10 Paradoxes That Will Boggle Your Mind

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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.

1. 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 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, 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 BOOTSTRAP PARADOX

The Bootstrap Paradox is a 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 Travellers—involves an author and his manuscript.

Imagine that a time traveller 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 traveller finds it, buys it, and takes it back to Shakespeare. Who, then, wrote Hamlet?

3. 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.

4. 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.

Invented by the British logician Philip Jourdain in the early 1900s, the Card Paradox is a simple variation of what is known as a “liar paradox,” in which assigning truth values to statements that purport to be either true or false produces a contradiction. An even more complicated variation of a liar paradox is the next entry on our list.

5. THE CROCODILE PARADOX

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 is no problem if the mother guesses that the crocodile will return him—if she is right, he is returned; if she is wrong, the crocodile keeps him. If she answers that the crocodile will not return him, however, we end up with a paradox: if she is 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 is 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, while "crocodility" is an equally ancient word for captious or fallacious reasoning

6. 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 sixteenth of the way there, and then a thirty-second of the way there, a sixty-fourth 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.

7. 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.

8. 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.

9. THE POTATO PARADOX

Imagine that a farmer has a sack containing 100 lbs of potatoes. The potatoes, he discovers, are comprised of 99% water and 1% 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%. But when he returns to them the day after, he finds his 100 lb sack now weighs just 50 lbs. How can this be true? Well, if 99% of 100 lbs of potatoes is water then the water must weigh 99 lbs. The 1% of solids must ultimately weigh just 1 lb, giving a ratio of solids to liquids of 1:99. But if the potatoes are allowed to dehydrate to 98% water, the solids must now account for 2% of the weight—a ratio of 2:98, or 1:49—even though the solids must still only weigh 1lb. The water, ultimately, must now weigh 49lb, giving a total weight of 50lbs despite just a 1% 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.

10. 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?

11 Masks That Will Keep You Safe and Stylish

Design Safe/Designer Face Covers/Its All Goods
Design Safe/Designer Face Covers/Its All Goods

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15 Facts About Babe On Its 25th Anniversary

James Cromwell in Babe (1995).
James Cromwell in Babe (1995).
Universal Pictures Home Entertainment

It's hard to believe that it has been 25 years since a tiny pink piglet named Babe stole the heart of audiences around the world, and turned many of them into lifelong vegetarians (more on that later). What’s almost even harder to believe is that the heartwarming story of a pig who wants to be a sheepdog was partially ushered into existence by George Miller, the same man who brought us the Mad Max franchise. Here are 15 things you might not know about the little piggy that could.

1. James Cromwell thought the original idea for Babe was silly.

When actor James Cromwell first heard about Babe, which is based on Dick King-Smith's novel, “I thought it sounded silly,” he told Vegetarian Times. “I was mostly counting the lines to see how much of a role the farmer had.”

2. Farmer Hoggett has just 16 lines in Babe.

But by that point, Cromwell was already sold on the script, intrigued by what he called the “sophisticated yet pure-of-heart piglet.” And he clearly made the right call: The part earned Cromwell an Oscar nomination for Best Supporting Actor.

3. It took 48 different pigs to play the role of Babe.

Because pigs grow quickly, the crew utilized four dozen Large White Yorkshire piglets throughout the course of filming, shooting six at a time over a three-week period. A total of 48 pigs were filmed, though only 46 of them made it to the screen.

4. Babe also featured one animatronic pig.

Animal trainer Karl Lewis Miller seemed almost embarrassed to admit that they did have one animatronic pig play Babe, too. This is the pig they used for wide shots—when there was at least 15 feet surrounding Babe all the way around, and no place for Miller to hide.

5. Babe is a girl.

While this is never explicitly stated in the movie, because a male pig’s private parts would have been visible on film, all of the pigs used for filming were females.

6. In all, there were 970 animals on the set of Babe.

Universal Pictures Home Entertainment

In an interview with Entertainment Weekly, Karl Lewis Miller—who had 59 people assisting him—said that, all told, there were 970 animals used for the film, though only 500 of them actually made it into the movie. This included pigs and dogs, of course, plus cats, cows, horses, ducks, goats, mice, pigeons, and sheep, too. Baa-ram-ewe indeed!

7. Babe is also Dexter from Dexter's Laboratory.

In addition to voicing Babe, voice actor Christine Cavanaugh—who passed away in December 2014—lent her vocal chords to more than 75 projects over the years, including the title role in Dexter’s Laboratory, Chuckie Finster on Rugrats, and Gosalyn Mallard on Darkwing Duck.

8. Babe was banned in Malaysia.

Not wanting to upset its Muslim community, to whom pigs are haram, Malaysia banned the family flick from screening in its theaters. But its proscription didn’t stick; the film was released on VHS about a year later.

9. Pork product sales dropped in 1995.

Universal Pictures Home Entertainment

In December 1995, just four months after Babe hit theaters, Vegetarian Times ran a story about the problems facing the pork industry. Among the factors contributing to the industry’s slump, according to writer Amy O’Connor, was “the motion picture Babe, featuring an adorable porcine protagonist and a strong vegetarian message.” She went on to note that, “This year, the U.S. Department Agriculture showed stagnant demand for pork, while retail sales of canned meats such as Spam hit a five-year low.”

10. Sales of pet pigs increased following the release of Babe.

In The Apocalyptic Animal of Late Capitalism, author Laura Elaine Hudson is unable to substantiate claims that pork sales dropped a full 25 percent in the U.S. following the release of Babe, as some sources claimed, but she did find that sales of pet pigs increased—as did, eventually, the number of abandoned pigs.

11. Babe turned many viewers into vegetarians.

Babe’s popularity—and its main character’s adorableness—led to many fans of the movie (particularly young viewers) adopting a vegetarian lifestyle. The practice became so widespread that it was dubbed “The Babe Effect,” and fans of the film who went meatless became known as “Babe vegetarians.”

12. James Cromwell is a "Babe vegan."

Among those individuals whose eating habits were altered by Babe was the movie’s human star. Though he had been a vegetarian decades before, Cromwell “decided that to be able to speak about this [movie] with conviction, I needed to become a vegetarian again.”

13. Mrs. Hoggett was aged up for Babe.

Magda Szubanski stars in Babe (1995).Universal Pictures Home Entertainment

Magda Szubanski, who plays the farmer’s wife Esme, was only 34 years old at the time of the film’s release. She logged lots of time in the makeup chair in order to pass as the wife of her then-55-year-old co-star.

14. Jerry Goldsmith was hired to score Babe, but was replaced.

Jerry Goldsmith wrote a good deal of the music for Babe, but he and George Miller’s ideas for what it should sound like did not mesh. So Goldsmith was replaced by Nigel Westlake.

15. Babe earned a Best Picture Oscar nomination.

Among Babe's seven Academy Award nominations (yes, seven) was a nod for Best Picture, which pit the pig film against an impressive lineup that included Sense and Sensibility, Il Postino, Apollo 13, and Braveheart (which took home the award). The film did win one Oscar: it beat out Apollo 13 for Best Visual Effects.

This story has been updated for 2020.