Why is motion logically impossible? Understanding Infinity Part #1

   Imagine this: You need to go from a place A to place B. To go from A point to B point, you must first travel half of the total distance. To travel the remaining distance, you must first travel half of what’s left over. No matter how small a distance is still left, you must travel half of it, and then half of what’s still remaining, and so on, infinitely. With an infinite number of steps required to get there, clearly you can never complete the journey. Thus logically making motion impossible, right?

  Well, this may sound stupid, and yes, even it sounded stupid to me at first! But to my surprise, and for sure, your surprise too, this illogical logical question is a very complex paradox in the history of physics and maths. It is known as the Zeno's Dichotomy Paradox. And believe me, this was not stated by any scientist of physicist, but by an ancient Greek philosopher Zeno of Elea, who is famous for inventing a number of paradoxes, arguments that seem logical, but whose conclusion is absurd or contradictory. 

  Zeno’s argument, at a cursory glance, might seem very silly, but this paradoxical argument has challenged some of the most respected and ancient ideas about space, time and more importantly motion. 

  Basically, if one were to add up to infinity, it would be a failure, as we would not be able to complete the task. None other than Aristotle goes on to say that "Not all infinites are the same. Some are actual infinites, while others are known as merely potential ones, which can go on as long as you like without any definite end-point. The easiest analogy for this is counting. A person can count as high as they want, but as soon as one feels that they have reached the highest number they possibly can reach, you can always add a one to it and make it bigger. That makes counting a potential infinity." What happens in Zeno’s paradox is similar to counting.

  Of the distance between A and B was to be 1 unit, you could reach that distance by the following series:

Series= ½+¼+⅛...

The value of this series will never become one, and that is exactly what we need to prove! So here's an explanation:

Series= ½+¼+⅛...

2(Series) = 1+½+¼+⅛...

(2(Series)- Series)=Series=1

  Well, although this sounds compelling at surface level, it is highly flawed. We all know that ½+¼+⅛... will never add upto one. After all 0.99... is still not 1. 

  Mathematically what tricks us the the infinite sum of ½+¼+⅛... if the distance between A and B is considered as one. This was the exact problem faced when more feasible methods like calculus were introduced in mathematics. Thus to close these infinity issues, limits were born.

  Let me make it more clear. Let's say there is a unit distance between A to B. Now I take 10 steps forward from A towards B. According to the paradox, the distance I covered should be as follows:

1/2 + 1/4 + 1/8 + 1/16 + 1/32 + 1/64 + 1/128 +1/256 + 1/512 + 1/1024 + 1/2048 = 2047/2048

As you can see, here, I reach very close to B but not at point B. And mathematically, I will never. To reach B, I must consider infinite amount of steps instead of 10 steps. This is where limits will be introduced. You see, at first I took 10 as the limit, and so I got a finite answer. Now as infinity is not a number, I must take limits approaching to infinity or better known limit n tending to infinity. Here, the value of 1/n will tend to zero. (Yes my college friends, it's the same limits you learnt!) Practically this will mean that closer I reach to B, the shorter the steps I will take and at a point I will take no steps since 1/n or the distance of my steps tends to 0. 

  Hey, I know it's hard to get away with disappointment when we don't have an satisfactory answer to a question. But, this paradox includes infinity. It's a hard topic, that's why I made a whole series to understand it properly. For now we learnt why are limits important and how crazy Zeno was, but we didn't find any solution to the paradox, well that's why it's called a paradox right?

  And I don't want you to walk away with disappointment. See, the mathematical line of reasoning is not good enough to show that motion is possible or impossible. As the paradox isn’t simply about dividing a finite thing up into an infinite number of parts, but rather also about the inherently physical concept of a rate, so we need to call another stream of research.

  Well with the help of physics, we won't be able to understand infinity, but you may get a solution for the paradox. As we all know that distance=(velocity)x(time). Now here, velocity is not finite always, but because velocity doesn't change unless an external unbalanced force acts on the body, we are able to reach B.

  In more simpler words, physically, according to Newton's first law, let's keep velocity constant. Now the point to understand here is that for each step, as the distance reduces by half by its previous step, so does time. If I cover 1/2 distance is step 1, I required 1/2 units time to cover that. And so on for the rest.

  So conclusively, motion is possible due to the relationship between distance, velocity and time. Yes, in order to cover the full distance from one location to another, you have to first cover half that distance, then half the remaining distance, then half of what’s left, etc. But the time it takes to do so also halves, and so motion over a finite distance always takes only a finite amount of time for any object in motion.

  And hey, if you didn't understand the whole thing, its okay! Infinity is a difficult concept. Even mathematicians and physicists have problem with it. So just sit back and wait for upcoming posts in this series to understand infinity better!

  By the way, I would love to hear your thoughts about Zeno's crazy paradox. Don't forget to mention them in the comments!

Posts before October 2021 have been marked as "Old Posts". Less likely, but they might have out dated or incorrect information, ugly looking bits of code, no labels, etc. Don't get me wrong, many of these posts are top-notch and interesting too.

I thought it would be better not to delete or revamp these posts, even if they suck. The bitter truth is that old works always suck, but I take that as a positive tool to convey that I am growing. Besides there's no better way to showcase my journey without these old, messy, poorly written posts!

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