The germination of this crisis appeared around 450 BC. Zhi Nuo noticed the contradiction caused by the understanding of infinity, and put forward four paradoxes about the finiteness and infinity of time and space:

"Dichotomy": An object moving to its destination must first pass through the midpoint of the journey, but to pass this point, it must first pass through the 1/4 point of the journey, and so on. The conclusion is that infinity is an endless process and movement is impossible.

"Achilles can't catch up with the tortoise": Achilles always has to reach the starting point of the tortoise first, so the tortoise must always run ahead. This argument is the same as the dichotomy paradox, except that it is not necessary to divide the required distance equally again and again.

"The arrow doesn't move": It means that the arrow must be in a certain position at any time during the movement, so it is stationary, so it can't be moving.

"Playground or parade": Two objects, A and B, move in opposite directions at the same speed. From the point of view of static C, for example, A and B both moved 2 kilometers in 1 hour, but from the point of view of A, B moved 4 kilometers in 1 hour. Exercise is contradictory, so exercise is impossible.

The contradiction revealed by Zhi Nuo is profound and complicated. The first two paradoxes challenge the view that time and space are infinitely separable, so motion is continuous, while the last two paradoxes challenge the view that time and space are infinitely inseparable, so motion is discontinuous. Zeno paradox may have a deeper background, not necessarily for mathematics, but they have set off a huge surprise in the mathematics kingdom. They show that the Greeks saw the contradiction between infinitesimal and infinitesimal, but they could not solve these contradictions. Therefore, infinitesimal has been excluded from Greek geometric proof. After years of hard work, calculus was finally formed in the late 7th century. Newton and Leibniz are recognized founders of calculus, and their achievements mainly lie in: unifying the solutions of various related problems into differential method and integral method; There are clear calculation steps; Differential method and integral method are reciprocal operations. Because of the completeness of operation and the universality of application, calculus became an important tool to solve problems at that time.