At first glance, the paradox suggests that one person is causing the rest of the bar to drink. In fact, all it's saying is that it would be impossible for everyone in the bar to be drinking unless every single customer were drinking. Therefore, there is at least one customer there i. The Banach-Tarski paradox relies on a lot of the strange and counterintuitive properties of infinite sets and geometric rotations. The pieces that the ball gets cut into are very strange-looking, and the paradox only works for an abstract, mathematical sphere: As nice as it would be to take an apple, cut it up, and reassemble the pieces so you have an extra apple for your friend, physical balls made of matter can't be disassembled like a purely mathematical sphere.
The key to the potato paradox is to closely look at the math behind the nonwater content of the potato. The potato starts at grams, so that means that it contains 1 gram of dry material. Another surprising math result, the birthday paradox comes from a careful analysis of the probabilities involved. The friendship paradox is caused by how, in most social networks, most people have a few friends, while a handful of people have a large number of friends.
Those social butterflies in the second group disproportionately show up as friends of people with smaller numbers of friends, and drag up the average number of friends-of-friends accordingly. The bootstrap paradox is the opposite of the classic grandfather paradox : Rather than going back in time and preventing oneself from going back in time, some information or object is brought back in time, becoming a "younger" version of itself, and enabling itself later to travel back in time.
One then has to ask: How did that information or object come into being in the first place? The bootstrap paradox is common in science fiction and takes its name from a short story by Robert Heinlein. One of the underlying assumptions in astronomy is that Earth is a pretty common planet in a pretty common solar system in a pretty common galaxy, and that there is nothing cosmically unique about us.
NASA's Kepler satellite has found evidence that there are probably 11 billion Earth-like planets in our galaxy. Given this, life somewhat like us should have evolved somewhere not overly far away from us at least on a cosmic scale.
But despite developing ever-more-powerful telescopes, we have had no evidence of technological civilizations anywhere else in the universe. Civilizations are noisy: Humanity broadcasts TV and radio signals that are unmistakably artificial. A civilization like ours should leave evidence that we would find.
Furthermore, a civilization that evolved millions of years ago pretty recent from a cosmic perspective would have had plenty of time to at least begin colonizing the galaxy, meaning there should be even more evidence of their existence.
Indeed, given enough time, a colonizing civilization would be able to colonize the entire galaxy over the course of millions of years. The physicist Enrico Fermi, for whom this paradox was named , simply asked, "Where are they? One resolution of the paradox challenges the above idea that Earth is common and posits instead that complex life is extremely rare in the universe. Another posits that technological civilizations inevitably wipe themselves out through nuclear war or ecological devastation.
A more optimistic solution is the idea that the aliens are intentionally hiding themselves from us until we become more socially and technologically mature. Yet another idea is that alien technology is so advanced that we wouldn't even be able to recognize it. For you. World globe An icon of the world globe, indicating different international options. Get the Insider App. Click here to learn more. A leading-edge research firm focused on digital transformation. Good Subscriber Account active since Shortcuts.
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Gus Lubin and Andy Kiersz. To go anywhere, you must go halfway first, and then you must go half of the remaining distance, and half of the remaining distance, and so forth to infinity: Thus, motion is impossible.
In any instant, a moving object is indistinguishable from a nonmoving object: Thus motion is impossible. If you restored a ship by replacing each of its wooden parts, would it remain the same ship? Can an omnipotent being create a rock too heavy for itself to lift? There's an infinitely long "horn" that has a finite volume but an infinite surface area.
A heterological word is one that does not describe itself. Does "heterological" describe itself? Pilots can get out of combat duty if they are psychologically unfit, but anyone who tries to get out of combat duty proves he is sane.
There is something interesting about every number. In a bar, there is always at least one customer for whom it is true that if he is drinking, everyone is drinking. A ball that can be cut into a finite number of pieces can be reassembled into two balls of the same size. 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. 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. 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.
Imagine a fletcher i. 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. In his final written work, Discourses and Mathematical Demonstrations Relating to Two New Sciences , 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. 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.
Imagine that a farmer has a sack containing lbs of potatoes. But when he returns to them the day after, he finds his lb sack now weighs just 50 lbs. How can this be true? 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.
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.
Just how much information can one statement actually imply anyway? BY Paul Anthony Jones.
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