I'm one of those who realized near the end of their Bachelor's in Business Mgmt that they really really love math. I'm interested in research in pure, theoretical math. I am confident in my ability to learn advanced math on my own. I admit this is based on a small sample of experiences I've had so far with self-studying. After a year of working, I decide to do it full time.
Since I'm behind my peers, I was thinking to just learn the fundamentals (Logic, Real Analysis, Abstract Algebra and Geometry) on my own, and then apply for a Master's. The reason I want to do these on my own is that I find I understand things better at my own pace taking the time to solidify the fundamentals. I took a year of math courses on an exchange program at London School of Econ and it felt like learning disconnected facts. Something like saying object x is round, black, with 4 holes etc. vs showing you a picture a black shirt button (which I find more efficient mentally). Concepts were introduced which were too new but no time was given to "form a picture" of the objects. As a result, it felt like very "syntactic". If I understand the basic concepts used over and over in math, rather than things built out of those facts like calculus, linear algebra, applied math and other "less abstract" branches, I feel I'd do well in my Master's course. I love taking the time explore mathematical concepts on my own, seeing how things fit. I find quite often I make my own concepts (simple ones though) where the picture feels incomplete, which is why I want to do research.
I was wondering how long I should continue this approach (i.e. when have I learned enough fundamentals)? More objectively, what should aim to master on my own, before applying to a master's program? (unless you disagree this is the right approach)
PS: I'm 22, and I have financial support at the moment while I get this done. I've started with Logic and Set Theory and becoming good at proofs. I find calculus and linear algebra intro books skip over too many of the more abstract underlying concepts, hence the bottom up approach.