To add: I dont know what the solution to this is. So lets say there are two types of approaching math. Approach one is formal/rigid (caring about Axiom of Choic

To add: I dont know what the solution to this is. So lets say there are two types of approaching math. Approach one is formal/rigid (caring about Axiom of Choice, Set theoretical foundations ..:-)) the other one still (partially) rigorous but more classical in spirit. A big question for me is, how is actually research done? Is it more the first or more the second approach? So what are possible solutions? I dont know. If you are trained in the first approach, you usually loose the ability to gloss over the gaps in the second approach. If you are trained in the second you cannot (or very difficult) come around the first one. And then there is physics which uses the sloppy (and often not correct) version of the second approach. I once asked Serge Lang about this (phrasing:'what about physicist getting the right result with wrong mathematics' (thinking about renormalization)). Lang replied:'this is God's way of calculation'.