1. An assertion seems to be definitely wrong, but it is actually right (paradox).
2. An assertion seems to be definitely right, but it is actually wrong (specious theory).
A series of reasoning seems impeccable, but it leads to logical contradictions.
Paradox is a bit like magic, which makes people want to know almost immediately after reading it: "How does this trick work?" When the skills are told to him, he will be unconsciously introduced into the profound and interesting mathematical world. Because of this, paradox has become a very valuable teaching method.
Paradox is a part of a broad and strictly defined branch of mathematics, which is famous for "interesting mathematics". This means that it has a strong game color. However, don't think that all great mathematicians despise the problem of "mathematics is interesting". Euler laid the foundation of topology by analyzing the mystery of crossing the bridge.
Leibniz also wrote about the pleasure of analyzing problems when he was playing a stick-inserting game (a game in which sticks were inserted in small squares). Hilbert proved many important theorems in cutting geometry. Von Newman laid the foundation of game theory. The most popular computer game "Life" was invented by the famous British mathematician Conway. Einstein also collected a whole shelf of books about math games and intellectual games.
Extended data:
Typical paradox:
1, barber paradox
In Saville village, the barber put up a sign: "I only cut the hair of those people in the village who don't cut their own hair." Someone asked him, "Do you cut your hair?" The barber was speechless at once.
This is a paradox: a barber who doesn't cut his hair belongs to the kind of person on the signboard. As promised, he should give himself a haircut. On the other hand, if the barber cuts his own hair, according to the brand, he only cuts the hair of people in the village who don't cut their own hair, and he can't cut it himself.
So no matter how the barber answers, he can't rule out the internal contradictions. This paradox was put forward by Russell in 1902, so it is also called "Russell paradox". This is a popular and story-telling expression of the paradox of set theory. Obviously, there is also an unavoidable problem of "self-reference".
2. Paradox of set theory
"R is the set of all sets that do not contain themselves."
People will also ask: "Does R include R itself?" If not, according to the definition of R, R should belong to R. If R contains itself, R does not belong to R..
Kurt G?del (Czech Republic, 193 1) put forward an "incomplete theorem", which broke the ideal that mathematicians thought that all mathematical systems could be deduced by logic at the end of19th century.
This theorem points out that any postulate system is incomplete, and there must be propositions that can neither be affirmed nor denied. For example, the negation of the "axiom of parallel lines" in Euclidean geometry has produced several non-Euclidean geometries; Russell's paradox also shows that the axiomatic system of set theory is incomplete.
3. The paradox of bibliography
A library compiled a dictionary of titles, which listed all the books in the library without their own titles. So will it list its own title?
This paradox is basically consistent with Barber's paradox.
4. Socrates paradox
Socrates (470-399 BC), an Athenian, is known as "Confucius in the West" and a great philosopher in ancient Greece. He was once opposed to the famous sophists Prut Golas and Gogis.
He established a "definition" to deal with the confusing rhetoric of sophists, thus finding out hundreds of miscellaneous theories. But his moral concept was not accepted by the Greeks, and he was regarded as the representative of sophistry when he was seventy years old. Twelve years after expelling Prut Goras and burning books, Socrates was also executed, but his theory was inherited by Plato and Aristotle.
Socrates famously said, "I only know one thing, and that is nothing."
This is a paradox, and we can't infer from this sentence whether Socrates doesn't know the matter itself. There are similar examples in ancient China:
5. "The words are the opposite."
This is what Zhuangzi said in Zhuangzi's Theory of Everything. Later Mohism retorted: If "everything is against the truth", isn't Zhuangzi's statement against the truth? We often say:
6. "There is no absolute truth in the world"
We don't know whether this sentence itself is "absolute truth".
7. Plato-Socrates Paradox
Plato (Plato, π λ? τ ω ν (about 427-347 BC), a great philosopher in ancient Greece, is also one of the greatest philosophers and thinkers in the whole western philosophy and even the whole western culture. He, Socrates, the teacher, and Aristotle, the student, are called the three great philosophers of ancient Greece.
Plato said, "Socrates' next sentence is wrong. "
Socrates said, "Plato is right."
No matter which sentence you assume is true, the other sentence will contradict it. These two sentences are self-evident, but as a whole, they also constitute the liar paradox.