Self study plan

Question

I don't want to go to college and I came up with plan in the following order:

0-Logic
1-Set Theory
2-Category Theory
3-Discrete Mathematics
4-Abstract Algebra
5-Linear Algebra
6-Geometry
7-Number Theory
8-Probability and statistics
9-Real Analysis
10-Complex Analysis
11-Topology
12-Functional Analysis

My background is a lot of programming and a fairly good amount of calculus and differential equations and linear algebra and geometry and a tiny bit of abstract algebra and category theory, but i would like to go deeper and get into pure mathematics.

assuming that i want to start clean as if i had an average mathematical background could you please comment on the order of the list above and if there is something you would change in it?

Answer 1

0-Logic
1-Set Theory
2-Category Theory
3-Discrete Mathematics
4-Linear Algebra
5-Real Analysis
6-(elementary) Number theory
7-Abstract algebra (after this you can go in algebraic number theory)
8-Complex analysis
9-Topology
10-Functional Analysis
11- Geometry (This subject involves a lot of mathematics, mostly everything if you go in higher levels)
12-Probability and statistics

This is the best I can tell you but math subjects are not disjoint sets. You can study Prob and stats after attaining maturity in real analysis and measure theory, but anyways I kept it at the end.

Answer 2

Before embarking on this (easily multi-year!) course of study, ask yourself why you want to learn each of the subjects. It is my experience that just trying to learn something, without a clear, immediate use, is little motivating, and can easily lead to going astray, exploring superficially interesting paths that turn out useless.

The list you give could well fill a undergraduate + graduate sequence in mathematics. Why study all this and not get some kind of official certification you did learn them? Joining some form of regular study (might be a MOOC, or some distance learning) will provide much needed guidance and feedback.

Answer 3

I don't know why you don't want to go to college. But if you can go forward the possible obstructions: go to collage. You should know that mathematics is something done by people in front of a blackboard: if you sit down alone in your room you will lose large part of the pleasure of this activity.

If you can't go forward the possible obstructions, I give you an advice: don't use static study plans. Choose a topic that you think can be interesting for you (e.g. Graphs Theory) and open a book or a survey article about that topic (e.g. the book by Bollobas). If you realise there is some background material you can't understand, ask to someone (on this website for example) fill the gaps and go forward... Be dynamic, don't do boring, pointless and painful things just because you feel it is the right thing that has to be done in order to have a good mathematical education.