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?

13 Great Rockumentaries Every Music (and Movie) Fan Should See

The Criterion Collection
The Criterion Collection

More people are watching documentaries these days, which likely means that more people are rocking their faces off with nonfiction. Far from Ken Burns’s soothing tones, these music-filled films demand amplification and an unseemly amount of perspiration.

Rock documentaries are tricky beasts. Though they often have the built-in advantage of following around famous people, they aren’t immune to boredom and eye-rolling faux depth. Keeping it simple by showcasing the music can be good, but it’s no way to be great. The best of the best manage to deliver a stellar soundscape, offer a backstage pass to the real humans who make it, and hold our ears even if we aren’t already devoted fans. If a little history gets made in the process, even better.

Grab a seat next to Penny Lane on the bus. Here are 13 of the best documentaries that every music—and film—fan should add to their Must Watch list.

1. WHAT’S HAPPENING! THE BEATLES IN THE U.S.A. (1964)

A singular piece of filmmaking where nonfiction talent met transcendent musical genius on the threshold of gargantuan stardom, this is the best Beatles documentary ever produced. Directed by legendary documentarians Albert and David Maysles, the film captures the band’s first frivolous jaunt through America, where they raised the screaming decibel level in The Ed Sullivan Show theater and goofed off in hotel rooms. It’s an explosion of youth before they changed music forever.

2. DON’T LOOK BACK (1967)

Another marriage of style, skill, and subject, Don't Look Back helped shape how the rockumentary genre could provide insights into the people who shape our popular culture. That so many iconic moments emerged from D.A. Pennebaker’s watershed work, which strolled with Bob Dylan through England in 1965, is a testament to the legendary musician's infinite magnetism. The cue cards, singing with Joan Baez in a hotel room on the edge of breaking up, the Mississippi voter registration rally, and on and on. Since it portrayed fame’s effect on the artist, the art, and the audience, most every other rock doc has been chasing its brilliance.

3. GIMME SHELTER (1970)

The rockumentary has evolved to be as diverse as the sonic landscape itself, which is why Popstar: Never Stop Never Stopping can send up the current scene just like This Is Spinal Tap! did in the 1980s. Still, 1970 feels like the year that defined the rockumentary. Another Maysles joint, this profound doc captured The Rolling Stones touring at a time when they were one of the biggest bands in the world and only getting bigger. The music is powerful and immediate, and the film closes with their appearance at the Altamont Free Concert, which turned deadly when—after a day of skirmishes between concertgoers and the Hell’s Angels acting as security—a fan with a gun was stabbed to death when he tried to get on stage during “Under My Thumb.”

4. WOODSTOCK (1970)

The other 1970 film that helped define the genre allowed thousands to claim they’d been to the biggest concert event of the generation without actually going. If rock ‘n’ roll emerged from unruly teenage years into conflicted young adulthood in the 1960s, nothing stamped that image in henna ink better than Woodstock and the documentary that accompanied it. The bands that appear are legendary: Crosby, Stills & Nash; The Who; Joe Cocker singing The Beatles; Janis Joplin; Jimi Hendrix; and many more. It’s a fly-by of the three days of peace and music that you could play on repeat with summery ease.

5. ZIGGY STARDUST AND THE SPIDERS FROM MARS (1973)

Rock doc royalty D.A. Pennebaker captured David Bowie’s final performance in his red-domed sci-fi persona at London's Hammersmith Odeon with a flair that captures the frenetic energy of the room. The crowd is as much a part of the moment as the band is, as the camera places you in the middle of a transitional moment in music history. To see Bowie that close up now is a wonder. And, naturally, the music is out of this world.

6. THE DECLINE OF WESTERN CIVILIZATION (1981)

Instead of following the famous, Penelope Spheeris’s debut dug its nails deep into the Los Angeles punk scene at the turn of the decade. Black Flag, The Circle Jerks, and other bands your parents have never heard of perform mosh pit-sparking anthems and show off their living conditions like a grungy proto-version of MTV Cribs. There’s a purity here missing from most music docs—a chronicle of people whose passion far, far outweighs their paychecks, and a screening that led the LAPD to request that the movie never be shown in LA again.

7. SIGN "☮" THE TIMES (1987)

Having Prince at the center of your concert doc is a shortcut to ensuring it’s one of the best of all time. There’s the music, of course. Hits like “Little Red Corvette” and “U Got the Look,” and Sheila E. beating the hell out of her drum kit. There’s also The Purple One's inexhaustible energy and stage presence. As a bonus, the film jumps between concert footage and (instead of candid hotel conversations) a sci-fi narrative where we get to go to Prince Planet. It’s a rocky, disorienting experience that could have only been held so tightly together by a master showman.

8. MADONNA: TRUTH OR DARE (1991)

It might be hard to explain to a younger audience just how dominant Madonna was as an artist coming out of the 1980s or the kind of landmark event this film represented because of her status. The travelogue of her Blonde Ambition Tour was like peeking into the insane world of the ultra-famous—not least because Madonna was dating Warren Beatty at the time and part of the film involves her hanging out with Al Pacino, Lionel Richie, and more. There are threats that the Canadian police will arrest her for simulating masturbation in her show, the Pope trying to get the tour canceled in Italy, and a slightly awkward return home to see family. All par for the course for someone whose personal life was carved up for public consumption.

9. RHYME & REASON (1997)

An unparalleled look into the lyricism and lifestyle of rap musicians from the genre’s rise through its global domination of the 1990s, the concert and party footage is fantastic, and the number of interviews is staggering. Peter Spirer spoke with more than 80 rap and hip-hop artists to craft a snapshot of what life was like for a group of musicians who discovered their voices could echo across the world as well as those who followed after to even greater success. Instead of going deep on one person behind the music, it’s a historical document of the culture itself as seen through the eyes of those at its very center.

10. THE DEVIL AND DANIEL JOHNSTON (2005)

For those who don’t know Daniel Johnston’s music, this doc is a crash course not only in its stripped-down, anti-folk vibes but the head it all comes spilling out of. Instead of romanticizing or ignoring his bipolar disorder, Jeff Feuerzeig’s movie engages with it directly, drawing beautiful gems from a troubled mind. An absolute masterpiece, it’s less a vision of a musician giving glimpses into his real life than it is a vision of a human being who makes music.

11. AWESOME; I F*CKIN’ SHOT THAT! (2006)

Rockumentaries follow two major formats: the raw concert doc that’s like a ticket to a show you couldn’t attend, and the profile where artists drop quotables in between performances. They’re safe and familiar, which is probably why the Beastie Boys gave both styles the middle finger in favor of a grand experiment. A year before YouTube launched, the rap trio gave 50 fans in their Madison Square Garden audience camcorders to capture the concert. The result is a genuine, fans’-eye-view of the experience, and a chaotic mashup of perspectives.

12. THE PUNK SINGER (2013)

It’s astonishing how much time and ground Sini Anderson’s portrait of Bikini Kill leader Kathleen Hanna covers. It’s so much that labeling her Bikini Kill’s leader is woefully reductive. Artist, pioneer, feminist, activist, and a dozen other titles swirl around Hanna’s sweat-covered brow as we get to know her both as an artist and as a person. It’s also a punk fever dream of riot grrrl greatness, featuring incendiary archival footage and excellent talks with members of Le Tigre, Bikini Kill, and Julie Ruin, as well as Carrie Brownstein and the Beastie Boys’s Adam Horovitz (who is also Hanna’s husband).

13. JANIS: LITTLE GIRL BLUE (2015)

A fairly recent addition to the pantheon, Amy J. Berg’s doc is a stirring tour of archival footage of the gravel-throated songstress. Narrated by musician Cat Power, instead of losing perspective to the fog of history, a blend of modern conversations and ghosts from the past offer fresh eyes and ears to create a heartsick celebration of one of music history's most beloved artists, whose career was cut woefully short.

20 Memorable Elvis Presley Quotes

Hulton Archive/Getty Images
Hulton Archive/Getty Images

More than 40 years after his death, Elvis Presley remains a rock ‘n' roll icon and has yet to be ousted from his position as “The King.” Yet the Tupelo, Mississippi-born, Memphis, Tennessee-raised superstar never took his fame for granted, nor did he forget his roots. Here are 20 memorable quotes about Elvis’s life and legacy.

ON AMBITION

“Ambition is a dream with a V8 engine.”

ON MAINTAINING YOUR VALUES

“It's not how much you have that makes people look up to you, it's who you are.”

“Values are like fingerprints. Nobody's are the same, but you leave 'em all over everything you do.”

ON THE MUSIC INDUSTRY

“I happened to come along in the music business when there was no trend.”

“I've never written a song in my life. It's all a big hoax.”

“I don't know anything about music. In my line you don't have to.”

ON THE ARMY

“After a hard day of basic training, you could eat a rattlesnake.”

“The army teaches boys to think like men.”

ON TRUTH

“Truth is like the sun. You can shut it out for a time, but it ain't goin' away.”

ON THOSE LEGENDARY DANCE MOVES

“Rock and roll music, if you like it, if you feel it, you can't help but move to it. That's what happens to me. I can't help it.”

“Some people tap their feet, some people snap their fingers, and some people sway back and forth. I just sorta do 'em all together, I guess.”

ON KEEPING POSITIVE

“When things go wrong, don't go with them.”

ON STARDOM

“If you let your head get too big, it'll break your neck.”

“I have no use for bodyguards, but I have very specific use for two highly trained certified public accountants.”

“The image is one thing and the human being is another. It's very hard to live up to an image, put it that way.”

“The Lord can give, and the Lord can take away. I might be herding sheep next year.”

ON LOVE

“Sad thing is, you can still love someone and be wrong for them.”

ON THE PITFALLS OF HOLLYWOOD

“I sure lost my musical direction in Hollywood. My songs were the same conveyer belt mass production, just like most of my movies were.”

ON GETTING OLDER

“Every time I think that I'm getting old, and gradually going to the grave, something else happens.”

ON LEAVING A LEGACY

“Do something worth remembering.”

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