As an applied mathematician, not quite. You build your model, but that's you, not the math. You're using the math and how well your model fits what is seen will help drive you to refine your model, but the model is not the math. It's just a model.
You are talking of mathematical logic and mathematics as a tool, and I am talking about the art of doing mathematics and mathematics as a discipline.
Mathematics, the art, includes building and refining a tractable model, applying mathematical logic, and interpreting the assumptions and results - all the crucial skills that fall within the domain of the mathematician.
As far as the mathematical logic itself is concerned, it is questionable whether it even makes sense to say it is wrong. Its application may be invalid or it may prove the inconsistency of the axioms. It could, however, be argued that the maths is wrong if the axioms are inconsistent.
If an applied mathematical result does not match physical observation, the mathematician would spend most of the time checking assumptions and approximations, very rarely axioms. The balance would shift for pure maths. So, of course it is valuable to rate the confidence in the various areas. But they all come under the discipline of mathematics.
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