[kmill]

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.

[xanderlewis]

Yes. It’s slightly disingenuous of me to suggest that any definition is as valid and ‘real’ as any other. Obviously, mathematicians care about some ideas more than others.

[kmill]

Yeah, during department teas you can hear mutters of "interesting" as ideas are exchanged and evaluated.

But, in my last comment I was just trying to temper my previous comment's claim about how important definitions are. At some point you get so used to a definition that even if you don't know a particular formulation word for word, you could still write a textbook on the subject because you know how the theory is supposed to go.