Mathematics Guidance for Hobbyists: Where to Start?

[mxc]

Hello,

I am seeking the opinion of mathematicians. I want a solid understanding of pure mathematics. I would like your opinions on where to start.

My background in formal mathematics courses ended with the standard calculus series. I pursued no further classes while attending college.

My plan is to start from the very basics and build up my knowledge through what would be considered a solid master's level degree of proficiency in the discipline.

My interests are mainly what would be considered "pure" math, but down the road I could see optimization and statistics being very interesting. I have no interest in mathematics as it applies to natural science.

So, does anyone have any recommendations on where to begin laying the foundation?

Please also note that I currently have a decent job and have no desire to go back to school. I am doing this purely as a hobby.

I'd like to thank the community in advance for your suggestions and guidance.

 

[IGU]

One reasonable attempt at compiling what you ought to know is here.

 

[KCC2]

Learn basics of proof methods and set theory (just learn the notation at first, come back to it later, if needed). Then start learning Analysis (epsilon delta definition of limit, continuity, Riemann Integral etc.), Topology, Vector Analysis, Measure and Integration theory (Lebesgue integral).

This seems to be the core (1st and 2nd year) for pure math students here. You can include other things according to your own interests. Like abstract algebra, logic, differential equations. I can't recommend books because I have used our department's own material which is not in English (and I haven't studied all this stuff yet).