1. Highly abstract: The abstraction of mathematics is different from that of other disciplines in object and degree. Mathematics is established and developed in an abstract way.

2. Strict logic: Mathematics has strict logic, and any mathematical conclusion must be strictly proved by logical reasoning before it can be recognized. Strict logic is not unique to mathematics.

3. Wide applicability: As a tool or means, mathematics is applied to almost any science and technology and all social fields.

Many mathematical objects, such as numbers, functions, geometry, etc. , reflecting the internal structure defined in its continuous operations or relationships. Mathematics studies the properties of these structures, for example, number theory studies how integers are represented under arithmetic operations. In addition, things with similar properties often occur in different structures, which makes it possible to describe their state through further abstraction and then axioms. What needs to be studied is to find out the structures that satisfy these axioms among all structures.

Therefore, we can learn from abstract systems such as groups, rings and fields. These studies (structures defined by algebraic operations) can form the field of abstract algebra. Because of its great universality, abstract algebra can often be applied to some seemingly unrelated problems, such as some ancient ruler drawing problems, which are finally solved by Galois theory, including field theory and group theory.

Another example of algebraic theory is linear algebra, which makes a general study of vector space with quantitative and directional elements. These phenomena show that geometry and algebra, which were originally considered irrelevant, actually have a strong correlation. Combinatorial mathematics studies the method of counting digital objects satisfying a given structure.

The study of space originates from Euclidean geometry, while trigonometry combines space with numbers, including the famous Pythagorean theorem, trigonometric function and so on. Nowadays, the study of space extends to high-dimensional geometry, non-Euclidean geometry and topology. Numbers and spaces play an important role in analytic geometry, differential geometry and algebraic geometry.

In differential geometry, there are concepts such as fiber bundle and calculation on manifold, and in algebraic geometry, there are descriptions of geometric objects such as solution sets of polynomial equations, which combine the concepts of number and space. There is also the study of topological groups, which combines structure and space. Lie groups are used to study space, structure and change.