10 Paradoxes That Will Boggle Your Mind

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iStock

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?

20 Surprising Facts About Pulp Fiction

Miramax
Miramax

On October 14, 1994, Quentin Tarantino's Pulp Fiction was released in theaters in America and a new Hollywood auteur was born. In addition to teaching Americans what a Quarter Pounder with Cheese is called in Europe, the film reignited the career of John Travolta (who received a Best Actor Oscar nomination for his work) and showed audiences a different side of Bruce Willis. In honor of the film's anniversary, here are 20 things you might not have known about Pulp Fiction.

1. THE FILM WAS RELEASED IN SOUTH KOREA, JAPAN, AND EVEN SLOVAKIA BEFORE IT ARRIVED IN AMERICA.

Tarantino’s film first played the Cannes Film Festival in May 1994. It was shown at other festivals around the world, from Munich to Locarno, before hitting American shores on September 23, 1994, at the New York Film Festival. The film was released in South Korea, Japan, and Slovakia before it officially opened in the U.S. on October 14, 1994. The feature rolled out across Asia and Europe throughout 1994 and 1995.

2. HONEY BUNNY WAS NAMED AFTER AN ACTUAL RABBIT.

Honey Bunny belonged to Linda Chen, who typed up Tarantino's handwritten script for Pulp Fiction. In lieu of payment, she asked Tarantino to watch her rabbit when she went on location; Tarantino wouldn't do it, and when the rabbit later died, he named Amanda Plummer's character after Chen's pet.

3. YOU CAN WATCH THE FILM CHRONOLOGICALLY ... KIND OF.

The narrative structure of the film plays out of sequence, but it’s easy enough to break it down into seven distinct sections (a prologue, an epilogue, two preludes, and three large segments) that can then be re-ordered into a chronological narrative (Hint: The first prelude, to the “Gold Watch” section, plays first. If that doesn’t help, here’s an infographic).

4. THE FILM CONTAINS 265 “F WORDS.”

Even that hefty number isn’t Tarantino’s highest (1992’s Reservoir Dogs used it 269 times). Still, the film was the big “f word” winner of 1994, as no other film released that year even came close to that amount of profanity.

5. VINCENT VEGA’S 1964 CHEVELLE MALIBU WAS STOLEN AFTER THE SHOOT.

John Travolta’s character in the film had a sweet ride—which, in real life, belonged to Tarantino—and it was such a hot rod that it was stolen soon after the film’s release. It wasn’t found for nearly two decades, when two cops happened on a pair of kids stripping an older car. After running the Vehicle Identification Number, they found it shared the number with a car in Oakland, which turned out to be Tarantino’s car.

6. THE MOVIE COST ONLY $8.5 MILLION TO MAKE.

Five million went to the actors’ salaries. It made that all back in its first week at the U.S. box office (the film pulled in $9.3 million the first weekend of release).

7. THE FILM WAS THE THIRD BIGGEST R-RATED EARNER OF 1994.

The film lost out on the title to True Lies ($146.2 million) and Speed ($121.2 million). The film’s earnings were strong enough to place it in the overall top 10 for the year, though 1994 was dominated by Forrest Gump, which made $329.6 million that year.

8. EVEN THOUGH THE FILM MADE OVER $100 MILLION, IT TOOK A LONG TIME TO GET THERE.

Even though Tarantino’s film ended up being a tremendous hit—especially considering that slim budget—it took some time to get there. The film was in release for 178 days before it finally pulled in 100 million domestic dollars. A little comparison? It took Harry Potter and the Deathly Hallows – Part 2 only two days.

9. VINCENT VEGA WAS WRITTEN FOR MICHAEL MADSEN ...

Tarantino specifically wrote a number of roles in the film for chosen actors (including Samuel L. Jackson, Harvey Keitel, Tim Roth, and Amanda Plummer), but nothing compared to his dedication to having Michael Madsen play Vince. Madsen, who knew of Tarantino’s plans and said he wanted to do the part, dropped out two weeks before the script was finished to star in Wyatt Earp.

10. ... WHICH COULD HAVE MADE HIM MR. BLONDE’S TWIN.

Tarantino has a long tradition of connecting characters in his various films—basically, the filmmaker is working with a number of sprawling family trees, and it’s always a treat to see how characters intersect—which would have made Madsen’s casting of Vince come with a surprising twist: it might have made him Mr. Blonde’s (Madsen’s character from Reservoir Dogs) twin, as it’s long been known that Vince and Blonde are brothers.

11. IT INSPIRED TOP GEAR’S STIG.

The mysterious, anonymous Stig was inspired by the mysterious, anonymous Gimp. The Gimp was even the original name for the Stig, until they couldn’t find a racing driver willing to use that name.

12. BUTCH WAS SUPPOSED TO BE A LOT YOUNGER.

Tarantino wrote the part as a young boxer, with Matt Dillon specifically in mind for the role, but when the actor took too much time considering the part, it was tweaked slightly to accommodate Bruce Willis (who was a little ticked that he wasn’t asked to play Vincent).

13. TARANTINO LOVES VINTAGE BOARD GAMES, AND IT SHOWS.

The filmmaker is an avid board game collector, which is why the film features Operation and The Game of Life. Tarantino convinced Travolta to come on board with an all-day Welcome Back, Kotter, Grease, and Saturday Night Fever board game marathon.

14. VINCENT’S PREFERRED READING MATERIAL IS REAL.

Vince loves reading pulp fiction books during his, ahem, private time, including Peter O’Donnell’s Modesty Blaise, a real pulp fiction novel based on O’Donnell’s '60s comic strip. Tarantino has long expressed interest in bringing that tale to the big screen, including giving his official license to the 2003 film (Quentin Tarantino Presents) My Name is Modesty.

15. DESPITE TARANTINO’S LOVE FOR UMA THURMAN, SHE WASN’T HIS FIRST PICK.

Other possible Mias? Isabella Rossellini, Julia Louis-Dreyfus, Meg Ryan, Alfre Woodard, Halle Berry, Daryl Hannah, Rosanna Arquette, Joan Cusack, and Michelle Pfeiffer. Tarantino’s original favorite was supposedly Pfeiffer.

16. THE ORIGINAL POSTER CAN FETCH YOU SEVERAL HUNDRED DOLLARS.

The first poster had Thurman smoking from a box of Lucky Strike cigarettes—but Miramax hadn’t licensed usage rights from Lucky Strikes, who threatened to sue. Rather than fight it, Miramax had the posters returned. Those that survived can now command big money.

17. JULES MAY HAVE BEEN WRITTEN FOR SAMUEL L. JACKSON, BUT HE ALMOST LOST THE PART.

Tarantino very much had Jackson in mind for the role of Jules, but when he auditioned Paul Calderon, he was so struck by the performance that he very nearly hired him. Jackson, desperate to get “his” role back, flew to Los Angeles and auditioned for Tarantino again.

18. CAPTAIN KOONS MIGHT HAVE A FAMOUS RELATIVE.

Well, famous in the Tarantino universe, anyway: It’s widely believed that Christopher Walken’s Captain Koons is a descendent of Django Unchained character Crazy Craig Koons, who is only mentioned by name in a Wanted poster.

19. ROBERT RODRIGUEZ DIRECTED PARTS OF THE FILM.

When Tarantino is on screen as Jimmie, someone else had to be behind the camera—and that someone was Robert Rodriguez. The pair later teamed up for a number of other projects, including From Dusk Till Dawn and Grindhouse.

20. TRAVOLTA DIDN’T REALLY INJECT THURMAN IN THAT SCENE.

The infamous scene in which Mia is stabbed with a very necessary adrenaline shot was stressful enough, so Tarantino took off some of the pressure: the needle was inserted, and then Travolta pulled it out. The scene was reversed in post-production so it looks as if Vincent Vega really is plunging that syringe into her. Movie magic!

Additional Sources: Short List; Box Office Mojo (1, 2)

This article originally ran in 2015.

Aaugh! 10 Facts About It's the Great Pumpkin, Charlie Brown

Warner Home Video
Warner Home Video

Lee Mendelson hadn’t planned on a career in animation. But when television sponsors saw the filmmaker’s documentary about cartoonist Charles Schulz, they asked if the two could team up to produce a Christmas special based on Schulz’s Peanuts strip. The result, A Charlie Brown Christmas, was seen by roughly half of all households watching television during its premiere on CBS on December 9, 1965.

Mendelson went on to produce other Peanuts primetime specials, but 1966’s It’s the Great Pumpkin, Charlie Brown remains one of the most endearing. As you prepare annual sympathy for poor ol' Chuck (“I got a rock”), check out some facts about naked composers, vomiting voice actors, and CBS’s bizarre ultimatum.

1. THE FUTURE OF ANIMATED PEANUTS SPECIALS DEPENDED ON IT.  


Warner Home Video

Mendelson and animator Bill Melendez had very high aspirations for A Charlie Brown Christmas. When they screened it prior to its premiere, however, they felt it didn’t live up to its potential—and CBS agreed. The network said it was the last Peanuts special they would buy. But after it delivered huge ratings, CBS changed their mind and asked for more. When the two delivered another hit—the baseball-themed Charlie Brown All-Stars—they thought they had earned the network’s confidence.

Instead, CBS told them they needed a special that could run every year, like A Charlie Brown Christmas. If Mendelson couldn’t provide it, they told him they might not pick up an option for a fourth show. Despite Schulz and his collaborators being annoyed by the network's abrasive attitude, they hammered out a story with a seasonal clothesline that could be rerun in perpetuity.   

2. THE VOICE OF VIOLET PUKED AFTER EVERY RECORDING SESSION.

It’s standard practice these days to use adult actors to mimic juvenile cartoon characters: adults are (presumably) better able to take direction and deliver a performance in line with the director’s wishes. But for many Peanuts specials, children were used to voice Charlie Brown, Lucy, Linus, and the rest. Anne Altieri, who portrayed both Violet and Frieda, was so nervous to be part of the show that she threw up every time she was done with a recording session.

3. IT WAS THE FIRST TIME LUCY SNATCHED THE FOOTBALL FROM CHARLIE BROWN.

In animated form, anyway. When Schulz, Mendelson, and Melendez were brainstorming scene ideas for the special, talk turned to the fact that Lucy’s habit of pulling the football away from Charlie Brown had never been seen in animation. They also decided it would be a good time to introduce Snoopy’s World War I Flying Ace. The joke had appeared in the strip, but Mendelson thought it would work even better in motion. He was right: the sequence with Snoopy in a doghouse dogfight is one of the most memorable in the Peanuts animated canon.

4. IT’S SECRETLY ABOUT SANTA.

The Great Pumpkin saga was adapted from Schulz’s newspaper strip, where he had conceived it as a metaphor for some of the hope (and disappointment) associated with Saint Nick. Schulz disliked the idea kids heard of a jolly fat man who delivered presents all over the world when he knew many families could only afford one or two gifts for the holidays. “The Great Pumpkin is really kind of a satire on Santa Claus,” he told Mendelson. “When [he] doesn’t come, Linus is crushed.”

5. THE MUSIC COMPOSER WAS FOUND NAKED BY COPS.


Warner Home Video

The jazzy scores of the early Peanuts specials were the work of composer Vince Guaraldi. When he was busy putting together “The Great Pumpkin Waltz” for the show, he decided to break for a shower. When he came out, he thought he heard noises outside and went to investigate, naked, and locked himself out in the process. Keyless, Guaraldi tried climbing a ladder to a second-floor window when cops spotted him. “Don’t shoot,” he said. “I’m the Great Pumpkin.” Police, who were many months away from getting the joke, let him back inside.  

6. A LISP ALMOST RUINED THE SHOW.

Kathy Steinberg was only four years old when she portrayed Sally for the first time in A Charlie Brown Christmas: her big break came when Mendelson, her neighbor, started work on the specials. While Steinberg had some limitations—like being too young to know how to read a script—things were going well until producers realized she was on the verge of losing a tooth. Fearing a lisp would ruin the voiceover work, they rushed to get her lines done. The day after finishing, the tooth fell out.  

7. KIDS SENT CHARLIE BROWN CANDY FOR YEARS.

One of the most poignant moments of any Peanuts cartoon comes when downtrodden Charlie Brown opens his Halloween goodie sack and discovers he’s been given rocks instead of candy. According to Schulz, this so angered viewers that for years his California office was inundated with sacks of treats addressed to the character.

8. THE ORIGINAL AIRINGS WERE SLIGHTLY DIFFERENT.

Production costs for the early Charlie Brown specials were subsidized by television sponsors Coca-Cola and Dolly Madison snack cakes: the brands appear at the beginning and end of the broadcast. The Coke “bug” appeared for several years before getting phased out. 

9. CBS GOT A LITTLE SALTY ABOUT LOSING THE RIGHTS.

After spending decades at CBS, the rights to three holiday Peanuts installments went up for grabs in 2000. Though CBS could make the first offer, it was ABC who made the winning bid. Privately, CBS executives were not at all pleased about the business decision to take the football away. “It's a shame that a few more dollars meant more to them than years of tradition and loyalty," one network employee anonymously told Variety

10. SOME SCHOLARS THOUGHT THE GREAT PUMPKIN WAS REAL.


Warner Home Video

A real myth, at any rate. Talking to the Schenectady Gazette in 1968, Schulz said that since the special began airing two years earlier, he had received a number of letters from academics wondering where the Great Pumpkin story had originated. “A number of professional scholars have written me about the origination of the legend,” he said. “They insist it must be based on something.” Schulz suggested they broach the topic with Linus instead.

This article originally ran in 2015.

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