1. Definition
Aristotle defined mathematics as "quantitative mathematics", which lasted until18th century. /kloc-since the 0/9th century, mathematical research has become more and more rigorous, and it has begun to involve abstract topics such as group theory and projection geometry that have no clear relationship with quantity and measurement. Mathematicians and philosophers have begun to put forward various new definitions.
Some of these definitions emphasize the deductive nature of a lot of mathematics, some emphasize its abstraction, and some emphasize some themes in mathematics. Even among professionals, the definition of mathematics has not been reached.
Whether mathematics is an art or a science has not even been decided. Many professional mathematicians are not interested in the definition of mathematics or think it is undefined. Some just said, "Mathematics is done by mathematicians."
The three main mathematical definitions are called logicians, intuitionists and formalists, each of which reflects a different school of philosophical thought. Everyone has serious problems, no one generally accepts it, and no reconciliation seems feasible.
2. Structure
Many mathematical objects, such as numbers, functions, geometry, etc., reflect the internal structure of continuous operation or the relationships defined therein. 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 for a class of structures 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 abstract systems such as groups, rings and domains.
These studies (structures defined by algebraic operations) can form the field of abstract algebra. Because abstract algebra has great universality, it can often be applied to some seemingly unrelated problems. For example, some ancient problems of drawing rulers and rulers were finally solved by Galois theory, which involved domain theory and group theory.
Another example of algebraic theory is linear algebra, which makes a general study of vector spaces 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 enumerating several objects satisfying a given structure.