Things get a bit messier once you're doing research mathematics — definitions don't just come from nothing, and a good definition is one that serves its theorems. Definitions can be "wrong" (they might be generalizable, they might have unexpected pathological examples, etc.), and it's the result of lots of hard work by lots of mathematicians throughout history that we have the definitions we enjoy the use of today.

But yeah, while studying math, I think it's similar to learning programming — don't blame the compiler for your mistakes, it's a well-tested piece of software.