What is your favorite paradox?
A paradox is a statement or group of statements that leads to a contradiction or a situation which defies intuition. It can also be an apparent contradiction that actually expresses a non-dual truth. Typically, either the statements in question do not really imply the contradiction, the puzzling result is not really a contradiction, or the premises themselves are not all really true or cannot all be true together. The word paradox is often used interchangeably with contradiction. Often, mistakenly, it is used to describe situations that are ironic. The recognition of ambiguities, equivocations, and unstated assumptions underlying known paradoxes has led to significant advances in science, philosophy and mathematics. But many paradoxes do not yet have universally accepted resolutions. Sometimes the term paradox is used for situations that are merely surprising. The birthday paradox, for instance, is unexpected but perfectly logical:
The year has 365 days. In a room there are 57 students. I bet $100 that at least 2 of them have the same birthday. Do you take the bet?
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As for the bet, I’ll take it. I know for a fact that none of the students in class I am currently moderating have the same birthday. And the class size is about 70 strong. do you want to know where to send the money?
I will say this, though, in a weird twist, I have twins in my class that were born on differnt days
I don’t take the bet, because I know the odds of two people in a 57 member group having the same birthday is very high, in fact it’s close to certain.
If you simply do the math, it becomes apparent that when the group reaches a certain number of people, the odds are in favour of two people having the same birthday. In fact, this chart shows just that, and that at 23 people there is a 50% chance that two will have the same birthday. 57 students seems to have a 100% chance based on that graph. I’m sure you picked that number precisely to make many hundreds of dollars :)
My favorite paradox is the Monty Hall paradox. I really enjoyed exploring it on my own and using computer simulations to prove that switching doors is the best strategy overall for winning the prize.
Harry Mudd’s paradox.
I lie all the time. In fact I am lying now.
One obvious example would be: This statement is false.
Another one is the Russell’s Paradox which goes like this:
Take the set of all sets that do not belong to themselves.
i.e. define A = { B is a set such that B does not belong to B }
There are two possibilities. Either A belongs to A or it does not.
If A belongs to A it means A should have the condition that all elements of A have. Meaning that A does not belong to A, which is a contradiction.
If A does not belong to A, then it satisfies the condition that elements of A satisfy. Therefore it should be an element of A, which is again a contradiction.
Another paradox goes like this: Your teacher tells you he will give you an “unexpected” exam sometime next week from Monday until Friday.
This exam cannot be on Friday, because on Thursday night we will realize he is going to give us the exam tomorrow and that would not be an “unexpected” exam. So we have established that the exam cannot be on Friday.
The exam cannot be on Thursday. Because we already know the exam is not going to be on Friday. So on Wednesday night we realize the exam is going to be on Thursday. This would not be an “unexpected” exam, either! If you repeat the same logic you can deduce that the teacher cannot give you an “unexpected” exam anytime during the week! :D
Here is a couple more Paradoxes that I remembered.
By mathematical induction I show there is no bald person. If somebody has only 1 single hair on his head, you would certainly call him bald. This proves the basis of our induction. If you call every person with N number of hairs of his head “bald”, then by adding one single hair you would clearly still be calling that person “bald”. Hence any person with N+1 hairs is also be called bald. Therefore by mathematical induction there is no bald!
Now, I am using mathematical induction to show all horses have the same color. If we only have 1 horse, obviously it has the same color as itself. Lets say for any N horses that I pick, they have the same color. I need to show all N+1 horses have the same color. Lets say these horses are numbered by 1, 2,..., N+1. By induction hypothesis horses # 1,..., N are all of the same color. Now look at horses 2, ..., N, N+1, these are also N horses and by induction hypothesis they should have the same color. But then horse #N has the same color as horse #N+1 and horse #N has the same color as horse #1, throu N-1. This shows all horses #1 throu #N+1 have the same color. Hence by mathematical induction ALL horses have the same color!!
@vulcanjedi that sounds like a variation of Epidemenides Paradox. He was born on the island of Crete around 600BCE. He’s famous for stating “Cretans Always lie.”
My favorite paradox was on Bill and Ted’s Excellent adventure yeah,remember THAT movie? where they had to get into the police station, but didn’t have the keys.
They resolved to go back in time…later in the future…and hide the keys so that, in the present, the keys would just magically appear. So they made a conscious contract, and poof there were the keys.
“But how do we know we’ll do it, dude” ... “I dunno”... “But we DID do it, dude!” “EXCELLENT!!”
In improvisational music:
If you give away space, then you get plenty of space. If you take space, then there is no space.
There are lots of good examples from the Zen lexicon that are paradoxical on a semantic level, but that resolve when understood on the transcendent level that Zen points to. These are used as a way of breaking the mind out of its dualistic mode. Just a sampling:
“Form is only Emptiness, Emptiness only Form”
“True self is no-self, our own self is no-self”
“If you assert that things are real, you miss their true reality; but to assert that things are void also misses reality”
How about the one that god is both omnipotent and omniscient, but humans still have free will
@TheLoneMonk – If you take the bet, this means you think that most of the time all 57 students have different birthdays. Still interested?
@Harp – Great examples. Thanks for sharing this!
There are a list I enjoy, so I will try to say one not everyone will have heard the Banach–Tarski paradox.
I always liked Zeno’s paradoxes. The first one goes sort of like this:
You can’t get there from here. In order to get from here to there, you have to cover half the distance first. But you have to cover half the distance between here and the midpoint, too, and so on, and so on. Therefore, movement is impossible.
http://mathforum.org/isaac/problems/zeno1.html
@evelyns_pet_zebra
If I’m not mistaken, that’s not a paradox, rather a logical contradiction.
@BonusQuestion , The horse color argument is not a paradox but a fallacy. It you take N=2, the argument falls apart.
This is one of my favorites. What is the smallest number that requires 21 or more words to describe it? If such a number exists it could be described as “The smallest number that requires 21 or more words to describe it.” But this is less than 21 words.
A good practical paradox is the wave/particle nature of light.
I won’t take your bet. I’d bet on duplication in a room of about 35.
I have positively never thought of ranking paradoxes in terms of my affection for them. I’ll have to think about that and get back to you.
“The smallest positive integer not definable in under eleven words.” (haven’t I just defined it with ten?) This is the Berry paradox.
The wave/particle nature of light is not a paradox.
I like the grandfather paradox, i.e.
A person invented a time machine, then went back in time and killed his grandfather before he had children, which means he would not have been born, and could not have invented the time machine, and therefore could not have gone back and killed his grandfather.
@DrBill – Yes, it’s a classic one and I like it too. How do you deal with it? Resort to the many worlds interpretation?
@Myndecho – That one is new to me. I looked it up. A really tough one indeed!
@Jeruba – You’re strong in math :-) Actually, many people take the bet because 57 is so much smaller than 365.
@DrBill “The wave/particle nature of light is not a paradox”
If not, then at least I’m not alone in my error
@Harp
Although the wave/particle nature is unique to light, the existence of light in the observable universe make it logically plausible, and therefore anything that can be proven, by definition cannot be paradoxical in nature.
You did make some good points with the link, but the only ‘paradox’ was that they could not explain how light acts like a wave and a particle, but they all agreed it does.
@DrBill
It’s a paradox in this sense that the questioner mentions in his details above: “It can also be an apparent contradiction that actually expresses a non-dual truth”. Really the same sense that applies to the Zen paradoxes I mentioned earlier.
@Harp
Under that definition, I would concur.
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The Nothing Paradox: “What does nothing look like?”
@mattbrowne @DrBill I also like the time machine paradox. Matt, I like the “alternate time-stream” version of it, used very effectively in many sci-fi stories.
Also, some of the most interesting paradoxes occur when you try to merge logic with theology.
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