The finite and infinite in mathematics reflect the finite and infinite in the real world. The development and change of the whole material world is the unity of finiteness and infinity.

Infinity first refers to the infinity of the material world and the infinity of the universe. Motion is the inherent property of matter, time and space are the way of existence of matter, and the infinity of the material world is manifested in the infinite duration of time and the infinite extension of space. Infinity in mathematics is the reflection of infinity in this material world.

Finite means that everything exists in a specific time and space, so it is always a period of time, with scales and boundaries. That is, everything is concrete. The finiteness in mathematics embodies this finiteness.

Finite and infinite are unity of opposites. They are opposites, different and interrelated. And transform each other under certain conditions. Infinity and finiteness in mathematics also reflect this point. For example, the integer set is an infinite set, and people cannot get a complete integer set. But every integer is finite. We can get any integer. Given any mathematical object, we can immediately determine whether it belongs to an integer set. In this way, an integer set is a complete set and a finite concept. Therefore, the integer set itself is the unity of opposites between infinity and finiteness.

Finite and infinite are opposite and different, and the properties of finite set and infinite set are qualitatively different. For example, a finite set cannot have a one-to-one relationship with any of its proper subset, while an infinite set can have a one-to-one relationship with one of its proper subset. For example, natural number set and one of its proper subset pairs, natural number set, can establish a one-to-one correspondence. Another example is that a finite set of good ordinal numbers and a finite set of natural numbers must have a maximum number and a minimum number. However, the infinitely good ordinal set does not have this property, and the real number set has neither a maximum number nor a minimum number.

Finite and infinite are closely related. There is no infinity without finiteness, and there is no finiteness without finiteness. Infinity cannot be completely proved or fully realized. This is not because infinity does not exist, but only because once infinity is completed and realized, it no longer becomes infinity, but becomes finite. But if all infinity becomes finite, infinity does not exist, and then finite does not exist. Because finiteness exists, infinity cannot be fully realized.

In fact, finite sum constitutes infinity, and infinity exists through finite. This situation is also reflected in mathematics. For example, an integer set is composed of concrete integers, and the infinity of this set is expressed by the sum of countless finite integers.

Finite and infinite can be transformed into each other under certain conditions. For example, matter is infinitely divisible, and the process of "division" is the process of mutual transformation between finite and infinite. "Zhuangzi Tian Xia Pian" said: "One foot pestle, half a day is endless", which expressed this process.

A foot away, take half a day, which is a process from finite to infinite. As far as the length of the foot is concerned, the process of splitting is infinite. No matter how small it is, it can always take half of its length. This is an infinite process. However, there is a limit to the division of pure quantity, and when this limit is reached, it will be transformed into a qualitative difference. As a certain quality, the concrete division can be exhausted, that is, when it is assigned to a certain joint point, the quality of "Qi" cannot be maintained. This joint point is a limit of "fen", which marks the transformation of the process of fen from infinite to finite. This joint point was reached on the thirtieth day, when the pestle was about one billionth of a foot in length, which was less than the order of magnitude of molecules and no longer became its pestle. It can be seen that the process of "dividing" is the process of unity of opposites between finite and infinite, quality and quantity.

In mathematics, the conversion between finite and infinite is often realized through limit. For example, the sum of a convergent positive series consists of infinite terms, but its limit value is a specific value. Conversely, the value of a sine function is a specific value in its domain. But it is expanded into a series of infinite terms. For example, derivatives and integrals are also special limits, so they are also powerful tools for finite and infinite transformations. Mathematics can solve many practical problems through finite and infinite transformation levers.

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