A paradox is a statement or problem that either appears to produce two entirely contradictory (yet possible) outcomes, or provides proof for something that goes against what we intuitively expect. Paradoxes have been a central part of philosophical thinking for centuries, and are always ready to challenge our interpretation of otherwise simple situations, turning what we might think to be true on its head and presenting us with provably plausible situations that are in fact just as provably impossible. Confused? You should be.
Table Of Contents
- 1. The Paradox of Achilles and the Tortoise
- 2. The Grandfather Paradox
- 3. The Bootstrap Paradox
- 4. The Ship of Theseus Paradox
- 5. and 6. The Sorites Paradox and The Horn Paradox
- 7. The Liar Paradox
- 8. The Pinocchio Paradox
- 9. The Card Paradox
- 10. The Crocodile Paradox
- 11. Newcomb’s Paradox
- 12. The Dichotomy Paradox
- 13. The Boy or Girl Paradox
- 14. The Fletcher’s Paradox
- 14. Galileo’s Paradox of the Infinite
- 15. The Potato Paradox
- 16. The Raven Paradox
- 17. The Penrose Triangle
- 19. Hilbert’s Paradox of the Grand Hotel
- 20. The Interesting Number Paradox
1. The Paradox of Achilles and the Tortoise
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 BCE. 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, 500 meters. 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 500-meter mark, the tortoise has only walked 50 meters further than him. But by the time Achilles has reached the 550-meter mark, the tortoise has walked another 5 meters. And by the time he has reached the 555-meter mark, the tortoise has walked another 0.5 meters, then 0.25 meters, then 0.125 meters, and so on. This process continues again and again over an infinite series of smaller and smaller distances, with the 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.
2. The Grandfather Paradox
All of us know that if you ever travel back in time, you should definitely not kill your own grandfather, lest you create some kind of temporal paradox-slash-rift in the space-time continuum. This problem, known as the Grandfather Paradox, presents the main problem of time travel: If you go back and prevent yourself from being born, how would you ever have been able to go back in time in the first place?
3. The Bootstrap Paradox
The Bootstrap Paradox is another paradox of time travel that questions how something that is taken from the future and placed in the past could ever come into being in the first place. It’s a common trope used by science fiction writers and has inspired plotlines in everything from Doctor Who to the Bill and Ted movies, but one of the most memorable and straightforward examples—by Professor David Toomey of the University of Massachusetts and used in his book The New Time Travelers—involves an author and his manuscript.
Imagine that a time traveler buys a copy of Hamlet from a bookstore, travels back in time to Elizabethan London, and hands the book to Shakespeare, who then copies it out and claims it as his own work. Over the centuries that follow, Hamlet is reprinted and reproduced countless times until finally a copy of it ends up back in the same original bookstore, where the time traveler finds it, buys it, and takes it back to Shakespeare. Who, then, wrote Hamlet?
4. The Ship of Theseus Paradox
Theseus was a mythical king and the hero of Athens. (He was the guy who slayed the Minotaur, amongst other feats.) He did a lot of sailing, and his famed ship was eventually kept in an Athenian harbor as a sort of memorial/museum piece. As time went on, the ship’s wood began to rot in various places. Those wooden pieces were replaced, one by one. As time went on, more pieces needed replacing. The process of replacing rotten planks with new ones continued, at least in modern versions of the paradox, until the entire ship was made up of new pieces of wood. This thought experiment asks the question: Is this completely refurbished vessel still the ship of Theseus?
Let’s take it a step further: What if someone else took all of the discarded, original pieces of wood and reassembled them into a ship. Would this object be Theseus’s ship? And if so, what do we make of the restored ship sitting in the harbor? Which is the original ship?
This paradox is all about the nature of identity over time, and has been the subject of philosophical discussions for thousands of years. It appears in other forms, such as the Question of the Grandfather’s Axe and Trigger’s Broom, both of which ask whether an object remains the same after all the aggregate parts have been replaced.
The idea even expands to questions of personal identity. If a person changes drastically over time, so much so that who they are no longer matches any part of who they once were, are they still the same person?
5. and 6. The Sorites Paradox and The Horn Paradox
Another paradox about the vague nature of identity is the Sorites Paradox. The premise is fairly simple. It generally involves a heap of sand. If you take away a single grain of sand from the heap, it’s still, almost certainly, a heap of sand. Now take away another grain. Still a heap. If we continue this enough times, eventually it will be down to one grain of sand, which is, almost certainly, not a heap anymore. When did the sand cease being a heap and start being something else?
The Sorites Paradox is all about the vagueness of language. Because the word heap doesn’t have a specific quantity assigned to it, the nature of a heap is subjective. It also leads to false premises. For example, if you try the paradox in reverse, you start with a single grain of sand, which is not a heap. Then, one could argue that one grain of sand plus another grain of sand is also not a heap. Then, two grains of sand plus another grain of sand is also not a heap. This continues until even the statement “a million grains of sand is not a heap” which, as we know, does not make sense.
The name of the paradox, Sorites, comes from the Greek word soros, which means “heap” or “pile.” It’s often attributed to Eubulides of Miletus, a logician from the 4th century BCE who was basically a paradox machine. Most of his paradoxes deal with semantic fallacies, like the Horn Paradox. If we accept the idea that “What you have not lost, you have,” then consider the fact that you have not lost your horns. Therefore, you must have horns. And yes, most of his paradoxes are just as infuriating.
7. The Liar Paradox
One of Eubulides of Miletus’s more famous paradoxes, the Liar Paradox, is still discussed today. It has a very simple premise but a very mind-boggling result. Here it is: This sentence is false.
Think about it for a moment. If the statement is true, then that means that the sentence is in fact false, as it claims. But that would then mean that the sentence is false. And if the sentence “this sentence is false” is false, then that means it’s true. But, if it’s true that it’s false, then—you get the picture. It goes on and on, forever.
8. The Pinocchio Paradox
The Liar’s Paradox has been discussed and adapted many times, eventually leading to the Pinocchio Paradox. It follows the same general structure, but with an added visual component. Imagine Pinocchio uttering the statement “My nose grows longer now.” If he’s telling the truth, then his nose should grow longer, like he said. But as we know, Pinocchio’s nose only grows if he’s telling a lie. Which means that if his nose did grow longer, then the statement would have been false. But if “my nose grows longer now” is false, then it should not have grown in the first place … Has your brain exploded yet?
This version of the paradox was created in 2001 by philosopher Peter Eldridge-Smith’s 11-year-old daughter. He summarized it neatly like this: "Pinocchio’s nose will grow if and only if it does not.”
9. The Card Paradox
Imagine you’re holding a postcard in your hand, on one side of which is written, “The statement on the other side of this card is true.” We’ll call that Statement A. Turn the card over, and the opposite side reads, “The statement on the other side of this card is false” (Statement B). Trying to assign any truth to either Statement A or B, however, leads to a paradox: If A is true then B must be as well, but for B to be true, A has to be false. Oppositely, if A is false then B must be false too, which must ultimately make A true. The Card Paradox is a simple variation on the Liar Paradox that was invented by the British logician Philip Jourdain in the early 1900s.
10. The Crocodile Paradox
Another variant of the Liar Paradox actually helped shape language in the 16th century. A crocodile snatches a young boy from a riverbank. His mother pleads with the crocodile to return him, to which the crocodile replies that he will only return the boy safely if the mother can guess correctly whether or not he will indeed return the boy. There’s no problem if the mother guesses that the crocodile will return him—if she’s right, he is returned; if she’s wrong, the crocodile keeps him.
If she answers that the crocodile will not return him, however, we end up with a paradox: If she’s right and the crocodile never intended to return her child, then the crocodile has to return him, but in doing so breaks his word and contradicts the mother’s answer. On the other hand, if she’s wrong and the crocodile actually did intend to return the boy, the crocodile must then keep him even though he intended not to, thereby also breaking his word.
The Crocodile Paradox is such an ancient and enduring logic problem that in the Middle Ages the word crocodilite came to be used to refer to any similarly brain-twisting dilemma where you admit something that is later used against you, and crocodility is an equally ancient word for captious or fallacious reasoning
11. Newcomb’s Paradox
Another place where having to make a choice pops up is Newcomb’s Paradox. Imagine that you walk into a room where there are two boxes. You can see that the first box contains $1000. But the second box is a mystery.
Before you came into the room, an omniscient entity made a prediction about the choice you will make. If it predicted that you’d take only the second box, that box would contain $1 million. But if it predicted that if you’d take both boxes, the second box would be empty, and you’d walk away with $1000 and two boxes.
So what to do? One side argues to take only the second box—this is an omniscient entity doing the predicting, after all. The other side would argue that the entity’s decision has already been made. Nothing you do now in that room will have any effect on the dollar values in the boxes, so might as well take the gamble. And people can be surprisingly split on what to do—in 2016, a nonscientific online poll by The Guardian—which called the paradox “one of philosophy’s most contentious conundrums”—found 53.5 percent chose just the second box and 46.5 percent chose both boxes.
12. The Dichotomy Paradox
Imagine that you’re about to set off walking down a street. To reach the other end, you’d first have to walk half way there. And to walk half way there, you’d first have to walk a quarter of the way there. And to walk a quarter of the way there, you’d first have to walk an eighth of the way there. And before that a 16th of the way there, and then a 32nd of the way there, a 64th of the way there, and so on.
Ultimately, in order to perform even the simplest of tasks like walking down a street, you’d have to perform an infinite number of smaller tasks—something that, by definition, is utterly impossible. Not only that, but no matter how small the first part of the journey is said to be, it can always be halved to create another task; the only way in which it cannot be halved would be to consider the first part of the journey to be of absolutely no distance whatsoever, and in order to complete the task of moving no distance whatsoever, you can’t even start your journey in the first place.
13. The Boy or Girl Paradox
Imagine that a family has two children, one of whom we know to be a boy. What, then, is the probability that the other child is a boy? The obvious answer is to say that the probability is 1/2—after all, the other child can only be either a boy or a girl, and the chances of a baby being born a boy or a girl are (essentially) equal. In a two-child family, however, there are actually four possible combinations of children: two boys (MM), two girls (FF), an older boy and a younger girl (MF), and an older girl and a younger boy (FM). We already know that one of the children is a boy, meaning we can eliminate the combination FF, but that leaves us with three equally possible combinations of children in which at least one is a boy—namely MM, MF, and FM. This means that the probability that the other child is a boy—MM—must be 1/3, not 1/2.
14. The Fletcher’s Paradox
Imagine a fletcher (i.e. an arrow-maker) has fired one of his arrows into the air. For the arrow to be considered to be moving, it has to be continually repositioning itself from the place where it is now to any place where it currently isn’t. The Fletcher’s Paradox, however, states that throughout its trajectory the arrow is actually not moving at all. At any given instant of no real duration (in other words, a snapshot in time) during its flight, the arrow cannot move to somewhere it isn’t because there isn’t time for it to do so. And it can’t move to where it is now, because it’s already there. So, for that instant in time, the arrow must be stationary. But because all time is comprised entirely of instants—in every one of which the arrow must also be stationary—then the arrow must in fact be stationary the entire time. Except, of course, it isn’t.
14. Galileo’s Paradox of the Infinite
In his final written work, Discourses and Mathematical Demonstrations Relating to Two New Sciences (1638), the legendary Italian polymath Galileo Galilei proposed a mathematical paradox based on the relationships between different sets of numbers. On the one hand, he proposed, there are square numbers—like 1, 4, 9, 16, 25, 36, and so on. On the other, there are those numbers that are not squares—like 2, 3, 5, 6, 7, 8, 10, and so on. Put these two groups together, and surely there have to be more numbers in general than there are just square numbers—or, to put it another way, the total number of square numbers must be less than the total number of square and non-square numbers together. However, because every positive number has to have a corresponding square and every square number has to have a positive number as its square root, there cannot possibly be more of one than the other.
Confused? You’re not the only one. In his discussion of his paradox, Galileo was left with no alternative than to conclude that numerical concepts like more, less, or fewer can only be applied to finite sets of numbers, and as there are an infinite number of square and non-square numbers, these concepts simply cannot be used in this context.
15. The Potato Paradox
Imagine that a farmer has a sack containing 100 pounds of potatoes. The potatoes, he discovers, are comprised of 99 percent water and 1 percent solids, so he leaves them in the heat of the sun for a day to let the amount of water in them reduce to 98 percent. But when he returns to them the day after, he finds his 100-pound sack now weighs just 50 pounds. How can this be true?
Well, if 99 percent of 100 pounds of potatoes is water then the water must weigh 99 pounds. The 1 percent of solids must ultimately weigh just 1 pound, giving a ratio of solids to liquids of 1:99. But if the potatoes are allowed to dehydrate to 98 percent water, the solids must now account for 2 percent of the weight—a ratio of 2:98, or 1:49—even though the solids must still only weigh 1 pound. The water, ultimately, must now weigh 49 pounds, giving a total weight of 50 pounds despite just a 1 percent reduction in water content. Or must it?
Although not a true paradox in the strictest sense, the counterintuitive Potato Paradox is a famous example of what is known as a veridical paradox, in which a basic theory is taken to a logical but apparently absurd conclusion.
16. The Raven Paradox
Also known as Hempel’s Paradox, for the German logician who proposed it in the mid-1940s, the Raven Paradox begins with the apparently straightforward and entirely true statement that “all ravens are black.” This is matched by a “logically contrapositive” (i.e. negative and contradictory) statement that “everything that is not black is not a raven”—which, despite seeming like a fairly unnecessary point to make, is also true given that we know “all ravens are black.” Hempel argues that whenever we see a black raven, this provides evidence to support the first statement. But by extension, whenever we see anything that is not black, like an apple, this too must be taken as evidence supporting the second statement—after all, an apple is not black, and nor is it a raven.
The paradox here is that Hempel has apparently proved that seeing an apple provides us with evidence, no matter how unrelated it may seem, that ravens are black. It’s the equivalent of saying that you live in New York is evidence that you don’t live in L.A., or that saying you are 30 years old is evidence that you are not 29. Just how much information can one statement actually imply anyway?
17. The Penrose Triangle
While most paradoxes are presented through a spoken or written philosophical prompt, some are visual in nature. Take, for example, the Penrose triangle. It’s an object that is described by one of its creators as “impossibility … in its purest form,” but you can build one and show it to people. Obviously it’s a trick of proportions and viewing angles, but even after you reveal the trick, people will still see it as an impossible triangle.
You might know variations of these “visual paradoxes” from their representations in the works of MC Escher, who is the poster child for mind-bending art. His Waterfall from 1961, for example, depicts an impossible object.
19. Hilbert’s Paradox of the Grand Hotel
Hilbert’s Paradox of the Grand Hotel is a famous thought experiment that is meant to show the counterintuitive nature of infinity. Imagine walking into a big, beautiful, hotel, looking for a room. How big? Infinitely big. This hotel has a countably infinite number of rooms. However, all the rooms are currently occupied by a countably infinite number of guests. (Countably infinite means you can one-to-one attach a natural number to everything in the set.) One might assume that the hotel would not be able to accommodate you, let alone more guests, but Hilbert’s paradox proves that this is not the case.
In order to accommodate you, the hotel could, hypothetically, move the guest in room one to room two. Simultaneously, the guest in room two could be moved to three, and so on, which would move every guest from their current room, x, to a new room, x+1. As there are infinite rooms, everyone would get a new room, and now, room one is totally vacant. Enjoy your stay.
What if we wanted to apply this idea to any number of finite guests? Let’s say 3000 people arrive and want rooms. No problem, just the repeat process but instead of x+1, simply do x+y—y, in this case, being 3000.
What if a countably infinite number of people line up behind you, each of which wants a room? There’s a solution to that, too. The pattern would now be 2x. Simply move the guest in room one to room two, the guest in room two to room four, and the guest in room three to room six, and so on. This would leave all the odd-numbered rooms open, so each new guest could take one of the newly vacated odd-numbered rooms and the previous patrons would all be moved to the next even room.
The basis of the Grand Hotel Paradox is the idea of counterintuitive results that are still provably true. In this example, the statements “there is a guest in every room” and “no more guests can be accommodated” are not the same thing because of the nature of infinity. In a normal set of numbers, such as the number of rooms in a normal hotel, the number of odd-numbered rooms would obviously be smaller than the total number of rooms. But in the case of infinity, this isn’t the case, as there are an infinite number of odd numbers, and an infinite number of total numbers.
This paradox was first introduced by philosopher David Hilbert in a 1924 lecture and has been used to demonstrate various principles of infinity ever since.
20. The Interesting Number Paradox
The interesting number paradox is debatably not a paradox at all, though it’s often called one. It basically goes to prove that all numbers are “interesting”—even the boring ones … which are actually interesting, of course, and not boring at all … because they’re boring.
Interesting, in this case, means it has something unique to it. For example, 1 is the first non-zero natural number; 2 is the smallest prime number; 3 is the first odd prime number. The list can go on and on, until you reach the first “uninteresting” number. It doesn’t have anything special or fascinating about it. But, being the first uninteresting number you stumbled upon, it is, in fact, unique, and therefore interesting.
This process can be repeated indefinitely, hypothetically. This idea was born out of a discussion between the mathematicians Srinivasa Ramanujan and G.H. Hardy. Hardy remarked that the number of the taxicab he had recently ridden in, 1729, was “rather a dull one.” Ramanujan responded that it actually was interesting, being the smallest number that is the sum of two cubes in two different ways.
This story combines a piece written in 2016 with a list adapted from an episode of The List Show on YouTube.