The Paradox Thread
 

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:

Solution: Monty Hall problem - Wikipedia, the free encyclopedia

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.

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.

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.

^ 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

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.

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....

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. :bringit:

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.

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.

Yeah I'm thinking I'm in over my head on this one, but lemme try and wrap my head around it. :thumb:

From how I always saw it, all the matter and everything that exists (dark matter included) was once packed tightly into a marble with a very high density, so I don't THINK that anything was created from the big bang, so much as spread out.

As for what dimension the actual big bang existed in, you got me on that one. Consult the link I posted above, because it's beyond my knowledge.

The whole pouring in from the universe is some crazy **** that's making my head spin. In other words, I love it! You talk about a layer of nothingness that the singularity had to go through. My question is, what's nothingness composed of? As far as the other Universes, where did they originate from? Where's the original Universe and how did it get it's start? Just some questions to throw back at ya and your theory. ;)

Which brings up the idea of black holes and their theorized portals to other parts of the Universe (or maybe other Universe's)?

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".

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.

Eventually they're gonna have to cancel Lets Make a Deal because too many clever assh*les are gonna end up winning the big prizes now that they figured this sh*t out.

I guess saying it "approaches infinity" may be a more appropriate term for it.

It's not even the defining the length of it matter. It's getting the precise measurements of it.

For example, a straight line's length CAN be defined.

A real jagged coastline, as is used in the coastline paradox, can't be because it's impossible to take in every little degree (and 1/8 of a degree, 1/16, etc.).

See where I'm going? Or do I just sound like an idiot? :p:

I agree, but where did the matter come from? It had to come from somewhere.

Think of nothingness as an infinitely large bowl of water, provided we're assuming the water isn't actually anything, for this mind experiment. We'll use the water as our version of absolute nothingness. Now, drop in some oil. This will be our space fabric, which contains whatever matter we put inside it, following the same rules as the fabric in which it exists. Since oil is not soluble in water, the oil exists as its own little drops in a medium of which it is both unaware, and unconcerned with, but that medium facilitates its existence during this given instance. All matter and items within these droplets of oil are strictly adhered to the properties and laws of the oil, and not the water facilitating it.

Now think of our universe. It is comprised of uncountable numbers of various formations comprised of matter, which we know as galaxies, stars, suns, planets, dust, energy, etc., and it's all existing in SOMETHING. A common medium. Something that can facilitate the validity of actually being there. "something to float in". But what's outside of it? Well, we call this water, nothing. Only, unlike the scenario I've used, actual nothing has no physical properties to restrain what's being put in it. There are no dimensional restraints, no viscosity levels... all that exists in it is the propensity for something to fill it.

Now where it really becomes mind-bending directly addresses your question, "what is nothing comprised of". Well, in short, it has to be... nothing. Nothing isn't comprised of anything. It's not a barrier. It absolutely can't be, because it would then be considered "something", and would have to be comprised of some type of matter. So if we assume that nothing is actually not comprised of anything, you come to the conclusion that there is nothing stopping our universe from expanding. What would be stopping it? If you put a large amount of "something" into "nothing", why would it NOT expand given the vast amounts of compressed "stuff"? Nothing is stopping it.

The critical part is that there can only be so much "stuff" in the first place, unless there's a steady leak... which we (assumingly) know there isn't. So you can conclude that there is a limited universe in which we live in. It "leaked in" or "exploded in" our canvas of nothingness, and expanded into the infinite void of propensity to be filled. Or you can just call it "space" in a non-traditional sense of the word... like pouring a gallon of paint onto a canvas the size of New York... At some point, the paint is going to stop spreading, because there isn't enough paint pressure left in the center of it to drive the outward force.

That makes logical sense to me, but the conflicting news is that science is actually witnessing the expansion accelerating. To me, that means one of two things... Either everything we assume about the creation of the universe is fundamentally wrong, or whatever leaked into our existence is still leaking. We can't create more energy than we already have. It's coming from somewhere other than what already exists for us.

Oh, and back to the topic of space and the Universe for a second. One theory that is recognized as a possible ending to the Universe is the "Big Crunch", which is basically the reverse of the big bang. Using this theory, one can postulate that the Universe has been on a never ending cycle of Big Bang/Crunches.



Also, random fun fact: some of what you hear in static is actually the echo of the big bang.

No I think we're actually saying the same thing. I was saying that by that argument nothing with mass can actually be defined by size. I used the term length because we were still talking about a coastline.

Also, to make another point, space (as we know it) is most definitely populated with a fabric of being. (Einstein called it Space-Time)
But what I think is important to notice about it is that light can travel through it, objects can travel through it, heat through it, radiation, gravity has effects through it... it's a medium through which something can travel and exist. Therefore, space itself is not nothing. We've known that for a long time, but you'd actually be surprised at how many people still think space is the absolute void of anything at all. And I think it's important to understand that the universe we live in and all the spaces in between are actually "something". And all that something had to come from somewhere.

The only problem I see with that theory is that you can measure the expansion of the universe, and using simple logic and knowledge of gravity, know that the further massive bodies are away from each other, the less gravitational force they exert on one another, so gravity couldn't be the cause of a crunch. I'm not aware of any other force that could cause a crunch, but I could be mistaken.

This is crazy to think about. To think that if time is infinite and will never end then maybe it never "began" and just always was.

http://i690.photobucket.com/albums/v...eadExplode.gif

Time won't exist unless there's something to move through it... Change rate gauges time. Time is just the movement between now and then. I don't think time is a separate, independent factor at all, like some cosmic clock that all realities are synched to.... I think it's just the progression of things changing. The perception of that change is merely dependent on the prospector.

Okay, I have a one. More a riddle/problem so to speak. It's so simple but it can fuck with your head. Especially for overthinkers.

There are 3 people. They have all rented a room at a hotel. The fee for the hotel room is $30. Therefore, they each contribute $10 to the cost of the hotel room. During their stay, their electricity is cut for a period time. To compensate for this, the hotel manager, takes $5 from the $30 dollars which he received and gives one his workers the $5 to pay the 3 people sharing the room. The employee thinking that $5 would be too cumbersome to split, decides to keep $2 for himself and gives each of the 3 people in the room $1. Now, since they received $1 dollar each, it is correct to say that they payed $9 dollars each for the room right? And the employee kept $2. Now doing the math, $9 x 3 = $27 +$2 = $29. Where did the other dollar go?

Ok I'm going to take a guess, and I may be completely wrong in doing so, so don't call me out as a total idiot if I'm wrong. Please? :shycouch:

Isn't the problem in the equation? In that it should be: $9 x 3 = $27 + $3 = $30 because the customers are getting $3 back, not $2?


So, the break down is:

The customers pay $27
.........$25 goes to the owner
..........$2 goes to the employee

The customers have $3 left over from the money they got back

Yeah, it's as simple as that. The equation in the problem is fallacious and adds in the employee's +$2 for no real reason.

The problem I think is in the wording of the riddle and how they phrase things to distort what the problem really is. This seems for the most part to be correct though.

Yea that's kind of like one of those riddles that starts with "Ok.. you're the bus driver....."

Cool thread. The impossibility of measuring, or infinity within, a coastline, or in this case a piece of string, is explained in this BBC documentary:

These aren't really paradoxes but are not worthy of their own thread, so I'm sticking them in here.

We don't really touch anything. Burn yourself on a hot coal, stroke a cat, cut your finger off with an axe, swim in the sea, you haven't touched any of those objects. What you really feel is electromagnetism.

Light is invisible. Shine a laser through a vacuum, it's invisible. All we can see are the things light strikes, not the light itself. Those battles in space full of laser beams in films like Star Wars and Star Trek, it wouldn't look like that.

Well this was an interesting bump.

I wish I was as inquisitive now as 3 years ago.

What killed your naturally inquisitive mind? The grind of modern life?

Gimme the answer for what does E equalls M C square? rest in peace, Albert Einstein.

If you're traveling the speed of light and you shine a flashlight ahead of you, what happens to that flashlights' beam of light?

Does light have the same principals of velocity that solid matter does?

It's a good question. My understanding is the flashlight wouldn't emit a beam of light and that neither photons nor particles of matter can exceed the speed of light. However, recent research seems to suggest that information (in the experiment in question, it's information shared between two entangled photons) can exceed the speed of light. And that begs the question what is the information their sharing made from?

Here's an article on it: http://en.wikipedia.org/wiki/Quantum_entanglement

If we could somehow understand it and employ it to move real life matter, interstellar travel may become a reality.

Well, thats the 2nd intelligent answer I've ever had after posing this question, and I've asked this for many years. I usually get the "Deer caught in headlights" look. The first was from a friend with a Phd in physics. However, it's my understanding that the beam of light would double in speed, but then again this really isn't a paradox is it.

I hate math.

I am driving my car at the speed of light and I turn on my headlights. What do I see?

Seems we're both wrong. The person travelling (close to) the speed of light would see a normal beam of light but it wouldn't exceed the speed of light.

I love science but I never really know what I'm talking about because all I can do is repeat second hand information and form a vague and shadowy understanding based on the experiences of others, never knowing whether what I heard or read is true or even likely to be true. But then again, I think everyday life is a bit like that too, whether we realise it or not.

Hey,

yeah personal stuff I'm facing. I've been too focused on internal crises (guess I'm still questioning things just on a smaller scale)

^ Good discussions above though, keep it up :)