Paradox is a logical term, which originally refers to propositions or discourses that will lead to logical contradictions. For example, the well-known "Han Feizi? The Chu people who sold spears and shields in Nan Yi claimed that his spear was very sharp and could pierce any kind of shield, while his shield was extremely hard and could not pierce any kind of spear. People ask, "Where are the spears with children and the shields with traps?" The Chu people were speechless. The discussion about spear and shield here is a paradox. In practical use, the meaning of the word paradox has been expanded, which often includes all kinds of questions or words that are contrary to human intuition, experience or objective facts. So it is sometimes called "paradox" and "paradox".

Paradox seems absurd, but it has had an important influence in the history of mathematical philosophy. Some famous paradoxes shocked talented philosophers and mathematicians and racked their brains for them, which also caused people to think hard for a long time. It can be said that the study of paradox has made great contributions to the deepening development of mathematical thought.

The earliest recorded paradox in the world is the famous sports paradox put forward by the Greek philosopher Zhi Nuo in the 5th century BC. China's third century BC "Zhuangzi? There are also several famous paradoxes in Tianxiapian. The proposition and solution of these paradoxes are all related to mathematics. The most shocking paradox in the history of mathematics is the "set theory paradox" put forward by the British philosopher Rosso in 1902, which almost shook the foundation of the whole mathematics building and triggered the so-called "third mathematics crisis". These serious topics are recorded and discussed in many books on mathematical methodology, books on the history of mathematics and related reading materials.

This article just wants to talk about some light topics. In fact, many mathematical paradoxes are very interesting, which can not only open your eyes, but also enjoy endless fun. Faced with all kinds of thoughtful, interesting and confusing questions, you must make some intellectual preparations, otherwise you may not be able to turn out in this paradox maze. Look at the following story, and you will believe this statement is true.

The first story happened to an investigator. The researcher was entrusted to three middle schools, A, B and C, to investigate the students' subscription to Mathematics for Middle School Students. He quickly figured out that the proportion of boys who subscribed in A school was greater than that of girls, and the survey of B school and C school also reached the same result. So he intends to write a brief report, saying that from the survey data of three schools, the proportion of boys subscribing to middle school students' mathematics is greater than that of girls. Later, he counted the students from three schools together, and something incredible happened. At this time, the statistical results he got surprised him. Among all the students who subscribe to middle school mathematics, the proportion of girls is greater than that of boys. How is that possible? Just like magic, less becomes more, less changes. Can you help him find the reason?

The next paradox seems simpler. Some people put it into the research category of game theory in mathematics.

An American mathematician came to a casino, stopped two gamblers at random and taught them a simple and profitable gambling method. The method is that two people take out all the money and count it. Whoever has less money will win all the money of those who have more money. Gambler A thinks that if I have more money on me than my opponent, I lose, but if his opponent has more money than me, I win more money than I bring, so I definitely win more than I lose. The amount of money we bring is random, and the possibility is 50-50, so this gambling method is beneficial to me and worth a try. The idea of gambler B coincides with that of A. So both of them happily accepted the mathematician's advice. It seems that this is really a profitable gamble.

The question now is, how can a gamble benefit both sides? Is it like a game of guessing the pros and cons of a coin with equal opportunities? Lose only 1 yuan, and win 2 yuan? It is said that this has always been a headache for mathematicians and logicians. Scientific American magazine has been seeking the answer to this question. In fact, it is not difficult to give a convincing explanation for this problem as long as it is carefully analyzed.

Let's look at another logical paradox. A math professor told his students that the exam would be held some day next week. What day is it? I didn't know it until the day of the exam, which was unexpected in advance. Students have strong logical reasoning ability. They think that according to the professor, there will be no exam on Friday, because if there is no exam on Thursday, the professor's statement that "it will be known on the day of the exam, which is unexpected" is wrong. So Friday's exam can be ruled out. Then the exam is only possible from Monday to Thursday. In this case, it is impossible to take an exam on Thursday, because if there is no exam on Wednesday, it will only be Thursday. In this case, it is not surprising. So Thursday's exam was also ruled out. You can say that it is impossible to take an exam on Wednesday, Tuesday and Monday for the same reason. After reaching the conclusion, the students were very happy. The professor's words led to contradictions. Let's relax. As a result, next Tuesday, the professor announced the exam, and the students were shocked. Why did strict reasoning fail? The professor really realized what he said, and no one could predict the time of the exam. Now please think about it. What's wrong with students' reasoning?

The paradox of motion has a long history. The paradox of ant and rubber rope introduced here is an interesting mathematical problem, which tests your intuition. The problem is this: an ant climbs from one end to the other at a constant speed of 1 cm per second along a rubber rope with a length of 100 m. Every 1 s, the rubber rope will stretch 100 m. For example, after 10 seconds, the rubber rope will be stretched to 1000 meters. Of course, this problem is purely mathematical, assuming that the rubber rope can be stretched at will and the stretching is uniform.

Ants will also climb tirelessly. When the rope is stretched evenly, the position of ants will naturally move forward evenly. Now, can ants finally climb to the other end of the rubber rope at this speed?

Maybe you will think that the poor distance of ants crawling is far less than the constant elongation of rubber rope, but it is getting farther and farther away from the finish line! But it's true that ants climbed to the finish line. Is that weird?