Mathematics, which originated in ancient Greece, is a subject that studies concepts such as quantity, structure, change and spatial model. By using abstract and logical reasoning, the shape and motion of objects are counted, calculated, measured and observed. The basic elements of mathematics are: logic and intuition, analysis and reasoning, individuality and individuality.

Mathematics, as an expression of human thinking, embodies people's aggressive will, meticulous logical reasoning and pursuit of perfection. Its basic elements are: logic and intuition, analysis and reasoning, individuality and individuality. Although different traditional schools can emphasize different aspects, it is the interaction of these opposing forces and their comprehensive efforts that constitute the vitality, availability and lofty value of mathematical science.

The most obvious feature of modern mathematics is deduction, which is a way from basic definitions and axioms to all theorems through logical reasoning. It is not accidental to adopt this method, but there is an inherent demand. If we want to make a set of concepts clear, we must explain them with simpler concepts, but these concepts need to be clarified again. If we go on like this, if we can't get a vicious circle that can't explain anything, it will extend indefinitely and reach an unknown front. The purpose of human knowledge is to organize their own external knowledge and understand the appearance and essence of things. Therefore, before falling into the abyss of unknowability, they will definitely stop at some concepts that we intuitively think are quite clear. We regard these concepts as the basis of theoretical development and will not explain their meanings, that is, we will put aside their specific contents for the time being. These concepts are called basic concepts. From then on, in the development of our theory, all concepts must be defined by these basic concepts, otherwise they cannot be adopted. If the basic concepts are not related to each other and obviously cannot be used to establish a set of meaningful theories, then in the narrative of connecting the basic concepts, we must pick out some of the most understandable ones as the starting point. These statements are called axioms. Since then, we have deduced all theorems from basic concepts and axioms by logical methods, and we don't think that all statements that can't be deduced by this program are correct propositions in this theory. All theories in modern mathematics are basically organized by this deductive method. More complex theories, besides their own basic concepts and axioms, often refer to the results of other theories. So strictly speaking, the basic concepts and axioms of those theories must also be included. But for the sake of brevity, we usually don't write all of them like this. For example, most theories refer to the concepts and theorems of set theory, and all mathematical theoretical systems must be based on logical systems, otherwise they cannot be derived.

Mathematics studies abstract concepts and uses abstract methods. The development of mathematics is reflected in the gradual deepening of abstraction.

But if we go deeper, the essence of mathematics is not conclusive. Below I will be divided into three parts, which will briefly explain the various viewpoints mentioned in the wiki link provided by @ Taowu.

General mathematics

Corresponding to the practical mathematics and logical mathematics on Wikipedia. Most ordinary people, researchers and many mathematicians hold these views. Under these viewpoints, mathematics is closely combined with reality, so its application is of course very extensive.

One of the more superficial is:

Mathematics is the need of production and life. For example, geometry is used to measure land, and mathematics is a tool.

The representative of this view ... Marx (if he really said so). So 1+ 1 = 2, because one apple and the other apple are two apples, which is the experience and law summarized from practice.

A more reliable idea is:

Mathematics is an immaterial and eternal objective existence and a natural law waiting to be discovered.

The questioner and most people have this idea. Many mathematicians, including some masters, also have this idea. So Pythagorean theorem is not only useful for surveying land, but also a universal law of right triangle, which is an object in nature.

There are other mathematicians, and many people who study computers think:

Mathematics is a part of logic and an axiomatic system.

This view is still very popular in practice, and it is really powerful. But many of these paradoxes can't stand the scrutiny of the following literary mathematics. In this view, the sum of numbers is axiomatic.

Literary mathematics

Corresponding to formalism on Wikipedia. Many mathematicians, many philosophers and myself hold this view.

Formalism holds that a mathematical system is a thinking the game with certain rules, which has nothing to do with the real world.

Different from previous views, this view is unprecedented abstract and open. We began to invent all kinds of abnormal rules and play all kinds of strange and inhuman games. In this view, Pythagoras theorem is correct under Euclid's geometric rules, but we can invent other non-Euclid geometry to make it incorrect; Number is an element in algebraic structure, and operation is the rule of the game.

This view has brought unprecedented development to mathematics, and also led to a serious disconnect between pure mathematics and reality. Whether it is useful or not is also worth studying for formalists. Although it is not directly applied to reality, if other disciplines actively digest it, they can still find a good home.

Congenital mathematics

What I want to mention is intuition. Many cognitive scientists and neuroscientists may hold this view. ...

Intuition says that mathematics is a human brain activity and mathematics is experienced.

Mathematical objects exist because you can build them in your brain. So some radical people will deny the existence of the concept of infinity. One of my cognitive teachers told us that mathematicians often feel inspired, but in fact they are just ideas gained from experience after learning a lot, and there is no free idea.

Actually, I think there is some truth in their views, but ... for example, Sheldon said she had great ideas, and Amy said she studied how these ideas came from.