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 Incredible Facts About Frederick Douglass

Photo Illustration: Mental Floss. Douglass: Glasshouse Images, Alamy. Backgrounds: iStock
Photo Illustration: Mental Floss. Douglass: Glasshouse Images, Alamy. Backgrounds: iStock

The list of Frederick Douglass's accomplishments is astonishing—respected orator, famous writer, abolitionist, civil rights leader, presidential consultant—even more so when you consider that he was a former slave with no formal education. In honor of his 201st birthday, here are 13 incredible facts about the life of Frederick Douglass.

1. He bartered bread for knowledge.

Because Douglass was a slave, he wasn't allowed to learn to read or write. A wife of a Baltimore slave owner did teach him the alphabet when he was around 12, but she stopped after her husband interfered. Young Douglass took matters into his own hands, cleverly fitting in a reading lesson whenever he was on the street running errands for his owner. As he detailed in his autobiography, Narrative of the Life of Frederick Douglass, he'd carry a book with him while out and about and trade small pieces of bread to the white kids in his neighborhood, asking them to help him learn to read the book in exchange.

2. He credited a schoolbook with shaping his views on human rights.

Engraving of Frederick Douglass, circa the 1850s.
Engraving of Frederick Douglass, circa the 1850s.
Hulton Archive, Getty Images

During his youth, Douglass obtained a copy of The Columbian Orator, a collection of essays, dialogues, and speeches on a range of subjects, including slavery. Published in 1797, the Orator was required reading for most schoolchildren in the 1800s and featured 84 selections from authors like Cicero and Milton. Abraham Lincoln was also influenced by the collection when he was first starting in politics.

3. He taught other slaves to read.

While he was hired out to a farmer named William Freeland, a teenaged Douglass taught fellow slaves to read the New Testament—but a mob of locals soon broke up the classes. Undeterred, Douglas began the classes again, sometimes teaching as many as 40 people.

4. His first wife helped him escape from slavery.

Portrait of Anna Murray Douglass, Frederick Douglass's first wife.
First published in Rosetta Douglass Sprague's book My Mother As I Recall Her, Wikimedia Commons // Public Domain

Anna Murray was an independent laundress in Baltimore and met Douglass at some point in the mid-1830s. Together they hatched a plan, and one night in 1838, Douglass took a northbound train clothed in a sailor's uniform procured by Anna, with money from her savings in his pocket alongside papers from a sailor friend. About 24 hours later, he arrived in Manhattan a free man. Anna soon joined him, and they married on September 15, 1838.

5. He called out his former owner.

In an 1848 open letter in the newspaper he owned and published, The North Star, Douglass wrote passionately about the evils of slavery to his former owner, Thomas Auld, saying "I am your fellow man, but not your slave." He also inquired after his family members who were still enslaved a decade after his escape.

6. He took his name from a poem.

He was born Frederick Augustus Washington Bailey, but after escaping slavery, Douglass used assumed names to avoid detection. Arriving in New Bedford, Massachusetts, Douglass, then using the surname "Johnson," felt there were too many other Johnsons in the area to distinguish himself. He asked his host (ironically named Nathan Johnson) to suggest a new name, and Mr. Johnson came up with Douglas, a character in Sir Walter Scott's poem The Lady of the Lake.

7. He was deemed the 19th century's most photographed American.

Portrait of Frederick Douglass
Wikimedia Commons // Public Domain

There are 160 separate portraits of Douglass, more than Abraham Lincoln or Walt Whitman, two other heroes of the 19th century. Douglass wrote extensively on the subject during the Civil War, calling photography a "democratic art" that could finally represent black people as humans rather than "things." He gave his portraits away at talks and lectures, hoping his image could change the common perceptions of black men.

8. He refused to celebrate Independence Day.

Douglass was well-known as a powerful orator, and his July 5, 1852 speech to a group of hundreds of abolitionists in Rochester, New York, is considered a masterwork. Entitled "What to the Slave is the Fourth of July," the speech ridiculed the audience for inviting a former slave to speak at a celebration of the country who enslaved him. "This Fourth [of] July is yours, not mine," he famously said to those in attendance. "Do you mean, citizens, to mock me, by asking me to speak to-day?" Douglass refused to celebrate the holiday until all slaves were emancipated and laws like the Compromise of 1850, which required citizens (including northerners) to return runaway slaves to their owners, were negated.

9. He recruited black soldiers for the Civil War.

The Union attack on Fort Wagner, Charleston, during the American Civil War. The fort was under attack from July 18 to September 7, 1863, by soldiers including the 54th Massachusetts Volunteer Infantry, the first African-American regiment in the U.S. Army.
The Union attack on Fort Wagner, Charleston, during the American Civil War. The fort was under attack from July 18 to September 7, 1863, by soldiers including the 54th Massachusetts Volunteer Infantry, the first African-American regiment in the U.S. Army.
Hulton Archive, Getty Images

Douglass was a famous abolitionist by the time the war began in 1861. He actively petitioned President Lincoln to allow black troops in the Union army, writing in his newspaper: "Let the slaves and free colored people be called into service, and formed into a liberating army, to march into the South and raise the banner of Emancipation among the slaves." After Lincoln signed the Emancipation Proclamation, Douglass worked tirelessly to enlist black soldiers, and two of his sons would join the 54th Massachusetts Regiment, famous for its contributions in the brutal battle of Fort Wagner.

10. He served under five presidents.

Later in life, Douglass became more of a statesman, serving in highly appointed federal positions, including U.S. Marshal for D.C., Recorder of Deeds for D.C., and Minister Resident and Consul General to Haiti. Rutherford B. Hayes was the first to appoint Douglass to a position in 1877, and Presidents Garfield, Arthur, Cleveland, and Benjamin Harrison each sought his counsel in various positions as well.

11. He was nominated for Vice President of the United States.

As part of the Equal Rights Party ticket in 1872, Douglass was nominated as a VP candidate, with Victoria Woodhull as the Presidential candidate. (Woodhull was the first-ever female presidential candidate, which is why Hillary Clinton was called "the first female presidential candidate from a major party" during the 2016 election.) However, the nomination was made without his consent, and Douglass never acknowledged it (and Woodhull's candidacy itself is controversial because she wouldn't have been old enough to be president on Inauguration Day). Also, though he was never a presidential candidate, he did receive one vote at each of two nomination conventions.

12. His second marriage caused controversy.

Frederick Douglass with Helen Pitts Douglass (seated, right) and her sister Eva Pitts (standing, center), circa the 1880s.
Frederick Douglass with Helen Pitts Douglass (seated, right) and her sister Eva Pitts (standing, center), circa the 1880s.

Two years after his first wife, Anna, died of a stroke in 1882, Douglass married Helen Pitts, a white abolitionist and feminist who was 20 years younger than him. Even though she was the daughter of an abolitionist, Pitts's family (which had ancestral ties directly to the Mayflower) disapproved and disowned her—showing just how taboo interracial marriage was at the time. The black community also questioned why their most prominent spokesperson chose to marry a white woman, regardless of her politics. But despite the public's and their families' reaction, the Douglasses had a happy marriage and were together until his death in 1895 of a heart attack.

13. After early success, his Narrative went out of print.

Narrative of the Life of Frederick Douglass, An American Slave, Written by Himself, his seminal autobiography, was heralded a success when it came out in 1845, with some estimating that 5000 copies sold in the first few months; the book was also popular in Ireland and Britain. But post-Civil War, as the country moved toward reconciliation and slave narratives fell out favor, the book went out of print. The first modern publication appeared in 1960—during another important era for the fight for civil rights. It is now available for free online.

This article originally ran in 2018.

The 15 Best Documentaries on Netflix Right Now

A still from Ava DuVernay's 13th (2016)
A still from Ava DuVernay's 13th (2016)
Netflix

Truth is not only stranger than fiction, it’s frequently more entertaining. Thanks to the Netflix acquisition team, the streaming service offers hundreds of documentaries that chronicle everything from riveting tales of true crime to stories about bare-knuckle fighters and custody battles over amputated legs. To help you sort through their formidable selection, we’ve selected 15 films currently streaming that will either make your jaw drop, bring a tear to your eye, or both.

1. Finders Keepers (2015)

If an appendage is removed from your body, are you still its lawful owner? That’s the question posed by this irreverent investigation of Shannon Whisnant, a junk trader who successfully bids on a storage locker and discovers the mummified remains of a severed leg. The stump once belonged to John Wood, a man injured in a plane crash. When his leg was amputated as a result, he decided to keep it as a memento, storing it in a grill inside the locker. The argument over who has rightful possession of this fleshy trophy is at the center of the film, which sees the men try to resolve their differences in a variety of ways, including an appearance on Judge Mathis.

2. Long Shot (2017)

Juan Catalan is that most compelling of true crime clichés: an innocent man being railroaded for a murder he didn’t commit. With law enforcement dismissing his alibi, his lawyers make a last-ditch effort to prove that Catalan was at a Los Angeles Dodgers game at the time of the assault. How they do that—and which famous comic actor plays a role—is best left to discover on your own.

3. Gringo: The Dangerous Life of John McAfee (2016)

Anti-virus software tycoon John McAfee was one of the internet’s biggest success stories. Flush with money, power, and a desire to reinvent himself, McAfee relocated to Belize, where his story began to take on echoes of Marlon Brando’s Colonel Kurtz in Apocalypse Now. When all of McAfee’s whims are tended to by locals, questions over a neighbor’s murder take on sinister connotations. Michael Keaton is set to play McAfee in a feature film version.

4. Brother’s Keeper (1992)

The bonds of brotherhood are explored in this arresting feature about siblings Delbert, Roscoe, and Lyman Ward, farmers in upstate New York who close ranks when police begin to suspect one of them murdered their other brother, William.

5. Jim & Andy: The Great Beyond (2017)

When Jim Carrey stepped into the role of the late comedian Andy Kaufman for director Milos Forman’s 1999 biopic Man on the Moon, he didn’t so much imitate Kaufman as become him. That process was documented in behind-the-scenes footage that was buried in studio vaults for years and revealed here for the first time. Executives feared people would consider Carrey—who alternately charms and antagonizes people on the set by never behaving as “Jim”—as being exceptionally difficult to work with. Perhaps, but Carrey’s modern-day reflections on inhabiting the eccentric Kaufman even when the film cameras weren’t rolling are a fascinating study of both the performer’s commitment and the nature of identity.

6. Amanda Knox (2016)

College student Amanda Knox seized headlines in 2007 and beyond for being the prime suspect in the murder of fellow student and roommate Meredith Kercher while both were studying in Perugia, Italy. The competency and motives of Italian police are examined in this documentary, which features the first time Knox has spoken at length about her trials (yes, there was more than one) and struggles in a foreign justice system. Plenty of ink was spilled in the American media over her suspected guilt: Knox’s unflinching stare into the camera as she tells her side of the story will likely persuade you to think otherwise.

7. Fyre: The Greatest Party That Never Happened (2019)

Sun. Models. Booze. Would-be mogul Billy McFarland promised a lot and delivered little more than cold cheese sandwiches in his 2017 music festival debacle, which collected a small fortune in admission and ancillary profits and then wound up leaving hundreds of guests stranded on an island to fend for themselves. Pairing Netflix’s examination of the debacle and its fallout with Hulu’s Fyre Fraud makes for a fine double feature (even if you might be left with more questions than answers).

8. The Power of Grayskull: The Definitive History of He-Man and the Masters of the Universe (2017)

Toy and nostalgia fans will get a kick out of this rewind to the early 1980s, when Mattel’s He-Man dominated retail stores and syndicated television. The feature examines the toy line’s origins—which involved dueling toy designers and a failed attempt to secure a Conan license—and its later incarnation as a low-budget 1987 movie. (Yes, Dolph Lundgren makes an appearance.)

9. 13th (2016)

Director Ava DuVernay delivers a powerful (and Oscar-nominated) indictment of the U.S. justice system and takes a closer look at how incarceration and sentencing feeds into widespread inequality. Peering through DuVernay’s lens, viewers may feel the scales of justice are tipped in favor of privatized and profiteering prisons.

10. Icarus (2017)

The cat-and-mouse game between drug testing agencies and cheating athletes is put under a microscope in director Bryan Fogel’s Oscar-winning documentary, which uncovers the lengths competitors will go to in order to push past their physical limits. As Fogel digs deeper into the world of pro cycling and its high-ranking political influences, you may discover that drugs are so pervasive that athletes aren’t necessarily looking to cheat—they’re simply looking to even the playing field.

11. Senna (2010)

Sports documentaries don’t come much better than this portrait of Ayrton Senna, a Brazilian Formula One racer who became a national hero for his obsessive commitment to being the best. That passion conflicts with the inherent danger of his sport, which undergoes a technological metamorphosis in the 1980s and 1990s that threatens the safety of drivers. Those risks are on display in the film’s kinetic, heart-in-throat race sequences.

12. The Seven Five (2014)

There are bad cops, there are dirty cops, and then there’s Mike Dowd, a Brooklyn officer who used his badge to siphon money from criminals and exploit the very community he was charged with protecting. Dowd’s downfall ushered in one of the biggest police corruption scandals of the 1990s. The film features Dowd’s unabashed account of his dirty deeds.

13. Voyeur (2017)

Acclaimed journalist Gay Talese stumbles upon what he thinks is the story of a lifetime: A Colorado motel owner named Gerald Foos who modified his guest rooms so he could spy on his occupants. Not all of Foos’s recollections of his voyeur’s playground hold up to scrutiny, and the film sometimes wonders who’s really in control of the narrative—the directors, Talese, or the enigmatic Foos.

14. The Battered Bastards of Baseball (2014)

In the 1970s, Kurt Russell’s father, Bing Russell, started a rogue minor league baseball team, the Portland Mavericks. Playing without any Major League affiliation, the ragtag team barnstormed their way through several seasons, with an electric group of MLB castoffs making up the roster. It’s a fun look at a group that rivals the Bad News Bears in dropping the ball.

15. Dawg Fight (2015)

Florida native Dhafir “Dada5000” Harris tries to keep gangs and drugs from destroying his neighborhood by hosting a series of bare-knuckle fighting events in his mother’s backyard. The action is raw, but Harris’s intentions are pure. In orchestrating violence rather than letting it explode on the streets, Harris provides an outlet for young men to find some peace.

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