Seeking education

1. This isn't easy. Look for something to steady your motivation. Question your reluctance to go through the formal process (i.e. go to a university), weigh it against the substantially higher risk of "dropping out" of self-directed study.

2. If you live near a decent university, look for ways to use it without attending it formally. In many places, you can audit the courses freely without anyone caring that you're not a paying student. Use the library, not just for the books, but to force yourself off internet and to study, if you have a problem with that. Knock on the door of a math professor during their visiting hours and ask them to advise you how to prepare a self-study program (don't trust any of their recommendations blindly). Or do the same with a PhD student. Don't feel you're intruding. Mathematicians as a rule feel that anyone without a college-level background in math is missing out terribly in life. They will nearly always be sympathetic to someone trying to gain that knowledge.

3. Try different ways of acquiring knowledge; look for the one that clicks with you. Some people are very strong at studying from a textbook. Others learn much better from a lecture, and there're many excellent university courses available as video online. If you're studying from a textbook, be sure to try and do the exercises. There are some very few people who can learn without doing exercises; you're almost certainly not one of them. Most mathematicians, including most brilliant mathematicians, aren't either.

4. Treat any textbook recommendation, here or elsewhere, as a weak hint. For whatever reason, even though the contents of undergraduate math textbooks on a given subject are almost completely determined by tradition (and trivial from a professional point of view), different textbooks seem to resonate strongly with different people. Even the most famous texts don't work for everyone. If you hate the text several people told you is the best, it may be wrong for you; try a different text (w/o compromising on content).

5. Go to the website of a top university. Study the degree requirements for a math major. They will have a number of required courses; find their course pages, look through the syllabi, note the textbooks the instructors have chosen. Compare with 2-3 other top universities. Use this data to design your own course of study based either on detailed lecture notes/videos of the actual courses, or on the textbooks they are based on, in the latter case make sure to cover everything in the course's syllabus.

   Example: start with http://math.berkeley.edu/programs/undergraduate/major/pure Examine the lower division required courses in detail: http://math.berkeley.edu/courses/choosing/lowerdivcourses For any given course, google its name and "berkeley" to find an even more comprehensive course page for a particular version in a particular year, e.g. http://math.berkeley.edu/~zworski/math54.html It might have the complete list of textbook chapters to study, exercises to do, homework/midterm examples, etc. Make use of those. You now have enough information to structure the study of this particular course for yourself, if you can muster the necessary mental discipline.

6. You cannot skip (basic) things you don't like. For example, you can't skip calculus, and just go on to study other things you like. You won't have the flexibility of abstract thinking that comes from running the epislon-delta formalism a few hundred times in your head, in different variations, and you'll run into a mental wall later on in any other subdiscipline of math. Basically all the material in the required courses list in the example above is something that is a) blindlinly obvious to any professional mathematician; b) should be, if not blindingly obvious, thoroughly familiar to you by the time you're done with your course of study. All of it is real core stuff, something that either lies directly in the foundation of most advanced math material, or has been shown by time to help you develop thought patterns necessary for most advanced math material.

Why? It's a long and difficult road you want to travel, and in my albeit limited experience, people without something to protect don't travel it very long.

if you wanted to educate yourself to graduate level in mathematics, but didn't actually want to go to university, what would you do?

If you want to really learn the stuff, and you're not outlandishly talented, it'd still take you at least a couple of years of full-time effort, likely more (and correspondingly even more in part-time effort), assuming you don't trip up and go in a wrong direction (e.g. form shallow understanding of advanced topics instead of building reliable foundation). This is unlikely to be achieved if you aren't led step-by-step by a college system, don't enjoy the process, and don't have something to protect.

MIT OCW has quite a few mathematics courses listed. Many of them rely on textbooks, but one might be able to find the books used or borrow them from libraries if one can't afford or just doesn't want to purchase them.

Khan Academy might be useful if one wants to build up a more basic foundation, but it looks like the highest level of their offerings consists of differential equations and linear algebra.

My first inclination is to say Khan Academy. Their stuff tends to be quite high quality, at least compared to the curriculum at a middling American university. Which really isn't saying much.

As I'm a bit further on in this path, so the details of the beginnings (introductions to proofs and logic) are a bit blurred to me, but I'll present a few books that I think are good for moving from basics to graduate level topics.

In number theory to move from basics to graduate level topics would be Hatcher's Topology of Numbers into Silverman and Tate into Koblitz.

For linear algebra I'd recommend Axler's book, for complex analysis I liked Churchill and Brown in that I could basically read through and do the exercises very quickly as an undergrad. I didn't have a great time with any of my real analysis texts so I don't think I can give a good recommendation for that.

Of course your mileage may vary but these are the books that I've enjoyed the most, in that they stay grounded in computations, give decent intuition with good geometric pictures, and tend to have approachable exercises compared with other books I've read.

Also: learn to program in python, and use SAGE to solve project Euler problems.

One good trick in addition to book study is to make friends with some mathematicians, and then prompt them to talk about mathematical topics. Mathematicians/scientists/programmers/etc usually LOVE to hold forth about technical topics and love to have an audience that will listen.

A tutor of some kind wouldn't hurt. Now, understand: I live with a bio-chem/chemistry major who loves math and physics and two of my friends are in mathematics. So, I know that I would have an easier time than most finding someone already with knowledge of the subject.

That said, it wouldn't hurt to look and see if you could find anyone to help. I feel you can learn any subject under the sun on your own with the right motivation and resources, but it would be easier if you had someone to correct you in mathematics because simple mistakes or misunderstandings build. It could be someone online you get to know or someone you meet (not an actual paid tutor, god no. Then you might as well just go to school).

This is how I would do it at least. Find someone to help ensure that my understanding of the material is right. I could research the material on my own, but a mental safety net never hurts.

Scattered thoughts:

• Use multiple textbooks if you can. Often a definition or theorem doesn't quite gel until I've read multiple presentations of it.

• Google is your friend. For example, this blog post is the best explanation of normal subgroups I've ever seen. When you combine it with this Math Exchange thread, this web page from John Baez, and Wikipedia, you can actually learn a lot about normal subgroups without even opening a textbook.

• This is probably exclusive to how I learn, but I find that going through a text book linearly is, in general, horribly boring. You are typically given a collection of defintions, which you are supposed to memorize and trust that they will eventually become important. Then you prove theorems about the properties of these minimally-motivated definitions, and if you persist long enough you'll eventually make it to an interesting result and see that the definitions were indeed useful. I can't learn like that; my eyes glaze over. Instead I skim places like Wikipedia for topics that seem interesting and not too far out of reach, and then I learn only the things that I need to understand the topic. Essentially, I try to learn like a package manager, installing only the depencies I need for the package I'm currently interested in. I mention this only because I spent a lot of time trying and failing to make my way systematically through math textbooks before I found this method, which seems to work for me.

It might be worthwhile to get to the level of something like "mathematics for electrical engineers", which is reasonably easy and would cover all the math presented or mentioned on this site, including QM. It is unreasonable to shoot for the graduate level in math, unless you actually have the goal of doing a PhD in math, or unless you are a math genius. The difference in effort between the two is at least an order of magnitude (there are many more topics to cover, and each topic is harder than the one before), while the payoff difference is negligible. Well, unless (another unless) your goal is to research AGI/FAI.

Given that, have you thought through the utility of learning all this math vs doing something else equally hard and long, and possibly boring?

I'm somewhere a long this path, so I can share my experiences. Something worth playing with is Coq, I enjoyed doing the most of course here. One benefit is that it uses a consistent notation to represent things and not too many unfamiliar symbols. It introduces things like the Peano axioms and other interesting stuff. It will also nail down commutativity, transitivity and other useful concepts. Not sure how useful it is, if you haven't done some programming.

For learning important symbols and terminology, I would look at Set theory. Knowing the difference between "member of" symbol and "subset of" symbol of is fairly basic stuff, I just picked it up by osmosis. Predicate calculus also crops up a fair bit, I got taught it in my computing degree. I've got Conceptual Mathematics because I found a fair amount of the maths around functional programming languages was expressed in terms on Category Theory. It was good for the first few chapters but my interest waned at some point and I find it hard to get back into,

I'm interested what other people would recommend.