Happy Birthday!

Another friend of mine have her birthday today!
Hooray for friends!

You won't see many of these posts. I have some friends and family that mean the world to me and some people I look up to, and I wish to commemorate all the laps around the sun that have elapsed since their separation with their mothers' anatomy; but their birthdays are all concentrated into a single short period of time!

So what did I decide to give her? Well, I am a big fan of "it's the thought that counts", because other than thinking there is not much that I can offer. With that in mind, I decided to give her what I believe she wants more than anything right now.
Are you ready?
Here it comes!

A SHORT RESPITE FROM ALL THE INSANITY IN THE WORLD!

This person is one of the most lucid people I know, yet she is frequently exposed night in and night out with shit that noone of her intellect should need to deal with.
Insanity in my mind originates from peoples' inability to make sense of the world, and that in turn can be attributed to the common belief there are things that cannot be made sense out of. After all, once you accept there are things that cannot be made sense out of, and you fail to immediately make sense out of something in turn, it is natural to think that is impossible to make sense of. In that spirit, I shall tackle some notorious paradoxes and expose them to the world!
All in her name ( of course, I don't want to give away her name in this blog ).


Zeno's Paradox

One day mighty Achilles decided to challenge the swiftest tortoise in the land to a race to the ocean and back! In his clemency, Achilles granted the tortoise a head-start of n seconds, so when Achilles started running, given the tortoise ran at a speed of m meters per second, he was already at a distance of m*n meters ahead! It took Achilles only p seconds to travel that distance, but on that time the tortoise had already run p*n meters! Achilles managed to overcome that distance even faster, but then the tortoise were already several nanometers away! At this rate, given that Achilles must first pass an infinite number of points before he can overtake the tortoise, how can he EVER hope to win the race?

I have a little trouble coming up with a sufficient rolution of this problem only because I never really understood wherein the problem lies, and thus what solution would satisfy it. Nevertheless it occurs to me that whenever the tortoise is ahead of Achilles, it will be because he hasn't surpassed the tortoise yet; he is not yet out of the temporal frame of reference wherein the tortoise is ahead, and that will never be subverted no matter how many fractions are added, because the fractions will always be too small to make a difference. Let me demonstrate this with a well known constant: e

In simple terms; e is the sum of the reciprocals of the factorials of all positive integers.
... that didn't turn out very simple, did it?
It's basically just an infinite addition of fractions, each fractions significantly lower than the last:
1 + ( 1/1 ) + ( 1/ ( 1*2 ) ) + ( 1/ ( 1*2*3 ) ) + ( 1/ ( 1*2*3*4 ) ).....
And so on. Essentially you add one more factor to the denominator every time, always divided into 1 and then added to the soup.
There is no limit to how many times you can add into this, and each addition comes closer to e. And yet, we know that e is not only less than 3, it's less than 2.7182818284590452353602877!
It's the same deal with this "paradox"; of course there are infinitely many infinitely fine instances where the tortoise is ahead; that's true for ALL moments! That doesn't mean that in all moments the tortoise is ahead.
Unless Achilles trips and cracks his skull, but as far as I am concerned that is not part of this paradox.


Proof the Easter Bunny Exists

 The way I see it, the existence of the Easter Bunny is predicated by the existence of an Existing Easter Bunny. So if we can find an Existing Easter Bunny, then that means the Easter Bunny MUST exist. Flawless reasoning! But doesn't that make the non-existence of the Easter Bunny quite absurd? Because there are exactly two cases; either the Existing Easter Bunny exists or she doesn't exist. But an Existing Easter Bunny that doesn't exist is clearly a logical contradiction! Since there are two cases; that the Easter Bunny exists or she doesn't exist, and as we know the latter to be necessarily false, we can simply deduce that the Easter Bunny MUST exist!

 So tell me then; what is the difference between an Existing Easter Bunny and an Easter Bunny? That an Existing Easter Bunny exists? How do we know the Easter Bunny exists? We don't!
 This is what we in the logic business call "circular reasoning":

If A then A

A

Therefore A

See, there are logical contradictions, meaning every possible conclusion is false, and then there is the exact opposite of that, a tautology, wherein all cases are necessarily true! A tautology is often considered worse than contradictions, because a contradiction at least, we now to be false. A tautology is worse than false, it is unknowable!
As soon as someone proposed the concept of an Existing Easter Bunny, they already accepted the existence of the Easter Bunny; so proving its existence after that is kind of moot.
The Easter Bunny herself have, of course no obligation to exist just because we say so!


Right Triangle Paradox

 Compare these triangles. Which one is bigger? Have you discovered they are equally big yet? Well, yeah, they are equally big. But they shouldn't be! Because the lower piece is clearly missing a square! I mean, whoa! Squares disappearing? In perfect geometry? How is that possible!? It can't be possible, but clearly it is!

The thing about this paradox is that it is actually two paradoxes in one hoax.
First, that adding a square-unit to an object doesn't change its size, and second that rearranging the components of an object doesn't change its shape. In order to debunk this paradox we need to disprove both.
First things first, does the object actually maintain its size?

Why, look at that! Once the triangles overlap you'll see they are not as similar after all! That little beer-belly might not seem big, but I bet it just about corresponds to the area of one square!
But that drives us to the next paradox. If one trianle is bigger, you'd think the components would also be differently sized, but upon examining it I find that not to be the case. So is there a paradox after all?
Wait a minute! Triangles don't have bellies! And look:

No matter how hard I try I cannot make a straight line match the hypotenuse!
That's no triangle! It's an imposter!
And look! It turns out it was Adam from that one episode of Catfish!

Now it all makes sense!

In all seriousness, if it doesn't behave like a triangle it is in all likelihood not a triangle. The red and blue shapes were designed to be intercompatible while still appearing as parts of a triangle. If you were to try and do that with the real deals it probably wouldn't turn out that way. In short, it's an optic illusion.
What can I say?
Seeing is not believing. Imperical testing is.

And these are my three paradoxes; all of them resolved!
My friend! There might be moments out there when life seem to consist out of paradoxes. Everything moves too fast, and you feel as if you can't keep up. The paradoxes you face may be much more perplexing than the ones I faced today; but that just makes them more worth solving!
I believe that for all the lies, delusions and empty words circulating across the world, there is still truth and beauty; there is still sanity. And I hope you discover that some day!

Congratulations!