Yes, we want to pick the axioms that seem to reflect the real world. I choose realism because I think that reflects the world around me, but I can't prove it so I take it as axiom. You (apparently) choose non-realism because it seems right to you. Your choice is every bit as logically sound as mine, but we each make our own choices. Some people (like bnonn, e.g.) choose some religiously motivated axioms. That's also logically sound, but reflects the world around us less well. I wouldn't say they depend on what we see, but rather that a good axiom consistently reflects what we see around us. That's an important distinction because in some situations you may have more than one system (e.g. realism/non-realism) that equally reflect what we see.