Mathematics is to study the relationship between quantity and space with symbolic language. The research object can be real or virtual, which is very abstract.

Science, on the other hand, studies the phenomena of natural materials and the laws of the occurrence, development and change of things, including not only individual objects, object systems and object subdivision, but also macro and micro phenomena. The object of scientific research must be true and have strong enthusiasm.

The relationship between them is complementary. On the one hand, mathematics is the pioneer of science. Mathematics provides a tool for scientific research, and mathematical reasoning provides a research direction for scientific exploration. Only when mathematics develops to a higher level can science rise to a new theoretical system.

For example, in the field of physics, when the development level of mathematics is in Euclidean geometry period, scientific research can only be based on static mechanics, and the corresponding scientific system is Ptolemy's "geocentric theory"; Newton invented calculus, so that scientific research can be carried out on the basis of dynamic mechanics, and the scientific system of Copernicus-Newton was gradually established.

When mathematics developed to the stage of non-Euclidean geometry, Einstein invented the general theory of relativity and established the scientific system of Einstein-Hawking.

On the other hand, scientific curiosity and exploration promote the continuous development of mathematics, and in the process of scientific development, it also puts forward some new topics for mathematics. Quantum mechanics was founded by a large number of scientists in the early 20th century. The mathematics used at the beginning of its establishment was linear algebra, which completely changed scientists' views on the composition of matter.

Quantum mechanics has been studied for almost a hundred years, especially quantum entanglement. Some phenomena can neither be described by algebra nor deduced by analysis. Due to the limitation of the development level of mathematics, there has been no major breakthrough so far, and it has not changed the public's cognition of material composition.

There is no breakthrough in quantum mechanics because there is no ready-made mathematical method available and it is urgent to invent new mathematics.