I don’t know how much I’ll be doing these posts where I post a video and then discuss my feelings on it but I have a few lined up. For various reasons I will be unavailable for website updates (most likely) in the coming weeks but it doesn’t look like I updated all that much anyways. There are many reasons, some emotional, some physical, and other times just a matter of bad time management. It seems fitting when discussing my inability to divide up problems and projects over my limited time that I’d bring up Zeno.
The example given in the video is that if you clap your hands you are committing to a series of movements that involve the halving of the distance between your hands, then halving that distance again, and then again. He mentioned that it goes as follows.
You travel .5 of the distance, then .25 of the total distance, then .125, and so on. Now he shows how you can prove that it can be solved mathematically but then mentioned that an infinite process having an “end” was the paradox that has puzzled mathematicians for thousands of years. Here is my dilemma, I don’t see the Paradox and I’m fairly certain I’m not smarter than this man or anyone who tried this problem in 2500 years. I’m a random schmuck who is in love with Sleeping Dogs [a review to come], Magic Cards, and occasionally day dreaming in the shower. I barely scratched by on a General Studies Degree, etc, etc.
With all that personal deconstruction out of the way let me explain why I don’t see this as a paradox.
It begins with 0 and it ends with 1.
There are infinitely many points between 0 and 1. However we have established a beginning and an end to this infinite system. It begins at 0 and ends at 1. In fact, if you halve the total distance an infinitely many times and sum them you will come out with the answer of 1. I’m going to try and prove that with excel, because we all know that excel is the most accurate math program ever devised.
What I did was took the distance of 1. I then halved it, then halved it again, then halved it again, and repeated this process 1,048,575 times. I then summed this value and was given 1. Now obviously excel has cheated a bit and rounded because it didn’t want me tapping my foot impatiently as it carried the 2’s and 3’s. But this helps a little to get my point across. Hopefully the rest I can do with words.
The video mentions that while it took a second to reach that first half point, it takes half that time to reach the quarter point, and half of THAT time to reach the eighth point, and so on. What this means to me is that while there are infinitely many points each is infinitely faster than the previous point. If you take this entire group of infinite points and group them into a single value that value will end up being 1. While there are infinitely many pieces to this pie that is entirely feasible because in mathematics you can have a pool of infinite. Just like you can have different sized infinities (plenty of proofs of this online and even videos on numberfiles). The infinite group that makes up 0-1 is smaller than the infinite group that makes up say 0-2. Because the sum total of all the infinite parts of 0-2 is 2. Just as the sum total of infinite parts from 0-1 is 1.
Basically infinity for me is not a looped system. You aren’t doing the math one problem at a time for the whole of eternity without end. Any infinite system with a defined edge (Like the sum of all the halves between 0 and 1) looks to me like a finished product. Like a piece of folded paper that you’ve cut in many ways to be unfolded into a magnificent snowflake. While you can examine it for eternity it is already done.
In reality [that is to say real life] even the above might not be the case, we might not be able to infinitely reduce time and space. It is sounding like even string theory has hit the smallest that things get. Beautiful little vibrating strings with infinitely many outcomes. There is nothing weird about this to me, the idea that there could be a smallest thing is entirely understandable and not at all unusual. We only expect there to be smaller things because our macro world is made up of micro parts. There is nothing saying that a certain point parts couldn’t just be. Heck, that’s the entire spine to most god fearing religions.
If we assume that time and space cannot be divided infinitely many times then we hit the excel answer even harder. At a certain point those every halving points become equal to 0 and no longer impact the mathematics. Just like an infinite loop being closed before the computer crashes, just in this case it is the natural state of things. Water cannot be infinitely compressed and time/space cannot be infinitely divided.
The harmony in this to me is entirely sensible and that is my problem. None of this seems hard to handle or incorrect and that’s how I know I’ve got something wrong. I don’t know what, there must be something I’m missing. But for me this paradox is not a paradox at all, its fascinating and beautiful and fun but not paradoxical. For me it sounds a bit like those riddles that you hear that play on your ability to overlook an obvious hint. Of COURSE he was standing on a block of dried ice…how could I have been so silly.
So that’s my thoughts on this particular paradox. I can see no reason why it is a problem at all and I wish I knew why I couldn’t.