CLOSE
iStock
iStock

10 Paradoxes That Will Boggle Your Mind

iStock
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?

nextArticle.image_alt|e
iStock
arrow
Animals
10 Biting Facts About Snapping Turtles
iStock
iStock

Here in the Americas, lake monster legends are a dime a dozen. More than a few of them were probably inspired by these ancient-looking creatures. In honor of World Turtle Day, here are 10 things you might not have known about snapping turtles.

1. THE COMMON SNAPPING TURTLE IS NEW YORK'S OFFICIAL STATE REPTILE.

Elementary school students voted to appoint Chelydra serpentina in a 2006 statewide election. Weighing as much as 75 pounds in the wild (and 86 in captivity), this hefty omnivore’s natural range stretches from Saskatchewan to Florida.

2. ALLIGATOR SNAPPING TURTLES CAN BE LARGE. (VERY LARGE.)

An alligator snapping turtle
NorbertNagel, Wikimedia Commons // CC BY-SA 3.0

Utterly dwarfing their more abundant cousin, alligator snappers (genus: Macrochelys) are the western hemisphere’s biggest freshwater turtles. The largest one on record, a longtime occupant of Chicago’s Shedd Aquarium, weighed 249 pounds.  

A monstrous 403-pounder was reported in Kansas during the Great Depression, though this claim was never confirmed.  

3. COMMON SNAPPERS HAVE LONGER NECKS AND SPIKIER TAILS.

Alligator snappers also display proportionately bigger heads and noses plus a trio of tall ridges atop their shells. Geographically, alligator snapping turtles are somewhat restricted compared to their common relatives, and are limited mainly to the southeast and Great Plains.

4. BOTH VARIETIES AVOID CONTACT WITH PEOPLE.

If given the choice between fight and flight, snapping turtles almost always distance themselves from humans. The animals spend the bulk of their lives underwater, steering clear of nearby Homo sapiens. However, problems can arise on dry land, where the reptiles are especially vulnerable. Females haul themselves ashore during nesting season (late spring to early summer). In these delicate months, people tend to prod and handle them, making bites inevitable.

5. YOU REALLY DON'T WANT TO GET BITTEN BY ONE. 

Snapping turtle jaw strength—while nothing to sneeze at—is somewhat overrated. Common snapping turtles can clamp down with up to 656.81 newtons (N) of force, though typical bites register an average of 209 N. Their alligator-like cousins usually exert 158 N. You, on the other hand, can apply 1300 N between your second molars.

Still, power isn’t everything, and neither type of snapper could latch onto something with the crushing force of a crocodile’s mighty jaws. Yet their sharp beaks are well-designed for major-league shearing. An alligator snapping turtle’s beak is capable of slicing fingers clean off and (as the above video proves) obliterating pineapples.

Not impressed yet? Consider the following. It’s often said that an adult Macrochelys can bite a wooden broom handle in half. Intrigued by this claim, biologist Peter Pritchard decided to play MythBuster. In 1989, he prodded a 165-pound individual with a brand new broomstick. Chomp number one went deep, but didn’t quite break through the wood. The second bite, though, finished the job.

6. SCIENTISTS RECENTLY DISCOVERED THAT THERE ARE THREE SPECIES OF ALLIGATOR SNAPPING TURTLES.

A 2014 study trisected the Macrochelys genus. For over a century, naturalists thought that there was just a single species, Macrochelys temminckii. Closer analysis proved otherwise, as strong physical and genetic differences exist between various populations. The newly-christened M. suwanniensis and M. apalachicolae are named after their respective homes—namely, the Suwannee and Apalachicola rivers. Further west, good old M. temminckii swims through the Mobile and the Mississippi.

7. THANKS TO A 19TH CENTURY POLITICAL CARTOON, COMMON SNAPPING TURTLES ARE ALSO KNOWN AS "OGRABMES." 

Snapping turtle cartoon
Urban~commonswiki via Wiki Commons // CC BY PD-US

Drawn by Alexander Anderson, this piece skewers Thomas Jefferson’s signing of the unpopular Embargo Act. At the president’s command, we see a snapping turtle bite some poor merchant’s hind end. Agitated, the victim calls his attacker “ograbme”—“embargo” spelled backwards.

8. ALLIGATOR SNAPPERS ATTRACT FISH WITH AN ORAL LURE …

You can’t beat live bait. Anchored to the Macrochelys tongue is a pinkish, worm-like appendage that fish find irresistible. Preferring to let food come to them, alligator snappers open their mouths and lie in wait at the bottoms of rivers and lakes. Cue the lure. When this protrusion wriggles, hungry fish swim right into the gaping maw and themselves become meals.

9.  … AND THEY FREQUENTLY EAT OTHER TURTLES. 


Complex01, WikimediaCommons

Alligator snappers are anything but picky. Between fishy meals, aquatic plants also factor into their diet, as do frogs, snakes, snails, crayfish, and even relatively large mammals like raccoons and armadillos. Other shelled reptiles are fair game, too: In one Louisiana study, 79.82% of surveyed alligator snappers had turtle remains in their stomachs.

10. YOU SHOULD NEVER PICK A SNAPPER UP BY THE TAIL.

Ideally, you should leave the handling of these guys to trained professionals. But what if you see a big one crossing a busy road and feel like helping it out? Before doing anything else, take a few moments to identify the turtle. If it’s an alligator snapper, you’ll want to grasp the lip of the upper shell (or “carapace”) in two places: right behind the head and right above the tail.

Common snappers demand a bit more finesse (we wouldn’t want one to reach back and nip you with that long, serpentine neck). Slide both hands under the hind end of the shell, letting your turtle’s tail dangle between them. Afterwards, clamp down on the carapace with both thumbs.

Please note that lifting any turtle by the tail can permanently dislocate its vertebrae. Additionally, remember to move the reptile in the same direction that it’s already facing. Otherwise, your rescue will probably turn right back around and try to cross the road again later. 

nextArticle.image_alt|e
Jenny Anderson, Getty Images for Tony Awards Productions
arrow
entertainment
10 Things You Might Not Know About Tina Fey
Jenny Anderson, Getty Images for Tony Awards Productions
Jenny Anderson, Getty Images for Tony Awards Productions

Tina Fey has transformed modern comedy more than just about anyone else. From the main stage of Second City to the writer’s room of SNL to extremely fetch comedy blockbusters, Elizabeth Stamatina Fey has built a national stage with a dry, eye-popping sarcasm and political satire where no one is safe. She has a slew of Emmys, Golden Globes, SAG, PGA, and WGA awards to prove it—plus a recent Tony nomination (her first). But, more importantly, she’s the closest thing we have to a national comic laureate.

Here are 10 facts about a fantastically blorft American icon.

1. SHE DID A BOOK REPORT ON COMEDY WHEN SHE WAS 11.

Fey got a very early start in comedy, watching a lot of The Mary Tyler Moore Show, Bob Newhart, and Norman Lear shows as a kid. Her father and mother sneaked her in to see Young Frankenstein and would let her stay up late to watch The Honeymooners. So it’s no surprise that she chose comedy as the subject of a middle school project. The only book she could get her hands on was Joe Franklin’s Encyclopedia of Comedians, but at least she made a friend. "I remember me and one other girl in my 8th grade class got to do an independent study because we finished the regular material early, and she chose to do hers on communism, and I chose to do mine on comedy," Fey told The A.V. Club. "We kept bumping into each other at the card catalog."

2. THE SCAR ON HER FACE CAME FROM A BIZARRE ATTACK THAT OCCURRED WHEN SHE WAS A CHILD.

Fey’s facial scar had been recognizable but unexplained for years until a profile in Vanity Fair revealed that the mark on her left cheek came from being slashed by a strange man when she was five years old. “She just thought somebody marked her with a pen,” her husband Jeff Richmond said. Fey wrote in Bossypants that it happened in an alleyway behind her Upper Darby, Pennsylvania, home when she was in kindergarten.

3. HER FIRST TV APPEARANCE WAS IN A BANK COMMERCIAL.

Saturday Night Live hired Fey as a writer in 1997. In 1995 she had the slightly more glamorous job of pitching Mutual Savings Bank with a radical floral applique vest and a handful of puns on the word “Hi.” In a bit of life imitating art, just as Liz Lemon’s 1-900-OKFACE commercial was unearthed and mocked on 30 Rock, the internet discovered Fey’s stint awkwardly cheering on high interest rates a few years ago and had a lot to say about her '90s hair.

4. SHE WAS THE FIRST WOMAN TO BE NAMED HEAD WRITER OF SNL.

Four years after that commercial and two after she joined Saturday Night Live’s writing staff, Fey earned a promotion to head writer. Up until that point, the head writers were named Michael, Herb, Bob, Jim, Steve. You get the picture. She acted as head writer for six seasons until moving on to write and executive produce 30 Rock. Since her departure, two more women (Paula Pell and Sara Schneider) have been head writers for the iconic show.

5. SHE’S THE YOUNGEST MARK TWAIN PRIZE WINNER.

Established in 1998, the Kennedy Center’s hilarious honor has mostly been awarded to funny people in the twilight of their careers. Richard Pryor was the first recipient, and comedians who made their marks decades prior like Lily Tomlin, Whoopi Goldberg, and George Carlin followed. Fey earned the award in 2010 when she was 40 years old, and the age of her successors (Carol Burnett, Bill Murray, Eddie Murphy, David Letterman ...) signals that she may hold the title of youngest recipient for some time.

6. SHE WROTE SATIRE FOR HER HIGH SCHOOL NEWSPAPER.

Fey was an outstanding student who was involved in choir, drama, and tennis, and co-edited the school’s newspaper, The Acorn. She also wrote a satirical column addressing “school policy and teachers” under the pun-tastic pseudonym “The Colonel.” Fey also recalled getting in trouble because she tried to make a pun on the phrase “annals of history.” Cheeky.

7. SHE MADE HER RAP DEBUT WITH CHILDISH GAMBINO ON "REAL ESTATE."

Donald Glover (a.k.a. Childish Gambino) first gained notice as a member of Derrick Comedy in college, and Fey hired him at the age of 23 to write for 30 Rock. Before jumping from that show to Community, Glover put out his first mixtape under his stage name. After releasing his debut album, Camp, in 2011, Gambino dropped a sixth mixtape called Royalty that featured Fey rapping on a song called “Real Estate.” “My president is black, and my Prius is blue!"

8. SHE VOICED PRINCESSES IN A BELOVED PINBALL GAME.

Between the bank commercial and Saturday Night Live, Fey has an intriguing credit on her resume: the arcade pinball machine “Medieval Madness.” Most of the game’s Arthurian dialogue was written by Second City members Scott Adsit (Pete Hornberger on 30 Rock) and Kevin Dorff, who pulled in fellow Second City castmate Fey to voice for an “Opera Singer” princess, Cockney-speaking princesses, and a character with a southern drawl. (You can hear some of the outtakes here.)

9. SHE USED MEAN GIRLS TO PUSH BACK AGAINST STEREOTYPES OF WOMEN IN MATH.

Tina Fey and Lindsay Lohan in 'Mean Girls' (2004)
Paramount Home Entertainment

There’s a ton of interesting trivia about Mean Girls, Fey’s first foray into feature film screenwriting. She bid on the rights to Rosalind Wiseman’s book that inspired the movie without realizing it didn’t have a plot. She initially wrote a large part for herself but kept whittling it down to focus on the teenagers, and her first draft was “for sure R-rated.” Fey also chose to play a math teacher to fight prejudice. “It was an attempt on my part to counteract the stereotype that girls can’t do math. Even though I did not understand a word I was saying.” Fey used a friend’s calculus teacher boyfriend’s lesson plans in the script.

10. SHE SET UP A SCHOLARSHIP IN HER FATHER’S NAME TO HELP VETERANS.

Fey’s father Donald was a Korean War veteran who also studied journalism at Temple University. When he died in 2015, Fey and her brother Peter founded a memorial scholarship in his name that seeks to aid veterans who want to study journalism at Temple.

"He was really inspiring," Fey said. "A lot of kids grow up with dreams of doing those things and their parents are fearful and want them to get a law degree and have things to fall back on, but he and our mom always encouraged us to pursue whatever truly interested us." Fey also supports Autism Speaks, Mercy Corps, Love Our Children USA, and other charities.

SECTIONS

arrow
LIVE SMARTER
More from mental floss studios