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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?

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15 Heartwarming Facts About Mister Rogers
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Though Mister Rogers' Neighborhood premiered 50 years ago, Fred Rogers remains an icon of kindness for the ages. An innovator of children’s television, his salt-of-the-earth demeanor and genuinely gentle nature taught a generation of kids the value of kindness. In celebration of the groundbreaking children's series' 50th anniversary, here are 15 things you might not have known about everyone’s favorite “neighbor.”

1. HE WAS BULLIED AS A CHILD.

According to Benjamin Wagner, who directed the 2010 documentary Mister Rogers & Me—and was, in fact, Rogers’s neighbor on Nantucket—Rogers was overweight and shy as a child, and often taunted by his classmates when he walked home from school. “I used to cry to myself when I was alone,” Rogers said. “And I would cry through my fingers and make up songs on the piano.” It was this experience that led Rogers to want to look below the surface of everyone he met to what he called the “essential invisible” within them.

2. HE WAS AN ORDAINED MINISTER.

Rogers was an ordained minister and, as such, a man of tremendous faith who preached tolerance wherever he went. When Amy Melder, a six-year-old Christian viewer, sent Rogers a drawing she made for him with a letter that promised “he was going to heaven,” Rogers wrote back to his young fan:

“You told me that you have accepted Jesus as your Savior. It means a lot to me to know that. And, I appreciated the scripture verse that you sent. I am an ordained Presbyterian minister, and I want you to know that Jesus is important to me, too. I hope that God’s love and peace come through my work on Mister Rogers’ Neighborhood.”

3. HE RESPONDED TO ALL HIS FAN MAIL.

Responding to fan mail was part of Rogers’s very regimented daily routine, which began at 5 a.m. with a prayer and included time for studying, writing, making phone calls, swimming, weighing himself, and responding to every fan who had taken the time to reach out to him.

“He respected the kids who wrote [those letters],” Heather Arnet, an assistant on Mister Rogers’ Neighborhood, told the Pittsburgh Post-Gazette in 2005. “He never thought about throwing out a drawing or letter. They were sacred."

According to Arnet, the fan mail he received wasn’t just a bunch of young kids gushing to their idol. Kids would tell Rogers about a pet or family member who died, or other issues with which they were grappling. “No child ever received a form letter from Mister Rogers," Arnet said, noting that he received between 50 and 100 letters per day.

4. ANIMALS LOVED HIM AS MUCH AS PEOPLE DID.

It wasn’t just kids and their parents who loved Mister Rogers. Koko, the Stanford-educated gorilla who understands 2000 English words and can also converse in American Sign Language, was an avid Mister Rogers’ Neighborhood watcher, too. When Rogers visited her, she immediately gave him a hug—and took his shoes off.

5. HE WAS AN ACCOMPLISHED MUSICIAN.

Though Rogers began his education in the Ivy League, at Dartmouth, he transferred to Rollins College following his freshman year in order to pursue a degree in music (he graduated Magna cum laude). In addition to being a talented piano player, he was also a wonderful songwriter and wrote all the songs for Mister Rogers' Neighborhood—plus hundreds more.

6. HIS INTEREST IN TELEVISION WAS BORN OUT OF A DISDAIN FOR THE MEDIUM.

Rogers’s decision to enter into the television world wasn’t out of a passion for the medium—far from it. "When I first saw children's television, I thought it was perfectly horrible," Rogers told Pittsburgh Magazine. "And I thought there was some way of using this fabulous medium to be of nurture to those who would watch and listen."

7. KIDS WHO WATCHED MISTER ROGERS’ NEIGHBORHOOD RETAINED MORE THAN THOSE WHO WATCHED SESAME STREET.

A Yale study pitted fans of Sesame Street against Mister Rogers’ Neighborhood watchers and found that kids who watched Mister Rogers tended to remember more of the story lines, and had a much higher “tolerance of delay,” meaning they were more patient.

8. ROGERS’S MOM KNIT ALL OF HIS SWEATERS.

If watching an episode of Mister Rogers’ Neighborhood gives you sweater envy, we’ve got bad news: You’d never be able to find his sweaters in a store. All of those comfy-looking cardigans were knitted by Fred’s mom, Nancy. In an interview with the Archive of American Television, Rogers explained how his mother would knit sweaters for all of her loved ones every year as Christmas gifts. “And so until she died, those zippered sweaters I wear on the Neighborhood were all made by my mother,” he explained.

9. HE WAS COLORBLIND.

Those brightly colored sweaters were a trademark of Mister Rogers’ Neighborhood, but the colorblind host might not have always noticed. In a 2003 article, just a few days after his passing, the Pittsburgh Post-Gazette wrote that:

Among the forgotten details about Fred Rogers is that he was so colorblind he could not distinguish between tomato soup and pea soup.

He liked both, but at lunch one day 50 years ago, he asked his television partner Josie Carey to taste it for him and tell him which it was.

Why did he need her to do this, Carey asked him. Rogers liked both, so why not just dip in?

"If it's tomato soup, I'll put sugar in it," he told her.

10. HE WORE SNEAKERS AS A PRODUCTION CONSIDERATION.

According to Wagner, Rogers’s decision to change into sneakers for each episode of Mister Rogers’ Neighborhood was about production, not comfort. “His trademark sneakers were born when he found them to be quieter than his dress shoes as he moved about the set,” wrote Wagner.

11. MICHAEL KEATON GOT HIS START ON THE SHOW.

Oscar-nominated actor Michael Keaton's first job was as a stagehand on Mister Rogers' Neighborhood, manning Picture, Picture, and appearing as Purple Panda.

12. ROGERS GAVE GEORGE ROMERO HIS FIRST PAYING GIG, TOO.

It's hard to imagine a gentle, soft-spoken, children's education advocate like Rogers sitting down to enjoy a gory, violent zombie movie like Dawn of the Dead, but it actually aligns perfectly with Rogers's brand of thoughtfulness. He checked out the horror flick to show his support for then-up-and-coming filmmaker George Romero, whose first paying job was with everyone's favorite neighbor.

“Fred was the first guy who trusted me enough to hire me to actually shoot film,” Romero said. As a young man just out of college, Romero honed his filmmaking skills making a series of short segments for Mister Rogers’ Neighborhood, creating a dozen or so titles such as “How Lightbulbs Are Made” and “Mr. Rogers Gets a Tonsillectomy.” The zombie king, who passed away in 2017, considered the latter his first big production, shot in a working hospital: “I still joke that 'Mr. Rogers Gets a Tonsillectomy' is the scariest film I’ve ever made. What I really mean is that I was scared sh*tless while I was trying to pull it off.”

13. ROGERS HELPED SAVE PUBLIC TELEVISION.

In 1969, Rogers—who was relatively unknown at the time—went before the Senate to plead for a $20 million grant for public broadcasting, which had been proposed by President Johnson but was in danger of being sliced in half by Richard Nixon. His passionate plea about how television had the potential to turn kids into productive citizens worked; instead of cutting the budget, funding for public TV increased from $9 million to $22 million.

14. HE ALSO SAVED THE VCR.

Years later, Rogers also managed to convince the Supreme Court that using VCRs to record TV shows at home shouldn’t be considered a form of copyright infringement (which was the argument of some in this contentious debate). Rogers argued that recording a program like his allowed working parents to sit down with their children and watch shows as a family. Again, he was convincing.

15. ONE OF HIS SWEATERS WAS DONATED TO THE SMITHSONIAN.

In 1984, Rogers donated one of his iconic sweaters to the Smithsonian’s National Museum of American History.

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5 Things You Might Not Know About Ansel Adams

You probably know Ansel Adams—who was born on February 20, 1902—as the man who helped promote the National Park Service through his magnificent photographs. But there was a lot more to the shutterbug than his iconic, black-and-white vistas. Here are five lesser-known facts about the celebrated photographer.

1. AN EARTHQUAKE LED TO HIS DISTINCTIVE NOSE.

Adams was a four-year-old tot when the 1906 San Francisco earthquake struck his hometown. Although the boy managed to escape injury during the quake itself, an aftershock threw him face-first into a garden wall, breaking his nose. According to a 1979 interview with TIME, Adams said that doctors told his parents that it would be best to fix the nose when the boy matured. He joked, "But of course I never did mature, so I still have the nose." The nose became Adams' most striking physical feature. His buddy Cedric Wright liked to refer to Adams' honker as his "earthquake nose.

2. HE ALMOST BECAME A PIANIST.

Adams was an energetic, inattentive student, and that trait coupled with a possible case of dyslexia earned him the heave-ho from private schools. It was clear, however, that he was a sharp boy—when motivated.

When Adams was just 12 years old, he taught himself to play the piano and read music, and he quickly showed a great aptitude for it. For nearly a dozen years, Adams focused intensely on his piano training. He was still playful—he would end performances by jumping up and sitting on his piano—but he took his musical education seriously. Adams ultimately devoted over a decade to his study, but he eventually came to the realization that his hands simply weren't big enough for him to become a professional concert pianist. He decided to leave the keys for the camera after meeting photographer Paul Strand, much to his family's dismay.

3. HE HELPED CREATE A NATIONAL PARK.

If you've ever enjoyed Kings Canyon National Park in California, tip your cap to Adams. In the 1930s Adams took a series of photographs that eventually became the book Sierra Nevada: The John Muir Trail. When Adams sent a copy to Secretary of the Interior Harold Ickes, the cabinet member showed it to Franklin Roosevelt. The photographs so delighted FDR that he wouldn't give the book back to Ickes. Adams sent Ickes a replacement copy, and FDR kept his with him in the White House.

After a few years, Ickes, Adams, and the Sierra Club successfully convinced Roosevelt to make Kings Canyon a national park in 1940. Roosevelt's designation specifically provided that the park be left totally undeveloped and roadless, so the only way FDR himself would ever experience it was through Adams' lenses.

4. HE WELCOMED COMMERCIAL ASSIGNMENTS.

While many of his contemporary fine art photographers shunned commercial assignments as crass or materialistic, Adams went out of his way to find paying gigs. If a company needed a camera for hire, Adams would generally show up, and as a result, he had some unlikely clients. According to The Ansel Adams Gallery, he snapped shots for everyone from IBM to AT&T to women's colleges to a dried fruit company. All of this commercial print work dismayed Adams's mentor Alfred Stieglitz and even worried Adams when he couldn't find time to work on his own projects. It did, however, keep the lights on.

5. HE AND GEORGIA O'KEEFFE WERE FRIENDS.

Adams and legendary painter O'Keeffe were pals and occasional traveling buddies who found common ground despite their very different artistic approaches. They met through their mutual friend/mentor Stieglitz—who eventually became O'Keeffe's husband—and became friends who traveled throughout the Southwest together during the 1930s. O'Keeffe would paint while Adams took photographs.

These journeys together led to some of the artists' best-known work, like Adams' portrait of O'Keeffe and a wrangler named Orville Cox, and while both artists revered nature and the American Southwest, Adams considered O'Keeffe the master when it came to capturing the area. 

“The Southwest is O’Keeffe’s land,” he wrote. “No one else has extracted from it such a style and color, or has revealed the essential forms so beautifully as she has in her paintings.”

The two remained close throughout their lives. Adams would visit O'Keeffe's ranch, and the two wrote to each other until Adams' death in 1984.

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