Mathematical 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.

The definition of intuitionism comes from mathematician L.E.J.Brouwer, who equates mathematics with some psychological phenomena. An example of the definition of intuitionism is that "mathematics is a psychological activity constructed one after another".

Intuitionism is characterized by rejecting some mathematical ideas that are considered effective according to other definitions. In particular, although other mathematical philosophies allow objects that can be proved to exist, even if they cannot be constructed, intuitionism only allows mathematical objects that can actually be constructed.

Formalism defines mathematics through mathematical symbols and operational rules. Haskell Curry simply defined mathematics as "formal system science". A formal system is a set of symbols, or marks, and there are some rules that tell how the marks are combined into formulas.

In the formal system, the word axiom has a special meaning, which is different from the ordinary meaning of "self-evident truth" in the formal system. Axiom is a combination of symbols contained in a given formal system, without using the rules of the system to deduce it.