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?

14 Famous People Who Survived the 1918 Flu Pandemic

National Archives and Records Administration, Wikimedia Commons // Public domain
National Archives and Records Administration, Wikimedia Commons // Public domain

Over a century ago, a deadly flu pandemic swept across the globe. The first cases of the so-called Spanish Flu—named because that’s where early news reports of the disease originated, though research has put its actual origin anywhere from China to Kansas to France—are traditionally dated to Kansas in March 1918. The disease ultimately infected some 500 million people, and estimates put the death toll anywhere from 20 to 50 million. The people on this list contracted the deadly flu and lived to tell the tale.

1. Walt Disney

Walt Disney sitting in a chair.
Hulton Archive // Getty Images

If Walt Disney hadn’t contracted the flu, we might never have had Mickey Mouse. Even though he was only 16 at the time, Disney lied about his birth year to sign up for the Red Cross Ambulance Corps at the tail end of WWI. Then he got sick. By the time he was ready to ship out, the war was over.

2. Mary Pickford

A close-up photo of silent film star Mary Pickford smiling.
General Photographic Agency // Getty Images

The silent film star was at the height of her fame when she fell ill; thankfully, Pickford’s bout with the flu was uneventful, but as the disease spread, many movie theaters were forced to close. Irritated theater owners in Los Angeles, claiming they had been singled out, petitioned for all other places that people gathered together (except for grocery stores, meat markets, and drug stores) to be forced to close as well. While stores were not forced to close, schools were and public gatherings were banned.

3. David Lloyd George

David Lloyd George sitting outside with his dog and reading a newspaper.
Ernest H. Mills // Getty Images

Weeks before the end of World War I, Lloyd, Prime Minister of the UK at the time, came very close to dying of the flu. He was confined to his bed for nine days, had to wear a respirator, and was accompanied by a doctor for over a month. Because it was thought that news of the Prime Minister’s illness would hurt the morale of the British people and “encourage the enemy,” his condition was kept mostly hidden from the press.

4. Franklin D. Roosevelt

Portrait of a young Franklin D. Roosevelt.
Hulton Archive // Getty Images

In 1918, Franklin D. Roosevelt was the Assistant Secretary of the Navy, and had been in Europe for two months before contracting the flu on the boat home. The New York Times described his illness as “a slight attack of pneumonia caused by Spanish influenza.” Roosevelt convalesced at his mother’s New York City home until he was well enough to head back to Washington, D.C.

5. Woodrow Wilson

Woodrow Wilson circa 1912.
Hulton Archive // Getty Images

Considering Woodrow Wilson was president of the United States and he was dealing with the end of WWI, early 1919 was a seriously inconvenient time to get sick. Not only did he get the flu, but he fell ill so violently and so quickly that his doctors were sure he had been poisoned. When Wilson was well enough to rejoin the “Big Three” negotiations a few days later, people commented on how weak and out of it he seemed.

6. Wilhelm II

Wilhelm II in his uniform.
Hulton Archive // Getty Images

While the German Kaiser was undoubtedly upset to get sick himself, he had reason to be happy about the flu epidemic, or so he thought. One of his military generals insisted—despite the fact that the surgeon general disagreed—that the illness would decimate the French troops, while leaving the Germans mostly unharmed. Since Germany needed a miracle to win the war, the flu must have seemed like a godsend. In the end, it ravaged all armies pretty much equally, and Germany surrendered.

7. John J. Pershing

John J. Pershing in uniform sitting on a horse.
Hulton Archive // Getty Images

While the great American general got sick himself, the flu gave him a much larger problem. His troops were dying at a faster rate from illness than from bullets. Soon there were more than 16,000 cases among U.S. troops in Europe alone. Pershing was forced to ask the government for more than 30 mobile hospitals and 1500 nurses in just over a week.

8. Haile Selassie I

Haile Selassie sitting in a chair drinking tea.
Keystone/Hulton Archive/Getty Images

The future emperor of Ethiopia was one of the first Ethiopians to contract the disease. His country was woefully unprepared for the epidemic: There were only four doctors in the capital available to treat patients. Selassie survived, but it's unknown how many people the flu killed in Ethiopia; it killed 7 percent of the population of neighboring British Somaliland.

9. Leo Szilard

A black and white photo of Leo Szilard in a suit and tie.
Department of Energy, Wikimedia Commons // Public Domain

You may not have heard of him, but the atomic scientist Leo Szilard might have saved the world. While he survived the flu during WWI (he was supposedly cured by spending time in a humid room, the standard treatment for respiratory illness at the time), what he should be remembered for is his foresight before WWII. When he and other physicists were discovering different aspects of nuclear fission, he persuaded his colleagues to keep quiet about it, so that the Nazis wouldn’t get any closer to making an atomic bomb.

10. Katherine Anne Porter

Author Katherine Anne Porter sitting in a chair wearing a hat with a bow on it.
Hulton Archive // Getty Images

The author turned her experience with sickness in 1918 into a short novel called Pale Horse, Pale Rider. The story is told by a woman with the flu who is tended to by a young soldier. While she recovers, he contracts the disease and dies.

11. Alfonso XIII

The King of Spain working at his desk.
Keystone/Hulton Archive/Getty Images

Alfonso was the King of Spain when the “Spanish” flu hit, and he was not immune to its outbreak. The flu was no worse in Spain than anywhere else, but unlike most journalists in other countries—who were under wartime censorship—the Spanish media actually covered the pandemic, leading to an unfair association that persists to this day.

12. Edvard Munch

A portrait of Edvard Munch standing in the snow.
Nasjonalbiblioteket, Wikimedia Commons // CC BY 2.0

Munch, the artist behind The Scream, had an apparent obsession with sickness and death long before he came down with the flu—he painted many works on the subject. But the flu obviously affected him especially: He painted a few self-portraits of both his illness and shortly after his recovery.

13. Lillian Gish

A portrait of Lillian Gish.
General Photographic Agency // Getty Images

The silent film star started feeling sick during a costume fitting and collapsed with a 104-degree fever when she got home. Fortunately, she could afford a doctor and two nurses to attend to her around the clock. While she recovered, it wasn’t all good news. Gish complained later, “The only disagreeable thing was that it left me with flannel nightgowns—have to wear them all winter—horrible things.”

14. Clementine Churchill

Clementine Churchill speaks at a microphone.
Arthur Tanner/Topical Press Agency/Getty Images

While Winston was in France in 1919, the Churchill household—including his wife Clementine and their nanny Isabelle, who was looking after their young daughter Marigold—contracted the flu. According to Churchill’s daughter Mary Soames, Isabelle grew delirious and took Marigold from her cot despite being sick herself. Clementine grabbed the child and was anxious for days about Marigold’s condition. Isabelle died of the flu, but Clementine and Marigold survived. (Sadly, Marigold would die from a bacterial infection that developed into sepsis in 1921.)

During World War II, Clementine served as a close adviser to Winston. She was also the “Chairman” of the Red Cross Aid to Russia Fund, which raised 8 million pounds during WWII and resulted in her being awarded the Soviet Order of the Red Banner of Labor, being made a Dame, and being given a 19th century glass fruit bowl from Stalin. Churchill’s Chief Staff Officer, General Hastings “Pug” Ismay, would later comment that without Clementine the “history of Winston Churchill and of the world would have been a very different story.”

7 Timeless Facts About Paul Rudd

Rich Fury, Getty Images
Rich Fury, Getty Images

Younger fans may know Paul Rudd as Ant-Man, one of the newest members of the Marvel Cinematic Universe. However, the actor has been a Hollywood mainstay for half his life.

Rudd's breakout role came in 1995’s Clueless, where he played Josh, Alicia Silverstone's charming love interest in Amy Heckerling's beloved spin on Jane Austen's Emma. In the 2000s, Rudd became better known for his comedic work when he starred in movies like Wet Hot American Summer (2001), Anchorman (2004), The 40-Year-Old Virgin (2005), Knocked Up (2007), and I Love You, Man (2009).

It wasn’t until 2015 that Rudd stepped into the ever-growing world of superhero movies when he was cast as Scott Lang, a.k.a. Ant-Man, and became part of the MCU.

Rudd has proven he can take on any part, serious or goofy. More amazingly, he never seems to age. But in honor of (what is allegedly) his 51st birthday on April 6, here are some things you might not have known about the star.

1. Paul Rudd is technically Paul Rudnitzky.

Though Paul Rudd was born in Passaic, New Jersey, both of his parents hail from London—his father was from Edgware and his mother from Surbiton. Both of his parents were descendants of Jewish immigrants who moved to England from from Russia and Poland. Rudd’s last name was actually Rudnitzky, but it was changed by his grandfather.

2. Paul Rudd's parents are second cousins.

In a 2017 episode of Finding Your Roots, Rudd learned that his parents were actually second cousins. Rudd responded to the discovery in typical comedic fashion: "Which explains why I have six nipples." He also wondered what that meant for his own family. "Does this make my son also my uncle?," he asked.

3. Paul Rudd loved comic books as a kid.

While Rudd did read Marvel Comics as a kid, he preferred Archie Comics and other funny stories. His English cousins would send him British comics, too, like Beano and Dandy, which he loved.

4. Paul Rudd wanted to play Christian in Clueless. And Murray.

Clueless would have been a completely different movie if Rudd had been cast as the suave Christian instead of the cute older step-brother-turned-love-interest Josh. But before he was cast as Cher’s beau, he initially wanted the role of the “ringa ding kid” Christian.

"I thought Justin Walker’s character, Christian, was a really good part," Rudd told Entertainment Weekly in 2012. "It was a cool idea, something I’d never seen in a movie before—the cool gay kid. And then I asked to read for Donald Faison's part, because I thought he was kind of a funny hip-hop wannabe. I didn’t realize that the character was African-American.”

5. Paul Rudd idolizes Paul Newman.

In a 2008 interview for Role Models, which he both co-wrote and starred in, Rudd was asked about his real-life role model. He answered Paul Newman, saying he admired the legendary actor because he gave a lot to the world before leaving it.

6. Before Paul Rudd was Ant-Man, he wanted to be Adam Ant.

In a 2011 interview with Grantland, Rudd talked about his teenage obsession with '80s English rocker Adam Ant. "Puberty hit me like a Mack truck, and my hair went from straight to curly overnight," Rudd explained. "But it was an easier pill to swallow because Adam Ant had curly hair. I used to ask my mom to try and shave my head on the sides to give me a receding hairline because Adam Ant had one. I didn’t know what a receding hairline was. I just thought he looked cool. She said, 'Absolutely not,' but I was used to that."

Ant wasn't the only musician Rudd tried to emulate. "[My mom] also shot me down when I asked if I could bleach just the top of my head like Howard Jones. Any other kid would’ve been like, 'F*** you, mom! I’m bleaching my hair.' I was too nice," he said.

7. Romeo + Juliet wasn’t Paul Rudd's first go as a Shakespearean actor.

Yet another one of Rudd's iconic '90s roles was in Baz Luhrmann's Romeo + Juliet, but it was far from the actor's first brush with Shakespeare. Rudd spent three years studying Jacobean theater in Oxford, England, and starred in a production of Twelfth Night. He was described by his director, Sir Nicholas Hytner, as having “emotional and intellectual volatility.” Hytner’s praise was a big deal, considering he was the director of London's National Theatre from 2003 until 2015.

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