The Paradox Thread

[midnight rain]

Right, so this is a thread for people to post paradoxes or any math related problems (or any puzzles really) that they find takes your brain to a whole new level. In essence, this thread will likely bore the pants off the vast majority of MBers.

I thought I'd start with a relatively well known one (especially if you've seen 21), the Monty Hall problem:

Originally Posted by Wikipedia

Suppose you're on a game show and you're given the choice of three doors [and will win what is behind the chosen door]. Behind one door is a car; behind the others, goats [unwanted booby prizes]. The car and the goats were placed randomly behind the doors before the show. The rules of the game show are as follows: After you have chosen a door, the door remains closed for the time being. The game show host, Monty Hall, who knows what is behind the doors, now has to open one of the two remaining doors, and the door he opens must have a goat behind it. If both remaining doors have goats behind them, he chooses one [uniformly] at random. After Monty Hall opens a door with a goat, he will ask you to decide whether you want to stay with your first choice or to switch to the last remaining door. Imagine that you chose Door 1 and the host opens Door 3, which has a goat. He then asks you "Do you want to switch to Door Number 2?" Is it to your advantage to change your choice?

Solution: Monty Hall problem - Wikipedia, the free encyclopedia

[Seltzer]

The Monty Hall problem is a pretty cool one. I think the easiest way to understand it is by realising:

1.) After choosing Door A, the probability that the car is behind Door B or Door C combined is 2/3.
2.) After the host reveals that Door B or C has a goat behind it, the 2/3 probability which was previously distributed over 2 doors is now shifted to one door (B or C, whichever one the host didn't open). So if we switch to this door, there's a 2/3 prob that the car will be behind it, but only a 1/3 prob that the car is behind Door A which we originally chose.



My favourite is Russell's Paradox which goes like this:

We define set S as the set of all sets which don't contain themselves as members. The paradox arises when you consider whether S should contain itself. If S contains itself as a member, that's a direct contradiction of its definition. And if S doesn't contain itself, then it should by its definition as the set of all sets which don't contain themselves.

The common non-mathematical analogy is the Barber's Paradox where we have a barber who shaves everyone who does not shave themselves... this is right-track's ideal town where no-one has a beard The paradox of course presents itself when we consider whether the barber should shave himself.

[Freebase Dali]

I think the simplest, and probably the easiest to understand in its impossibility, is the following:


The sentence below this one is false.
The sentence above this one is true.


Simple, but quite a mindfuck if you really follow it around for long enough in that loop. The infinity of impossibility is really emphasized here in a very simple way.
Overall, I think that paradoxes underline our ability to comprehend that there are things we're not able to comprehend, but that we are also tenacious enough to try and solve the unsolvable anyway. That, in itself, is the real paradox.

Thinking about the prime mover is one of my favorites. "What created the thing that created everything?" The question itself, logically, is infinite and a paradox. If we are to logically assume that in order for something to exist, it must be made to exist by a previously existing force, then our entire understanding breaks down at a point if you're rewinding in causation. But all this assumes we're operating under a unified logic, which is mostly the case.
The intriguing question is whether the paradox is unsolvable, or if we're putting square pegs in a round hole because it's the only hole we see.

[Sljslj]

I consider myself a fairly intelligent person and seems to me the Monty Hall problem isn't solved in the way everyone says.
So, there's two left, right? One is the car, the other isn't, therefore, you've got a 50% chance of selecting the car. I would think the use of thirds would be eliminated (and replaced with use of halves) once there are only two options. Can someone explain to me why this is wrong?

EDIT: Nvm I get it.

[midnight rain]

^ yeah those two are both mind****s.

Another good one is the Coastline Paradox: it basically shows that a coastline of a landmass is infinite in length, as the lower the measurements you take to measure the coastline, the closer it gets to infinity.

Coastline paradox - Wikipedia, the free encyclopedia

[Sljslj]

Here's one that has caused many long conversations between friends and I (in fact, I'm only thinking of it now because I just had this conversation with someone):

How can the universe end, but how can it not? If it just ends at some point, then there's absolutely nothing beyond it? Makes no sense. I also don't see how it could on forever, meaning there's an infinite amount of matter. Makes no sense. Maybe it's just the square peg and round hole thing like Freebase was saying.

Sorry it's so hard to figure out what the **** I'm talking about, it's hard to word it in a non-confusing way.

[midnight rain]

Originally Posted by Sljslj

Here's one that has caused many long conversations between friends and I:

How can the universe end, but how can it not? If it just ends at some point, then there's absolutely nothing beyond it? Makes no sense. I also don't see how it could on forever, meaning there's an infinite amount of matter. Makes no sense. Maybe it's just the square peg and round hole thing like Freebase was saying.

Sorry it's so hard to figure out what the **** I'm talking about, it's hard to word it in a non-confusing way.

It's not infinite in that sense.

Think of the Big Bang. All matter started at one point and then exploded, pushing all matter outward. So the Universe is constantly expanding. What's weird is it's speeding up in it's expansion.

A true mind**** is that the Earth is the center of the observable Universe, as is every point in the Universe....

[Sljslj]

I'm sorry I have to use weird wording again, but...

Expanding into what? (rhetorical question) There has to a space beyond it for it to expand into.
Understand? Probably not. I tried to think of an analogy so you can better understand what I'm saying, but I failed.
Think if you have a balloon and... wait, no... so if you have this room and then... umm, FUCK! I'm sure I'll figure something out if I sleep on it.

[midnight rain]

Originally Posted by Sljslj

I'm sorry I have to use weird wording again, but...

Expanding into what? (rhetorical question) There has to a space beyond it for it to expand into.
Understand? Probably not. I tried to think of an analogy so you can better understand what I'm saying, but I failed.
Think if you have a balloon and... wait, no... so if you have this room and then... umm, FUCK! I'm sure I'll figure something out if I sleep on it.

Haha I get you. I think that's a question that transcends human knowledge (at least for the next thousand years or so).

Since I didn't know the answer to your question either, I Googled and this is the closest I got to an answer: Curious About Astronomy: What is the universe expanding into?

Not exactly satisfactory.

[Freebase Dali]

Originally Posted by Tuna

It's not infinite in that sense.

Think of the Big Bang. All matter started at one point and then exploded, pushing all matter outward. So the Universe is constantly expanding. What's weird is it's speeding up in it's expansion.

A true mind**** is that the Earth is the center of the Universe, as is every point in the Universe....

I like your brain. Let me play with it for a moment.

If you're to think of the big bang as an explosion, it doesn't make much sense if you think about it. Not so much because of the uniformity of temperature and all that, as that's been sorta explained by inflation theory, but in the actual manner it "banged". The distribution of energy, later to cool and form matter, later to be affected by gravity and form our universe, would have to have banged INTO something. Nothing isn't a viable option (bear with me here). But first think about the universe itself... We already know that empty space is not the absence of everything. It's filled with something. Science calls it dark matter at the moment, but it's just a name for something we can't see but logically know is there. That something NEEDS to be the fabric in which all matter exists... we can really just chalk existence up to "taking advantage of having something to float in" if you really think about it.
Something can't exist in nothing. Not if we're using science and logic. The fabric of nothing (or probably better termed, the fabric of what we can't see or measure at this time) had to come from somewhere. But where?
Was it always here? If so, what created it?

What if, during the singularity and resultant big bang, nothing was created. And by nothing, I mean the undefinable shit we're floating around in. But not created out of a necessity of being there, but pouring in through a leak from somewhere else? Another universe perhaps?
What if the singularity was actually a hole being opened from the layer between a universe and absolute nothingness and existence itself poured into and created a pocket universe that we now live in, like water being injected into a bowl of jello through a syringe...

I know this all sounds crazy, and people are probably wondering if I'm on acid or not, but I just wanted to throw some creative ideas out there to you, as they're all in realms of paradox-land, and see what you might toss back. I love that shit.