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

[jibber]

Originally Posted by Tuna

^ 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

the coastline itself isn't infinite, but the length is undefinable. Although maybe that's what you meant by infinite.

[Freebase Dali]

Originally Posted by jibber

the coastline itself isn't infinite, but the length is undefinable. Although maybe that's what you meant by infinite.

I think I see what you're saying, right? Zoom in to a yard of a particular portion of coastline... then down to an inch... now further to a centimeter... go further down to an atom of one particular grain of sand comprising the coast line. Keep going past the Neutron. Quarks. Further in, theoretical probabilities. Space between space between space. Theoretically, you could go forever into the space between space. It could never be accurately measured.

[jibber]

Originally Posted by Freebase Dali

I think I see what you're saying, right? Zoom in to a yard of a particular portion of coastline... then down to an inch... now further to a centimeter... go further down to an atom of one particular grain of sand comprising the coast line. Keep going past the Neutron. Quarks. Further in, theoretical probabilities. Space between space between space. Theoretically, you could go forever into the space between space. It could never be accurately measured.

Yeah that's what I'm saying. I agree that it could never be measured but I disagree with the term infinite because the coastline does end, but the EXACT measurement just isn't discernible by any means we have. But, if you use that logic for a coastline then really isn't the measurement of everything on the planet "infinite?" Replace 'every strand of hair and skin cell' with 'grain of sand' in that paragraph and you have an argument for why a person's height is "infinite".

[midnight rain]

Originally Posted by jibber

the coastline itself isn't infinite, but the length is undefinable. Although maybe that's what you meant by infinite.

In a way, though, it is infinite in length. I mean you can just keep getting more accurate and accurate with more precise measurements and that just expands the length of the coastline. This makes me question any distance ever measured and the concept of 'distance' as a whole.

[jibber]

Originally Posted by Tuna

In a way, though, it is infinite in length. I mean you can just keep getting more accurate and accurate with more precise measurements and that just expands the length of the coastline. This makes me question any distance ever measured and the concept of 'distance' as a whole.

Yeah well that's kind of what I said in my post above. The coastline is just an arbitrary example of an argument that says nothing can really be accurately measured. I just take issue with the word "infinite" to define it rather than unmeasurable. The actual measure of an object/thing/whatever may be infinite, but that doesn't necessarily mean that the thing itself is. It just means that we don't have the means to define that end.