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?

10 Facts About Christopher Marlowe

A stone in memory of Christopher Marlowe at Kings School, Canterbury
A stone in memory of Christopher Marlowe at Kings School, Canterbury
John K Thorne, Flickr // Public Domain

Christopher Marlowe is more than a footnote in William Shakespeare’s life, even though that’s the position he’s most often relegated to, especially in fiction. It’s obvious why: Shakespeare is the most famous English playwright, and Marlowe is merely one of the most famous English playwrights. Plus, since Marlowe was a contemporary of Shakespeare's, he ends up bursting onto the scene in cameo appearances during tales focused on the Bard.

The other reason? We simply don’t know that much about him.

Born in 1564, Marlowe led a brief, extraordinary life even before you get to all the mythology and conspiracy theories that have grown up surrounding him. He offered a memorable poetic voice that paved the way for Shakespeare while crafting stories of outsized personalities forever flying too close to the sun (or the Devil).

Here are 10 facts about a man we should know more about.

1. Christopher Marlowe achieved a lot in a short time.

Rupert Everett was almost 40 when he portrayed Marlowe in Shakespeare in Love, but Marlowe only lived to age 29. Marlowe built on the work of Thomas Norton and Thomas Sackville among others, and his unrhymed, iambic pentameter—specifically the wildly popular and oft-imitated Tamburlaine the Great—represented an evolution in style that became an accepted structure in Renaissance English theatre. It’s what Shakespeare used, and what you probably learned about in high school literature class.

2. Christopher Marlowe wasn’t going to graduate Cambridge until the government intervened.

A portrait of an unknown 21-year-old man said to be Christopher Marlowe, discovered at Cambridge in 1952
A portrait of an unknown 21-year-old man said to be Christopher Marlowe, discovered at Cambridge in 1952
Wikimedia // Public Domain

In 1587, Marlowe had the Elizabethan equivalent of too many absences from his master’s program at Cambridge University, and there were rumors that he was preparing to go to France to become a Catholic priest. Cambridge officials considered refusing to award his degree, but the Privy Council (Queen Elizabeth’s advisers) sent them a letter denouncing the rumor and explaining that Marlowe had been operating to “the benefit of his country” and had done “her Majesty good service.”

3. Christopher Marlowe might have been a spy.

The "good service" he was doing for Her Majesty? The Privy Council never explained. Nevertheless, the secretive work, the religious nature of the rumors during an era when England persecuted Catholics, and the fact that Queen Elizabeth’s spymaster, Francis Walsingham, often recruited young men attending Cambridge, have created the foundation for the theory that Marlowe was part of a spy network. At the very least, Marlowe did some undisclosed work for the government, which got him a helping hand that explained his school absences.

4. Christopher Marlowe was arrested for counterfeiting coins in Holland.

In 1592, about five years after the wild success of Tamburlaine, Marlowe was arrested for counterfeiting coins in the Dutch town of Vlissingen. This was a crime punishable by death, but Marlowe seems to have walked away with no, or very light, punishment. Naturally, some think this supports the idea that Marlowe worked as a spy.

5. Christopher Marlowe translated ancient poetry.

In addition to his plays (he wrote at least four, and some say seven), Marlowe also wrote poetry—"The Passionate Shepherd to His Love" and "Hero and Leander" most notably. In the former, a shepherd woos a lover by glorifying nature, and the latter retells a Greek myth where a man swims a narrow sea to reach the woman he loves. Marlowe also translated ancient works, including the first book of the Pharsalia, a Roman epic by Lucan about Caesar facing Pompey the Great in battle, and Ovid’s books of love poetry, Amores.

6. Christopher Marlowe was arrested for holding heretical views.

In 1593, the English government had a largely welcoming attitude to Protestant immigrants, so authorities were livid when anti-immigrant tracts began being posted on the streets of London. One that was judged to "exceed the rest in lewdness" alluded to two of Marlowe’s plays and was signed “Tamburlaine.” As part of a sweep targeting suspicious characters, authorities arrested and then tortured Marlowe’s friend and fellow playwright Thomas Kyd, who asserted that an unorthodox religious tract found in his room belonged to Marlowe. A warrant was issued, and Marlowe presented himself to the Privy Council, who told him to check in with them every day with them until further notice. He died 10 days later.

7. Christopher Marlowe's death inspired conspiracy theories.

The official story is that Marlowe was killed on May 30, 1593 while arguing about money in a boarding house with an associate named Ingram Frizer, and that very well may be the truth. But the strange circumstances around the event are numerous: Marlowe had been arrested for being an "atheist" only 10 days prior but received no real punishment for it; Frizer (and the two other men there) had all been employed by spymaster Walsingham; and even contemporaries doubted the plausibility of the coroner’s report. The list of people who apparently might have had cause to want Marlowe dead is long (right up to the queen herself), but the most fanciful theory is that the whole event was faked so that Marlowe could escape a very real death if convicted for religious heresy.

8. Christopher Marlowe pushed against anti-LGBT bigotry in his work.

Some scholars think Marlowe may have been gay, but (like so many other elements of his life) there is no conclusive evidence. However, there is concrete evidence that he treated same-sex relationships differently than other writers of the time. In other work of the same period, gay characters were usually villains, but Marlowe wrote about Edward II’s relationship with Piers Gaveston with humanity and beauty in Edward II. Some experts believe the play upheld conventional views on gay relationships by “punishing” Gaveston with death and killing Edward II in a way that evokes sodomy, but, even if so, Marlowe still covered the topic throughout the play with greater complexity and consideration than his contemporaries.

9. Westminster Abbey installed a window memorializing Christopher Marlowe in 2002.

The Poets’ Corner of Westminster Abbey is home to the graves of over 100 poets and writers, starting with Geoffrey Chaucer, who was buried there in 1400. Marlowe is buried in an unmarked grave in St. Nicholas's Church in Deptford, London, but shares a memorial in the form of a window at Poet's Corner with Elizabeth Gaskell, Oscar Wilde, and more. The space was donated by The Marlowe Society, who included a question mark next to his death date.

10. Shakespeare paid tribute to Christopher Marlowe in verse.

There would be no Shakespeare without Marlowe. Honoring the young trailblazer after his death, Shakespeare included one of Marlowe’s lines from Hero and Leander in As You Like It (“Who ever lov’d that lov’d not at first sight?”) and had a character possibly allude to Marlowe’s killing. There are also nods in Hamlet and Love’s Labour’s Lost. Of course, Shakespeare’s highest homage came in how often he echoed Marlowe’s poetic style and dramatic themes. (Though definitely not written by Shakespeare, there’s also a 1981 rock ‘n’ roll musical tribute to Marlowe that’s set in the 16th century but somehow also included miniskirts.)

10 Bold Breaking Bad Fan Theories

Bryan Cranston as Walter White and Aaron Paul as Jesse Pinkman in Breaking Bad.
Bryan Cranston as Walter White and Aaron Paul as Jesse Pinkman in Breaking Bad.
Ben Leuner, AMC

It’s been nearly six years since Breaking Bad went out in a blaze of gunfire, but fans still haven’t stopped thinking about the award-winning crime drama. What really happened to Walter White in the series finale? What’s the backstory on Gus Fring? And what did Jesse Pinkman’s doodles mean?

While El Camino, Vince Gilligan's new Breaking Bad movie, offers definitive answers to at least one of these questions, these fan theories offer some alternative answers—even if they strain the limits of logic and sanity along the way. Read on to discover the surprising source of Walt’s cancer diagnosis, and why pink is always bad news.

1. Walter White picks up traits from the people he kills.

Walter White is an unpredictable guy, but he’s weirdly consistent on one thing: After he kills someone, he kind of copies them. Remember how Krazy-8 liked his sandwiches without the crust? After Walt murdered him, he started eating crustless PB&Js. Walt also lifted Mike Ehrmantraut’s drink order and Gus Fring’s car, leading many fans to wonder if Walt steals personal characteristics from the people he kills.

2. Gus Fring worked for the CIA.

Gus Fring (Giancarlo Esposito) and Juan Bolsa (Javier Grajeda) in Breaking Bad
Giancarlo Esposito and Javier Grajeda in Breaking Bad.
Ursula Coyote, AMC

Who was Gus Fring before he became the ruthless leader of a meth/fried chicken empire? Well, we know he’s from Chile. We also know that any records of his time there are gone. And we know that cartel kingpin Don Eladio refused to kill him when he had the chance. Since Don Eladio has no qualms about eliminating the competition, Gus must have some form of protection. Could it be from the U.S. government? A detailed Reddit theory suggests that Gus was once a Chilean aristocrat who helped the CIA install the dictator Augusto Pinochet in power. Once Pinochet became a liability, Gus went to Mexico at the CIA’s behest to infiltrate a drug cartel. His alliance with U.S. intelligence kept him alive even as his work got more violent, and helped him bypass the normal immigration issues you'd typically encounter when you’ve murdered a bunch of people.

3. Madrigal built defective air filters that gave Walter white cancer.

Madrigal Electromotive is a corporation with varied interests. The German parent company of Los Pollos Hermanos dabbles in shipping, fast food, and industrial equipment … including air filters. According to one fan theory, Gray Matter—the company Walter White co-founded with Elliott Schwartz—purchased defective air filters from Madrigal and installed them while Walt still worked at the company. The filters ultimately caused Walt’s lung cancer, pushing him into the illegal drug trade and, eventually, business with Madrigal.

4. Color is a crucial element in the series.

Marie Schrader (Betsy Brandt) and Hank Schrader (Dean Norris)
Betsy Brandt and Dean Norris as Marie and Hank Schrader in Breaking Bad.
Ben Leuner, AMC

Color is a code on Breaking Bad. When a character chooses drab tones, they’re usually going through something, like withdrawal (Jesse) or chemo (Walt). Their wardrobe might turn darker as their stories skew darker—like when Marie ditched her trademark purple for black while she was under protective custody. Also, pink signals death, whether it’s on a teddy bear or Saul Goodman’s button down shirt.

5. Breaking Bad and The Walking Dead exist in the same universe.

Breaking Bad and The Walking Dead both aired on AMC, but according to fans, that’s not all they have in common. There’s an exhaustive body of evidence connecting the two shows—and one of the biggest links is Blue Sky. The distinctively-colored crystal meth is Walt and Jesse’s calling card on Breaking Bad, but it’s also Merle Dixon’s drug of choice on The Walking Dead. Coincidentally, his drug dealer (“a janky little white guy” who says “bitch”) sounds a lot like Jesse.

6. Walter white froze to death and hallucinated Breaking Bad's ending.

Bryan Cranston in the 'Breaking Bad' series finale
Ursula Coyote, AMC

In her review of the Breaking Bad series finale “Felina,” The New Yorker critic Emily Nussbaum suggested an alternate ending in which Walt died an episode earlier, as the police surrounded his car in New Hampshire. He could’ve frozen to death “behind the wheel of a car he couldn’t start,” she theorized, and hallucinated the dramatic final shootout in “Felina” in his dying moments. This reading has gained traction with multiple fans, including SNL alum Norm Macdonald.

7. Jesse’s superheroes are a peek into his inner psyche.

In season 2 of Breaking Bad, we discover that Jesse Pinkman is a part-time artist. He sketches his own superheroes, including Backwardo/Rewindo (who can run backwards so fast he rewinds time), Hoverman (who floats above the ground), and Kanga-Man (who has a sidekick in his “pouch”). The characters are goofy, just like Jesse, but they may also reveal what’s going on in his head. Backwardo represents Jesse’s tendency to run from conflict. Hoverman reflects his lack of direction or purpose, while Kanga-Man hints at his codependency.

8. Madrigal was founded by Nazi war criminals.

Walter White (Bryan Cranston) and Uncle Jack (Michael Bowen) in 'Breaking Bad'
Bryan Cranston and Michael Bowen in Breaking Bad.
Ursula Coyote, AMC

This might be one of the wilder Breaking Bad theories, but before you write it off, consider Werner Heisenberg: The German physicist, who helped pioneer Hitler’s nuclear weapons program, is the obvious inspiration for Walt’s meth kingpin moniker. While Heisenberg only appears in name, there are plenty of literal Nazis on the show. Look no further than Uncle Jack and the Aryan Brotherhood, who served as the Big Bad of season 5. At least one Redditor thinks all these Nazi references are hinting at something bigger, a conspiracy that goes straight to the top. The theory starts in South America, where many Nazis fled after World War II. A group of them supposedly formed a new company, Madrigal, through their existing connections back in Germany. Eventually, a young Chilean named Gus Fring worked his way into the growing business, and the rest is (fake) history.

9. Walter white survived, but paid the price.

Lots of Breaking Bad theories concern Walt’s death, or lack thereof. But if Walt actually lived through his seemingly fatal gunshot wound in “Felina,” what would the rest of his life look like? According to one Reddit theory, it wouldn’t be pretty. The infamous Heisenberg would almost certainly stand trial and go to prison. Although he tries to leave Skyler White with information to cut a deal with the cops, she could also easily go to jail—or lose custody of her children. The kids wouldn’t necessarily get that money Walt left with Elliott and Gretchen Schwartz, either, as they could take his threats to the police and surrender the cash to them. Basically it amounts to a whole lot of misery, making Walt’s death an oddly optimistic ending. (This is one theory El Camino addresses directly.)

10. Breaking Bad is a prequel to Malcolm in the Middle.

Bryan Cranston in the series premiere of 'Breaking Bad'
Bryan Cranston in the series premiere of Breaking Bad.
Doug Hyun, AMC

Alright, let’s say Walt survived the series finale and didn’t stand trial. Maybe he started over as a new man with a new family. Three boys, perhaps? This fan-favorite theory claims that Walter White assumed a new identity as Malcolm in the Middle patriarch Hal after the events of Breaking Bad, making the show a prequel to Bryan Cranston’s beloved sitcom. The Breaking Bad crew actually liked this idea so much they included an “alternate ending” on the DVD boxed set, where Hal wakes up from a bad dream where "There was a guy who never spoke! He just rang a bell the whole time! And then there was another guy who was a policeman or a DEA agent, and I think it was my brother or something. He looked like the guy from The Shield."

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