① Dichotomy paradox
Conclusion: Exercise is impossible. If you want to reach the finish line, you must first reach1/2 of the whole journey; To reach 1/2, you must first reach 1/4 ... whenever you want to reach a point, there is always a midpoint that needs to be reached first, so you will never reach the finish line.
Zhi Nuo, an ancient Greek philosopher, put forward a series of philosophical paradoxes about the inseparability of motion, and the dichotomy paradox is one of them. It was not until the end of 19 that mathematicians gave a formal description of the infinite process problem, similar to the case of 0.999 ... equal to 1.
So how exactly did we get to our destination? The dichotomy paradox will only magnify the problem like an empty valley. In order to solve this problem properly, we must rely on the derivative theory of the 20th century, such as whether matter, time and space are infinitely separable.
Brain hole: You deserve the pleasure of being infinitely divided into 16 inch cheesecake, but you can't finish it.
(2) The arrow does not move
Summary: An arrow can't move. At any time in flight, it has a temporary position, which shows that a moving arrow is a set of all immobile.
Zhi Nuo is another famous paradox. He thinks the unit of time is an instant. In fact, just because motion will not happen at any particular moment does not mean that motion will not happen. Hui Shi, a representative figure of sophistry in the Warring States Period, once said, "The shadow of a bird never moves."
"The arrow doesn't move" actually implies the viewpoint of quantum mechanics. Under the background of special relativity, objects are different at rest and in motion. According to the theory of relativity, observers will have different feelings about objects moving at different speeds and hold different views about the world around them.
Brain hole: I saw the beautiful girl's heartbeat for 3 seconds, and her request to call was rejected. Ahem, the arrow doesn't move, I don't move.
(3) Theseus ship.
Summary: If the wood on the theseus ship is gradually replaced until all the wood is not the original wood, is the ship still the original?
The famous paradox of identity in ancient Greece triggered various discussions among Heraclitus, Socrates and Plato. In the modern Enlightenment, two great British philosophers, Thomas Hobbes and John Locke, also tried to answer this question. The answer is always right or wrong, and it is difficult to make a final decision.
Brain hole: human cells are updated every seven years. Seven years later, there is another you in the mirror.
④ Torricelli trumpet.
Summary: An object with limited volume can have unlimited surface area.
/kloc-geometric paradox in the 0/7th century. Torricelli, an Italian mathematician, rotated the part of y= 1/x where x≥ 1 around the X axis and got the small numbers above (note: only a part of the numbers are shown in the above figure). Then he came to the conclusion that the surface area of this trumpet is infinite, but its volume is π.
Brain hole: It turns out that there are times when a flat chest may not save the country.
⑤ Paradox of interesting numbers
Overview: 1 is a non-zero natural number, 2 is the smallest prime number, 3 is the first odd prime number, 4 is the smallest composite number and so on; If you can't find the interesting feature of this number, it is the first boring number, which is also very interesting.
Therefore, Nathaniel Johnston, a researcher in the field of quantum computing, defined these interesting integers as a whole and arranged them into a sequence, such as prime numbers, Fibonacci numbers, Pythagoras numbers and so on.
Based on this definition, Johns proposed in his blog in June 2009 that the first number that did not appear in the sequence was 1 1630. After updating the monthly series of 20 1 1 in 2003, he said that 14228 was the smallest boring number.
Brain hole: n frogs have n mouths, 2n eyes and 4n legs, and plop into the water ... Do you remember what a sequence is?
⑥ Ball and vase
Summary: Suppose there are infinite balls and a vase. Now we have to do a series of operations, and each operation is the same: put 10 balls in the vase, and then take out 1 balls. So, after countless times of this operation, how many balls are there in the vase?
The answers are varied. The most direct thing is infinity, and some mathematicians think that every ball will be taken out. Logicians James M. Henle and Thomas Mazzucco suggested that the number of balls in a vase can be arbitrary in the end, and there is even a specific construction method.
1976, Sheldon Ross introduced this problem in his "The First Lesson of Probability Theory", so it was called "Ross-Littlewood Paradox".
Brain hole: the ultimate edition of the probability problem of hitting the ball in the dark bag of primary school Olympic mathematics.
⑦ potato paradox
Summary:100g of potato contains 99% water. If you squeeze out 2%, the remaining 98% water will weigh only 50 grams. That is to say, 100 g of potatoes contains 1 g of dry matter. When 98% water is left, 1 g will correspond to 2% content, so a potato with 98% water weighs 50 grams.
Brain hole: science students laugh to internal injuries.
8 drinking paradox
Summary: This happened in a bar: if someone was drinking, everyone was drinking. At first glance, it seems that one person drinking causes everyone to drink.
In fact, if at least one person in the bar doesn't drink, then according to the material conditions in mathematics, for those who don't drink, some people are drinking, and everyone of these people is drinking, and the situation still holds.
The Drinking Paradox is famous for Raymond Smolian's book. What's the name of this book? )。
Pet-name ruby barber paradox
Summary: the barber in the small town boasted, "I only shave all the people in the city who don't shave themselves." So the question is, does the barber shave himself? If he shaves himself, he breaks his promise to shave only those who can't shave themselves.
If he doesn't shave himself, he must shave himself, because his promise says that he will only shave those who can't shave themselves. Neither of these assumptions makes sense.
The famous Russell paradox was put forward by British mathematician Professor Brandt Russell at the beginning of the 20th century. This paradox proves that the set theory of19th century is flawed, which almost changed the research direction of mathematics in 20th century.
Brain hole: a female hairdresser who doesn't shave can't.
Attending grandfather paradox
Summary: If you take Doraemon's time machine back to before your grandparents met, what would happen if you killed your grandfather? If you kill your grandfather, then you will never be born; If you were not born, how could you go back and kill your grandfather?
Grandfather paradox seems to have put an end to the possibility of man-made manipulation of fate, and the past cannot be changed. Grandpa will certainly survive the murder of his grandson. Another possibility is that you have entered another parallel universe, a world you have never lived in, but your grandparents are also here.
This paradox about time travel originated from robert heinlein's short stories and recently appeared in Nolan's Interstellar.
Brain hole: If we go back to before World War II and kill Hitler, we can successfully prevent the outbreak of World War II. However, if there is no World War II, what is the reason to go back and assassinate Hitler? Time travel itself eliminates the purpose of travel and is questioning yourself.