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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?

Martin Sleziak
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Bread
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    Linear algebra should come before abstract algebra. Real analysis should be much higher on the list, in one of the top three spots (along with linear algebra and logic). Also, I'd paralelize the list a bit. Don't attempt to learn all of logic before going on to set theory. – 5xum Feb 17 '16 at 10:38
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    @5xum i really wanted abstract algebra before linear algebra the concepts in abstract algebra are fairly simple, i don't mind that but i thought they teach it that way at college because there are more consumers to linear algebra than abstract algebra – Bread Feb 17 '16 at 10:41
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    Please do linear first... And do linear with a lot of practice – Bhaskar Vashishth Feb 17 '16 at 10:50
  • Cool name. I like it, plain. – Em. Feb 17 '16 at 11:21
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    Anyway, if you mean typical courses in probability and statistics, you can really take those in any place on the list since you say you are comfortable with calculus. If you mean probability and statistic from a measure theory and real analysis sense, then you're gonna have to push that back towards the end of the list, after real analysis. – Em. Feb 17 '16 at 11:23
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    Linear algebra is literally everywhere, whether or not you like it. The first task for you is to study the fundamentals of analysis (rigorous treatment of calculus) and algebra (groups, rings, etc.). Real analysis and complex analysis should follow immediately. You will learn a fair bit of topology in analysis. These are the absolutely crucial ones. Studying logic and axiomatic set theory in their own rights is not necessary. Category can wait. Discrete math is not so closely related to other branches. Functional analysis is fundamental to me, don't really know how important it is to others. – Ningxin Feb 17 '16 at 11:27
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    Geometry relies on all of topology, analysis, and algebra. Number theory can also wait. These are more of ultimate goals, rather than the first classes. – Ningxin Feb 17 '16 at 11:36
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    @Bread Reading the sentence "concepts in abstract algebra are fairly easy" made me laugh out loud. If you think that, then you are in for a rough time, because concepts in abstract algebra are easy in the same way as a tornado is gentle. – 5xum Feb 17 '16 at 11:42
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    @Bread For a longer discussion about why linear algebra should come before abstrac algebra, see my answer here: http://math.stackexchange.com/questions/1494531/textbook-on-abstract-algebra-a-specific-request/1494540#1494540 – 5xum Feb 17 '16 at 11:54
  • @5xum in my list abstract algebra only includes Groups,Rings,Fields,Modules,Galois Theory in depth nothing more, are we talking about the same abstract algebra or you had something more than that in mind? – Bread Feb 17 '16 at 12:06
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    @Bread To understand rings, it is highly advisable to first have basics in linear algebra (see the answer I linked before). Even more so for modules, and then there's Galois theory. Galois theory is one of the hardest subjects of undergraduate mathematics in most university courses. It often doesn't even make it into an undergrad level. It is non-intuitive a.f. and very very very far from easy. – 5xum Feb 17 '16 at 12:09
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    The real issue is, these subjects shouldn't really be put together in such a linear order. For example, the debate of "linear then abstract, or abstract then linear?" should really be answered, learn some linear algebra, then learn some abstract algebra, then learn some linear algebra again. – xxxxxxxxx Feb 17 '16 at 12:10
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    @MorganRodgers I agree 100% on that. I'd say abstract algebra is within reach after you learn and understand diagonalization, Jordan's forms and inner products. – 5xum Feb 17 '16 at 12:14
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    ^Thank you all that was really helpful – Bread Feb 17 '16 at 12:17
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    Don't start with logic or category theory. Start with abstract/linear algebra and analysis. Good analysis books will have enough set theory (and topology) to get started. It's a mistake to think that because logic underlies mathematical thinking, the formal study of logic will be useful for the rest of math. And the methods of mathematical logic can become very mathematical, meaning that they can be hard to appreciate if you haven't encountered them first in more ordinary areas of math. – David Feb 17 '16 at 14:26
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    Why this question has been closed? I mean: this guy needs help and this community can help him. That's stupid. Next time I will leave... – Antonio Alfieri Feb 17 '16 at 14:42
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    @Bread Your preference for abstract algebra first (or concurrently with linear algebra) is not unreasonable at all. Artin's Algebra does both at once. – David Feb 17 '16 at 14:48

3 Answers3

3
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.

Bhaskar Vashishth
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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.

vonbrand
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    I agree... This is a multi-year educational program. If you want to do it as an hobby it's ok, but if you want to do it seriously: why do you want to stay alone at your home? It's pointless, I think... – Antonio Alfieri Feb 17 '16 at 14:25
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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.

Antonio Alfieri
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