I am not a mathematician but, as such, I think I have a pretty good background in mathematics. I have a good understanding and experience with calculus, differential equations, linear algebra, and probability theory. I also have interest in abstract algebra concepts, though I wouldn't say I am much experienced there. What I think I lack, in comparison to mathematicians, is a deeper feeling for the big picture. I am not even sure how to formulate my question properly, so I'll just share my raw reasoning and hope that my problem will be understood from it. Mathematics is used (or can potentially be used) in almost any field. It can give intuition about almost any concept and help with the solution of almost any real world problem. However, how much of the mathematical answers we get depend on the axioms we start with? Say we discover that a particular system's behavior is completely governed by a set of differential equations. We happen to be able to solve the system analytically and are now able to predict the evolution of the system. How confident are we in the predictions of our model? Does the fact that every single mathematical concept has been ultimately derived from an axiomatic system (like ZFC) introduce any uncertainty in our expectations for the future of the system, assuming no errors were made in the process of solving the system? Do mathematicians "remember" the primary assumptions on which the truth of all those statements rely? Or do mathematical axioms have a different role in mathematics and don't really have any real world implications? Does choosing a different axiomatic system ever lead to any change in any expectations regarding any real world event?