In later centuries, after the ancient Greeks first brought their paradoxes flourished in all strata of society, delighting and infuriating millions of people. Some of them are problems with illogical answers, others - unsolvable problems. We chose the ten most interesting and little-known.
Maxwell's Demon
Named after the Scottish physicist of the 19th century, first proposed the idea, "Maxwell's demon" - a thought experiment in which James Clerk Maxwell tried to violate the second law of thermodynamics. Newton's laws remain inviolable, so the fact of their possible violations led to a paradox.
There is a box filled with uncertain gas temperature. In the middle of the box there is a wall. Daemon opens a hole in the wall, allowing only fast (average) molecules penetrate into the left side of the box. Thus, the daemon creates two separate areas: hot and cold. Separation temperature allows, in turn, generate energy, allowing the flow of molecules to flow from hot to cold regions through a heat engine. At first glance, such a system would violate Newton's second law, which states that the entropy of an isolated system can not change.
However, the second law says and what the demon will not be able to do it without losing their energy every minute. This denial was first proposed by Hungarian physicist Leo Szilard. The point of this argument is that the daemon will generate entropy simple measurement which molecules move faster than average. In addition, the movement and the movement of the demon doors will also generate entropy.
Thomson lamp
James Thomson was a British philosopher who lived in the 20th century. His most notable contribution was the paradox known as "Thomson lamp" puzzle associated with the phenomenon of super-task. (Goal - is counting infinite sequences that occur in a specific order in a finite time).
The problem is as follows. There is a lamp with a button. Pressing the button turns on and turns off the light. If each press of the button will take half the time than the previous one, whether the light is on or off after a specified period of time?
Due to the nature of infinity, it is impossible to know whether the light is on or off, as the last touch of a button simply will not. For any time, at least for two minutes at least ten, the switch will have to click an infinite number of times. Supergoals were first proposed by Zeno of Elea, and Thomson has brought this problem to the paradox. Some philosophers like Paul Benacerraf still claim that machines like lamps Thomson at least logically possible.
Two envelopes problem
Lesser known cousin "Monty Hall" - "two-envelope problem" - is explained as follows. Man shows you two envelopes. He says that one is a certain amount of dollars, and in the other - twice. You need to choose an envelope and check the contents. Then you can choose: keep the envelope or take other. What will give you more money? Provided that you do not know exactly how much money is in your or another envelope.
Initially your chance to take an envelope with a large amount of money is 50/50, or 1 to 2. Most common mistake made when calculating the better option is the following formula, where Y - the value of an envelope in your hand: 1/2 (2Y) + 1/2 (Y / 2) = 1,25 Y. The problem with this solution is that you need to make an infinite number of choices, because that is how you will get more money. In this paradox. Have been many solutions, but none of them has been widely adopted.
The paradox of a boy or girl
Suppose a family has two children. Given that the probability of having a boy is 1/2, what are the chances that another child is also a boy? Intuitively, again 1/2, but it is not. Correct answer - 1/3.
There are four options for a family with two children: the elder brother and younger sister (MD), the older brother with his younger brother (BB), the elder sister younger brother (DM) or an older sister with her younger sister (DD). We know that the option of DD is impossible because a family already has one boy. Thus, the only possible options for MD and BB DM. The probability of 1/3. You can still argue about the twins, but technically they are not born at the same time.
Dilemma crocodile
Kind of paradox of the liar, which popularized the ancient Greek philosopher Evbolid. "Dilemma crocodile" was as follows. Crocodile stolen a child from its parent and then the parent says that return of the child if the parent correctly guesses, return whether or not the child crocodile. If a parent says "you return my child," all right and the baby back. But if a parent says, "You will not return my child," the paradox.
The paradox is that if the crocodile returns the child, he breaks his word, as a parent is not guessing. However, if the crocodile will not return the child, he also breaks his word, as a parent guessing. Apparently, the child destined to remain in the jaws of a crocodile, as the couple never agree. Pseudosolution this paradox - secretly inform the third party in the true intention of the crocodile. Then the crocodile keep his promise, regardless of the answer.
Young Sun paradox weak
This astrophysical paradox arose when we realized that our Sun is about 40% brighter than it was nearly four billion years ago. However, if that were true, the Earth should have been getting a lot less heat in the past, and hence the planet's surface would be completely frozen. First raised scientist Carl Sagan in 1972, the young Sun paradox weak stumped the entire scientific community, because the geological evidence suggests that our planet's oceans is almost always covered.
As possible solutions have been proposed greenhouse gases. But their level was supposed to be in the hundreds or thousands of times higher than it is now. Plus there is plenty of evidence that this was not. Possibly played a role kind of "planetary evolution." According to this theory, the conditions of the Earth (such as the chemical composition of the atmosphere) changed with the development of life.
Hempel's paradox
Also known as the "paradox of the ravens," Hempel's paradox - it is a question of the nature of evidence. He begins with the statement "all ravens are black" and logically contrapositive statement "all things are not black - not crows." Then the philosopher argues that whenever seen crow - and all crows are black - the first assertion is confirmed. In addition, whenever not see a black object like a green apple, confirmed the second assertion.
The paradox arises because each green apple also provides evidence that all crows are black, as are two hypotheses are logically equivalent. The most widely used "solution" of the problem will be the agreement that each green apple (or white swan) gives evidence that crows are black, but with the proviso that the amount of evidence is so small that it would be irrelevant.
The paradox of the barber shop
In July 1894 in Mind (British scientific journal) Lewis Carroll, author of "Alice in Wonderland", offered a paradox known as the "paradox of the barber shop." It looks like this. Uncle Joe and Uncle Jim went to the barber shop, discussing three hairdressers - Carr, Allen and Brown. Uncle Jim wanted his clipped Carr, but was not sure that Carr works. One of the three barbers worked because barbershop was open. They also knew that Allen never leaves the barber shop without Brown.
Uncle Joe claimed he could logically prove that Carr works because it should always work, because Brown will not work without Allen. However, the paradox is that Allen could be inside, and Brown could be home. Uncle Joe argued that this leads to two contradictory statements, and therefore should be inside Carr. Modern logic proved that technically it is not a paradox. The only thing that matters - if Carr does not work, then works Allen, and who cares about Brown?
Galileo's paradox
Better known for his work in astronomy, Galileo also dabbled in mathematics and brought the paradox of infinity and squares of natural numbers. He first said that there are some positive integers which are squares, and some that are not. Thus, he suggested that the sum of these two groups should be greater than the sum of the squares only group. Looks sensible.
Nevertheless paradox arises because of any natural number is a square, and each square - a natural positive number that will be its square root. It turns out that there is a one-to-one at the squares of natural numbers and the concept of infinity. This confirms the idea that a subset of the infinite number can be as large as the set of infinite numbers, from which it follows that subset. While it may seem that it is not.
Problem sleeping beauty
Sleeping Beauty went to bed on Sunday and a coin is tossed. If you roll a "tails", the princess wakes up on Monday, giving an interview and then goes to bed, taking sleeping pills. If the coin falls on the "Eagle", the princess wakes up on Monday and Tuesday, each time giving an interview and then goes to sleep. Regardless of the outcome, it wakes up in the medium and the experiment terminated.
The paradox arises when you try to figure out how it should answer the question: "How do you think the coin fell?". Even if we consider that the probability of determining coin half is far from clear that Sleeping Beauty has to say really. Some argue that the actual probability of 1/3, because it does not know which was the day when she woke up. There are three possibilities: tails on Monday, the eagle and the eagle on Monday Tuesday. So, it needs to say "eagle".