Problem description:

What is the paradox? Are there any interesting stories or unsolved mysteries?

Analysis:

Paradox theory (bèi) is a paradoxical, unreasonable and ambiguous theory. But paradox is not nonsense, it contains philosophy in absurdity and inspires people. Along the reasoning route it guides, you will feel that you have embarked on a narrow path of flowers and plants. At first, you will feel logical, and then you will unconsciously fall into the quagmire of contradictions. Once the contradiction is exposed, it is memorable and ridiculous. After careful consideration, people's ability to understand problems has been improved.

Some people divide paradoxes into two categories. One is the paradox between logic and mathematics, which consists of the concepts in logic and mathematics. The other is the paradox of philology, which consists of naming, truth and falsehood. Attach importance to the first kind of paradox in mathematical research. The usual form of this paradox is: if a proposition is admitted to be correct, it will be inferred to be wrong; If it is considered incorrect, the inference is correct.

Now let's take the simplest "liar paradox" as an example, which was put forward by the Greek philosopher Euclid in the 4th century BC.

The original proposition is: "What I said is a lie."

If you think what he said is true, then according to the content of the sentence itself, what he said is a lie. If you think what he said is a lie, then since what he said is a lie, the result of the analysis should be the truth. No one can tell whether he is telling the truth or not (Figure 149).

A similar paradox was put forward as early as the 6th century BC, which was put forward by Epi Mannix, a Crete philosopher. He said, "Every word of Crete is a lie." Is this sentence itself true or false? If we think it is true, then Epi Mannix himself is from Crete, and his words should be deceptive. If we think it is a lie, it means that the Cretes are telling the truth. Of course, this proposition should be denied. So no matter how you look at it, it's hard to justify it. But the difference between this paradox and the previous one is that it can only deduce negative results from positive premises, but not positive results from negative premises, so it is not a typical paradox.

Paradox is interesting to read, but it often annoys scientists. Because rigorous science should be true and reliable. Mathematics, in particular, is based on strict logical reasoning, and there is no room for any contradictory propositions or conclusions. For example, the assertion that "three points not in a straight line determine a plane" is correct, then a plane cannot be determined by just two words, three points are in a straight line or four points are not in a straight line. But the paradox destroys this rigor, which reflects that mathematical science is not monolithic, and its building still has cracks. There are still contradictions, imperfections and inaccuracies in some of its concepts and principles, which need further discussion and solution by scientists. Mathematics is developed in the process of constantly discovering and solving contradictions. Although paradox has brought happiness to many people from ancient Greece to today, people usually classify it as "interesting mathematics", but those great scientists and mathematicians always take it very seriously. In fact, some great progress in modern logic and * * * theory is the direct result of trying to solve the classical paradox.