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I have been looking for a new hobby and I have chosen mathematics. I was wondering what the path would be to self-teach (books, online courses/videos...free preferred) oneself to go from basic mathematic skills to the ability to understand and write moderate to advanced proofs.

Though I completed many maths courses (e.g., calculus, differential equations, linear algebra) at uni it was nearly two decades ago and I think it would be fun to start at the beginning with a new appreciation of learning it for the sake of interest and not as a stepping stone to a degree/job. Thusly, if answers could assume an average secondary education level of maths skills that would be great.

I'm not an advanced mathematician by any means, but I think I do have the ability to understand quite abstract maths and follow or produce advanced proofs.

Given that you've covered uni maths courses before (20 years ago), I would suggest that you do a quick review of the fundamentals at a uni level. Linear Algebra, Calculus etc.

Then, a good pathway into abstract reasoning would be a topic called "Real Analysis". You can think of it as re-doing Calculus on a much more rigorous foundation. Things like Limits, Derivatives and Integrals are spelt out in black and white details without having to appeal to intuition.

Following Real Analysis, I would then suggest moving on to Abstract Algebra.

Please let me know what you think. I'm happy to give you more of my opinions on this.

Cheers,

Generally one needs to start with basic subjects (such as calculus, algebra, combinatorics...) and build on them, connect them in different ways, etc... As you progress you should be able to notice what is more or less interesting to you and be able to "go your own way". On the other hand, there are lots of ``small bits'' of information that you only get by interacting with other people; other points of view on the same subject and so on. Depending how serious you are, the best way to go would be to attend math classes at a university. Seminars are also possible, if you have at least covered the basics. Alternatively, come to StackExchange (or MathOverflow) and read questions and answers!

The website How to Become a Pure Mathematician (or Statistician) has several good reference suggestions. Of course You do not want to take it as a "to-do list", and follow the "stages" as in there, but mainly take it as a list of possible topics to tackle, with an explanation for their order, etc...

I would strongly suggest starting with Spivak'sCalculus, as it is a formal approach to something you have already studied. It is not an easy read for newcomers, but just your mental maturity and the fact that you have studied the topic (even if 20 years ago) should make it digestible. It is a terrific book which starts with an axiomatic and algebraic approach to the real numbers (so it gives you a taste of algebra), then goes on to define functions in a set-theoretic way. The next chapters are pretty much what you would expect on a (very good) Calculus book. The Epilogues introduce you to some algebraic an analytic techniques.

Again, you do not need to read everything (for example, skip the chapter "Planetary Motion" if you're not interested, or any applied exercises if you don't care about physics).

If you want to read more on set theory, go to Halmos' Naive Set Theory. For algebra, take Lang's Undergraduate Algebra, or even Lang's Algebra if you're not scared of it being a "Graduate book* (you shouldn't be: If it isto hard to read just try another book and go back to this one later).

At some point you will think that all groups, rings, vector spaces, etc... have a lot of similar properties. You may want to look at some Model Theory (see Hodges) or some Category Theory (see MacLane's Categories for the working mathematician).

Of course, this would be a possible take if you want to study Algebra. If you want Analysis or Geometry you'd take a different path, and there are "standard references" in all these areas (this site has lots of reference requests).

Most importantly, solve exercises. Some books have suggested solutions (Spivak has a secondary book with suggested solutions), which you should use to compare yours with. When proving something, sometimes you get a different answer than the one suggested, but both could be right at the same time. I would say that looking at the solution before solving the problem yourself should be a last resort. You should even just skip the problem you cannot solve and go to the next ones, read a couple more chapters, and later (even a few days later) go back to it and try again. After some point you should be able to say it yourself if your proof is right or not. If you got a solution but are not sure if it is right or not, you can look for the same question here, or simply ask a "proof-verification" question.

If you're looking for free videos to learn real analysis and abstract algebra and see proofs, I would recommend Winston Ou's lectures on Rudin's Principles of Mathematical Analysis and Benedict Gross' lectures on abstract algebra (Artin textbook), which are available on youtube.