So, why is mathematics excluded from science? According to its research objects, science can be divided into natural science, social science, thinking science, and philosophy and mathematics that run through these three fields. From this classification, we can find that mathematics and philosophy are listed separately, which shows their differences from other sciences. What we call "science" here mainly refers to natural science.
Natural science explores the mysteries of nature and the universe. It is a science to explore the law of movement and change in the material world by means of experimental operation and data collection. Although there are many theories in natural science, they are empirical and concrete in nature. For example, geologists study the structure and composition of a certain rock. The study of natural science originated from the collection and arrangement of empirical facts. Without these basic materials, the theory of natural science is passive water.
Mathematics and natural science are different in origin. Mathematical knowledge comes from the abstract thinking of human brain, such as making a circle with a fixed point as the center and a fixed length as the radius on a plane, which is easy to imagine geometrically, but it is impossible to find such an "ideal" circle in real life.
Of course, this does not mean that the origin of mathematics is divorced from human life practice. On the contrary, mathematics, like natural science, is closely related to the needs of real life and is not a mathematician's fantasy. Only natural science is developed by collecting and studying experience and material facts; Mathematics requires us to transform the actual situation into abstract models in our minds, such as points, lines and surfaces in geometry, and use these abstract concepts in our thinking to promote the development of research step by step.
Summary: After understanding why mathematics is not science, I believe you will have a further understanding and experience of what is science and what is mathematics.