A where-from and how-to study pure mathematics question

Question

I've tried to find answers to my question on this community as well as several others,but couldn't find a satisfactory answer,so here I am. I thank in advance, to anyone who decides to give time to my question.

Background: I am trying to learn pure mathematics on my own from books and any other resources I can find online. I used to do contest mathematics an year or so ago,but have lost touch with almost everything over the past year. Now,I decided to study maths again and am hooked on the maths again but a little lost. I've read A short introduction to mathematics by Timothy gowers. And am currently planning to start the book what is mathematics? By courant and Robbins. I have the following plan for my studies after this book,

Start by the high school precalculus,then proceed to learn calculus and then further(this is one of the things where I am lost). The books I plan to study from are(possibly in this order)

1. Basic mathematics by serge lang, along with the book series by I.M. Gelfand
2. Precalculus mathematics in a nutshell by George finlay Simmons
3. Euclid's elements,along with the new mathematical library series from mathematical association of america.
4. Calculus by Spivak.
5. A course of pure mathematics by hardy.

After that I pretty much have no idea of what things are and in what order am I supposed to study further. My plan is to study these books and continue further studies with the help of the further readings in these books,use online resources like Khan academy, mit open course ware,etc.(not a very good plan,I know).

So,my questions are:

1. How much do you think is the effectiveness of my plan?How much in depth of mathematics can I go with that?
2. In what order should I study further mathematics?
3. What are some of the books and resources that I can study from?
4. Also I would like to how much time do I have to invest daily into my studies (5 hrs?8 hrs? Etc.) So that I can complete undergraduate level mathematics in about 2 years or so(please tell me if that's not enough time).

Thanks again for your time. Also please forgive my mistakes as I'm quite new to this site as well as the world of mathematics. Any answer or advice would be appreciated.

Answer 1

If your plan is go from having a highschool level understanding of mathematics to an undergraduate degree level understanding from self-study, then 2 years is extremely optimistic. Depending on your work ethic I would say such a venture would take closer to 4-5 years or maybe even longer.

An undergraduate degree in mathematics covers a massive range of topics. By the time you finish an undergratuate degree you will have covered number theory, analysis (real and complex), group theory, linear algebra, differential geometry, topology, combinatorics, graph theory and more.

Now I don't want to discourage you because learning mathematics on your own is certainly do-able. In fact I think your current plan is good as a starting point. But realistically the resources you've listed here will only cover basic plane geometry, introductory calculus and introductory algebra. That's fine though, because these are the things you should be starting with. If you get a good understanding of these things then it will put you in a good position to start learning the more advanced stuff I listed above.

As for the order you should learn things. I think this is good as a rough outline:

1. Get comfortable with the material you've listed already: plane geometry, calculus, and basic algebra.
2. Move on to number theory and real/complex analysis. They are closest to what you would've been studying already and good gateways into more advanced topics.
3. Study group theory and linear algebra. They are more abstract and at first it can be difficult to really see how they fit in. But they are important topics, so it's worth making an effort with them.
4. Once step 2 and 3 are completed then things get a bit looser and depends on your interests. If you're more interested in geometric style things then start on differential geometry, metric spaces or topology. If you like algebra then study commutative algebra or more advanced group theory. You get the idea.

Like I said, don't take these steps as gospel, I've thrown them together based on my own experience learning mathematics, so some topics that others consider to be vital for understanding pure mathematics are probably missing. So following what I've suggested here will not give you an equivalent education to an undergraduate degree. This is what I think you should do if you want an approximation to an undergraduate understanding of pure mathematics. To be honest by the time you get to step 4 you almost certainly won't need this guide. You'll have a clearer picture of the landscape of mathematics, what you need to learn and also what you're interested in.

With regards to how much you should study, it really depends on you. I think 2 or 3 hours a day is a perfectly reasonable amount of time to dedicate to studying if you've got time.

Other than that my only advice is similar @Alexey Burdin's comment. Don't give up, mathematics is hard so don't get frustrated if you don't understand something! There isn't a mathematician on earth that understands everything first time.