> I usually feel that I need to reinterpret algebraic manipulation that I have done to understand what significance they have. So in a way, I allowed myself to be a mechanical executer through the rules of algebra/calculus I've used, and then I re-engaged my intuition to understand what is happening.
That's a sensible approach, but it works because you're executing mechanical in a well-defined problem domain, so there's an intuition attached to each term used, and you can build an overall intuition. You might as well use random mechanical transformations over an algebra and they wouldn't make any sense, just like if you attached random dictionary words forming grammatically correct sentences.
I see usage of formal languages as mathematicians and philosophers making "very precise sentences", i.e. being extremely precise with the meaning attached to each term so that it remains consistent among all the transformations of discourse (in fact, there's a formal theory of meaning which does exactly that).
But the main value of such exercise is in knowing what you want to express, not in the fact that you're being extremely precise while doing it; the latter can help you convince others of what you're saying, but they need to agree to the original axioms of your theory, otherwise they'll just be able to point very precisely where they think you are wrong.
That's a sensible approach, but it works because you're executing mechanical in a well-defined problem domain, so there's an intuition attached to each term used, and you can build an overall intuition. You might as well use random mechanical transformations over an algebra and they wouldn't make any sense, just like if you attached random dictionary words forming grammatically correct sentences.
I see usage of formal languages as mathematicians and philosophers making "very precise sentences", i.e. being extremely precise with the meaning attached to each term so that it remains consistent among all the transformations of discourse (in fact, there's a formal theory of meaning which does exactly that).
But the main value of such exercise is in knowing what you want to express, not in the fact that you're being extremely precise while doing it; the latter can help you convince others of what you're saying, but they need to agree to the original axioms of your theory, otherwise they'll just be able to point very precisely where they think you are wrong.