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What exactly is the program in mathematical modeling used for? Why do I feel I don't have to?
Computers were originally invented for calculation. . .

Those fancy things, such as interfaces, later appeared, and the application field of computers became broad.

But computing is still the core.

Depending on the complexity of the object and model you are modeling, you don't have to use a computer to realize it. If you design more complicated calculation methods, such as Newton iteration method, or you need to solve differential equations, matrix operations, etc. You need a computer to solve them.