Mathematics is not only an important "tool" or "method", but also a way of thinking, that is, "rational thinking in mathematical way";

Mathematics is not only a science, but also a culture, that is, "mathematical culture";

Mathematics is not only some knowledge, but also a quality, that is, mathematical quality.

Popular term for mathematical quality: what remains after all the mathematical knowledge you have learned has been excluded or forgotten.

For example: look at the starting point of the problem from a mathematical point of view; Organized thinking, rigorous thinking and strong ability to verify; Concise, clear and accurate expression; Sense and ability of logical reasoning when solving problems and summing up work; Reasonable quantification and simplification of work, careful planning, etc.

I give three examples of attaching importance to mathematics culture and quality:

The first example: Plato, the great philosopher of ancient Greece (about 427 BC-347 BC) once founded a philosophy school and posted a list at the school gate, stating that people who don't know geometry are not allowed to enter his school. This is not because the courses offered by the school need to be based on geometry knowledge. On the contrary, the courses offered by Plato's school of philosophy are all about sociology, politics and ethics, and the issues discussed are also about society, politics and morality. Therefore, such courses and topics do not need to directly use geometric knowledge or geometric theorems as learning or research tools. It can be seen that Plato asked his disciples to be familiar with geometry first, not focusing on the instrumental nature of mathematics, but based on the cultural nature of mathematics. Because Plato knew the importance of the cultural concept and quality of mathematics. He fully realized that mathematical training based on mathematical cultural character plays an extraordinary role in cultivating one's sentiment, training one's thinking ability and improving one's comprehensive quality level. Therefore, Plato believes that it is difficult for people without strict mathematical training to discuss in depth the courses and topics he has set up.

The second example: As we all know, people who are engaged in the profession of lawyers are highly respected in British society. However, British lawyers have to take many advanced mathematics courses in universities, which is neither because British law requires calculus calculation nor because English law courses should be based on advanced mathematics knowledge, but only because they believe that through strict mathematics training, they can have firm, objective and fair character and form rigorous and accurate thinking habits, which is of great help to his career success. In other words, they fully realize that the study and training of mathematics is by no means a simple pragmatic knowledge transfer, but they are well aware of the decisive role of mathematical cultural concepts and literacy in cultivating first-class talents.

The third example: the West Point Military Academy in the civilized world has been established for nearly two centuries, which has trained a large number of senior military commanders, and many American famous soldiers have also graduated from the West Point Military Academy. In the military teaching plan, in addition to taking some mathematics courses that play an important role in actual combat (such as operational research, optimization technology and reliability methods, etc.). ), requiring students to take many advanced mathematics courses that are not directly linked to actual combat. The reason why the West Point Military Academy requires students to take these mathematics courses is of course based on the cultural character of mathematics. That is to say, they fully realize that only through strict mathematical training can students combine that special vitality with the high flexibility of military action, so that students can have the ability to grasp military action and adaptability, thus laying a solid foundation for their relaxation on the battlefield.

Mathematicians in ancient and modern times have many opinions on mathematics, so I won't introduce them here.

Philosophical theory of mathematics: Mathematics is a kind of philosophy. Philosophical theory originated from ancient Greece, and the representative figures are Aristotle (384-322 BC), Euclid and others. Aristotle once said, "New thinkers regard mathematics and philosophy as the same." Indeed, many mathematicians in ancient Greece were also philosophers.

Isaac Newton (1642- 1727) also said in the preface of Mathematical Principles of Natural Philosophy that he regarded this book as a work of philosophical mathematical principles and "raised as many mathematical problems as possible within the scope of philosophy". This can also be regarded as a philosophical theory of mathematics.

Someone concluded that "the withdrawal of philosophy from a discipline means the establishment of this discipline;" When mathematics enters a discipline, it means that the maturity of this discipline is "an insight into mathematics and philosophy!" In addition, mathematics has the following statement:

Symbolism means that mathematics is a high-level language and a symbolic world.

"Scientific theory" means that mathematics is a precise science and "mathematics is the queen of science".

"Instrumentalism" means "Mathematics is the source of all other knowledge tools".

"Set theory" means that the contents of all branches of mathematics can be expressed in the language of set theory.

"Model theory" means that mathematics is the study of various models, such as calculus is the model of object motion, probability theory is the model of accidental and inevitable phenomena, Euclidean geometry is the model of real space, and non-Euclidean geometry is the model of non-Euclidean space. "Activity theory" means that "mathematics is one of the most important activities of human beings".

"Art theory" means "Mathematics is an art".