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?

15 Long-Lost Words To Revive This Christmas

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Nog. Tidings. Wassail. Every time Christmas rolls around it brings with it its own vocabulary of words you barely hear the rest of the year. But while words derived from ancient English ales (like the nog in eggnog) and Middle English greetings (wassail is thought to derive from a Germanic phrase meaning “good health!”) are one thing, some choice festive words haven’t stood the test of time, and are basically unknown outside of the dustiest corners of the dictionary.

Here are 15 long-lost and long-forgotten words to get you through the holiday season.

1. Ninguid

Derived from Latin, a landscape that is ninguid is snow-covered. And if that’s what your walk to work looks like over the festive period, you might also need to know that to meggle is to trudge laboriously through snow. (A peck-of-apples, meanwhile, is a fall on ice.)

2. Crump

That crunching sound you make walking on partially frozen snow is called crumping.

3. Hiemate

Hibernate is sleeping throughout the entire winter; hiemate is to spend winter somewhere.

4. Yuleshard

As another word for the festive period, Yule comes via Old English from jol, an ancient Scandinavian word for a series of end-of-year festivities. A yuleshard—also called a yule-jade (jade being an insult once upon a time)—is someone who leaves a lot of work still to be done on Christmas Eve night.

5. Yule-Hole

And the yule-hole is the (usually makeshift) hole you need to move your belt to after you’ve eaten a massive meal.

6. Belly-Cheer

Dating from the 1500s, belly-cheer or belly-timber is a brilliantly evocative word for fine food or gluttonous eating.

7. Doniferous

If you’re doniferous then you’re carrying a present. The act of offering a present is called oblation, which originally was (and, in some contexts, still is) a religious term referring specifically to the presentation of money or donation of goods to the church. But since the 15th century it’s been used more loosely to refer to the action of offering or presenting any gift or donation, or, in particular, a gratuity.

8. Pourboire

Speaking of gratuities, a tip or donation of cash intended to be spent on drink is a pourboire—French, literally, for “for drink.” Money given in lieu of a gift, meanwhile, has been known as present-silver since the 1500s.

9. Toe-Cover

A cheap and totally useless present? In 1940s slang, that was a toe-cover.

10. Xenium

A gift given to a houseguest, or a gift given by a guest to their host, is called a xenium.

11. Scurryfunge

Probably distantly related to words like scour or scourge, scurryfunge first appeared in the late 18th century, with meanings of “to lash” or, depending on region, “to scour.” By the mid-1900s, however, things had changed: perhaps in allusion to scrubbing or working hard enough to abrade a surface, scurryfunge came to mean “to hastily tidy a house” before unexpected company arrive.

12. Quaaltagh

Quaaltagh was actually borrowed into English in the 1800s from Manx, the Celtic-origin language spoken on the Isle of Man—a tiny island located halfway between Britain and Ireland in the Irish Sea. It was on the Isle of Man that festive tradition dictates that the identity of the first person you see (or the first to enter your house) on Christmas or New Year morning will have some bearing on the events of the year to come. And in Manx culture, the person you meet on that early-morning encounter is called the quaaltagh.

13. Lucky-Bird

We’re more likely to call them a first-footer these days, but according to old Yorkshire folklore the first person across the threshold of your home on New Year’s morning is the lucky-bird. And just like the quaaltagh, tradition dictates that the identity of the lucky-bird has an important bearing on the success of the year to come: Men are the most fortuitous lucky-birds; depending on region, either dark-haired or light-haired men might be favored (but dark-haired is more common). Other regional variations claimed the man had to be a bachelor, had to bring a gift of coal (though by the 1880s whisky was increasingly popular), and/or had to have a high arch on the foot. People with a suitable combination for their region could “become almost professional,” according to the Leeds Mercury Weekly Supplement.

14. Apolausticism

Derived from the Greek word for “to enjoy,” apolausticism is a long-lost 19th-century word for a total devotion to enjoying yourself.

15. Crapulence

Once all the festive dust and New Year confetti has settled, here’s a word for the morning after the night before: crapulence, as the Oxford English Dictionary puts it, is an 18th-century word for “sickness or indisposition resulting from excess in drinking or eating.”

14 Facts About Celiac Disease

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Going gluten-free may be a modern diet trend, but people have been suffering from celiac disease—a chronic condition characterized by gluten intolerance—for centuries. Patients with celiac are ill-equipped to digest products made from certain grains containing gluten; wheat is the most common. In the short-term this can cause gastrointestinal distress, and in the long-term it can foster symptoms associated with early death.

Celiac diagnoses are more common than ever, which also means awareness of how to live with the condition is at an all-time high. Here are some things you might not know about celiac disease symptoms and treatments.

1. Celiac an autoimmune disease.

The bodies of people with celiac have a hostile reaction to gluten. When the protein moves through the digestive tract, the immune system responds by attacking the small intestine, causing inflammation that damages the lining of the organ. As this continues over time, the small intestine has trouble absorbing nutrients from other foods, which can lead to additional complications like anemia and osteoporosis.

2. You can get celiac disease from your parents.

Nearly all cases of celiac disease arise from certain variants of the genes HLA-DQA1 and HLA-DQB1. These genes help produce proteins in the body that allow the immune system to identify potentially dangerous foreign substances. Normally the immune system wouldn't label gliadin, a segment of the gluten protein, a threat, but due to mutations in these genes, the bodies of people with celiac treat gliadin as a hostile invader.

Because it's a genetic disorder, people with a first-degree relative (a sibling, parent, or child) with celiac have a 4 to 15 percent chance of having it themselves. And while almost all patients with celiac have these specific HLA-DQA1 and HLA-DQB1 variations, not everyone with the mutations will develop celiac. About 30 percent of the population has these gene variants, and only 3 percent of that group goes on to develop celiac disease.

3. Makeup might contribute to celiac disease symptoms.

People with celiac disease can’t properly process gluten, the protein naturally found in the grains like wheat, rye, and barley. Patients have to follow strict dietary guidelines and avoid most bread, pasta, and cereal, in order to manage their symptoms. But gluten isn’t limited to food products: It can also be found in some cosmetics. While makeup containing gluten causes no issues for many people with celiac, it can provoke rashes in others or lead to more problems if ingested. For those folks, gluten-free makeup is an option.

4. The name comes from 1st-century Greece.

A 1st-century Greek physician named Aretaeus of Cappadocia may have been the first person to describe celiac disease symptoms in writing [PDF]. He named it koiliakos after the Greek word koelia for abdomen, and he referred to people with the condition as coeliacs. In his description he wrote, “If the stomach be irretentive of the food and if it pass through undigested and crude, and nothing ascends into the body, we call such persons coeliacs.”

5. There are nearly 300 celiac disease symptoms.

Celiac disease may start in the gut, but it can be felt throughout the whole body. In children, the condition usually manifests as bloating, diarrhea, and abdominal discomfort, but as patients get older they start to experience more “non-classical” symptoms like anemia, arthritis, and fatigue. There are at least 281 symptoms associated with celiac disease, many of which overlap with other conditions and make celiac hard to diagnose. Other common symptoms of the disease include tooth discoloration, anxiety and depression, loss of fertility, and liver disorders. Celiac patients also have a greater chance of developing an additional autoimmune disorder, with the risk increasing the later in life the initial condition is diagnosed.

6. Some patients show no symptoms at all.

It’s not uncommon for celiac disease to be wrecking a patient’s digestive tract while showing no apparent symptoms. This form of the condition, sometimes called asymptomatic or “silent celiac disease,” likely contributes to part of the large number of people with celiac who are undiagnosed. People who are at high risk for the disease (the children of celiac sufferers, for example), or who have related conditions like type 1 diabetes and Down syndrome (both conditions that put patients at a greater risk for developing new autoimmune diseases) are encouraged to get tested for it even if they aren’t showing any signs.

7. It’s not the same as wheat sensitivity.

Celiac is often confused with wheat sensitivity, a separate condition that shares many symptoms with celiac, including gastrointestinal issues, depression, and fatigue. It’s often called gluten sensitivity or gluten intolerance, but because doctors still aren’t sure if gluten is the cause, many refer to it as non-celiac wheat sensitivity. There’s no test for it, but patients are often treated with the same gluten-free diet that’s prescribed to celiac patients.

8. It's not a wheat allergy either.

Celiac disease is often associated with wheat because it's one of the more common products containing gluten. While it's true that people with celiac can't eat wheat, the condition isn't a wheat allergy. Rather than reacting to the wheat, patients react to a specific protein that's found in the grain as well as others.

9. It can develop at any age.

Just because you don’t have celiac now doesn’t mean you’re in the clear for life: The disease can develop at any age, even in people who have tested negative for it previously. There are, however, two stages of life when symptoms are most likely to appear: early childhood (8 to 12 months) and middle adulthood (ages 40 to 60). People already genetically predisposed to celiac become more susceptible to it when the composition of their intestinal bacteria changes as they get older, either as a result of infection, surgery, antibiotics, or stress.

10. Not all grains are off-limits.

A gluten-free diet isn’t necessarily a grain-free diet. While it’s true that the popular grains wheat, barley, and rye contain gluten, there are plenty of grains and seeds that don’t and are safe for people with celiac to eat. These include quinoa, millet, amaranth, buckwheat, sorghum, and rice. Oats are also naturally gluten-free, but they're often contaminated with gluten during processing, so consumers with celiac should be cautious when buying them.

11. Celiac disease can be detected with a blood test.

Screenings for celiac disease used to be an involved process, with doctors monitoring patients’ reactions to their gluten-free diet over time. Today all it takes is a simple test to determine whether someone has celiac. People with the condition will have anti-tissue transglutaminase antibodies in their bloodstream. If a blood test confirms the presence of these proteins in a patient, doctors will then take a biopsy of their intestine to confirm the root cause.

12. The gluten-free diet doesn’t work for all patients.

Avoiding gluten is the most effective way to manage celiac disease, but the treatment doesn’t work 100 percent of the time. In up to a fifth of patients, the damaged intestinal lining does not recover even a year after switching to a gluten-free diet. Most cases of non-responsive celiac disease can be explained by people not following the diet closely enough, or by having other conditions like irritable bowel syndrome, lactose intolerance, or small intestine bacterial overgrowth that impede recovery. Just a small fraction of celiac disease sufferers don’t respond to a strict gluten-free diet and have no related conditions. These patients are usually prescribed steroids and immunosuppressants as alternative treatments.

13. If you don’t have celiac, gluten probably won’t hurt you.

The gluten-free diet trend has exploded in popularity in recent years, and most people who follow it have no medical reason to do so. Going gluten-free has been purported to do everything from help you lose weight to treat autism—but according to doctors, there’s no science behind these claims. Avoiding gluten may help some people feel better and more energetic because it forces them to cut heavily processed junk foods out of their diet. In such cases it’s the sugar and carbs that are making people feel sluggish—not the gluten protein. If you don’t have celiac or a gluten sensitivity, most experts recommend saving yourself the trouble by eating healthier in general rather than abstaining from gluten.

14. The numbers are growing.

A 2009 study found that four times as many people have celiac today than in the 1950s, and the spike can’t be explained by increased awareness alone. Researchers tested blood collected at the Warren Air Force Base between 1948 and 1954 and compared them to fresh samples from candidates living in one Minnesota county. The results supported the theory that celiac has become more prevalent in the last half-century. While experts aren’t exactly sure why the condition is more common today, it may have something to do with changes in how wheat is handled or the spread of gluten into medications and processed foods.

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