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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10 Facts About Sagamore Hill, Theodore Roosevelt's Home

Theodore Roosevelt's Long Island home has 23 rooms and more books than you can count.
Theodore Roosevelt's Long Island home has 23 rooms and more books than you can count.
J. Stephen Conn, Flickr // CC by NC 2.0

Fleeing Manhattan for the country is a tradition that wealthy New Yorkers have partaken in for centuries—and our 26th president, Theodore Roosevelt, was no exception. Starting when he was a teen, TR and his family would retreat to Long Island for the summer, and as an adult, he built his own home there: Sagamore Hill, which became his permanent home after his presidency. In honor of what would be TR’s 162nd birthday, here are 10 facts about Sagamore Hill, of which Roosevelt once wrote, “there isn't any place in the world like home—like Sagamore Hill.”

1. Sagamore Hill was built near where Theodore Roosevelt spent his childhood summers.

Oyster Bay on Long Island, New York, first served as a refuge for a sickly TR in his youth. He’d hike, ride horses, row, and swim—generally engaging in the “strenuous life” and beginning his lifelong love affair with nature. The family home was known as Tranquility, and was situated two miles southwest from the future Sagamore Hill mansion.

2. Theodore Roosevelt bought the land for Sagamore Hill in 1880.

The same year he married his first wife, Alice Hathaway Lee, Roosevelt purchased 155 acres on the north shore of Long Island for $30,000 to build a home. Situated on Long Island Sound, the site is home to a wide variety of habitats, from woodlands to salt marshes, as well as plenty of ecological diversity, thus giving Roosevelt much to observe and document.

3. Sagamore Hill wasn't supposed to go by that name.

The home that would become Sagamore Hill was originally going to be named Leeholm, after Roosevelt's wife Alice. However, following her tragic death shortly after giving birth to their daughter, the property was renamed Sagamore—according to Roosevelt, after Sagamore Mohannis (today more commonly known as Sachem Mohannes), who was chief of a tribe in the area over 200 years earlier. Sagamore is an Algonquian word for "chieftain."

4. Theodore Roosevelt had very specific ideas for the layout of Sagamore Hill.

Among his "perfectly definite views" for the home, he would later recall, were "a library with a shallow bay window opening south, the parlor or drawing-room occupying all the western end of the lower floor; as broad a hall as our space would permit; big fireplaces for logs; on the top floor a gun room occupying the western end so that north and west it [looks] over the Sound and Bay." Long Island builder John A. Wood began work on the Queen Anne-style mansion (designed by New York architecture firm Lamb and Rich), on March 1, 1884. It was completed in 1885, with Roosevelt's sister, Anna, taking care of the house (and new baby Alice) while Roosevelt was out west in the Dakota Badlands, nursing his grieving heart.

5. Theodore Roosevelt delivered campaign speeches from the porches of Sagamore Hill.

Theodore Roosevelt addresses a crowd of 500 suffragettes from the porch of his Sagamore Hill home around 1905. Hulton Archive/Getty Images

It was one of Roosevelt’s greatest wishes for the Sagamore Hill home to possess "a very big piazza ... where we could sit in rocking chairs and look at the sunset," and so wide porches were built on the south and west sides of the house. Roosevelt would use the piazza to deliver speeches to the public, and it was here that he was notified of his nominations as governor of New York (1898), vice president (1900) and president (1904).

6. Sagamore Hill was Theodore Roosevelt's "Summer White House."

Roosevelt became the first president to bring his work home with him, spending each of his summers as president at Sagamore Hill. He even had a phone installed so he could conduct business from the house. But by 1905, Edith had had enough of TR usurping the drawing room—which was supposed to be her office—to hold his visitors [PDF], and of his gaming trophies and other treasures taking up space. So the Roosevelts constructed what would become the North Room. "The North Room cost as much as the entire house had," Susan Sarna, curator at Sagamore Hill, told Cowboys & Indians magazine in 2016. "It is grandiose." Measuring 40 feet by 20 feet, with ceilings 20 feet high, it was constructed of mahogany brought in from the Philippines. The addition brought the total number of rooms at Sagamore Hill from 22 to 23.

7. Theodore Roosevelt met with foreign leaders at Sagamore Hill.

Roosevelt stands between Russian and Japanese dignitaries in Portsmouth, New Hampshire, in 1905. On September 5, they signed the Treaty of Portsmouth, ending the Russo-Japanese War and earning Roosevelt the 1906 Nobel Peace Prize; he was the first American to win a Nobel Prize of any kind.Photos.com/iStock via Getty Images

In September 1905, Roosevelt brokered peace talks between Russian and Japanese dignitaries, which led to end of the Russo-Japanese War. But before the peace talks (which took place on a yacht in the Navy yard at Portsmouth, New Hampshire), Roosevelt met the negotiators—from Japan, Takahira Kogorō, ambassador to the U.S., and diplomat Jutaro Komura; and from Russia, diplomat Baron Roman Romanovich von Rosen and Sergei Iluievich Witte—at Sagamore Hill. TR earned a Nobel Peace Prize for his efforts.

8. Sagamore Hill has a pet cemetery.

Roosevelt’s love of animals was passed down to his six children, who adopted a veritable menagerie, including cats, dogs, horses, guinea pigs, a bear, and a badger. A number of those beloved companions ended up in Sagamore Hill's pet cemetery; among them is Little Texas, the horse TR rode on his charge up Kettle Hill during the Spanish-American War.

9. Life at Sagamore Hill was lively.

The atmosphere at Sagamore Hill was a boisterous one. According to the National Park Service, Massachusetts Senator Henry Cabot Lodge complained about how late they stayed up, how loud they talked, and how early they woke up. Eleanor Roosevelt, Roosevelt’s favorite niece, too, recalled a constant barrage of activity during her visits. The children partook in all manner of outdoor activities, and Roosevelt was known for abruptly ending his appointments in order to join them.

10. Theodore Roosevelt died at Sagamore Hill.

Roosevelt passed away on January 6, 1919 at Sagamore Hill. Edith died there on September 30, 1948, and five years later, Sagamore Hill was opened to the public. In 2015, a $10 million renovation of the house was completed; 99 percent of what can be seen at the home today is original—including thousands of books, extensive artwork, and yes, 36 pieces of taxidermy.

Shortly before Roosevelt died, he asked Edith, “I wonder if you will ever know how I love Sagamore Hill?” and thanks to the extensive work done to restore his home, we all can.