Why did the discovery of irrational numbers trigger the first mathematical crisis?

About the 5th century BC, hippasus of Pythagoras School found that the right side of an isosceles right triangle and its hypotenuse were incommensurable. The discovery of this incommensurable measure and Zeno's paradox together triggered the "first mathematical crisis". Because of this mathematical discovery, hippasus was thrown into the sea by the Pythagorean school and was punished by "drowning". Because he actually created such a thing in the universe to deny the Pythagorean creed: all phenomena in the universe can be attributed to integers or the ratio of integers. Pythagoras School is one of the oldest schools of philosophy in ancient Greece. It is said that this school has two maxims that best summarize their ideological characteristics: "What is the smartest? Only the number "What is the best"? Only harmony. " The great contribution of Pythagoras school to mathematics is to prove Pythagoras theorem, but it is also found that the hypotenuse of some right-angled triangles cannot be expressed as integers or the ratio of integers (incommensurability), such as right-angled triangles with all right-angled sides of 1. It is found that the right side of a right triangle and its hypotenuse are not commensurable, which puzzles Pythagoras school. It not only violated the Pythagorean creed, but also impacted the belief that "all quantities can be expressed by rational numbers" held by the Greeks at that time. So people usually call this contradiction discovered by hippasus hippasus Paradox. The discovery of this paradox shocked the western mathematics community at that time, and also caused the renewal of ancient Greek mathematics concepts. This "crisis" shows that intuition and experience are not necessarily reliable, but reasoning is reliable. From then on, the Greeks began to pay attention to from calculation to reasoning, from arithmetic to geometry, and thus established a geometric axiom system. In fact, this is a great revolution in mathematical thought! Since the discovery of irrational numbers, Zhi Nuo of Elias School has "prominently raised the issues of discreteness and continuity". According to the general view of western academic circles, besides hippasus's paradox, Zeno's paradox is also the main factor leading to the "first mathematical crisis". It is said that Zhi Nuo, like his teacher parmenides, was originally a Pythagorean scholar. Zhi Nuo put forward the "four paradoxes" about sports, not so much to defend his teacher's point of view as to attack Pythagoras' mathematical theoretical basis. These four paradoxes can be summarized as follows: the first is the dichotomy paradox; The second is the paradox that Achilles can't catch up with the tortoise; The third is the paradox of "the arrow does not move"; The fourth is the "playground paradox". Zhi Nuo's conclusion about the "four paradoxes" of sports obviously contradicts people's intuition, but it was considered hard to refute at that time. In fact, they seem to have nothing to do with each other, but their overall idea reveals the contradictory nature of the movement extremely profoundly. In the face of Zeno's paradox, the scientific theories at that time, including mathematical theories, seemed to fall into an insoluble contradiction dilemma. At that time, people had two opposing views on time and space: first, space and time were infinitely separable; Second, space and time are composed of inseparable fragments (just like movies). According to the first view, movement is continuous; According to the second view, the movement will be a series of small jumps. Zeno paradox is put forward for the above two viewpoints. Zeno paradox puts forward how to understand finiteness and infinity, indirectness and continuity, time and space, motion and stillness from the perspective of ideological history, but it does not solve these problems. Of course, when Zhi Nuo denied the contradiction of sports, he objectively revealed the contradiction of sports. Aristotle refuted Zhi Nuo's four arguments one by one within the scope of experience, thinking that they were completely wrong. However, he did not reveal the contradiction of movement conceptually. On the contrary, Zhi Nuo discovered the contradiction of movement through conceptual argumentation, which made his thoughts more profound. So Hegel praised him as the founder of conceptual dialectics. Does the discovery of paradox mean the collapse of mathematical theory? The answer is no, we think that non-contradiction is very important for a theoretical system, but non-contradiction is not a sufficient condition for a theoretical system to have truth. As the French philosopher Pascal said, "contradiction is not a false sign, and it is not a true sign without contradiction." It should be pointed out that hippasus's paradox and Zeno's paradox discovered in history both originated from the theoretical system of Pythagoras School, and the contradictions they revealed were extremely profound. In fact, hippasus's paradox and Zeno's paradox not only directly triggered the "first mathematical crisis", but also stimulated the development of mathematics and logic. After the first mathematical crisis, two classic works were published: one was Euclid's Elements of Geometry, the first classic work on mathematics; The second is the first classic work of logic-Aristotle's Theory of Tools. These two classic works marked the birth of axiomatic geometry and logic, and became major events in the history of mathematical development.