Paradox is defined by Wikipedia as follows:
A paradox is a statement that apparently contradicts itself and yet might be true (or wrong at the same time).
There are several types paradoxes. I will briefly describe each type of paradox with some examples for each one.
Visual Paradoxes:
These paradoxes are a bunch of figures/diagrams which show that our visual perception cannot always be trusted. These paradoxes usually involve some sort of visual trick which plays with our minds. For example,
[1] The Infinite staircase: The diagram shows a figure of a man who climbs a staircase for an infinite amount of time without finding an end for it.
The colored pieces of this puzzle can be rearranged to form two "13 by 5 right triangles" that have different surface areas. This is a visual paradox that can be explained mathematically.
Logical Paradoxes:
Logic is a powerful tool; it can be used to discern and to discover truth. Armed with the laws of logic and a few simple, plausible, and apparently harmless assumptions, philosophers can construct proofs of the most absurd conclusions. These proofs can give us pause; should we believe the unbelievable? This is the power of a paradox.
These paradoxes are largely constructed by some flawed logic like making references to infinity, dividing by zero or some seemingly logical argument.
1. The Zeno's Paradox:
The Paradox of Achilles and the Tortoise is one of a number of theoretical discussions of movement put forward by the Greek philosopher Zeno of Elea in the 5th century BC. It begins with the great hero Achilles challenging a tortoise to a footrace. To keep things fair, he agrees to give the tortoise a head start of, say, 500m. When the race begins, Achilles unsurprisingly starts running at a speed much faster than the tortoise, so that by the time he has reached the 500m mark, the tortoise has only walked 50m further than him. But by the time Achilles has reached the 550m mark, the tortoise has walked another 5m. And by the time he has reached the 555m mark, the tortoise has walked another 0.5m, then 0.25m, then 0.125m, and so on. This process continues again and again over an infinite series of smaller and smaller distances, withthe tortoise always moving forwards while Achilles always plays catch up.
Logically, this seems to prove that Achilles can never overtake the tortoise—whenever he reaches somewhere the tortoise has been, he will always have some distance still left to go no matter how small it might be. Except, of course, we know intuitively that he can overtake the tortoise. The trick here is not to think of Zeno’s Achilles Paradox in terms of distances and races, but rather as an example of how any finite value can always be divided an infinite number of times, no matter how small its divisions might become.
It is just like saying, " If you eat half of a pie, and then eat half of the remaining pie, and half of that remaining pie and so on, will the pie ever run out?"
2. The Liar Paradox (Eubulides or Epimenides Paradox)
This is a well known paradox written by the great stoical logician Chrysippos. The poet, grammarian and critic Philetas of Cos was said to have died of exhaustion attempting to resolve it.
A Cretan sails to Greece and says to some Greek men who are standing upon the shore: "All Cretans are liars." Did he speak the truth, or did he lie?
A week later, the Cretan sailed to Greece again and said: "All Cretans are liars and all I say is the truth." Although the Greeks on the shore weren't aware of what he had said the first time, they were truly puzzled.
If someone says "I always lie", are they telling the truth? Or are they lying?
3. The Barber's Paradox:
Suppose there is a town with just one male barber; and that every man in the town keeps himself clean-shaven: some by shaving themselves, some by attending the barber. It seems reasonable to imagine that the barber obeys the following rule: He shaves all and only those men in town who do not shave themselves.
Under this scenario, we can ask the following question: Does the barber shave himself?
Asking this, however, we discover that the situation presented is in fact impossible:
- If the barber does not shave himself, he must abide by the rule and shave himself.
- If he does shave himself, according to the rule he will not shave himself
4. The Grandfather Paradox:
This is one of the reasons why time travel is supposed to be impossible.
The grandfather paradox is a proposed paradox of time travel first described by the science fiction writer Nathaniel Schachner in his short storyAncestral Voices and by Rene Barjavel in his 1943 book Le Voyageur Imprudent (Future Times Three). The paradox is described as follows: the time traveler goes back in time and kills his grandfather before his grandfather meets his grandmother. As a result, the time traveler is never born. But, if he was never born, then he is unable to travel through time and kill his grandfather, which means the traveler would then be born after all, and so on.
Some one line Paradoxes to keep you thinking!
- Nobody goes to the restaurant because it is too crowded.
- Don't go near the water until you learn how to swim.
- Can a man drown in the fountain of eternal life?
- Your mission is not to accept the mission. Do you accept?
- If I ask you out on a date, would your answer be the same as your answer to this question?
- This sentence is false.
- If everything is possible, is it possible for something to be impossible?
- What happens if Pinocchio says "my nose will now grow"?
- If you make a new years resolution to not keep any new years resolutions would you keep it?
Consider the function f(x)=1x with x∈[1,+∞)
Rotating its graph around the x-axis we obtain a 3D horn-shaped object calledGabriel's horn.
Using calculus, we can calculate the volume and the surface of this solid of revolution and find something paradoxical:
The volume of the solid is:
V=limt→∞π∫t1(1x)2dx=limt→∞π(1−1t)=π
while its surface:
S=limt→∞2π∫t11x1+1x4−−−−−√>
>limt→∞2π∫t1(1x)=limt→∞2πlnt=+∞
Therefore, Gabriel's horn has a finite volume, while having an infinite surface.
More details can be found on the Wikipedia page Gabriel's Horn.
Rotating its graph around the x-axis we obtain a 3D horn-shaped object calledGabriel's horn.
Using calculus, we can calculate the volume and the surface of this solid of revolution and find something paradoxical:
The volume of the solid is:
while its surface:
Therefore, Gabriel's horn has a finite volume, while having an infinite surface.
More details can be found on the Wikipedia page Gabriel's Horn.
Unexpected Hanging paradox:
Self Reference Paradoxes:
These usually have to do with some self reference in set theory which is usually a flawed argument which seems completely legitimate when we first look and analyse it. They were extremely popular during the time when "The Foundations of Mathematics" were being laid out by mathematicians like Georg Cantor. This breed of paradoxes are usually called Russell's paradoxes.
[1] The Pinocchio Paradox:
It has already been stated in the answers on this page by Prasad Fadke. Taking it directly from there.
[2] Opposite day:
Mathematical Paradoxes:
These paradoxes are the ones which have been given to us by the world of mathematics. They are usually more difficult to comprehend and have a solid mathematical basis behind them.
These are the most interesting paradoxes as they are fully resolved without logical flaws and have a sound basis. They reveal astonishing results.
[1] The Birthday Paradox:
The birthday paradox concerns the probability that, in a set ofn randomly chosen people, some pair of them will have the same birthday. By thePigeonhole principle, the probability reaches 100% when the number of people reaches 367 (since there are 366 possible birthdays, including February 29). However, 99.9% probability is reached with just 70 people.
[2] The Monty Hall Problem:
[3] The Banach Tarski paradox:
The catch here is that the pieces themselves are not "solids" in the usual sense, but infinite scatterings of points.
Cosmological paradoxes:
These are based on space and our observations of the cosmos around us.
[1] The Olber's paradox:
Why is the night sky dark if there is an infinity of stars, covering every part of the celestial sphere?
[2] Fermi's Paradox:
Fermi's paradox is to do with the high probability of alien life versus our lack of contact with them.
For more details on the Fermi paradox visit this link: The Fermi Paradox - Wait But Why
There are plenty of other categories of paradoxes like
[1] Time paradoxes: If I go back to the past and kill my Grandfather, then will I be born in the present?
[2] Philosophy paradoxes: If god is simple then how did he create something as complex as the universe, if he is complex then we cannot understand him and hence cannot comment about him.
[3] Artificial intelligence: Will the AI destroy us for not believing that we can make them stronger than we currently are.
[4] History: We learn from history that we do not learn from history.
[5] Perception: Like this impossible cube.
Do visit List of paradoxes for more examples.
References:
A judge tells a condemned prisoner that he will be hanged at noon on one weekday in the following week but that the execution will be a surprise to the prisoner. He will not know the day of the hanging until the executioner knocks on his cell door at noon that day.
Having reflected on his sentence, the prisoner draws the conclusion that he will escape from the hanging. His reasoning is in several parts. He begins by concluding that the "surprise hanging" can't be on Friday, as if he hasn't been hanged by Thursday, there is only one day left - and so it won't be a surprise if he's hanged on Friday. Since the judge's sentence stipulated that the hanging would be a surprise to him, he concludes it cannot occur on Friday.
He then reasons that the surprise hanging cannot be on Thursday either, because Friday has already been eliminated and if he hasn't been hanged by Wednesday night, the hanging must occur on Thursday, making a Thursday hanging not a surprise either. By similar reasoning he concludes that the hanging can also not occur on Wednesday, Tuesday or Monday. Joyfully he retires to his cell confident that the hanging will not occur at all.
The next week, the executioner knocks on the prisoner's door at noon on Wednesday — which, despite all the above, was an utter surprise to him. Everything the judge said came true.
Self Reference Paradoxes:
These usually have to do with some self reference in set theory which is usually a flawed argument which seems completely legitimate when we first look and analyse it. They were extremely popular during the time when "The Foundations of Mathematics" were being laid out by mathematicians like Georg Cantor. This breed of paradoxes are usually called Russell's paradoxes.
Does the set of all those sets that do not contain themselves contain itself?
[1] The Pinocchio Paradox:
It has already been stated in the answers on this page by Prasad Fadke. Taking it directly from there.
For those who don't know about our dear boy, Pinocchio was a wooden puppet blessed with life. However he had one curse(or boon): His nose grew whenever he lied.
So the paradox is:
What will happen if Pinocchio says," My nose will grow".
If his nose grows, that means he must have lied, but that makes his statement a truth!
If his nose doesn't grow, that means he hasn't lied, but that makes his statement a lie!
[2] Opposite day:
"It is opposite day today." Therefore it is not opposite day, but if you say it is a normal day it would be considered a normal day.
Mathematical Paradoxes:
These paradoxes are the ones which have been given to us by the world of mathematics. They are usually more difficult to comprehend and have a solid mathematical basis behind them.
These are the most interesting paradoxes as they are fully resolved without logical flaws and have a sound basis. They reveal astonishing results.
[1] The Birthday Paradox:
The birthday paradox concerns the probability that, in a set of
Suppose you're on a game show, and you're given the choice of three doors: Behind one door is a car; behind the others, goats. You pick a door, say No. 1, and the host, who knows what's behind the doors, opens another door, say No. 3, which has a goat. He then says to you, "Do you want to pick door No. 2?" Is it to your advantage to switch your choice?
[3] The Banach Tarski paradox:
The Banach–Tarski paradox is a theorem in set-theoretic geometry, which states the following: Given a solid ball in 3‑dimensional space, there exists a decomposition of the ball into a finite number of disjoint subsets, which can then be put back together in a different way to yield two identical copies of the original ball. Indeed, the reassembly process involves only moving the pieces around and rotating them, without changing their shape.
The catch here is that the pieces themselves are not "solids" in the usual sense, but infinite scatterings of points.
Cosmological paradoxes:
These are based on space and our observations of the cosmos around us.
[1] The Olber's paradox:
Why is the night sky dark if there is an infinity of stars, covering every part of the celestial sphere?
[2] Fermi's Paradox:
Fermi's paradox is to do with the high probability of alien life versus our lack of contact with them.
The apparent size and age of the universe suggest that many technologically advanced extraterrestrial civilizations ought to exist. However, this hypothesis seems inconsistent with the lack of observational evidence to support it.
For more details on the Fermi paradox visit this link: The Fermi Paradox - Wait But Why
There are plenty of other categories of paradoxes like
[1] Time paradoxes: If I go back to the past and kill my Grandfather, then will I be born in the present?
[2] Philosophy paradoxes: If god is simple then how did he create something as complex as the universe, if he is complex then we cannot understand him and hence cannot comment about him.
[3] Artificial intelligence: Will the AI destroy us for not believing that we can make them stronger than we currently are.
[4] History: We learn from history that we do not learn from history.
[5] Perception: Like this impossible cube.
Do visit List of paradoxes for more examples.
References:
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