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I do not agree with your first statement. Mathematics is about developing and studying tools to solve problems. Proofs are only one aspect of that.

> Mathematics is about developing and studying tools to solve problems.

There clearly exist methods for solving problems that have nothing to do with mathematics. This should falsify your thesis.

I think maybe you made a mistake, because the exchange reads like this:

A: Knives are for eating.

B: Eating is only a part of it. Knives are for handling food.

A: You can handle food with a fork. Your claim is false.

Specifically, "math is for solving problems" does not imply "problems are only solved by math".