On Paradoxes

One time in middle school, I saw someone trade away their pencil for two sticks of chewing gum. One minute later, I saw that same person complain about how they had no pencil. <Insert trite but appropriate meme here>

This other time, I gave a beggar $5. Later that day, I found $20 on the street.

My point is, life is weird. And nothing better shows the weirdness of life than a paradox (well, two).

I recently stumbled on the Two Envelopes paradox, which basically says the following:
There are two unmarked, identical envelopes, each with some amount of money inside. One envelope has twice the amount of money than the other. You get to choose one of the envelopes and take all the money inside. Once you choose an envelope, before you open it, you get a chance to switch envelopes.

Here's the problem: Mathematically speaking, it is always in your best interest to switch envelopes. The logic is as follows:

Assume the envelope you chose has $100 in it. Then the other envelope either contains $50 or $200. There's a 50% chance of either one. You, being the math genius you are, know that your expected payoff is equal to (Probability1*Payoff1)+(Probability2*Payoff2). In other words, the expected payoff for switching envelopes is .5*50+.5*200 = 25+100 = $125. The envelope you're holding is worth $100 -- so it's in your interest to switch.

But... you can apply the same logic you just used, again. It's always mathematically beneficial to switch envelopes.

It gets weirder when you realize that the paradox is only possible because you have the opportunity to switch envelopes. If you didn't have this opportunity, then the probability of  you getting the envelope with more money in it is a straight up 50%. But when you're allowed to switch, you become trapped in an infinite cycle where it's always better for you to do so.

Another paradox I find to be interesting is the Sorities Paradox. (This one is named after a Greek philosopher, so you know it's a Big Deal.) It says the following:

1.) Fifty million grains of sand is clearly a heap of sand.
2.) If you remove a single grain of sand from this heap, it will still be a heap of sand.
3.) One grain of sand is not a heap of sand.

The trouble is that, given (1) and (2), you can keep removing single grains of sand from a heap of sand, forever and ever, and you'll always have a heap of sand -- so (3) can't possibly be true. But (3) is clearly true. So where does a "heap" of sand end?

Similarly, I can "prove" that there are no big numbers. We all know that 1 is a small number. If you take 1 and add 1 to it, we get 2, which is also clearly a small number. So if you add 1 to a small number, you get a small number. Repeat the process infinitely, and you see that there are no big numbers. But there are. 10^1000 is obviously an enormous number.

Weirder still, I can use this exact same logic to prove that there are no small numbers. If we start with 10^1000 and subtract one, it's the same thing, except reversed. How does this work?

Referring back to the original sand question, people have tried to claim that a "heap" of sand isn't actually a thing. Therefore, Premise 1 is flawed; therefore, everything about the paradox falls apart. In the same way, you could claim that "small" and "big" numbers are not real boundaries. I guess this explanation works, but I'm not satisfied with it. I mean, 1 is obviously a small number, and 10^1000 is obviously a big number... Right? It seems like a bit of a cop-out to me.

Anyway, the point of all this is that sometimes, we have to be happy with weirdness. We have to embrace it. Because if we don't recognize and come to terms with how strange the world is, we might just switch envelopes forever.

-Me