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What is the guiding significance of mathematics to other sciences?
Very meaningful, and philosophy. Mathematics and philosophy are two important subjects. To learn philosophy well, you can look at the problem from a height, and you will find that the principles of any subject and its branches in the world are the same. As the saying goes, "things are the same as me" is philosophy.

The significance of mathematics is that it is the foundation of all disciplines. Because disciplines are interrelated and homologous, and these relationships are based on mathematical logic, you will find that almost all disciplines have introduced mathematical mechanism analysis.

These inner things will not be discovered until they are learned to a certain extent. Maybe you can't understand them yet. When you get to a certain height, you will find that what I said is right.

These are my own personal feelings, not copied from other places.