I have recently made the decision to study Physics seriously. However, in the past, I've had some difficulty with the subject because of my primarily mathematical background. I find that sometimes even if my reasoning is axiomatically correct, it doesn't lead to the right answer for simple reasons like I made a wrong assumption, or I hadn't assumed something I should have. So, I have three questions:
1) Is there some system or method I can use to determine what assumptions I should be making? This is important to me because it seems the problems I've encountered traditionally seem to be open-ended in this regard when they're not simple plug-and-chug questions.
2) There's also the question of definitions. I am uncomfortable with the fact that most definitions are not rigorous enough; this usually leaves some room for interpretation and consequently, it's unclear how they may be applied to certain situations. How do I know if the interpretation I've made of the definition is accurate?
3) Thirdly, when I'm reasoning with a problem, I'm not sure if at each step, what I've concluded is accurate or would actually happen in reality. This applies especially to problems where I'm asked to make simplifying assumptions (such as, "assume the surface is friction-less" or "assume the rod supporting the beam is mass-less") which simply wouldn't be applicable in the real world. So, I'm not really sure what to model my reasoning after in these cases; how can I get around this? Also, is solving such problems really helpful as far as learning actual Physics goes?
I can only speak for classical Physics, as I haven't progressed much further. Is Modern Physics the same in this regard, or is it more axiomatic? Finally, if there really isn't any systematic approach to this, then how can we really call the process Science? Isn't the scientific method all about finding agreement?
Thanks for all the help in advance!