Paradox
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“”A paradox, a most ingenious paradox! We’ve quips and quibbles heard in flocks, but none to beat this paradox!
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| —W. S. Gilbert, The Pirates of Penzance |
A paradox is
- A statement or argument which leads to a contradiction in its own foundation or premise, such as "The next sentence is false. The previous sentence is true."
- A statement that, while true, defies common knowledge by leaving out important details. For example "Alex was born February 29th, 1984. He has had 4 birthdays."
- Sometimes used to refer to a linguistic fallacy.
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[edit] Famous or interesting paradoxes
- The grandfather paradox — If you go back in time and kill your grandfather (particularly before your own father was born), your father would never be born and you would never be born. Therefore you couldn't go back in time to kill your grandfather. A generalised version of this is any situation whereby going back in time prevents you from going back in time originally[1].
- The omnipotence paradox — Asserts that a truly omnipotent being (e.g. God) cannot exist, since some of the actions it could perform would challenge its own omnipotence - e.g. creating a stone so heavy that it could not lift it, creating a more powerful being than itself, destroying itself, etc.
- Zeno's Paradox — Zeno of Elea made three bizarre paradoxes that could be best described as taking mathematics far too literally. In his most famous one, a faster object (such as a hare) cannot overtake a slower object (such as a tortoise) providing the slower one has a head start. This is because by the time the faster object catches up, the slower one will have moved on to a new location. When it has caught up again, the slower object would have again moved (although less than before). This continues to infinity, proving that motion is impossible. Zeno, however, didn't account for the fact that the time interval between successive catchings-up gets shorter and shorter. The two rates cancel each other out, and as a result, the world functions as observed. This is also an example of a 'false' description. In the paradox, static measurements between the hare and tortoise are taken at times. However the time and distance measurements don't describe the system; both the hare and tortoise are moving.
- The Liar's Paradox — Exist two forms : the first, said by a Cretan is "All the Cretan are liars" and is usually solved by saying that liar can say truth, or more easily by saying that the speaker lie (the ancient Greeks viewed this statement as paradoxical, but it is just false). The other form is : "this phrase is false" (or "Next phrase is true. Last phrase is false."), which is trickier to manage.
- The Unexpected Fire Drill -- A fire drill that by definition can never happen, because it is announced as "A surprise fire drill will be called next week, but we aren't saying on what day." It obviously cannot happen on the Friday, because if Friday dawns and there's been no fire drill, it won't be a surprise. So that's Friday out. If Thursday dawns and there's been no fire drill, it lacks any element of surprise because we already know it can't be on Friday. And so on.[2]
- Russell's paradox: The set of all sets which are not members of themselves. Is this set a member of itself? If it is, then it is not. If it is not, then it is.
- The St. Petersburg Paradox: I am running a betting game where I toss a fair coin. If it comes down heads, you receive triple your stake and I invite you to play again. If it comes down tails, you lose. Simple probability says that you get back, on average, $1.50 for each $1 you bet. A naive strategy to maximise your winnings would mean that you would never stop playing, betting everything you have at each step. The paradox arises because this behaviour is certain to lose you all the money you bet - the probability of getting tails at some point in the infinite game is 1. That this is one of the simpler paradoxical results in probability theory says a lot about statistics. And statisticians.
- The Barber's Paradox: Suppose there is a town with just one barber, who is male. In this town, every man keeps himself clean-shaven, and he does so by doing exactly one of two things: shaving himself; or being shaved by the barber.
Also, "The barber is a man in town who shaves all those, and only those, men in town who do not shave themselves." From this, asking the question "Who shaves the barber?" results in a paradox because according to the statement above, he can either shave himself, or go to the barber (which happens to be himself). However, neither of these possibilities are valid: they both result in the barber shaving himself, but he cannot do this because he shaves only those men "who do not shave themselves".
[edit] Impossible objects
One kind of paradoxical concept is something called an "impossible object". Impossible objects are drawings which, due to their construction, could not possibly exist in the Euclidean 3-dimensional world without an amount of cheating. Impossible objects typically make sense when only a part of them is viewed, but together, its impossibility is apparent. Examples of the impossible object include:
- The Penrose triangle, or tribar which includes three rectangular bars arranged in a triangle, each at right angles to each other. Any single vertex when viewed looks like a right angle, but viewing all three together presents the most well-known of all the impossible figures.
- The poiuyt. If you only look at the top half, it appears to be a fork with two prongs. If you only look at the bottom half, it appears to be three columns. Viewed as a whole, it can't really exist.
[edit] Obligatory trivia section
Paradox was also the name of a database programme for Windows which was released by Borland. Icons used for the programme were often visual puns on the name, such as two ducks ('pair of ducks') or two wooden jetties ('pair of docks').
[edit] See also
[edit] External links
- Wikipedia has a list of paradoxes; some are intriguing but many are only described as paradoxes because they disobey common sense rather than actual reality, mathematics, or probability.
[edit] Footnotes
- ↑ That, or whatever you do while you travelled to the past has no consequences to whatever the place you came from. Science fiction writers typically handle this by arguing that at every moment, timestreams split infinitely according to, say, a given butterfly flapped its wings once, twice, or just glided for a few seconds longer. Then, the timestream in which you arrived in the past would instantly no longer be the timestream which led to your birth, the development of time travel, and your decision to become a time traveller rather than, say, a Taoist monk.
- ↑ This is a lighter version of the Unexpected hanging paradox [1]