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Paradoxes

mendhak

Gone But Not Forgotten
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Dec 11, 2003
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Hello. :)

I'm looking to collect paradoxes for an article I will write soon, and I'm pretty sure you guys would have plenty to share. Basically, the paradox can be scientific or non-scientific in nature, no matter. To give you an idea of what I'm looking for, here's what I've got so far:

The Shampoo-User's Paradox
Ever notice those shampoo bottles that come with these instructions:
1) Wet hair
2) Massage shampoo into scalp
3) Leave for two minutes
4) Rinse
5) Repeat

This would finish the shampoo off in a single shower!!

The Quit-Smoking Paradox
(This one happens to me )
Often, some "How to quit smoking tips" will suggest that everytime you can successfully resist the urge to not smoke a cigarette, put some money in a little box. At the end of the month, take the money, and reward yourself.

I tried this, and the problem was that at the end, I would use the money to buy extra cigarettes, which would once again bring me to the same level of addiction I was in the first place.



The Compulsive Liar's Paradox
If a compulsive liar tells you he's lying, is he telling the truth?

The Gamblers Paradox:
a compulsive gambler would gamble so much, odds would have it he win a ocuple times, and most likely, one big thing, or a bunch of small things. the money won would go back into his gambling.



The Javascript Paradox #1
A user goes to a certain page, say index1.html, and it redirects him to index2.html. index2.html, in turn, redirects to index1.html. The user is stuck.


The Javascript Paradox #2
A user visits a certain page, index1.html. The scripting is written such as to launch a new browser with the same page upon exit of the first page.


the Arrow
An arrow in flight is really at rest. for at every point in its flight, the arrow must occupy a length of space exactly equal to its own length. after all, it cannot occupy a greater length, nor a lesser one. but the arrow cannot move within this length it occupies. it would need extra space in which to move, and it of course has none. so at every point in its flight, the arrow is at rest.and if it is at rest at every moment of its flight, then it follows that it is at rest during the entire flight.



Mailer Daemon Paradox
Two friends set their respective email accounts to "vacation mail," which would respond with an automated message to every message sent to them.

Then, one friend mails the other to say he's going on vacation. The email is responded to with a vacation mail, which in turn is responded to with a vacation mail. This continued, until either or both accounts fill up, after which Mailer Daemon's are exchanged, and responded to. (Not a practical situation, but could have been possible)

Heap of Sand
One grain of sand does not make a heap. Two grains of sand do not make a heap either. Therefore, three grains do not make a heap either. This can go on, until say, 10000 grains of sand, but that would still not make it a heap of sand.


God's Stone
Can god create a stone so heavy that even he cannot lift it?

Olber's Paradox
In a Newtonian Universe, the stars, being infinite in number, would be distributed randomly throughout space. This would logically imply that every direction we look, our line of sight would fall upon a star's surface. Stars that are farther away, would appear individually dimmer, but there are more of them. Nearby clusters of stars would contain several bright stars, and more distant clusters would have dimmer stars, and these would cancel each other out. Hence, the amount of light reaching us, on average, is the same in every direction. Although the light from these would be feeble compared to, say, a light bulb, but considering the Universe is infinite, all these would add up, and the sky would be as bright as the surface of a star.


So I'd appreciate anything you have to contribute. Already existing ones, ones you've made up, even links.

TIA.
 
Re The Arrow,
theres something called The Archers Paradox, when the arrow leaves the bow it bends around the riser (handle),then flexes back the other way, in all it oscilates quite a few times before it flys striaght, so it never flys "straight" to the target, but sort of wobbles its way there, which is why arrows have to be "spined" to the draw weight of the bow, if I try to use one of the wifes or kids arrows in my bow,.well stand very clear!.:D they are nowhere near "strong" enough for mine, (also possiblity of arrow shattering)
 
Michael Watson said:
Re The Arrow,
theres something called The Archers Paradox, when the arrow leaves the bow it bends around the riser (handle),then flexes back the other way, in all it oscilates quite a few times before it flys striaght, so it never flys "straight" to the target, but sort of wobbles its way there, which is why arrows have to be "spined" to the draw weight of the bow, if I try to use one of the wifes or kids arrows in my bow,.well stand very clear!.:D they are nowhere near "strong" enough for mine, (also possiblity of arrow shattering)

Hastings aye? Surely people from hastings would have learned by now that Arrow's are dangerous? http://www.battle1066.com/btpt5b.shtml
 
Fermi's paradox;
if aliens exist, they would be here.

(cue UFO believers saying 'but they are here' )
 
"Always" and "never" always render the statements in which they are contained untrue, and should therefore never be used.

By the same token, all generalities are false.
 
And remember, there's only one catch, and that's Catch-22...
There was only one catch and that was Catch-22, which specified that a concern for one's safety in the face of dangers that were real and immediate was the process of a rational mind. Orr was crazy and could be grounded. All he had to do was ask; and as soon as he did, he would no longer be crazy and would have to fly more missions. Orr would be crazy to fly more missions and sane if he didn't, but if he was sane he had to fly them. If he flew them he was crazy and didn't have to; but if he didn't want to he was sane and had to. Yossarian was moved very deeply by the absolute simplicity of this clause of Catch-22 and let out a respectful whistle.
"That's some catch, that Catch-22," he observed.
"It's the best there is," Doc Daneeka agreed.
 
The Shampoo-User's Paradox

"Always" and "never" always render the statements in which they are contained untrue, and should therefore never be used.

I'm always amused by the information on over the counter medicines that commands you to "always read the label".

"The Arrow" and the "Heap of Sand" are classical paradoxes - Zeno's and Sorites respectively, and have been curdling brains since ancient times.

Eubulides was another greek bloke who liked that kind of thing, his most famous being the Liar paradox where Epimenides, a Cretan, says that all Cretans are liars.

All the best more modern paradoxes seem to be mathematical equations and therefore baffle me long before the logic bomb explodes ;)

Although there is always Bertrand Russell's paradox
http://plato.stanford.edu/entries/russell-paradox/
 
A paradox is "An assertion that is essentially self-contradictory, though based on a valid deduction from acceptable premises", so I'm not sure that all of those are actually 'true' paradoxes in the logical sense, though some of them show paradoxical behaviour.

The only one that I'd say definitely isn't paradoxical at all is the shampoo-ing instructions, it's an iteration that presumes that the shampoo user knows when to stop.

There are the famous ones from sci-fi time travel, the grandfather paradox and the less common bootstrap paradox (eg a person or object appears for the first time in 2004 having come from 1800, then travels back to appear for the first time in 1800, then goes forward to where they first appeared in 2004, thus they/it exists but never starts or finishes anywhere).


Edit:ed for an excess of pedantry(!)
 
Does the set of sets that do not belong to themselves belong to itself.....

Russell's Paradox is lovely :)

Ex Falso Quod Libet (from the flase, anything follows) is a provable sequent in Classical Logic:

There are no shampoo bottles
so
If there is a shampoo bottle
Oxfordshire is leafy & quaint

or

-P : P --> Q

(may be a better formulation out there somewhere, but that captures the essense)

edit: searching dusty tomes, found a more conventional, but blindingly obvious formulation:

If their are no Goths but the girl at that table has intense black make-up and a Marlyn Manson crop-top
Then
I am the eggman, we are the eggmen

or

-P & P : Q

Relevance Logicians reject this, but they all live with their Mums ;)

Whip out those truth tables.... :)
 
Oh and before the joys of javascript you used to be able to get the same result by writing P.T.O. on both sides of a piece of paper :D
 
Q: What happens when the irresistible wind meets the immovable object?
 
Anyone mention the Racetrack (or Dichotomy) paradox.

One can never reach the end of a racecourse, for in order to do so one would first have to reach the halfway mark, then the halfway mark of the remaining half, then the halfway mark of the final fourth, then of the final eighth, and so on ad infinitum. Since this series of fractions is infinite, one can never hope to get through the entire length of the track (at least not in a finite time).

But things get even worse than this. Just as one cannot reach the end of the racecourse, one cannot even begin to run. For before one could reach the halfway point, one would have to reach the 1/4 mark, and before that the 1/8 mark, etc., etc. As there is no first point in this series, one can never really get started (this is known as the Reverse Dichotomy).

This one always screws my head up.
 
The Arrow (etc)

The arrow is not required to :
"move within this length it occupies"
It is required to displace the length it occupies one atom at a time (purely analogue).
The essence of movement is that an object enters a different portion of 3D space every instant that it can be said to be movement, not that an object must move within a specific zone identical to its own dimensions.
The basic premise of this one is flawed.

A compulsive liar could never tell you he was lying, he must always lie, and he must also say he is telling the truth.
If someone tells you they are lying, the can neither be a compulsive liar, nor a compulsive truth-teller, but (like most people) they must be between these extremes.
Is it impossible to discern a compulsive liar from a compulsive truth teller ?

The stars may be individually dimmer, and there may indeed be more of them, but the space between them is sufficiently massive to cancel any summation of their brilliance. If i was to take a torch and walk away from you until you could only just see the glow, then leave the torch at that point, and shine a second torch from a position as distant again, then you would simply not see the second torch, and the first would not appear brighter.
A photon has finite energy, and space, not being a perfect vacuum, robs photons of their energy as they travel through it.

A heap of sand is an undefined unit. You can get big heaps, and small heaps, therefore, as soon as you have enough sand so that one grain sits on top of another (theoretically possible with just two grains), then you have a heap of sand, just a very, very small one.

If there is a god, and he is all-powerful, then he can do anything, so he would be able to lift any stone he made. If his power is finite, then he may be able to make a stone he could not lift, just as a man could.
You might as well ask:

"What happens when an unstoppable force meets an unmovable object ?"

-from Iain Bank' "Walking on Glass" (?)
 
"What happens when an unstoppable force meets an unmovable object ?"

The answer is, you can only have one or the other - but not both. By definition, if there is an unstoppable force (or an "irresistable wind") then there can be no unmovable object. Or you may prefer the solution the other way round. Definition of terms is the clue to resolving most paradoxes. I think ;)
 
Are all paradoxes just tricks of language then?

Does this mean that, paradoxically, there are no paradoxes...?
 
Dr Poo, as the question implies both are in existence at once, the answer is taken to be "the immovable object moves, the force unstoppable force stops"
 
The majority of paradoxes are tricks of language rather than tricks of reality - very often they disappear like smoke in the wind when everyone is clear on what exactly they mean by their statements.

The javascript ones in the top post are just bugs rather than paradoxes- the infinite loop problem is a real donkey because it often doesn't explicitly fail, it just sits there eating resources until you kill it manually or your system runs out of memory.
 
Take the Father and Son paradox of time travel. The son lives in a time when time travel exists, gets in a time machine and goes back in time and kills his father before he ('the son') was born. So, in fact, the son would never be born in the first place.

Now, is this a paradox? Or just wild speculation on the nature of time travel? Of course we can't say because we don't understand time travel (at least not the backwards in time stuff)... once you understand all the components of a paradox -it think ceases to be one. :p
 
Racetrack Spoiler

Now i'm in the swing of things, I would like to talk about this one as well.
The statement is supposing that the 'horse' is a point mass that is moving logarithmically between two extremes referenced to a central ('halfway') point, when in fact this is not how a horse runs a race at all.
Firstly, the horse is not a point mass, it's about 2m long.
Secondly, and most importantly, the horse is not moving logarithmically about a midpoint, it is moving linearly through the range of the course. The start and finish are not extreme limits, but rather the horse moves through both.

It is true to say that a point mass moving exponentially towards an axis will never meet it, but the racing horse does not fit this scenario, therefore there is no paradox here.

Breakfast is absolutely correct, what we are discussing here are not true paradoxes, just word-tricks.
Dr Poo is on the right lines with the time travel thing, I think we should move towards cold, hard, physics to get some true paradoxes.

One idea is the existence of the soul.
If you had a teleporter that copied all your molecules, and made an exact replica of you in the next room, would you still be you ?
Would your 'soul' get copied?, Can it get copied ?
Would the person in the next room be a person at all ?
 
Oh, and by the way, Vacation emails only go out to each person once, so succesive emails sent after the initial auto-response remain un-answered.
 
This is kind of dumb, but on the elevator (lift) in my building, there is a sign that says "In case of fire, do not become alarmed. Press button marked Alarm and await help".
 
Fuzzy Logic has kind of resolved most of the so-called 'sorites' pardoxes (for instance, Xeno's arrow & Achilles and the Hare).

Kind of, in the sense that the transformation from one value to another has been very nicely modelled, but the attribution of values is still arbitrary.

Would love to share my thoughts on the implications unresolved paradoxes have for logic, but you'll all hit me with poo-smeared sticks. I'll save it for my forthcoming 'Alexius Abolishes Faith in Reason' thread. :)
 
Re: The Arrow (etc)

joester said:
The stars may be individually dimmer, and there may indeed be more of them, but the space between them is sufficiently massive to cancel any summation of their brilliance. If i was to take a torch and walk away from you until you could only just see the glow, then leave the torch at that point, and shine a second torch from a position as distant again, then you would simply not see the second torch, and the first would not appear brighter.
A photon has finite energy, and space, not being a perfect vacuum, robs photons of their energy as they travel through it.
Sorry, as an explanation for Olbers' Paradox that wouldn't work;

if the interstellar medium were to absorb the photons it would heat up; the interstellar medium would then radiate due to black-body radiation, which would mean the total energy that reaches the eye would stay the same in an infinite universe.

No, the real answer is that space is expanding, there fore we can only see a tiny portion of it; most of the universe is expanding away from us faster than light, so we can never see it.

Like Fermi's paradox, this paradox has a real answer; it is only a paradox until that answer is found.
Fermi's paradox still has no answer yet.
 
"which would mean the total energy that reaches the eye would stay the same in an infinite universe."

...but it would take an infinite amount of time
:)

In any case this is hardly a paradox, just an observation of what an idelaised infinite universe filled with stars might look like.
 
Well then, let me say in reply to Eburacum:

If you are suggesting that the energy transfer from photon, to matter, to heat, and back to light again, would be 100% efficient, and that no energy would be lost as either potential or kinetic energy, then yes, your argument works.
But energy transfer is never 100% efficient, therefore all of the light energy emitted by the supposedly infinite 'stack' of light sources would not reach us, as there would also be an infinite amount of loss to other forms/stores of energy.

I agree that the universe is probably finite, but I don't think this is the reason that space is not a infinitely bright light.
 
Looks like we still do not have any real paradoxes, just wordplay.

Zeno et al were dispatched (by calculus, I would say rather than fuzzy logic) long ago - simply because an infinite series can have a finite sum.

Maybe all paradoxes are simply a form of oxymoron - dark light, hot ice, square triangles, military intelligence - with no meaning in the real world.

Apparent paradoxes such as time travel are caused by our lack of knowledge about the field. Sometimes it takes an Einstein to explain why, if you hold a told in front of you and run forward, the light fromt he torch come out at the same speed as if you were standing still.
 
Well I just had a look in the OED.
I won't quote, but if you have a look for yourself you will see that Wembley is correct.
Paradoxes are either oxymorons, sound absurd when put a certain way (or put in a different context), or merely disagree with 'a preconceived notion' - I.e. we can't understand them because of a lack of knowledge.

So there may well be true paradoxes, things that are impossible, but still happen, and things that are certain, that don't happen.
But until we know absolutely everything, we won't know for sure if there are any true paradoxes.

Now if you'll excuse me, I must lather-up again (says so here).....
 
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