the infinity principle. You can think of a curved shape as being made of an infinite number of infinitesimally small simpler motions or shapes. This is one of the greatest ideas in history. Everything to do with movement, with things that are constantly changing can be studied using calculus. So you can make sense of complicated things by breaking them into infinitely many simpler things. Solve the problem for the simpler things and just add them up and BOOM- you just used infinity to calculate a finite quantity!
To sum up…
The reason why you are able to read this article without your device just permanently switching off once its battery drains is calculus. Electricity has enabled the entire modern world.
So, to answer one of the questions…does infinity exist? Well, in one of the senses of math…absolutely no question about it! We have a symbol for it, we now know how to manipulate that symbol, we can all agree on the conclusions that we reach when doing so, and we can also solve practical questions. So, from a pure math point of view, infinity does have the certificate of existence
Hold up! If the math says that infinity exists, then that should mean that infinity is somewhere out there right? Does infinity physically exist? Is it a place? All of these questions to be answered in the next part.
Till then, stay curious!
