I currently have an engineering-style education in mathematics. We covered quite a lot of material (e.g. real and complex analysis, some probability theory and graph theory), but more often than not we stated theorems without formal proofs, and, what's worse, there was a significant amount of hand waving (moving limits around without proper justification, and the like).
Now I am coming to the realization that this approach is akin to going on steroids to build your muscles: while it lets you cover a lot of ground in a relatively short time, it creates more problems than it solves in the long term.
What I would like are suggestions to 'convert' my mathematics knowledge into proper knowledge, using a self-study approach complemented with questions on here. What I have in mind are questions like (but not limited to):
- Where should I start? (set theory?, predicate logic?).
- What follows? (real analysis? complex analysis?).
- Should I work through whole books, solving every problem?
- What are the core topics that any self-respecting mathematician should know well?