Paradox: refers to a contradictory proposition, which implies two opposite conclusions, both of which can be established. (paradox: confusion, conflict; Open: talk, talk. )
There have been many mathematical paradoxes in history. Mathematical logic is the research method of mathematics, so many logical paradoxes also belong to mathematics. Here are some interesting mathematical paradoxes:
Becker paradox
/kloc-In the 7th century, Newton and Leibniz independently founded calculus, but their definition of infinitesimal in calculus was not clear, which led to the second mathematical crisis.
1734, the British Archbishop Becker refuted the calculus theory (which is anti-scientific in nature), pointed out the famous Becker paradox, and exposed the biggest defect of calculus at that time:
The solution of the second mathematical crisis was not completely solved until19th century, when many mathematicians, such as Polka, Cauchy, Abel and Cantor, established more rigorous mathematical definitions.
Russell's paradox
The famous Russell Paradox (also called Barber Paradox) directly led to the emergence of the third mathematical crisis.
/kloc-at the end of 0/9, with the perfection of set theory, the second mathematical crisis was solved, and mathematicians danced. At the 1900 international congress of mathematicians, poincare, a great French mathematician, even declared that mathematics had reached an absolutely strict level!
Unexpectedly, three years later, the British mathematician, logician and philosopher Russell put forward the famous Barber Paradox, which shocked the whole mathematical world:
The popular explanation of Russell's paradox: all the people in the city shave at a skilled barber, who says, "I only shave for the people who shave themselves in this city"! So, someone said to the barber, do you shave yourself?
Analysis: if he doesn't shave himself, then he belongs to the "person who doesn't shave himself". According to him, he will shave himself; If he shaves himself, he belongs to the "person who shaves himself". According to him, he should not shave himself.
The appearance of Russell paradox shows that set theory itself is incomplete; It was not until 1908 that mathematicians established an axiomatic system, which made set theory fundamentally avoid Russell paradox.
Unexpected paradox
A student union president announced that there will be a meeting next Monday to Friday afternoon, but you can't know in advance when the meeting will be held, because I won't inform you until 8 o'clock that morning.
If we analyze this passage carefully, we will find that there are contradictions that make the meeting impossible. Can you see the problem?
the crocodile paradox
This is an ancient Greek story: a crocodile snatched a child from a mother, and the mother begged, please let my child go, and I will promise anything you want.
So the crocodile proudly said, yes, then guess, will I eat your child? If you're right, I'll give you back the baby!
Mother thought for a moment and said, I think you will eat my child!
Crocodile pondered for a while, froze, and thought: if I ate the child, it means you guessed right, and I should give it back to you; If I don't eat your children, you are wrong, and I will eat your children again!
Sphere-dividing paradox
Paradox means self-contradictory proposition, but in some mathematical paradoxes, it also means some mathematical propositions, but this proposition is contrary to people's common sense, such as the paradox of dividing the ball.
The paradox of dividing the sphere, a theorem strictly proved in mathematics, can be described as: a three-dimensional sphere must have a way to be divided into finite parts, and then two identical spheres (the same radius, the same density ... all the properties are the same) can only be formed by rotation and translation.